Statistical approach to the optimisation of the technical analysis trading tools: trading bands strategies

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1 Saisical approach o he opimisaion of he echnical analysis rading ools: rading bands sraegies Mara Ryazanova Oleksiv To cie his version: Mara Ryazanova Oleksiv. Saisical approach o he opimisaion of he echnical analysis rading ools: rading bands sraegies. Humaniies and Social Sciences. École Naionale Supérieure des Mines de Paris, 8. English. <NNT : 8ENMP584>. <pasel-545> HAL Id: pasel-545 hps://pasel.archives-ouveres.fr/pasel-545 Submied on May 9 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Saisical Approach o he Opimisaion of he Technical Analysis Trading Tools: Trading Bands Sraegies Direceur de hese: Alain Galli Jury: Delphine Lauier Yarema Okhrin Michel Schmi Frederic Philippe Rapporeur Rapporeur Examinaeur Examinaeur

3 Dedicaed o my faher Igor Ya. Oleksiv - he nobles person I know. ii

4 Acknowledgmen I wan o hank my advisor Alain Galli for all his houghful guidance. Many hanks o he Members of he Jury - Delphine Lauier, Yarema Okhrin, Michel Schmi and Frederic Philippe for heir ime and valuable feedback. I am graeful o Margare Armsrong for her kind menoring. My special hanks go o: CERNA saff for giving me his opporuniy, which made his research work possible professors Maisonneuve, Schmi and Préeux of Ecole des Mines for heir classes in probabiliy heory and sochasic processes Mahieu Glachan, my fellow docoral sudens and all paricipans of he research aeliers, for heir consrucive criiques Jerome Olivier and Frederic Philippe of Banque BRED for sharing heir business perspecives and providing informaional suppor for cerain elemens of his hesis ha were compleed during my projec wih hem. Sesaria Ferreira for he grea adminisraive and logisical help. The five years I spen in Paris working on his hesis were more han jus he ime of my professional growh - i was a magnificen personal advenure. I hank my dear friends who shared his journey wih me: Larysa Dzhankozova, Anne-Gaelle Geffroy, Yann Ménière, Sevdalina and Panayo Vasilev, and Bike and Doga Cagdas. Many hanks o my parens-in-law, Elena and Yuri Razanau, for always having he words of encouragemen in difficul momens. I am hankful o my husband Aliaksei Razanau for his loving suppor, and o my son Mark who has been my greaes source of inspiraion. There are no words o fully express my indebedness o my parens, Tamara and Igor Oleksiv, wihou whom none of my achievemens would have been possible. iii

5 Absrac In his hesis we have proposed several approaches o improve and opimize one of he mos popular echnical analysis echniques - rading bands sraegies. Pars I and II concenrae on he opimizaion of he componens of rading bands: he middle line (in he form of he moving average) and bandlines. Par III is dedicaed o he improving of he decision-making process. In Par I we proposed he use of kriging mehod, a geosaisical approach, for he opimizaion of he moving average weighs. The kriging mehod allows obaining opimal esimaes ha depend on he saisical characerisics of he daa raher han on he hisorical daa iself as in he case of he simulaion sudies. Unlike oher linear mehods usually used in finance, his mehod can be applied o boh equally spaced daa (in our conex, radiional ime series) and daa sampled a unequal inervals of ime or oher axis variables. Par II proposes a mehod based on he ransformaion of he daa ino a normal variable, which enables he definiion of he exreme values and, herefore, he bands values, wihou consraining assumpions abou he disribuion funcion of he residuals. Finally, Par III presens he applicaion of disjuncive kriging mehod, anoher geosaisical approach, for more informaive decision making abou he iming and he value of a posiion. Disjuncive kriging allows esimaing he probabiliy of cerain hresholds being reached in he fuure. The resuls of he analysis prove ha he proposed echniques can be incorporaed ino successful rading sraegies. iv

6 Cee hese propose des approches pour ameliorer e opimiser un des insrumens les plus populaire d analyse echniques bandes de rading. Les paries I e paries II se concenren sur l opimizaion des composanes des bandes de rading: ligne cenrale (represenée par la moyenne mobile) e lignes des bandes. La parie III es dédiée à l amélioraion du processus de prise de decision. Dans la parie I on proposes d uilizer la méhode de krigeage, une approche geosaisique, pour l opimizaion des poids des moyennes mobiles. La mehode de krigeage perme d obenir l esimaeur opimal, qui incorpore les characerisiques saisiques des donnees. Conrarmen aux mehodes classiques, qui son uilisées en finance, cee mehode peux ere appliquée à deux ypes des données: echanillonées à disance régulière ou irrégulière. La parie II propose une mehode, basée sur la ransformaion des données en une variable normale, qui perme de definir les valeurs exremes e en consequence les valeurs des bandes sans imposiion des conraines de la foncion de la disribuion des residus. Enfin, la parie III presene l applicaion des mehodes de krigeage disjoncif, une aure mehode geosaisique, pour les decision plus informaive sur le iming e ype de posiion. Le krigeage disjoncif perme d esimer les probabiliés, que cerain seuils seron aeins dans le fuur. Les resulas d analyse prouven que les echniques proposées son promeeuses e peuven ere uilisées en praique. v

7 Resume L analyse echnique consise en l ensemble des insrumens, modèles graphiques e règles de rading, qui son fondées sur la hypohese que les prix passés peuven êre uilisés pour aniciper les prix fuurs. Les règles e modèles son souven developées par les raders-echniciens. L analyse echnique es largemen ignorée par les raders-fondamenalises, qui définissen leurs sraégies par les valeurs fondamenales (comme des macro- e microindicaeurs). Ses idées son aussi rejeées par la majorié de représenans academique, qui n accepen pas cee approche comme méhode pour la prevision des prix fuurs. Les chercheurs on des difficulés pour acceper cee mehode pour la raison suivane : L analyse echnique es fondée sur la hypohese que les prix passés peuven êre uilisés pour aniciper les prix fuurs. Cee idée conredi l hypohèse des marchés efficaces («efficien marke hypohesis») sur laquelle la majorié des modèles financiers classique es basée. L aure problème avec l analyse echnique es sa naure empirique: les règles de rading son souven dérivées d observaions empiriques pluô que de modèles mahémaiques. En plus, c es pluô la règle que l excepion pour les raders de déclarer que cerains parameres de ceraines sraégies son opimaux sans aucune référence à des condiions, hypohèses e crières d opimizaion. Finalemen, les barrières linguisiques crées par le jargon e la erminologie echnique uilisée par les raders e chercheurs compliquen encore le dialogue enre les deux paries. En conséquence, ce suje es insuffisammen développé dans les recherches scienifiques, qui évoquen pluo le scepicisme que l inérê. Nore vi

8 moivaion éai donc de conribuer à la recherche pour essayer de combler le fossé enre praiciens e scienifiques. Cee hèse propose des approches pour améliorer e opimiser un des insrumens les plus populaires de l analyse echnique, les bandes de rading. La bande de rading es la ligne placée auour de l esimaeur de endance cenrale. Quare composanes des bandes peuven êre définis : () la série de prix ; () la ligne cenrale ; (3) la bande haue (supérieure) ; (4) la bande basse (inférieure). Deux ypes de sraégies peuven êre définis pour ce insrumen : () «rend-following» ; e () «conrarian». Pour la sraegie «rendfollowing», les bandes serven de confirmaions du signal de endance éabli. Au conraire, les bandes définissen les insrumens qui son rop chers ou moins chers pour la sraegie «conrarian». Cerainmen, les posiions prise en conexe de ces ypes de sraegie son opposées. Les raders uilisen différens ypes des bandes. Les classemen des bandes peu êre défini par les idées/hypohèses concepuelles en ce qui concerne les prix pour lesquels les bandes son définies. Par example, les bandes de rading, les plus simples obenues par le déplacemen parallèle de la ligne cenrale en hau e en bas, supposen la volailié consane des prix. Les bandes de Bollinger essaien d incorporer la naure sochasique de la volailié des prix. Les aures ypes de bandes prennen en compe la disribuion saisique des residus calculés sur la base de prix. L avanage de bandes de rading consise en la possibilié d opimiser l insrumen par ses composanes. D abord on peu opimiser la ligne cenrale, comme esimaeur opimal de la endance (la parie I). Ensuie les bandes son opimisées de façon à ce qu elles coniennen K% des résidus (la parie II). vii

9 Enfin, la parie III es dédiée à l amélioraion du processus de prise de decision. Dans la hèse on considère rois ypes des bandes qui corresponden aux groupes présenés plus hau. En première parie on esime la ligne cenrale opimale sous la forme de la moyenne mobile krigée (KMA), la deuxième e la roisième paries uilisen respecivemen la moyenne mobile simple (SMA) e moyenne mobile exponenielle (EMA) en an que ligne cenrale. La parie II a examiné les bandes définies par les caracérisiques saisiques des données (par exemple, variance). La parie III a analysé les sraégies de rading pour les bandes créées par le déplacemen parallèle de la ligne cenrale en hau e en bas. Le choix de différens ypes de la moyenne mobile (KMA e SMA) pour les deux premières paries es jusifié par la nécessié d évier de mélanger des effes de l'amélioraion des bandes de rading provoquées par les opimisaions de ses composans. Quan au choix des bandes en parie III es expliqué par l'énorme popularié de ce ype de bandes chez les raders. Dans la parie I "Opimisaion de l indicaeur de la moyenne mobile: méhode de krigeage" on propose d uiliser le krigeage, une approche geosaisique, pour l opimisaion des poids des moyennes mobiles (MA). Cee méhode perme d'opimiser la srucure des poids pour une longueur prédéfinie de la fenêre sur lequelle la moyenne mobile es calculée. La méhode de krigeage perme d obenir l esimaeur opimal, qui incorpore les caracérisiques saisiques des données, elles que la covariance (auocovariance). Cela perme d'obenir des esimaeurs opimaux qui dépenden des caracérisiques saisiques des données pluô que des valeurs des données hisoriques comme dans le cas des éudes de simulaion. Conrairemen aux méhodes classiques, qui son uilisées en finance, cee méhode peux êre appliquée à deux ypes de viii

10 données: échanillonées à disance régulière ou irrégulière. Cee approche propose de définir le meilleur esimaeur de la moyenne comme une somme pondérée des observaions dans un voisinage, qui coïncide avec la définiion de la moyenne mobile. La méhode d opimisaion se base sur la minimisaion de la variance d esimaion. Nous avons vu que la meilleure moyenne mobile krigée (KMA), esimée sur les données régulières a une srucure des poids spécifique pour cerains modèles de covariance: les plus grand poids son aachés à la première e la dernière observaion, alors que ous les aures poids son faibles. En conséquence, KMA oscille auour de la courbe de SMA. La volailié e l'ampliude des oscillaions es une foncion indirece de la longueur du voisinage uilisé pour le KMA : le KMA sur un voisinage plus long es moins volaile e coïncide plus avec la courbe de SMA. Par conséquen, des sraégies de rend-following, qui son basées sur les KMA e SMA prendron des posiions différenes pour des voisinages cours e les même posiions pour des voisinages grands. La srucure des poids ne dépend pas de la longueur de la fenêre mais du modèle de covariance. Ce dernier a un impac sur les valeurs des coefficiens de pondéraion proches des bordures de la fenêre : le moins régulier es le modèle de variogramme à l origine le plus les poids de KMA son proches des coefficiens de pondéraion du SMA. Par example, le modèle effe de pépie amène aux poids opimaux qui corresponden aux poids de la moyenne mobile simple. La srucure des poids des échanillons à maille irrégulière es plus variable, elle dépend de l'écar enre les échanillons de la variable uilisée pour subordonner ix

11 les prix ou la disance enre les observaions de prix emporaires: plus l'écar es grand, plus la srucure des poids es volaile. L analyse du KMA en conexe de sraégies de rading monre que le KMA perme d obenir des résulas posiifs e inéressans. Les résulas de l'applicaion de sraégies «rend-following» définie par les croisemens de moyenne mobile e de courbe des prix monren que pour la majorié des insrumens considérés KMA génère des résulas plus élevés que les moyennes mobiles simples ou exponenielles. En plus, le profi maximal eai obenu pour des KMA sur de peis voisinages. Les moyennes mobiles radiionnelles sur des voisinages cours produisen normalemen beaucoup de faux signaux e de ce fai son moins renables. Malgré son caracère volaile, KMA ne génère pas plus de ransacions que les moyennes mobiles radiionnelles de même longueur. Par conséquen, il semble que la naure erraique de la courbe de KMA ne condui pas nécessairemen à générer plus de faux signaux pour les sraégies de rend-following. L applicaion des sraégies de rading pour les échanillons irréguliers monre que les différens ypes de moyennes mobiles calculées sur l échanillon ajusé (pour avoir un échanillon régulier) pourrai conduire à des résulas moins efficaces, que si on calcule la moyenne mobile opimale pour l échanillon irrégulier par la méhode de krigeage. La deuxième parie "Une alernaive aux bandes de Bollinger: les bandes, basées sur les données ransformées", propose une nouvelle approche pour opimiser les bandes de rading. x

12 Bandes de Bollinger on éé proposése au débu des années 98 e resen rès populaires parmi les professionnels de nos jours. Bollinger a proposé d'uiliser la moyenne mobile simple comme une ligne cenrale: m = n P i i= n+ avec { P }, i - la prix, n - longeur de la fenere. > i Les bandes son définies par l équaion suivane : ( ) b = m ± kσ, Pi SMA i= n+ avec σ =, k = cons > - paramères de Bollinger. n Les conclusions suivanes peuven êre dérivées pour les bandes de Bollinger :. Les bandes de Bollinger son ajusées pour la volailié des prix, comme la définiion des bandes incorpore l'écar-ype, en an que mesure de la volailié.. Les bandes supérieures e inférieures au même momen de emps son placés à disance égale de la moyenne mobile, c'es-à-dire ces bandes son symériques. Touefois, cee disance peu êre différene à différens momens du emps, en raison de l'évoluion de la naure de la volailié des prix. 3. La disance enre les bandes se rédui avec la diminuion de la volailié des prix e s élargi avec l augmenaion de celle ci. 4. L'usage des SMA comme la ligne cenrale es jusifiée par le fai que le SMA es la moyenne saisique des prix des sous-échanillons - la même valeur qui es uilisée pour les calculs de l'écar-ype de prix. En oure, ceraines recherches monren que la subsiuion de la SMA par xi

13 la moyenne mobile plus rapide ne produi pas de résulas plus élevés (Bollinger, ). Nous avons égalemen monré dans la parie qui moyenne mobile opimale krigée (KMA), en moyenne, coïncide avec la SMA : KMA coïncide avec le SMA pour les grandes longueurs de la fenêre. L'inérê de l'inroducion de la moyenne mobile exponenielle (EMA) au lieu du SMA pourrai consiser en méhode récurrene de son calcul (à l'heure acuelle EMA peu êre calculée comme une somme pondérée du prix acuel e de la valeur précédene de l EMA). Touefois, à ce égard SMA peu égalemen êre programmée avec des formules récurrenes, mais il a besoin d accumuler plus de données à chaque insan que pour le calcul d'ema. La méhode radiionnelle de Bollinger es saisiquemen jusifiée pour les cas de prix au minimum localemen saionnaires e avec une disribuion symérique. Les bandes, basées sur les données ransformées, fournissen un moyen simple mais puissan pour l opimisaion des bandes. La méhode, es basée sur la ransformaion des données en une variable normale, qui perme de définir les valeurs exrêmes e en conséquence les valeurs des bandes sans imposiion des conraines sur la foncion de la disribuion des résidus. Du poin de vue héorique les bandes opimales devraien conenir K% des données (par exemple, K% = 9%); oues les observaions qui se rouven en dehors des bandes son considérés comme exrêmes. Les bandes ne son pas faciles à définir pour la disribuion asymérique ou mulimodale e exigen un emps considérable pour la procédure d opimisaion. Nore méhode perme d'obenir les bandes dans un cadre plus simple e moins inensif en ermes de calculs. Pour cee procédure, les données brues (résidus) son d abord ransformées en variables normales. Pour la variable normale la disribuion es xii

14 connue e bien definie ; en conséquence les inervalles qui coniennen K% des données son connus. Ensuie, les bandes pour les données brues son obenues par ransformaion de l'inervalle (bandes) pour la disribuion normale en uilisan d'une foncion d anamorphose calibrée précédemmen. Les DT bandes coniennen le même pourcenage de données que l inervalle pour les données normales. Nous avons examiné noammen les résidus R i = Pi SMA, SMA = P i n i i [ n +; ] pour calibrer la foncion de ransformaion. Nore objecif principal éai de reser dans le conexe de la héorie des bandes de Bollinger qui uilisen ces résidus pour le calcul de l'écar-ype des données. En même emps, ces résidus provoquen la forme specifique de ces DT bandes : les bandes formen un escalier qui change de marche si il ya une variaion imporane de prix. Cependan, les DT bandes son moins sensibles à des mouvemens non significaifs des moyennes mobiles. En plus, il semble que les DT bandes peuven ere uilisées pour définir d aures signaux de rading comme les vagues d Ellio e niveaux Suppor/Resisance., On a analysé des sraégies différenes dans la parie II. Les sraégies de bande de Bollinger, comme oues les sraégies conrarianes envoien de faux signaux au cours de la endance présene dans les données à cause des erreurs qu'elles fon dans la définiion de la "vraie" valeur de l insrumen. Pendan les endances des marchés la «vraie» valeur augmene ou diminue, par conséquen, des signaux de "surévaluaion" ou "sous-évaluaion" d insrumen son fausses. Saisiquemen, cela implique que les paramères de nos bandes ne reflèen pas la vraie disribuion de probabilié, qui n'es pas consane. En raison de ces xiii

15 faux signaux les raders ne se basen pas uniquemen sur les signes envoyés par les bandes de Bollinger, mais les examinen en combinaison avec d'aures signaux d analyse echnique, dans le bu de confirmer la sur-évaluaion / sousévaluaion ou de prédire les mouvemens fuurs des prix (par exemple, inversion de endance). Dans cee parie nous avons examiné le momenum, comme l'un des signaux de confirmaion pour les sraégies basées sur les bandes de Bollinger. Le même signal es uilisé pour la confirmaion des sraégies pour les DT bandes. En oure, nous uilisons aussi les signaux de «Ellio» e «Suppor / Résisance» pour confirmer les signaux des DT bandes. À la suie, quare sraégies différens son analysées: () les sraégies de base, qui se fonden uniquemen sur les signaux envoyés par les bandes, () les sraégies confirmées par le momenum, (3) les sraégies confirmées par les signaux d «Ellio» ; e (4) les sraégies confirmées par les signaux d «Ellio» e des «Suppor / Résisance». Les résulas de simulaions rading pour quare insrumens différens monren que les DT bandes génèren plus de profis que les bandes classiques de Bollinger en sraégie confirmée par l indicaeur de momenum; les rajecoires de profil profis/peres on une pene plus posiive e ascendane. Les majoriés des sraégies gagnanes incorporen les DT bandes. En pariculier, les sraégies marchen bien pour rois des quare insrumens. En plus, la sraégie de DT bandes éai encore renable en présence des cerains coûs de ransacion e de slippage. Enfin, les DT bandes pourraien êre uiles dans la définiion d'aures règles d analyse echniques - les vagues d'ellio e les niveaux de suppor / résisance. En conséquence, les nouvelles DT bandes ne son pas seulemen mieux jusifiée d'un poin de vue saisique e plus simples xiv

16 dans leur applicaion, mais elles permeen égalemen générer des profis plus imporans. La parie III "Le krigeage disjoncif en finance: une nouvelle approche pour la consrucion e l'évaluaion des sraégies de rading» présene l applicaion des méhodes de krigeage disjoncif (DK), pour des décisions plus informaive en ermes de iming e ype de posiion. Comme beaucoup des sraégies de rading son basées sur des signaux envoyés par la rupure de cerains seuils, ces problème demande plus d aenion. Le krigeage disjoncif, une aure approche géosaisique, perme d esimer les probabiliés, que cerain seuils seron aeins dans le fuur. En pariculier, nous voulons prédire la probabilié condiionnelle P ( Z + Δ < zc Z Z,..., Z m, ) sur la base des dernières observaions disponibles dans cerains voisinages. Du poin de vue saisique, nous avons besoin de connaîre la disribuion ( T ) Zα Z à (n+)-dimensions, qui es compliquée, voire impossible à esimer à parir de données empiriques. La méhode de krigeage disjoncif implique seulemen la connaissance de la disribuion bidimensionnelle { Z( i ) Z( j )}, i < j n { Z( T ) Z( j )}, i < j n, e, comme une condiion nécessaire pour les calculs de la prédicion d une foncion non linéaire. Il s'agi d'une hypohèse moins srice que la connaissance de la disribuion à (n+)-dimensions. La méhode de krigeage disjoncif es basée sur l'hypohèse qu une foncion non linéaire de ceraines variables aléaoires peu êre développée en ermes de faceurs d un polynôme: xv

17 Quand f [ Y () ] = f + f H [ Y () ] + f H [ Y () ] + f H [ Y () ] =... k = Y () es une variable normale e les couples Z( ), Z( ), i < k k { } j n on une disribuion bivariable gaussienne, nous pouvons uiliser des polynômes orhogonaux d Hermie pour le développemen de la funcion nonlinéaire. Grace à l'orhogonalié des polynômes de Hermie, le krigeage disjoncif de la foncion non linéaire es rédui au krigeage des polynômes de Hermie. Pouran l'applicaion de la méhode de krigeage disjonciif à données financières demande quelques ajusemens en raison de la paricularié de celles ci. Un des problèmes es la non-normalié de la variable analysée. En ce cas la variable e les seuils son ransformés en variables normales. Le principal problème es pouran la non-saionnarié qui exige la ré-esimaion des paramères de la méhode, noammen de la foncion d anamorphose. Nous avons proposé la méhode qui perme d ajuser la foncion de ransformaion principale (basique) à la volailié locale des données. i j Deux ypes de probabiliés disjoncives peuven êre définis e évalués. Les probabiliés disjoncives poncuelles son les probabiliés esimées par krigeage disjoncif en des poins pariculier ; ils reflèen la probabilié qu un cerain seuil sera dépassé à un cerain momen de emps. Cee probabilié peu êre esimée, mais ne peu pas êre validée. Le krigeage disjoncif d un inervalle reflèe la probabilié qu un cerain seuil sera dépassé sur un inervalle de emps fuur. Ce ype de probabilié peu êre validé par les fréquences empiriques - la proporion des observaions lorsque le prix a éé au-dessous d un cerain seuil. xvi

18 Ces probabiliés on éé esimées pour quare insrumens différens. Les résulas son cohérens. L'inervalle DK probabilié (calculé pour la foncion d anamorphose consammen ajusée à la volailié locale) démonre une bonne prédicion en ermes de iming e des valeurs en comparaison avec les fréquences empiriques. Nous avons égalemen monré que seule la longueur de l'inervalle pour lequel la prévision a éé faie e la longueur de l'échanillon uilisé pour l'ajusemen de la foncion d anamorphose on un impac sur la prévision par la méhode de krigeage disjoncif. La pouvoir de la prédicion de la méhode de krigeage disjoncif a éé évalué aussi par la comparaison des résulas des sraégies de rading, qui incorporen cee probabilié krigée. Nous avons consrui deux ypes des sraégies: () sraégie de krigeage disjoncif, où la décision sur la posiion d enrée es faie sur la base des probabiliés krigées, e () la sraégie aléaoire, où la décision sur la posiion d enrée es faie au hasard (avec probabilié de.5). Nore éude révèle que la sraégie de krigeage disjoncif produis des résulas posiifs pour l'inervalle coninu des seuils. La sraégie aléaoire produi les bénéfices à naure aléaoire. La disribuion de profi per ransacion monre que le krigeage disjoncif perme de diminuer le nombre de ransacions avec les peres e augmener la nombre de ransacion avec les profis, si on compare avec la disribuion de la sraégie aléaoire, qui produi une disribuion symérique pour les gains des ransacions. En conséquence, nous avons monré commen cee méhode peu êre appliquée / ajusée pour les données financières d une manière coninue que xvii

19 par rappor à la sraégie aléaoire, la sraégie de krigeage disjoncif améliore le processus de prise de décision. Dans ce ravail, nous sommes concenrés sur des éudes d applicaion d analyse echnique e ses sraégies à un seul insrumen. Les recherches fuures devraien envisager l opimisaion des sraégies pour un porefeuille d'insrumens. En pariculier, une aure méhode géosaisique mulivariable, elle que cokrigeage peu êre uilisée pour l'esimaion de la moyenne du porefeuille e la prévision de sa valeur. Afin de séparer les effes de l'amélioraion de la ligne cenrale (par l'inroducion de la KMA) e l'amélioraion de bandes (par l'inroducion de la DT bandes), nous n'avons pas examiné les DT bandes qui inègren la KMA comme la ligne cenrale. Il serai pariculièremen inéressan d analyser des sraégies, fondées sur les DT bandes e KMA cour. L approche des DT bandes indique les direcions suivanes de recherche seraien à envisager. L'ajusemen de la foncion d anamorphose à la volailié locale, réalisée dans la parie III pour la méhode de krigeage disjoncif, peu êre appliquée à la définiion des DT bandes. L'éude de la relaion enre la renabilié de la sraégie e la valeur de paramère K% uilisée pour la définiion des DT bandes permera augmener les profis des sraégies définies sur la base de ces bandes. Cee approche crée des nouvelles possibiliés à l'amélioraion d'aures insrumens e règles d'analyse echnique. Par exemple, la sraégie confirmée xviii

20 par l indicaeur de momenum es fondée sur la définiion des seuils opimaux pour ce indicaeur. Dans cee analyse les seuils n éaien pas opimisés, mais l approche de ransformaion des donnée uilisée pour la définiion des DT bandes peux êre uilisée pour définir les seuil de momenum. Cela pourrai conduire à des seuils asymériques de momenum. L'aure exemple es l'applicaion des DT bandes à la définiion d'aures indicaeurs echniques, comme les vagues d'ellio e de suppor / résisance. Enfin, l'applicaion de la méhode de krigeage disjoncif aux données financières peu encore êre améliorée par un meilleur ajusemen de la foncion cumulée de la disribuion uilisée pour la ransformaion de données aux changemens de moyenne locale ou à l asymérie de disribuion. Les résulas d analyse prouven que les echniques proposées dans cee hèse son promeeuses e peuven êre uilisées en praique. Ils indiquen aussi de nombreux domaines de recherche pour l'avenir. xix

21 Conens General Inroducion. Par I : Moving average opimizaion: kriging approach... Inroducion... Moving average as a rading insrumen..... Definiion of moving average Types of he moving average.. Kriging mehod: Theory... Geosaisic insrumens and erminology: shor overview..... Saionary and inrinsic saionary funcions Variogram.... Simple kriging: predicion of he process wih zero mean Simple kriging: predicion of he process wih known mean Ordinary kriging: predicion of he process wih unknown mean Ordinary kriging: esimaion of he unknown mean Kriging non-saionary variable Universal kriging: rend esimaion..6.. Kriging inrinsic funcion IRF-. 3. Peculiariies of he kriging mehod applicaions in finance 3.. Daa non-saionariy. 3.. Daa sampling peculiariies 4. Kriging resuls: non-saionary, evenly spaced ime-series daa. 4.. Variogram analysis and opimal kriging weighs: Bund Trading resuls: kriged moving average versus simple moving average Bund DAX Bren X insrumen.. 5. Kriging resuls: bounded, evenly spaced ime-series daa MACD and rading sraegies. 5.. MACD sraegy: opimal signal line Resuls of he rading sraegy, based on he MACD indicaor and signal line Bund DAX Bren X insrumen. 6. Kriging resuls: unevenly spaced daa Unevenly spaced ime dependen price series Unevenly spaced price series subordinaed o cumulaive volume.. 7. Conclusions. 8. Appendices I... Par II. An Alernaive o Bollinger bands: Daa ransformed bands Inroducion.... Classical Bollinger bands conceps Bollinger bands definiion.. Bollinger bands and rading sraegies...3. Theoreical framework for Bollinger bands Daa ransformed bands Daa ransformed bands: algorihm descripion.... Daa ransformed bands: empirical observaions...3. Daa ransformaion bands: oher echnical rules descripion Sraegies descripions

22 3.. Sraegies 3.. Choice of he sreaegy parameers Analysis of he oucomes 4. Trading oucomes for differen insrumens Bund DAX Bren Insrumen X Conclusions Appendices Par III. Disjuncive kriging in finance: a new approach o consrucion and evaluaion of rading sraegies Inroducion..... Theory Disjuncive kriging..... Disjuncive kriging: normal random process Disjuncive kriging: non-normal random process.4. Disjuncive kriging: case of regularized variable Disjuncive kriging: paricular case of he variable wih exponenial covariance model... Peculiariies of he disjuncive kriging applicaion o he financial daa.... Calibraion of he ransform funcion: kernel approach..... Daa used for he esimaion of he CDF: locally adjused ransform funcion. 3. Examples of he DK probabiliies under saionariy assumpion 3.. Poin DK probabiliies: DAX case Inerval DK probabiliies for global ransformaion funcion: Bund case. 4. Examples of he DK probabiliies under local saionariy assumpion Framework for he esimaion and analysis of he inerval DK probabiliies 4.. Inerval DK probabiliies: DAX insrumen Impac of he window lengh 4... Impac of he inerval lengh Impac of he DCI lengh Impac of he bandwidh parameer 5. Applicaion of he DK mehod o he sraegy consrucion 5.. DK rading sraegy 5.. Random walk (or benchmark) sraegy Sraegies oucomes DAX Bund Bren X insrumen. 6. Conclusions. 7. Appendices III General Conclusions. Bibliography xxi

23 General inroducion Technical analysis is a holy debaed opic among researchers and raders. I has is devoed supporers, so called echnicians or echnical analyss, as well as he opponens who do no accep is mehods. The debae akes place no only beween differen groups of raders (fundamenaliss vs. echnicians ), bu also beween he represenaives of academic circles and raders (echnicians). The discussion beween differen ypes of raders is explained by differen principles and relaionships ha are used for predicing fuure prices. Fundamenaliss base heir predicions and, hus, heir sraegies on he marke fundamenals, such as macro-indicaors and micro-indicaors. Macro-indicaors evaluae general marke siuaion; inflaion, ineres rae, unemploymen rae, invenories, consumer confidence index, ec. are he examples of such indicaors. Micro-indicaors represen he characerisics of an insrumen, for which he predicion is made; for example, for he predicion of he price movemens of a paricular sock, he raders analyze company s revenues, asses, balance shee, ec. In heir urn, echnical analyss believe ha prices incorporae all informaion available in he marke (i.e. macro-, micro-indicaors, expecaions, ec.). Therefore, arguably, i is sufficien o use he exising price observaions o make predicions abou fuure price movemens. Thus, echnical analysis echniques are predominanly based on he price daa. When i comes o he academic audiences, mos of hem refuse o accep echnical analysis as a consisen price forecas mehod. As noed Lo e al. (), many academic researchers who easily accep fundamenal facors believe ha he difference beween fundamenal analysis and echnical analysis is no unlike he difference beween asronomy and asrology 3. Taking ino accoun ha he echnical analysis exiss for more han years, such resisance of he researchers is quie puzzling, and we believe ha here may be an explanaion o his phenomenon. Firs, he echnical analysis heory is based on he assumpion ha he pas price observaions can be used o predic he fuure price movemens. This assumpion conradics he efficien marke hypohesis (EMH) 4 ha is he cornersone of he financial heory and on which many financial models are based. A he same ime many deparures from EMH are observed in he real markes due o over- or under-reacion, cerain marke anomalies (such as size effec), behavioral effecs. Bernard and Thomas (99), Banz (98), Roll (983), Chan, Jegadeesh, Lakonishok (996), Huberman and Regev () are he examples of such research. Treynor, Ferguson (984) demonsraed heoreically ha pas prices, combined wih oher informaion, can predic he fuure price movemens. Lo and MacKinaly (988, 999), showed ha pas prices can be used as a forecas for fuure prices. Finally, Grosman and Sigliz (98) argue ha mere presence of he rading and invesmen aciviy and he possibiliy o earn profis in financial markes undermines he credibiliy of EMH. Despie he exisence of such anomalies, he supporers of EMH sill believe ha he invesmen opporuniies occur only in he shor-erm, and hey are eliminaed in Nowadays pure echnicians or fundamenalis among raders rarely exis. Technicians generally do follow he financial news and he macro-indicaors, while fundamenaliss apply some of he echnical analysis echniques. Some researchers hough believe in he predicion power of he echnical analysis. Furher we will provide hese works in general lieraure review. 3 Lo, A. W., Mamaysky, H. and J.Wang.. Foundaions of Technical Analysis: Compuaional Algorihms, Saisical Inference, and Empirical Implemenaion, The Journal of Finance, Vol. LV, #4 (Augus, ), pp , p The EMH saes ha he more efficien he markes are, he more random he price movemens in hese markes are. As a resul, i is impossible o use he pas prices o predic he fuure movemens under his hypohesis. Lieraure review on he EMH can be found in Lo (7).

24 he long run. As a resul, a presen here is no consensus regarding he validiy of he EMH in real markes. The second poin is ha mos echnical analysis echniques have been developed on he basis of he empirical observaions, raher han derived or modeled mahemaically. For example, he majoriy of char paerns, such as Suppor/Resisance, Head-and-Shoulders, ec. were he resuls of regular observaions of he price behavior. Evidenly, he experimen and observaion laid he foundaion for many major invenions in physics, mechanics, chemisry and engineering. The key difference beween he scieniss and echnicians appears o manifes iself in he way hey rea he observed resuls: conrary o he scieniss, echnicians frequenly do no boher o prove or explain heir observaions, bu ake hem for graned. The hird explanaion is driven by he fac ha he echnical rading rules are ofen unjusifiably presened as opimal. We can frequenly see he raders making claims abou opimal values of cerain echnical parameers (for example, he moving average lengh) wihou any addiional suppor or explanaions how and for wha daa ype (insrumen, daa frequency, ec.) hese values were obained. Obviously, such saemens raise los of skepicism from he scieniss. Finally, he language barriers creaed by he usage of he echnical jargon on one side and saisical erms and ess names on he oher complicae he assimilaion of he new ideas by boh sides (Lo e al., ). In addiion, he researchers frequenly misakenly believe ha he echnical insrumens are only abou charing, disregarding he mahemaical conceps ha are used in building he echnical sraegies (for example, moving average, momenum, ec.) Thus, we can conclude ha he absence of boh he scienific represenaion of he mehod and of a formal analysis of he mehod s predicion power creaes a misundersanding beween he echnical raders and academic researchers. As defined by Nefci (99), echnical analysis is a broad class of predicion rules wih unknown saisical properies, developed by praciioners wihou reference o any formalism 5. On our par, we believe ha echnical analysis should be viewed more as a bank of empirical observaions of he financial markes ha can be furher used by he researchers o develop models or well-defined saisical rading echniques. We also believe ha all he academic research performed o-dae in his field, is a necessary inpu in narrowing he gap beween he heoreical and pracical finance. According o some researchers, echnical analysis sudies can be spli ino he following groups: Trend sudies This group represens he indicaors ha idenify he rend and he rend breaks. Among he mos popular indicaors are moving averages, suppor and resisance levels, ec. Direcional sudies This group conains he indicaors ha define he lengh and srengh of curren rend/forecas. Among hese indicaors are DMI, Parabolics, range oscillaor, ec. Momenum sudies This group concenraes on he measuremens of he velociy of he price movemens. The examples of such indicaors are Sochasic, Momenum, Rae of 5 Nefci, S. N. Naïve Trading Rules in Financial Markes and Wiener-Kolmogorov Prediciion Theory: A sudy of Technical Analysis, Journal of Business, Volume 64, Issue 4 (Ocober, 99), pp , p.549.

25 change, MACD, Trix and CCI. Momenum insrumens are frequenly used for he definiion of he rend breaks ha ofen follow he slowness of he price velociy. Volailiy sudies This group presens he rading rules ha incorporae insrumens volailiy and he noion of exreme values. The examples of such rules are rading bands, among which he mos popular case is he Bollinger bands. Volume sudies In he conex of he echnical analysis, volume is he second (afer price) imporan daa elemen ha measures he rading aciviy in he markes. Volume iself as well as he indicaors ha incorporae volume informaion (for example, volume weighed moving averages) complees his group. Each group of echnical sudies conains indicaors and char paerns. Indicaors cover all buy/sell rules ha are formulaed on he basis of he well-defined mahemaical expressions (for example, rading rules based on he moving average). In conras, he chars canno be explicily defined by formulas; hey are he graphical paerns defined subjecively by a rader. Therefore, one rader can recognize a specific char as paricular price paern, while he oher rader would see no paern a all. The examples of such chars are Suppor/Resisance levels, Head and Shoulders and Triangles. I should be noed ha researchers ry o program he char paerns by algorihms and esimaion mehods (see Lo e al., ), however, he resuls of he char paern recogniion depends on he algorihm iself. The financial lieraure ha exiss in he field, can be spli ino he following groups:. Developmen of he scienific framework and formalizaion of he echnical analysis.. Evaluaion of he predicion power of echnical rading rules, as well as heir comparison wih oher predicion mehods. 3. Analysis of he saisical properies of echnical indicaors and heir rading oucomes. 4. Opimizaion of he exising indicaors/rules/sraegies; heir improving; developmen of he new echnical rading insrumens. For he firs group of sudies 6, he cornersone work is he paper by Nefci (99) Naïve Trading Rules in Financial Markes and Wiener-Kolmogorov Predicion Theory: A sudy of Technical Analysis. I proposes he general approach ha defines which echnical rules can be formalized and which canno. According o his framework, a well-defined rule should be a Markov ime, i.e. i should use only informaion available up o curren momen for is consrucion 7. Mos of he predicion echnique ha is used for financial marke forecass lies in he Wiener-Kolmogorov predicion heory framework, according o which ime-varying vecor auoregressions (VARs) should yield he bes forecass of a sochasic process in he leas square error (MSE) sense 8. However, his framework is no suiable for he forecas of he non-linear series. For example, Nefci (99) defines a leas wo cases when linear models canno produce plausible forecass such as () producing sporadic buy and sell signals (non-linear problem by is naure); and () predicing some paricular paerns, such as sock exchange crashes. Consequenly, any oher sysems of forecass ha can predic non-linear ime series can improve he forecas proposed by he Wiener-Kolmogorov framework. Similar conclusions are obained in Brock, Lakonishok and Lebaron (99). According o Nefci (99), i may be he case ha echnical analysis informally ries o analyze he informaion capured by he higher order momens of asse prices. In fac he paerns and rules of he echnical analysis can be 6 See also Rode, Friedman, Parikh, Kane (995) for formalizaion of he echnical analysis. 7 The mehod will be presened furher in more deails. 8 Nefci, S. N. Naïve Trading Rules in Financial Markes and Wiener-Kolmogorov Prediciion Theory: A sudy of Technical Analysis, Journal of Business, Volume 64, Issue 4 (Ocober, 99), pp , p

26 characerized by appropriae sequences of local minima and/or maxima 9, ha lead o nonlinear predicion problems (Nefci, 99). As he resul, he believes ha echnical analysis can improve he forecass of he fuure price movemens. Anoher par of he formal academic research ha can diminish he number of skepics abou he echnical analysis is he formalizaion of he echnical indicaors and insrumens hemselves. While here is a lo of lieraure devoed o he descripion, definiion or calculaions of he echnical indicaors (Murphy, 999; Achelis, ), here are few works ha explain or jusify he mehod from heoreical sandpoin; he examples are Bollinger () on he Bollinger bands, Ehlers ([38]) on moving average. Group of papers ry o explain he predicabiliy of he echnical indicaors in he conex of he microsrucure heory hrough he relaionship beween echnical analysis and liquidiy provision. The researchers believe ha he echnical analysis may indirecly provide informaion capured in limi-order books o make predicions abou fuure price movemens. Osler (3) provided he explanaion of he predicion power of such echnical rading rules as Suppor/Resisance, proving he following hypohesis: he clusers of ake-profi and sop-loss orders are he reasons why he rules succeed in predicing fuure price movemens. Kavajecz, Odders-Whie (4) relaed he moving average indicaors (price moving averages of differen lengh) o he relaive posiion of deph on he limi order book. The second group of sudies ha measure he predicive properies of he echnical analysis is bes represened in he financial lieraure. The majoriy of he papers in his field are devoed o he saisical (economeric) analysis of he predicion power of he echnical rules, while comparing hem wih oher (non-echnical) predicors or variables. The early works in he field of he echnical analysis did no find he superior predicion properies of he echnical rules comparing hem wih he Buy-and-Hold sraegy; as he resul hese works suppored he EMH heory (Alexander, 96, 964; Fama and Blume, 966; James, 968; Van Horne and Parker, 967; Jensen, Beningon, 97). More recen work by Allen, Karjalainen (999) and Raner and Leal (999) has also found lile evidence in favor of he echnical analysis. A he same ime oher research provides he evidence in favor of he echnical analysis. Brock, Lakonishok and Lebaron (99) showed ha 6 echnical rading rules applied o Dow Jones Indusrial Average over 9 years over-perform he sraegy of holding cash. Sullivan, Timmermann, Whie (999) shows ha some of he echnical rules considered in Brock e al. (99) are acually profiable even afer using he boosrap mehod o adjus for he daa-snooping biases. Levich, Thomas (993) found ha some moving average and filer rules were profiable in he foreign exchange markes. Osler, Chang (995) also found he evidence of he profiabiliy of he head-andshoulders paerns in foreign exchange markes. Lo, Mamaysky, Wang () showed ha he same echnical chars provide incremenal informaion abou fuure price movemens by comparing uncondiional disribuion of he socks reurns wih he condiional disribuion of he reurns (condiional on he presence of he char paern). Blume, Easley, O Hara (994) demonsraed ha he raders who use informaion conained in he marke saisics such as prices and volume do beer han he one who do no use i. In his conex, echnical analysis is a componen of rader s learning process. Blanche-Scallien e al. (5) compare he resuls of he echnical rules o he sraegies, based on he mahemaical model. Under cerain assumpions (prices follow one-dimensional Brownian moion, rader s wealh uiliy is represened by he logarihmic funcion) hey show ha MA rule can ouperform he sraegies based on he mahemaical models in case of severe misspecificaions of he model parameers. As for he hird group of sudies, Acar, Sachell (997) analyzed he saisical properies of he reurns from he rading rules, based on he moving averages of he lengh. They showed ha 9 Nefci, S. N. Naïve Trading Rules in Financial Markes and Wiener-Kolmogorov Prediciion Theory: A sudy of Technical Analysis, Journal of Business, Volume 64, Issue 4 (Ocober, 99), pp , p.55. 4

27 in he case when he asse price disribuion is a Markovian process, he characerisic funcion (and, herefore, he disribuion funcion oo) of he realized reurns could be deduced. The opimizaion of he echnical rading rules has a crucial imporance for he raders, who use his approach in he consrucion of heir sraegies. Boh researchers and raders conribue o his field of sudies. For some insrumens ha are more popular, many opimizaion approaches exis, while for he oher he niche is largely underdeveloped. For example, many works devoed o he opimizaion of he rules based on he moving averages, momenum (Gray, Thomson, 997). Cerainly, he choice of he opimizaion echniques largely depends on he ype of he echnical indicaor. However, here is one approach ha is applied o many differen echniques a simulaion of he rading sraegy based on he available hisoric daa samples. According o his approach he opimal parameers correspond o he global/local maximum or minimum of he rading oucomes (profi/losses, Sharpe raio, number of rades, ec.). The example of such opimizaion approach can be found in Williams (6). Alhough his approach is universal, as i can be applied o all rules ha are used in he rading sraegies, he mehod has is drawbacks. The oucomes are dependen on he hisorical daa used for opimizaion; hus, he parameers opimal for he sudied daa se migh be no longer opimal for a new daa sample. Finally, he developmen of he new insrumens is a very dynamic field ha is consanly enlarging boh wih he new ypes of he exising insrumens and oally new ones. For example, Arm ([], V.8:3) proposed he volume-weighed moving average, Chande ([8], V.:3) developed he volailiy adjused moving average, Chaikin and Brogan in heir ime inroduced Bomar bands (Bollinger, ). As we can see, he majoriy of he papers in he field of echnical analysis are devoed o he analysis of is predicion power wihin he conex of EMH. Despie a significan number of papers on his opic, here is sill no consensus wheher echnical analysis has superior predicion power over oher predicion mehods. Therefore, we will accep he hypohesis, similar o one in Grosman and Sigliz (98), ha survival of he echnical analysis among raders for he pas years can be considered a proof ha i can be inegraed ino profiable rading sraegies, a leas for some paricular insrumens; oherwise he raders would have sopped using i. A he same ime, fewer researchers concenrae on he developmen of he heoreical framework for he analysis and opimizaion of he echnical rules, alhough here is a pool of users (raders, echnicians) who creae he demand for his ype of research. Thus, in his hesis, we wan o concenrae on he opimizaion and developmen of he new rading echniques based on he exising echnical sraegies. The echnical indicaors/rules chosen for he analysis will be formalized and explained from he poin of view of he saisical heory. Conrary o many exising works in he field ha use rading simulaions o define he opimal parameers, we wan o use he opimizaion approaches, based on he saisical characerisics of he daa in he firs place. We will use he rading simulaions in all pars of his hesis, mainly o compare he opimized indicaor/sraegy wih oher non-opimal (in he conex of his work) indicaors/sraegies. I is obvious ha an exhausive analysis of all echnical rules is quasi-impossible: he se of rading rules is exremely large and i expands consanly wih he developmen of he new rules and paerns. We decided o concenrae our analysis on such popular echnical analysis echniques as rading bands. While being par of he volailiy sudies, rading bands frequenly incorporae (in heir consrucions or heir sraegies) oher echniques of he echnical analysis from such group of sudies as rend and momenum (see classificaion above). Besides, we will prove ha he 5

28 mehod iself is well defined wihin he framework developed by Nefci (99), briefly presened furher. Trading bands are lines ploed around a measure of cenral endency, shifed by some percenage up and down (upper and lower bands) (Bollinger, ). The schemaic represenaion of he concep is given in Figure. Trading bands have four key componens (see Figure ): () price (quoes), () mid-line, (3) upper band, and (4) lower band. The way hese componens are defined implies he exisence of differen bands ypes, such as envelopes and channels (Bollinger,, Murphy, 999). Upper band Cenral endency Lower band Figure. Schemaic represenaion of rading bands Despie he differences in consrucing he bands, he sraegies based on hem are quie similar. Touching/breaching upper/lower bands give rader informaion on he direcion of price movemens or on relaive price levels (wheher he insrumen is oversold or overbough), which are used as signals in sraegy consrucion. As a resul, boh rend following and conrarian sraegies can be defined on he basis of he rading bands. For example, le some moving average represen he mid-line in he bands. Suppose prices crossed he upper band of he rading bands afer coninuous flucuaions wihin he upper/lower bands and he mid-line crossing. For he rend following sraegy, he bands are he confirmaion of an esablished rend: he firs signal ha he upward rend had been esablished happen a he crossover of he moving average and price curve. Therefore, a breach of he Breaching he upper bands implies ha he price has been previously breaching he moving average line from below. 6

29 upper band can be used as a confirmaion signal of an upward rend. In his conex, rading bands allow o eliminae false signals generaed by he moving average rading rule. Breaching he rading bands in he conex of he conrarian sraegy confirms ha he insrumen is mis-priced. Therefore, breaching he upper band means ha currenly he insrumen is overpriced and he price should reurn o is average (moving average line) in fuure. As he same rading bands can be used in he sraegies ha lead o he opposie rading decisions, raders frequenly use rading bands in combinaion wih oher echnical rading signals ha confirm he presence or absence of a rend. In case of he rend-following sraegy, hese exra rules give addiional confirmaion signals ha he rend has been esablished, while in he case of he conrarian sraegies hey allow avoiding rending paerns, where conrarian sraegy sends false signals. As have been menioned already, i can be proven ha some ypes of he rading bands are well defined. According o Nefci (99), echnical rading rule is well defined if i is a Markov ime. Le { X } be an asse price; I - sequences of informaion ses (sigma-algebras) generaed by and oher daa ses observed up o ime. X Definiion A random variable τ is a Markov ime if he even A { < } = τ is measurable. Simply speaking, i means ha a rule/indicaor is well defined if for making a decision or is calculaion i uses only informaion available up o he curren momen, bu no he one ha anicipaes he fuure. For example, he firs momen of ime when prices increase % from he iniial level a = is Markov ime: on he basis of available hisory of prices up o momen, we can deermine wheher his even has happened or no. As a resul, he Markov ime approach eliminaes many echnical rules ha anicipae he fuure, among which many char paerns. The definiion of a echnical rule as a Markov ime implies () possibiliy o quanify he rule, () feasibiliy of he rule, (3) possibiliy o invesigae rule s predicive power. However, he fac ha he rule is well defined canno jusify is usage. In order o be used, a rule should produce (buy/sell) signal a leas once, i.e. he probabiliy ha he signal is generaed a leas once should be equal o one (see Definiion ). I Definiion A Markov ime τ is finie if ( τ < ) = P. In addiion, a rule should have a leas he same predicive power as oher well-defined forecasing echniques. As a resul, a rading rule gives a consisen forecas of he fuure price movemens if i is a finie Markov ime ha has a leas he same predicive power as more formalized forecasing mehods. For example, Nefci (99) showed ha he moving average rading rules are finie Markov imes and in some cases have higher predicive power han he linear forecas mehods, such as AR or ARMA models. The rading posiions aken wihin each sraegy would be he opposie: for he rend-following sraegy a long posiion is appropriae, while for he conrarian sraegy a shor posiion should be aken. 7

30 Furher we provide he proof ha he rading bands are Markov imes. } Les define { X as some random process ha represens price ime series; I as a sequences of informaion ses (sigma-algebras) generaed by and oher daa ses observed up o ime. Trading band is defined as following: middle line: ; m () () upper bands: m + δ, > ; δ () () lower bands: m + δ, <. Proposiion δ () () Le funcions,, δ be I -measurable. Les define he following variables: Then, m δ Y Y Z X = X m () = δ () () Z () () () = δ Z (3) enry generaed imes { τ i }: enry inf [ enry : () () > τ Y Y ] are Markov imes. exi τ : generaed imes { } are Markov imes. enry τ i, τ =, (4) i = i enry [ > τ : Z Z ] exi τ i = inf i, (5) Proof. Noe ha, () (), Y are -measurable. This implies ha he producs Z Z and Z Y () () enry exi () () Y Y are also I -measurable. and are defined as he firs enry of Y Y and Z Z in he inerval ( ; ] R respecively. Then, according o he Theorem (Shiryayev, 985), which saes ha he firs enry of he process τ i enry exi Markov ime, he and are Markov imes. τ i I τ i τ i { X } in some defined inerval is always Proposiion saes ha if esimaes of he middle line and rading bands are defined on he basis of he informaion available up o momen, he rading rules based on hese rading bands is well-defined. For example, moving average MA ha is ofen used as he middle () () line is I -measurable. Consan scalars δ, δ ogeher wih he I -measurable middle line define bands, which are -measurable. Finally, Bollinger bands, wih he middle line MA = n X n+ i n i= X and () I δ = δ = kσ () deviaion of he { }, are also -measurable. i n+ i I = n i= w X i n+ i, where k > some scalar, σ - experimenal sandard Noe ha he rading sraegy defined in Proposiion is as follows: a posiion is opened when he price breaches one of he bands; his posiion is closed, when he price crosses he middle line. 8

31 Proposiion demonsraes ha here are rading bands sraegies ha can be considered as well enry defined. The quesion is now wheher he signals hey send are finie, i.e. P( τ < ) =, exi P( τ < ) =. I is obvious ha he disance beween he bands defines wheher hese rading bands will send finie signals. In paricular, if he disance beween he wo bands is exremely large he enry signal migh never be generaed. The enry signals will be generaed for he saionary process X } and unbiased esimaor of is mean m, if he following inequaliy holds: { () () ( m + < X < m + δ ) < < P δ (6) The expression (6) can be considered as crieria for he choice of he disance δ + δ beween he bands ha can generae any enry signals. As for he exi signals, Nefci (99) showed ha he rading rules, based on he cross-over of price and MA curves, generae finie X is saionary and m-dependen price process. signals in he case when price process { } () ( ) The majoriy of he rading bands used by raders incorporae moving average as middle line. Moving average allows consrucing well-defined rading bands; herefore, we will narrow our analysis o his paricular ype of bands. Such rading bands are he funcion of four differen parameers or vecors of parameers: { } n ( n { w }, d d ) TB = f,,, i i n where n, w i - parameers of he moving average: lengh of he moving window and weighs i aached o he price in he window (see Ch.. for more deails); d U, d L - disance beween moving average and upper and lower bands respecively. Noe ha upper and lower disances can be differen, as well as each disance iself can be a funcion of ime: du d L, d ( ) = d( ). Despie he large number of he bands parameers, heir opimizaion can be simplified:. Search for he opimal parameers for middle line and bands can be separaed due o he differen roles ha hey play in he definiion of he rading bands sraegy. For example, wihin he conex of he conrarian sraegy moving average represens he mean value of he insrumen, while bands define he exreme values of he prices condiionally on he curren mean. We do no need o know he bands value in order o evaluae price mean, while we need a curren mean value (and probabiliy disribuion funcion) o define he exreme values, and hus, bands. As a resul, he band opimizaion problem can be spli: firs, bes middle line esimaor (is parameers) is obained, hen opimal bands parameers are searched for.. According o Proposiion 3, he opimal moving average should be he bes esimaor o he local mean (or rend). This allows choosing clearer and more objecive opimizaion crieria - minimizaion of he mean squared error: E ( Mˆ m ) min, where m - rue rend/mean, Mˆ - esimaor of he rue mean. U L 9

32 As a resul, he firs par of he hesis will be devoed o opimizaion of he moving average, while he second par will concenrae more on he developmen of he opimal bands (disances). Finally, i should be noed ha opimizaion of he sraegies, based on he echnical analysis, is frequenly subsiued by he problem of he parameer opimizaion. However, he decisionmaking around aking or exiing a posiion is a non-linear problem, while he parameer opimizaion frequenly involves linear mehods. Therefore, we will consider some non-linear approaches o opimize he decision-making in addiion o he opimizaion of he parameers of he echnical indicaors. Wihin he analysis of he rading bands, in addiion o moving average we will also consider some oher echnical analysis echniques such as Momenum, Moving Average Convergence Divergence (MACD), Bollinger bands, Suppor and Resisance paerns and Ellio waves. This hesis is spli ino hree pars. Par I concenraes on he search of he opimal rend esimaor. Par II proposes he approach o opimize bands. Finally, Par III is devoed o he opimal decision-making. The firs par Opimizaion of he moving average indicaor: kriging mehod considers kriging as a mehod o esimae he mid-line in he bands. Kriging approach, a geosaisical echnique, defines he opimal mean esimaor as a weighed sum of he observaions in some close neighborhood; in his respec he kriged esimaor coincides wih he definiion of he moving average. Unlike oher linear mehod usually used in finance, his mehod can be applied o boh equally spaced daa (in our conex, radiional ime series) and daa sampled a unequal inervals of ime or oher axis variable. The laer is he case of he insrumens ha are no regularly raded, or subordinaed price processes obained by changing he ime coordinae o oher random variable coordinae. The kriging mehod is based on he saisical characerisics of daa such as covariance (auocovariance) funcion. This allows obaining opimal esimaes ha depend on he saisical characerisics of he daa raher han on he hisorical daa iself as in he case of he simulaion sudies. This mehod opimizes he weighs srucure for a given lengh of he moving window. We will see ha he opimal kriged moving average (KMA) esimaed on he equally spaced daa sample has a specific weigh srucure for cerain covariance models: he larges weighs are aached o he firs and las observaion, while all he oher weighs are low. As a resul, his KMA coincides in lag wih he simple moving average wih all equal weighs (furher referred o as SMA), bu is more volaile. Moreover, hese border weighs values depend only on he covariance model. The weighs srucure for he subordinaed (unequally spaced) samples exhibis non-sable paerns ha largely depend on he discrepancy in he values of he variable used o subordinae he price curve or he disance beween he ime-observaions of price: he larger he discrepancy he more volaile he weighs srucure is. The comparison of KMA wih he radiional ypes of moving averages reurned ineresing resuls. The resuls of applying rading sraegies based on he crossovers of moving average and price curves show ha for he majoriy of considered insrumens KMA generaes higher resuls han simple or exponenial moving average. Moreover, he global maxima of he KMA-based Cerainly, in real-life applicaions he resuls will sill depend on he hisorical daa used for he evaluaion of he saisical properies of he variable. However, he resuls are more dependen on he esimaion accuracy han on he daa iself.

33 sraegies are achieved a shor lenghs of he moving window, where radiional moving averages normally generae more false signals, and hus, less profiable. Despie is volaile naure, KMA does no generae more ransacions han radiional moving average of he same lengh, and herefore, does no seems o generae more false signals. The second par An Alernaive o Bollinger bands: Daa ransformed bands describes he new approach o opimizing he bands. We propose a mehod ha enables he definiion of he exreme values and herefore bands values, wihou consraining assumpions abou he disribuion funcion of he residuals. From heoreical sandpoin he opimal bands should conain K% of he price daa (for example, K%=9%); all daa poins ha lie ouside of he bands are considered exreme. However, such an inerval and, hus, bands are no easy o define for asymmerical or mulimodal disribuion and require ime-consuming opimizaion procedures. Our mehod allows obaining he bands in a more sraighforward and less-inensive procedure. For his purpose he raw daa (residuals) are ransformed firs ino sandard normal variable. For his disribuion he inervals ha conain K% of he daa are known. Aferwards he daa-ransformed bands (DT bands) are obained by backward ransformaion of he inerval for normal disribuion by he means of a previously calibraed ransformaion funcion. The obained DT bands exhibi a peculiar sair-like paern; he bands change he level only if here is a significan price change. As for he rading oucomes, confirmed DT bands sraegies generae more profis han he classical Bollinger bands ha are more monoonous and upward sloping. As a resul, he new DT bands are no only more jusified from he saisical poin of view and sraighforward in heir applicaion, bu also hey allow generaing higher profis. The hird par Disjuncive kriging in finance: a new approach o consrucion and evaluaion of rading sraegies presens he disjuncive kriging mehod for more informaive decision making abou he iming and he value of a posiion. Frequenly he raders would like o know in advance ha cerain hresholds/bands would be breached. Disjuncive kriging (DK), anoher geosaisical mehod, allows esimaing he probabiliy ha some hresholds will be reached in he fuure. We demonsrae how his mehod can be applied/adjused o he financial daa on a coninuous basis and show ha in comparison o he random-walk hypohesis, DK improves he rading decision-making. The general conclusions can be found in he las secion of his hesis.

34 Par I. Moving average opimizaion: kriging approach Inroducion Predicion of fuure insrumens value movemens, as well as esimaion of a rend plays an imporan role in he analysis of financial daa. Tradiional approaches o rend esimaions are linear filers ha can idenify such feaures as rend, seasonaliy, noise, ec. A moving average (MA) is an example of such filers, which is used for he idenificaion and exracions of series rends. As for he rading applicaions, a MA is he mos widely used echnical analysis echniques. Many research works found ha some moving average and filer rules are profiable (Brock, Laconishok, Lebaron, 99; Sullivan, Timmermann, Whie, 999; Levich, Thomas,993). By is consrucion mehod, he MA is a weighed average. Two principal parameers should be idenified before MA calculaions: () he lengh of a sub-sample, for which he MA is esimaed; () he MA weighs, aached o each observaion in he sub-sample. In addiion, he manner in which he sub-sample is chosen should be decided in advance (for example, i can precede he esimaion poin, or i can include he esimaion poin, ec.). These parameers are responsible for wo characerisics of he MA: () is smoohness; and () a lag, by which he MA is lae in he predicion of he price movemens. As many MA rading rules are based on he relaive posiioning of he price and MA curves, he smoohness of he MA curve is considered o be have direc impac on he number of false signals generaed by he rule: smooher MA sends less false signals. For he same reason, he lag is responsible for he speed of he rading signal: for smaller lag he signal abou rend reversion is sen more rapidly. The dilemma is ha he smooher MA implies larger lag beween he price and MA curves and vice versa. The rade-off beween hese characerisics lays in he basis of many MA opimizaion procedures. The research in he field of he MA opimizaion can be spli in he following groups: () heoreical sudies; () simulaion sudies. The firs group of sudies involves he search of opimal parameers, which is based on some heoreical approach or exising relaionships in he marke. For example, Achelis () believed ha he lengh of he MA should fi peak-o-peak cycle of a securiy price movemen 3. Gray, Thomson (997) used he compromise crierion beween he smoohness and lag o opimize he MA parameers. Ehlers (hp:// proposed a mehod o calculae he opimal weighs for he MA as a funcion of he lag ha a rader can olerae 4. Di Lorenzo, Sciarrea (996) also defined he opimal MA parameers as a funcion of MA lag, which hey define a he level ha minimizes he number of false signals generaed by MA. As he resul hey ried o develop adapive moving average ha akes ino accoun he ransacion coss and price volailiy. Chande ([8], V.:3) developed a Variable index dynamic average ha also incorporae he noion of sochasic volailiy in he definiion of he exponenial MA weighs srucure. Arms ([], V.8:3) exploied he relaionship beween price and volumes in he consrucion of a volume-adjused moving average. Arringon, [6], V.:6) ried o incorporae daa saisical characerisics, such as exreme values, ino definiion of he opimal MA lengh for a Variable lengh moving average. Deailed analysis of he linear filers can be found in Gençay, Selçuk, Whicher (). Some rules are based on he relaive posiioning of he wo MA of differen lenghs. 3 See Appendix A for more deails. 4 See Appendix A for more deails.

35 The simulaion sudies search for he opimal MA parameers hrough he simulaion of a rading sraegy or a rule for he hisoric daa and analysis of he rading oucomes. Then he MA parameers ha maximize hese oucomes (or a leas generae profis) are considered as opimal. The examples of such works are Williams (6), Brock, Laconishok, Lebaron (99), Sullivan, Timmermann, Whie (999). Besides, all he exper judgmens of raders regarding opimal MA ypes or parameers mos likely are based on he experimenal applicaions of hese indicaors o he hisoric daa. The disadvanage of he approach is ha he opimal parameers are condiional on he rading rule and sraegy for which he simulaions were performed. The objecive of our analysis is o inroduce an opimisaion mehod ha akes ino accoun he saisical characerisics of he daa. We would also like o concenrae more on he opimisaion of he MA weighs, as many exising papers in boh groups of sudies are devoed o he opimisaion of he MA lengh. Opimisaion of he MA weighs is more complicaed ask, as i involves he search of n inerrelaed values. We propose a kriging mehod, a geosaisical approach, o opimise he moving average weighs. The goal of geosaisics is o provide quaniaive descripions of he naural variables disribued in space or in ime and space 5. Soil properies or ore grades in a mineral deposi are he examples of such naural variables. The principal objecive of he geosaisics is he reconsrucion of a phenomenon over domain on he basis of values observed a limied number of poins 6. Kriging mehod is used for he rend esimaion and can be applied o he problem of MA opimisaion. The kriged esimaor of a rend a some poin is a weighed sum of he values in he near neighbourhood o his poin, which evolves he direc comparison wih he classical MA indicaor. One of he objecives of he kriging esimaion procedure is o find he opimal weighs of he linear esimaor of a variable mean. There are several differences beween financial and geosaisical daa. Financial daa is mainly sampled in ime. Financial daa samples are much larger han he geosaisical daa ses. The objecive of he financial analysis is mainly no a reconsrucion of he phenomenon, bu a predicion of he fuure price movemens, for which he filering of he rend is done. For he ime series daa a formal analogy of he mehod ha uses pas values o predic fuure one are AR, ARMA, ARIMA models. The references in he domain are Greene (7), Box, Jenkins, Reinsel (8). Financial daa is usually reaed as ime-series daa wih values sampled a he a regular ime inervals as when daa is unevenly spaced, mos of he mehods used in he ime signal processing canno be applied. However, in realiy, financial daa is documened only a he momens when a ransacion akes place. As he resul, for less-acively raded insrumens he daa is no equally spaced. The daa sampled a very high frequency (for example, second) will also mos likely be unequally spaced. Finally, when he ime coordinae is changed o anoher variable coordinae (for example, volume), he daa subordinaed o anoher process would mos likely be unevenly spaced. In he case of he unevenly spaced financial daa, he geosaisic mehods will bring beer resul han he classical ime-series mehods. However, even for he equally spaced daa he difference exiss beween kriging mehod and ime-series models: he kriging approach does no demand compleely specified model of he 5 Chiles, J.-P. and P. Delfiner. Geosaisics. Modeling Spaial Uncerainy, John Wiley and Sons, Inc., 999, p.. 6 Chiles, J.-P. and P. Delfiner. Geosaisics. Modeling Spaial Uncerainy, John Wiley and Sons, Inc., 999, p. 5. 3

36 process as in he case of he ime series model; Only second-order properies are modelled for linear kriging. The analysis of he kriging mehod is made in he following way. Firs, we presen a definiion and brief descripion of he MA as a rading insrumen. Chaper presens he kriging mehod. Chaper 3 discusses he peculiariies of mehod applicaion o he financial daa. Chaper 4, 5 and 6 analyse he resuls of he kriging mehod applicaion o he equally and unequally spaced hisoric daa. Chaper 7 summarizes he obained resuls. Moving average as a rading insrumen Moving averages (MAs) are one of he mos widely used echnical indicaors by he raders. MAs lay in he basis of he many echnical rules and sraegies. They are also used o consruc new echnical indicaors, such as Moving Average Convergence Divergence (MACD). Furher in he Chaper we presen he definiion and ypes of he MA. The rading sraegies, which incorporae MA, are discussed in more deails in Ch.4, 5 and 6 when applied o he hisoric daa.. Definiion of he MA Le s { } represen discree ime series (sampled a equal ime inervals); while { x } x N are he observed ime series wih N observaions: x represens he firs observaion and x N - he las observaion. A linear filer convers ime-series { x } ino { y } N by some linear N ransformaion (Gençay, Selçuk, Whicher, ). The oupu { y } is he resul of he convoluion of he vecor x wih a coefficien vecor w : i= y = w o x = w x (I..) i i Many applicaions are no feasible for i < as i implies he usage of fuure values. Therefore some resricions migh be imposed wih respec o i-parameer: x i= y = w x (I..) The filer (I..) is called a causal filer, while (I..) a non-causal filer. The oher classificaions are based on he impulse response of he filers:. Finie impulse response (FIR) filer. Infinie impulse response (IIR) filer Furher, we wil consider only FIR filers, defined as following: i i y = n i= w x i n+ i (I..3) The majoriy of he moving averages used in finance are he represenaives of he group of causal FIR filers. MAs can be consruced for any financial series. However, mos frequenly 4

37 { x } ime-series are eiher insrumen prices or he indicaors, derived from price (for N example, Momenum, Moving average convergence divergence indicaor, ec.). The example of he simple MA, a paricular ype of he MA wih all equal weighs ( w i = n, where n is he lengh of he rolling window used for he MA calculaions as in (I..3)) is given in Figure price SMA price, SMA observaions Figure.. Bund quoes (3/7/3-7//6, frequency 3 minues) and simple moving average of he lengh n=5 observaions From he expression (I..3), we can see ha MA has wo parameers involved in is definiion: n size of he rolling window 7 (sub-sample) used for MA calculaions: [ n +; ] w - vecor of he weighs for each observaion in rolling window; { i } i n In all echnical analysis applicaions he following wo consrains are imposed on weighs: n w i i= = (I..4) w > (I..5) i None of hese consrains are general filers requiremens, bu hey can be jusified in some of he cases. The consrain (I..4), or universaliy condiion, assures ha he MA can be considered as an unbiased esimaor of an insrumen mean, which is imporan when mean is unknown. Posiive weighs (consrain (I..5)) assure ha he MA, as an esimaor of he mean of some always posiive variable (e.g. price), does no reach negaive values; his migh happen if negaive weighs are aached o some exreme observaions. Noe ha while he universaliy condiion (I..4) is frequenly applied by geosaicians, he consrain (I..5) is no used in he geosaisical applicaions. 7 Furher, he erm window indicaes he inerval of daa used for a moving average calculaions. 5

38 MA defines a rend by smoohening he daa, i.e. removing he higher frequency componens from price daa. Daa smoohening of higher degree allows defining more clearly he exising (long-erm) rend in he daa. However, he negaive side of he srong daa smoohening is he eliminaion of shor-erm rends ogeher wih noise. As he resuls, he filered daa lag he original daa. In financial applicaions he lag mean ha MA does no anicipae he urning poin in he rend - he signal abou he change in he rend comes afer he rend reversal has acually aken place. MA canno predic local maximum or minimum of he price funcion, bu only confirm i. The smooher is he daa, he larger is he lag beween MA and price curves. As have been menioned in he inroducion, his rade-off lies in he basis of some opimizaion approaches.. Types of he moving average The weighs vecor defines he ype of MA. Many ypes of MAs exis in he financial applicaions. We can spli hem in wo large groups:. MA wih fixed (consan) weighs. MA wih variable weighs MA wih fixed weighs assigns he same weighs vecor o all moving windows; weigh value is only a funcion of he posiion wihin moving window: w i, = f ( i), where i relaive posiion of he weigh wihin esimaion (rolling) window; i = corresponds o firs observaions in he window, i = n - o he las observaions. Taking ino accoun ha he inerval i [ : n] is he same for all ime momens, he weighs are considered as fixed. The examples of he MAs wih fixed weighs are given in Appendix B. The mos widely used examples of his MA group are: simple moving average (SMA): MA = n i= w n n n i P i = P i = i= n n i= P i { } where P i i - price series; n - lengh of he window; exponenial moving average (EMA): ( α ) EMA EMA = αp + where { P i } i - price series; α - parameer of he EMA: α =, n - lengh of he SMA + n ha has he same lag. MA wih variable weighs assigns differen weighs vecor in differen ime momens; weigh is a funcion of oher variables (indicaors) a momen : w i, = f ( T ), where T - is he vecor of indicaors dependen on momen. The raders ry o adjus he MA o volailiy, rading aciviy, ec. Among hese MAs are Volume Adjused Moving Average, Variable index dynamic average (VIDYA), Variable-lengh moving average (VLMA) (see Appendix C for more deails on hese MAs). 6

39 Kriging mehod: Theory Kriging is par of he geosaisics used o esimae values of a random variable a some poin, where i is no observed. Kriging can also be used o define a rend in he daa, deerminisic or random. Maheron inroduced he erm kriging in 97. The mehod emerge as he improvemen of he moving average echnique developed by D.G. Krige, a Souh African mining engineer, for esimaing of gold grades; i was named afer him (Armsrong, 4). Kriging is a linear predicion mehod used o obain unbiased and efficien (in erms of a variance) spaial esimae/predicor Z ( x ) or he mean of he random process Z, from he available observaions: ( Z ( x ), Z( x ),..., Z( x n )). For example, in he mining conex, he kriging is used o find he grade a some chosen poin of space, aking ino accoun he informaion abou he grade available a he oher poins were he sampling has been done (Maheron, 97). Kriging mehod aribues he weighs o he grades of available sample deposi poins, creaing he weighed average esimae. The weighs are chosen in a way ha minimizes he esimaion variance. The opimal weighs ake ino accoun he geomeric form of he deposi, he posiioning of he available samples. Inuiively, we migh assume ha he sample poins disan from he predicion poin should have less weigh han he closer one. However, here are more complex phenomena in mining ha in some cases conradics he inuiion. Therefore, kriging mehod akes ino accoun no only geomerical form of he deposi and how he sampling poins are posiioned, bu also he saisical characerisics of a random variable Z, such as covariance (or variogram) funcions. The disincive feaures of he kriging approach are he following:. Kriging mehod is used for spaial esimaion. This implies he following daa peculiariies: daa frequenly canno be defined as pas and fuure ; daa is frequenly unevenly sampled; daa has coninuous raher han discree locaion indexing space 8.. Conrary o rend esimaion mehods ha need he predefined deerminisic funcion, kriging esimaes of he rend is based on he saisical characerisics of he daa. 3. Besides deerminisic rends, kriging can also esimae he random rend in he daa. 4. The kriged esimaes avoid sysemaic error, caused by he difference beween samples empirical and rue saisical characerisics. Maheron (97) explains ha he hisogram of he real grade of he deposi conains less exreme values and more inermediae values han he experimenal hisogram buil on he analysed sample, which ofen cause underesimaion of a mean of he deposi grade. The kriging procedure allows avoiding he underesimaion error. The kriging approach is developed wihin he scope of second-order saisical models ha use only mean and covariance (or variogram 9 ) model. Conrary o oher rend esimaion mehods, kriging approach is subjec o fewer consrains:. None assumpion abou disribuion properies of he random variables are made.. Kriging does no define a priori he funcion ha represens he rend. 8 Chiles, J.-P. and P. Delfiner. Geosaisics. Modeling Spaial Uncerainy, John Wiley and Sons, Inc., 999, p The erm is presened in more deails furher in Ch... 7

40 3. Some of he kriging ypes (universal kriging) allows addressing he problem of random rends. 4. Kriging allows represening rend in he form of a weighed average ha can be easily ranslaed ino a MA. 5. Almos all kriging ypes provide no only he esimaes of he rend, bu also he variance of he esimaors ha allows building he confidence inervals for he esimaors. The following ypes of he kriging exis:. Simple kriging (SK). Simple kriging is applied when he mean of a process is consan and known. The process should be saionary; covariance model is used o derive he kriging esimaes. ( x) = m Y ( x) Z known +. Ordinary kriging (OK). Ordinary kriging is applied when he mean is consan, bu unknown. In his case he process is inrinsic ; variogram model is used o derive he kriging esimaes. ( x) = m Y ( x) Z + 3. Universal kriging (UK) Universal kriging is applied when mean is variable and unknown. The variabiliy of rend can have eiher deerminisic or random naure. Z ( x) m( x) + Y ( x) ( x, w) m( x, w) + Y ( x w) =, m(x) - deerminisic rend Z =,, m( x, w) - random rend The applicaion of he simple kriging in our conex is limied due o a small amoun of financial insrumens, which can be considered as saionary wih known mean. The excepions are he echnical indicaors derived from he price daa, such as Momenum or MACD ha are oscillaors by naure and hus flucuaes around -line. Ordinary and universal kriging has much more applicaions possibiliies as hey can be applied o boh second-order saionary and non-saionary daa. Furher we presen each of he mehods in more deails. We sar, however, his chaper wih he descripion of some of he geosaisical erminology and conceps, used furher in kriging applicaions.. Geosaisic insrumens and erminology: shor overview Kriging involves some common saisical definiion like saionariy, covariances, as well as some insrumens less known for a wider saisical communiy, such as inrinsic saionariy and variograms. The erm is presened in more deails furher in Ch... Furher he descripion of he heoreical approach is based on he G.Maheron s book La héorie de variables régionalisées, e ses applicaions, Les Cahiers du Cenre de Morphologie Mahémaique de Fonainebleau, Fascicule 5, 97, pp.7-86, adjused for noaion o our paricular case. We will avoid addiional ciaions o simplify he reading of he ex. 8

41 .. Saionary and inrinsic saionary funcions Any random variable Z can be characerized by is probabiliy disribuion. However, an esimaion of he probabiliy disribuion funcion is someimes complicaed and imeconsuming. Thus, saisicians frequenly subsiue probabiliy analysis by he calculaions of he saisical momens, such as mean and variance-covariance marix. Random funcions are more complicaed saisical eniies, as hey represened by he ses of random variables: Z ( ), Z( ),..., Z( n ). The main problem wih he esimaion of heir saisical characerisics lies in he lack of observaions: in real life we ofen have a single realizaion for each variable: z ( ), z( ),..., z( n ). Thus, in order o make some valuable saisical inferences abou he random funcion from hese realizaions, i is ofen assumed ha variable is saionary. Saionariy imposes he invariance of a join probabiliy densiy funcion and, herefore, all is momens under emporal (or spaial) shif (ranslaion), i.e. for any value h, probabiliy disribuion of he variables Z ( ), Z( ),..., Z( n ) is he same as of he variables Z( + h), Z( + h),..., Z( n + h). For he emporal daa i means ha all momens of he random variable does no depend on he ime a which he variables are observed, bu only on he disance beween hem. Simply speaking, informaion abou he process [is] he same no maer where i is obained. However, even under saionariy assumpions a join probabiliy disribuion is ofen difficul o esimae; herefore, he saisicians has inroduced second-order saionariy - he invariance of he firs and second momens under ranslaion (see Definiion.). Definiion. A random funcion { Z ( ) } is saionary of order wo if is mean and covariance do no depend on ime, bu only on he disan beween he variable, i.e., h : Ε [ Z ( ) ] = m (I..) Cov Z, Z + h = Ε Z Z + h m = C h (I..) [ () ( )] [ ( ) ( )] ( ) The covariance (I..) has he following properies:. C( ) = σ - variance of he random variable. C( h) = C( h) 3. C( h) C( ) Noe ha he covariance for saionary variable or order wo is bounded. Unforunaely, he group of he second-order saionary processes is no very large. In financial applicaions hese processes are even more rare. For example, sandard Brownian moion does no belong o he group of second-order saionary process. Therefore, he larger group of he random funcions was inroduced o enlarge he group of random processes for which many geosaisical mehods, including kriging, can be applied. This group consiss of inrinsic funcions (see Definiion.). Definiion. A random funcion { ( ) Z } is inrinsic if is incremens are saionary; i.e., h Ε[ Z ( + h) Z( ) ]= (I..3) Anselin L. Variogram analysis, presenaion 9

42 Var [ ] = γ ( h) [ Z( h) Z( ) ] = Ε ( Z( + h) Z( ) ) + (I..4) The group of he inrinsic processes is significanly larger. In paricular, i conains he group of he saionary processes of order wo, i.e. each saionary process of order wo is inrinsic saionary (see Box.. for he proof); bu no every inrinsic saionary process is saionary of order wo (see Box.. for he example). Box. The proof ha each process, saionary of order wo is inrinsic saionary process X,, T is saionary of order wo. Then his implies he following Suppose some process { [ ]} equaliies: Les consider is incremens: Therefore, Var Cov Ε ( X ) = m Var( X ) = v ( X X ) C( h),. + h = >, h > : X + h X Ε ( X + h X ) = + h X = Var X + h + Var X Cov X + h, X = v. Then he following equaliies hold: ( X ) ( ) ( ) ( ) C( h) = γ ( h) { X, [, T ]} is inrinsic. Box. Inrinsic saionary process: Sandard Brownian moion A random process { W, [, T ]} is a sandard Brownian moion if:. >, s > : W + s W N(, s) (normally disribued wih -mean an s - variance). >, s > : W + s W is independen of W. 3. W is a coninuous funcion of ime and W =. As we can see from definiion he incremens of he sandard Brownian moion >, s > : W + s W are saionary of order :. Ε [ W + s W ] =. Var[ W + s W ] = s 3. Cov[ W W, W ] = + s, [ ]} { } Therefore, { W, T is inrinsic saionary. However, we know ha covariance of he process W, E[ W Wτ ] = min(,τ ) is ime-dependen. Thus, he process iself { W, [, T ]} is no saionary of order. As he resul, we have showed ha an inrinsic process is no necessary a saionary process of order wo. Inrinsic random funcions of order k (IRF- k ) are he generalizaion of he inrinsic funcions. The IRF- k is a random funcion wih saionary incremens of order k 3. IRF- is he inrinsic funcion wih saionary incremens. IRF- k enlarge he group of he processes for which he kriging mehod can be applied. The analogy of hese funcions in he ime series is he ARIMA 3 Chiles, J.-P. and P. Delfiner. Geosaisics. Modeling Spaial Uncerainy, John Wiley and Sons, Inc., 999, p. 45.

43 processes: ARIMA of order is, in fac, IRF- d. There are sill differences beween ARIMA and IRF- k models. Firsly, ARIMA models are discree, while IRF can be boh coninuous and discree. Secondly, ARIMA models are compleely specified, while IRF should only be second-order model. Finally, ARIMA are one-dimensional, while IFR can be defined in n R. d ( ) In his work we concenrae mainly on saionary and IRF- models. Alhough IRF-k models allow working wih non-saionary daa, he experience has shown ha in he pracical applicaions oo much informaion abou variable is los when he esimaion procedures use is saionary incremens. Similar conclusions were he incenive for he developmen of coinegraion... Variogram Definiion of he inrinsic funcions is based on he variance of incremens. This variance is called variogram, a concep widely used in geosaisics. Definiion.3 The following funcion ( h) γ is called semi-variogram, or less formally variogram: [ ] γ ( h) = Var[ Z( + h) Z( ) ] = Ε ( Z( + h) Z( ) ) (I..5) Expression (I..5) indicaes a significan advanage of he variogram over he covariance: is definiion does no involve variable mean ha is usually unknown and should be esimaed. For saionary variables he following relaionship exiss 4 beween variogram and covariance: ( h) = C( ) C( h) = σ C( h) γ (I..6) This relaionship is represened graphically in Figure.. A variogram has he following characerisics:. γ ( ) =.. I can be disconinuous jus afer he origin (so called nugge effec). 3. Variogram is bounded for he saionary variables of order wo and end o be increasing for non-saionary variable. Several parameers characerize a variogram (see Figure.3). Sill is a level, by which he variogram is bounded. A lag, a which he variogram is sabilized around he sill level, called range. The range indicaes he lag a which here is no more correlaion beween samples (no auocorrelaion for he ime dependen random funcions). If disconinuiy is presen a he origin, i is called nugge effec. 4 The formulae (I..5) can be rewrien as following: γ ( h) = Ε ( Z( + h) Z( ) ) = Ε Z + h = [ C( ) C( h) ] = σ C( h) [ ] [( ( ) m) + ( Z( ) m) ( Z( + h) m) ( Z( ) m) ] =

44 Appendix D presens he mos frequenly used variograms ( h) ( ) γ and covariance C h models. Variance Variogram h Covariance Figure.. Relaionship beween variogram and covariance for saionary variable. Variance sill nugge effec range Figure.3. Variogram parameers h

45 According o Chilès, Delfiner (999), here are wo ways of fiing model o he empirical variogram:. Manual fiing. Auomaic fiing Frequenly geosaisician fi he model manually as variograms are non-linear in heir parameers, such as range (Maheron, 968). Auomaic fiing can be performed by he leas square echnique: ordinary leas square, generalized leas square or weighed leas square (Cressie, 99). As many observaions are available, furher in he applicaions we will use manual approach o variogram fiing. I should be sressed ha missing observaions canno be ignored in a variogram esimaions. If daa is missed due o a regular absence of he rading aciviy (for example, week-ends or nigh hours) his missing daa can be ignored. In order o avoid an overnigh effec (he absence of he overnigh daa) an empirical variogram can be subsiued by he average daily variogram. However, if daa is missing due o he irregular aciviy (for example, holiday), he missing daa should be reaed as non-available and differences in he variogram formulae ha incorporae hese observaions should no be aken ino accoun.. Simple Kriging: predicion of he process wih zero mean Yn + n+ H n+ n + Y () Y H ( s) K The kriged esimaor is he orhogonal projecion of he on a Hilber space H s, i.e. K Yn K is a unique elemen of he s, such ha Y is orhogonal o all oher Y. + () H () s In our case, he Hilber space represens he linear span of he available known poins. ) and variance- Y a Le s consider some random funcion Y ( ) wih zero expecaion ( Ε[ Y ( ) ] = 5 covariance marix σ uv = cov[ Y ( u), Y ( v) ]. Le Y α be realizaions of he random variable ( ) α = {} i Y () i n some experimenal poins. Les represen he kriged esimaor of he a poin n+ as a linear combinaion of Y : Y K α n λk, i i= α ( n+ ) = λky α = Y i (I..7) α The weighs vecor λ K minimizes he esimaion variance var ( Y Y ) [ ] [ ] = Ε ( Y Y ) n+ K n+ K 6 : 5 The variable can be a funcion of any variable (ime, volume, ec.). We use index ha corresponds o he ime coordinae, as his is he case mos ofen addressed by he echnical analysis. However all kriging mehods can be applied o oher cases. Ε Y YK =, and 6 Noe ha [( () ( ))] n n n Ε[ ( Y Y ) ] = Ε( Y ) E( Y Y ) + Ε( Y ) = σ λ Ε( Y Y ) + λ λ Ε( Y Y ) = σ n+ n+ K n i= λ σ i n+, i n+ + n n i= j= i n+ λ λ σ j K i, j K n+ i= i n+ i i= j= i j i j =. 3

46 n n n [( Y n Y K ) ] = σ n λiσ n, i λiλ jσ i, j i= Ε (I..8) i= j= Taking parial derivaives of he funcion (I..8) wih respec o he weighs vecor following sysem of n-equaions wih n-unknowns: α λ K gives he β K αβ = σ α, n+ λ σ, (I..9) where i-h equaion is: n j= λ σ = σ j i, j n+, i The sysem (I..9) is regular and have unique soluion if marix σ αβ is sricly posiive definie, which is usually assumed for heoreical variance-covariance marix. Kriging variance, defined in (I..8) akes he following form 7 : n [( Yn+ YK ) ] = σ n+ σ = Ε λ σ (I..) K.3 Simple kriging: predicion for he process wih known mean Le Z be some random funcion wih known mean Ε ( Z ) = m and variance-covariance marix ij [ Z ] σ = cov Z,. i j i= Les define some new random variable Y as following: i n+, i Y = Z Ε ( Z ) = Z m Y has zero mean and σ ij variance-covariance marix. ( n+ K Then he opimal esimaor of Z ) is he kriged esimaor of he following form: Z ( Z m) Z n + K α = m + Yn+ = m + λk (I..) K n+ α α where λ K are he kriged weighs ha saisfies kriging sysem (I..9) for Y. The variance of kriging is he same as in (I..) 8 : 7 α β α The equaion (I..9) implies ha: λ λ σ λ σ, or λ λ σ = λ σ. Then subsiuing his expression in (I..8) leads o (I..). K K αβ = K α, n+ n n i= j= i j i, j n i= i n+, i 4

47 K [( Z n Z n+ ) ] = Ε ( Yn+ YK ) n [ ] = σ n+ σ = Ε λ σ (I..) K +.4 Ordinary kriging: predicion of he process wih unknown mean Ordinary kriging is applied o he random funcion Z, when is mean is unknown. Suppose Z ( ) is saionary, i.e. is mean m is unknown, bu consan: Ε [ Z ( ) ] = m. σ uv = cov[ Z( u), Z( v) ] represens is variance-covariance marix. Le Z α be realizaions of he random variable Z ( ) a some experimenal poins α = {} i. A kriged esimaor of he Z ( ) a poin is a linear i n n+ Z α combinaion of : i= i n+, i Z K n λi i= α ( n+ ) = λk Zα = Z i (I..3) [ ] α The opimal weighs λ K should minimize he esimaion variance var ( Z ( n+ ) Z K ( n+ )). As K mean is unknown he kriged value Z n + should be an unbiased esimaor of he Z ( n+ ), i.e. he mean of he esimaion error should be zero: [( ( ) Z ( ) Ε n+ K n+ )]= Z (I..4) Taking ino accoun ha Z n = and Ε Z m condiion (I..4) akes he following form: Ε n λi i= Ε ( ( + )) m ( K ( n+ )) = n Z n+ K n+ λ i = (I..5) i= [( ( ) Z ( ))] = m The equaliy (I..5) holds only if m = (he case of simple kriging), or n i= λ = (I..6) i Condiion (I..6) is called an universaliy condiion. As he resul, he kriging problem can be reformulaed as he following opimizaion problem: 8 The equaion (I..) implies ha ( Z K n ) = m K K var[ ( Z )] = Ε ( Z Z ). [ n+ n ] Z n+ n+ + [ )] K Ε + ; herefore Ε ( Z n + Z n + = and 5

48 min λ ( var[ ( Z( ) Z ( ))]) n i= n+ λ = i K n+ (I..7) If universaliy condiion (.6) holds hen Ε [( ( n+ ) Z K ( n+ ))] = [( Z ( ) Z ( ))] = Ε ( Z( ) Z ( )). n+ K n+ n+ [ ] var K + n Z and Wih respec o his we can rewrie (I..7) as following: n n n min σ n+ λiσ n+, i + λiλ jσ i, λ i= i= j= n = λi i= j (I..8) The soluion o he problem (I..8) should saisfy he following sysem of equaions: n j= λ σ = σ j n ij j= i, n+ λ = j + μ (I..9) where μ - is a Lagrange muliplier. The variance of kriging is 9 : n+ + n σ = σ μ λ σ (I..) K i=.5 Ordinary Kriging: Esimaion of he unknown mean () Le Z be some random funcion, saionary of order wo. Is mean m is unknown, bu consan: Ε [ Z () ] = m. σ uv = cov[ Z( u), Z( v) ] represens is variance-covariance marix. Le Z α be realizaions of he random variable Z ( ) a some experimenal poins α = {} i. According i n o Maheron (97) he esimaor of he m as a linear combinaion of he available observaions : Z α i n+, i m * α = λ Z m α = n i= λ Z m i i (I..) 9 α β α α The expression follows from (I..9): λ λ σ λ σ + μ λ = λ σ + μ. αβ = α, n + α α, n + α 6

49 α α Noe ha he weighs vecor is no he same as vecor weighs, used o define he () predicor of he funcion Z. λ m In order o assure he unbiased mean esimaor, we impose he universaliy condiion: n i= λ =. To obain he efficien esimaor, we minimize he variance of he esimaion error: var * * [( m m )] = var ( m ) m i n n [ ] = i= j= m i m j ij λ K λ λ σ (I..) As he resul, he search of he opimal esimaor (I..) is reduced o he following opimizaion problem: min λ n n i= j= n m λi i= m m λ i λ j σ i, j = (I..3) The opimal weighs α λ m should saisfy he following sysem of equaions: n m λ j j= n where μ m - is a Lagrange muliplier. j= The corresponding kriging variance is : σ λ ij m j = μ = m (I..4) σ K = μ m (I..5).6 Kriging non-saionary variable There are several approaches in he geosaisics o he kriging of a non-saionary variable:. Universal kriging. Kriging an inrinsic funcion IRF-. 3. Kriging an inrinsic funcion IRF-k. α β Noe ha he following expression follows from (I..4): λ λ σ αβ = μ m λα = μ m. α 7

50 We will furher presen only firs wo approaches. As have been menioned already in Ch..., he kriging of he IRF-k leads o he loss of some informaion abou he principal non-saionary variable. In addiion, he main objecive of ou sudy is o define a drif of our (non-saionary process). The problem is ha in general IRF-k has no uniquely definable drif, for he excepion of he case when he analyzed process ( x) process Y s ( x) plus some polynomial drif: ( x) = Ys ( x) + l Z Al x (I..6) l Z can be represened as he saionary This represenaion (I..6) coincides wih he universal kriging model wih random coefficiens (Chilès, Delphiner, 999)..6. Universal kriging: rend esimaion Z () be some random funcion, which is non-saionary wih unknown mean m. Le σ = cov[ Z( u) Z( v) ] represens is variance-covariance marix: uv, Ε Ε[ Z( ) ] = m( ) [ Z( ) Z( )] = m( ) m( ) + σ ( ), (I..7) The universal kriging (UK) provide he bes linear esimaor of he rend consider he funcion m = Ε[ Z() ]. We can m ( ) o be regular and coninues or irregular and a random variable iself. There are he following hypohesis on which he esimaion mehod is based:. The m() funcion is esimaed locally and i can be approximaed by he following expression: f l () k l () = a f () m l= l where a funcion, chosen a priori and fixed hrough all applicaions (for example, ime polynomial); a l - unknown coefficien corresponding o l-funcion (migh be a funcion of.. We suppose ha he covariance (variogram) beween wo poins of ime can be esimaed locally as γ (, ) = ωγ ( ) and i is deforming slowly. The parameers of he covariance (variogram) funcion should be esimaed and furher conrolled, which is quie complicaed ask. However, o simplify hings, we suppose ha he parameers are known a priori. 3. Z() is some random non-saionary funcion ha saisfy condiions (I..7). Le Zα be he realizaion of he random variable Z a some experimenal poins., ( ) {} i i n α = Suppose ha Z() can be represened as he following process: Chiles, J.-P. and P. Delfiner. Geosaisics. Modeling Spaial Uncerainy, John Wiley and Sons, Inc., 999, p. 7. We define he universal mehod for he ime coordinae, hough as in he previous definion of he kriging - variable can be subsiued by he spaial x-coordinaes. 8

51 where E Y [ ()] (, w) Y (, w) + m( w) Z =,, (I..8) m (, w) - rend of he series, random funcion; Y (, w) - random funcion, such ha =, some Y are correlaed. Alhough i is no necessary, les assume ha Y (, w) and ( w) m, are independen, in order o divide he srucural effecs on process Z ( ) from each of hem. Furhermore, m (, w) should by is naure have much more larger effec on he Z ( ) and much more smaller volailiy han Y (, w). The regulariy and coninuiy of he m (, w) is aained by he following assumpion: ( ) l i (, w) a ( w) f () = a f (), V, f () = m l = l i (I..9) i= where a l w are random variables ha reflec he coefficiens of he rend; f l are polynomials of he order l ha defines he form of he rend. Les assume for simpliciy reason ha he rend is described by he linear funcion: ( w) = a ( w) + a ( w), V m, o (I..3) () For he rend esimaion, les assume ha α S V, wih S bounded. Z are he values of Z ( ) a he experimenal poins α Les assume ha he esimaor of he rend m() a one poin V is he following: α ( ) = λ Z = λ Z( ) = λ Y ( ) + λ m( ) * M UK α ( n) i α i i i α i i i α i i, (I..3) The opimal weighs vecor α λ UK minimizes he esimaion variance: E * α β ( M m ) = λ σ αβ = wih σ = cov( Y ( ), Y ( ) = E( Y ( ) Y ( ) ij i j λ λ λ σ, (I..3) i j i i α( n) j α( n) The universaliy consrain for he minimizaion problem (I..3) is: α λ (I..33) l l α fα = f j ij 9

52 Le Σ represen variance covariance marix of Y, wih he elemens σ ij ;... simple funcions in he form: f = [...],... F = n n... Then he minimizaion problem (I..3)- (I..33) in marix represenaion is: F - he vecor of Σ F F λ = μ f (I..34) The UK variance of he mean esimaor: UK * [( M m ) ] = ' f = f ( F' Σ F ) σ = Ε μ (I..35) f I should be menioned here ha for a finie case he UK model could be regarded as a linear regression model wih he correlaed residuals of he following (marix) form (Chilès, Delphiner, 999): Z = Fa + Y Then he generalized leas square (GLS) soluion for he opimal (unbiased, efficien) esimaor of he coefficiens vecor a will be: The covariance of he esimaed residuals is: a = ( F Σ F ) F Σ Z * * * ( Fa )( Z Fa ) = Σ F( F Σ F ) F Ε Z (I..36) The expression (I..36) shows ha he covariance of he esimaed residuals is a biased esimae of he covariance of he rue residuals Y..6. Kriging inrinsic funcion IRF- The problem wih he inrinsic funcion IRF- is ha he consan erm of he drif coefficien a canno be deermined, since he random funcion is deermined by is incremens (variogram). In order o avoid his problem, we can suppose ha over some limied domain (for example, ime-inerval) for some very large A >, funcion A γ ( h) represen he covariance. Subsiuing he new covariance ino he opimizaion problem (.3)-(.33) we will obain similar equaion sysem ha does no depend on A: Γ F F λ = μ f (I..37) 3

53 γ αβ where = [ ] Γ is he variogram marix., which is sricly condiionally negaive definie. This condiion is me for a valid variogram models. Conrary o he UK approach wih covariance, he kriging variance of he mean esimae canno be defined. 3 Peculiariies of he kriging mehod applicaions in finance As have been menioned already kriging mehod is developed o confron he spaial daa. In his chaper we discuss how he difference in he financial and spaial daa migh have impac on he financial applicaions of he kriging mehod. Mean value of geosaisic daa is esimaed for some sample ha forms a close neighbourhood o he esimaion poin. For he financial daa, only pas observaions are available, herefore he close neighbourhood is formed by he sub-sample ha precedes he esimaion poin. If we choose some n value for he lengh of he sub-sample ha precedes and conain he esimaion poin and consider i as he close neighbourhood, hen he kriged esimaor of a variable mean a each momen of ime will coincide wih he definiion of he weighed MA in he (I..3). There are wo principal peculiariies of he financial daa:. The majoriy of financial samples are non-saionary due o he presence of rends in he daa.. Many financial variables are sampled a equal disances. This is rue in paricular for a low frequency daa (for example, daily, monhly, annual observaions). 3. Daa non-saionariy As have been shown in Chaper, all kriging approaches are based on he second-order momens of he process Z or residuals Y ha supposed o be known. In real life, however, he covariance (variogram) should be esimaed firs and a valid model should be fi o he esimaes. In he case of he saionary insrumen, he raw variogram can be easily esimaed and fied. However, he presence of a drif in he daa inroduces a bias ino he esimaes of he raw variogram. Therefore, in order o obain he bes possible esimae of he rue variogram he drif should be removed from daa and he variogram of he residuals should be esimaed (Chilès, Delphiner, 999). The problem is ha he rend is usually unknown; hus, is esimaes should be used o define residuals. Le say he realizaions of a random funcion { Z } i are available a he experimenal poins x α. This process is represened as following: ( x) m( x) Y ( x) Z = + (I.3.) Les define he residuals R( x ) = Z( x ) mˆ ( x ), where ( ) x α. Les consider a variogram of he residuals: α α α x α mˆ is he rend esimaor a poins 3

54 γ R ( xα, xβ ) Ε( xβ xα ) = γ ( xβ xα ) Cov( Z β Zα, mˆ β mˆ α ) + Var( mˆ β mˆ α = ) (I.3.) The expression (I.3.) can be simplified if only mˆ ( x) is an opimal linear rend esimaor ( x) Then: γ R * * ( x x ) γ ( x x ) Var( m m ) α, β α = β α β (I.3.3) m *. Expression (I.3.3) shows ha even if he opimal rend esimaor is chosen he residual variogram migh sill underesimae he rue variogram. The good news is ha his bias is small a shor disances, bu can be significanly increased a large disances (Chiles, Delfiner, 999). Chiles, Delphiner (999) believe ha despie he presence of a rend i is always possible o reurn o he sandard srucural analysis of he saionary case. For example, if rend m( x) is mild, han he esimaed empirical variograms of Z on several daa sub-samples will no differ significanly a he shor disances. As kriging ofen is applied o raher close neighbourhood of daa, he empirical variogram can be acceped as a good esimae of he rue variogram. A he siuaion when we can assume he saionariy of he residual erm Y x and a some sufficienly large sub-sample of he available observaions he rend is equal o zero: m( xν ) =, xν xα, hen he empirical variogram esimaed on his sub-sample can again be acceped as a good esimae of he rue variogram. As he resul, we have he following soluions o address daa non-saionariy:. Assume ha a process Z can be represened as in (I.3.) wih saionary residuals Y. Then he covariance/variogram can be esimaed by using one of he wo approaches: a. Esimae and eliminae presen rend in he daa; hen use he variogram of he residuals as an esimae of he rue variogram. b. Esimae he empirical raw variogram on he sub-sample, where no rend is observed.. Assume ha process Z is non-saionary; use inrinsic model o fi he raw variograms and apply approach.6. o obain he opimal weighs esimaes. As for he soluion (a), he esimaion of a rend should be done before he applicaion of he kriging mehod. As have been shown in (I.3.3), in order o minimize he error when acceping he empirical residuals variogram for he rue variogram, we should apply he linear mehod o he rend esimaion. For example, line or polynomial can be fi o he daa by leas square mehods. However, curve fi will demand he subjecive choice of he polynomial. We propose o esimae rend as a moving average of predefined lengh. In paricular, we propose o subrac EMA of predefined (medium) lengh ( EMA, ) and evaluae he variogram model for he price residuals } T ({ Ri : Ri = Pi EMAi ). The mehod is no opimal in saisical erm, bu i has he n+ i following advanages:. EMA is he Markov ime indicaor in he sense ha only available hisoric daa is used for is calculaions.. EMA, as mos popular echnical indicaor, is inroduced in many rading sofware making hem very easy o use. 3. EMA has only one parameer o choose is lengh. Our main crierion is o choose such lengh, which guaranees he convergence of he residuals variogram a some no very large range values. From he rading poin of view, EMA should reflec a medium-erm ( ) 3

55 rend. The EMA lengh cerainly depends on he daa frequency and raders ime horizon. For example, for -second frequency and inraday rader medium erm can correspond o he wo hours (*6*6=7) window lengh; while for he 3-minues frequency i migh corresponds o one week MA lengh. We also propose o use only one simple funcion f = in he rend, implying consan (wihin moving window) unknown rend. In he empirical geosaisical applicaions he number of funcions in he rend is usually limied o one or wo, as more funcions were no improving significanly he resuls. 3. Daa sampling peculiariies The way he financial daa is sampled has direc effec on he resuls of he kriging mehod applicaion. On one hand, non-regular sampling ypical for insrumens ha are no raded frequenly, or for subordinaed processes, jusifies he usage of he kriging approach a he place of he usual filering mehods. On he oher hand equal-space sampling has an ineresing impac on he srucure of he opimal kriging weighs. In fac, Caselier, Laurenge (993) showed ha in he case of regular sampling he opimal weighs for he mean esimaor has quie similar behaviour. Under assumpion of one simple funcion in he rend, we have relaively high weighs for firs and las observaions in he window, and relaively low in he absolue erms (someimes negaive) he res of he weighs. They have derived he following close-form soluion for he case of he exponenial variogram model: λ = λn = N ( N ) b b λi =, i : < i < N, (I.3.4) N b ( N ) b = e a where N is a lengh of he window (lengh of he sample of observaions used in he kriging h procedure), a - range parameer of he variogram γ ( h) = σ e a. The expression (I.3.4) shows ha opimal weighs have he same srucure independenly on he lengh of he sample N or he variogram parameers; hey have impac only on he absolue values of he weighs. The examples of he weighs srucures for differen ypes of he variogram models are proposed in Figures The following models are considered: α h Sable model: ( ) = a γ h σ e ; Fracal model: γ ( h) α = σ h ; 33

56 Spherical model: ( h) 3 3h h σ, h a = 3 γ a a ; σ, oherwise All figures.4-.8 show similar resuls: concenraion of he principal weighs for he firs and las observaions in he sub-sample window, wih comparaively lower and relaively sable oher weighs. The weighs srucures for he sable model for differen parameers are given in Figures Figures.4-.5 represen he impac of he range parameer a on he weighs values, while fixing α a wo levels: α < and α >. Figure.4 shows he increased weighs volailiy wih increase in he range parameer for α =. 5 : very high range values corresponds o very high border weighs (heir sum is close o ), which are compensaed wih relaively large negaive weighs (# and #9) and more or less sable and small (negaive or posiive) oher weighs. Figure.5 presens much smooher parabolic weighs srucure, convexiy of which increases wih he increase in he range parameers (he border weighs also increases wih range parameer). Figure.6 show he ransformaion of he weighs srucure from he volaile ype (as in Figure.4) o he parabolic ype (as in Figure.5) wih he increase of α -parameer from o a= a=5 a= a=5 a= weighs value weighs # (# corresponds o mos recen observaion) Figure.4. Weighs, which correspond o differen a variable in sable model: window lengh=, α =.5. 3 For he range parameer a > for he uni sampling frequency, he oscillaion of he weighs for he gaussian model or α = is much more severe. See Caselier, Laurenge (993) for more examples. 34

57 weighs value a= a=5 a= a=5 a= weighs # (# corresponds o mos recen observaion) Figure.5. Weighs, which correspond o differen a variable in sable model: window lengh=, α =.5. weighs value alpha=. alpha=.5 alpha=.7 alpha= alpha=. alpha=.5 alpha=.7 alpha= weighs # (# corresponds o mos recen observaions) Figure.6. Weighs, which correspond o differen α variable in sable model: window lengh=, a =. 35

58 weighs value alpha=. alpha=.5 alpha=.7 alpha= alpha=. alpha=.5 alpha= weighs #: # corresponds o mos recen observaion Figure.7. Weighs, which correspond o differen α variable in fracal model: window lengh=, a =. Figure.7 presens he weighs srucure for he fracal variogram model. Fracal model corresponds o he IRF-k funcions. The weighs srucure resembles he case of sable model for α <. Figure.8 considers he case of he spherical variogram model. The case when a = implies he opimal MA in he form of he simple moving average; as daa is considered o be sampled a uni disance a = implies he absence of he correlaion and he closeness of he daa o he whie noise. For he oher range values, he same persisence of he large weighs for firs and las observaion in window is observed. 36

59 a= a=5 a= a=5 a= weighs value weighs #: # corresponds o mos recen observaion Figure.8. Weighs, which correspond o differen a variable in spherical model: window lengh=. As he resul, we can see ha independenly of he window size and variogram models 4, kriged MAs are no significanly differen from SMA in he erm of lag, while some of hem are less smooh (more volaile). In fac relaive volailiy of he weighs depends only on he variogram model (no on he lengh of he window), in paricular on he regulariy of he variogram model a he origin. Varying parameer α (in sable or fracal models) has impac on he form of he weighs curve and adds significanly o he volailiy of he kriged MA. We can expec ha kriged MA for sable and spherical models are more or less smooh; a he same ime he kriged MA for fracal model supposed o be very volaile and unsable (boh borders weighs are larger han ). As he resul, we can conclude ha in he case of regular sampling (one-dimension), kriging opimal weighs have he similar srucure: for he excepions of he border weighs (firs and las observaion), oher weighs are quie low in he absolue erms. The «border» behavior of he weighs in is urn is defined by he covariance srucure of he variable and depends on he regulariy of he variogram a he origin, i.e. by he presence of he range. Pure nugge model gives arihmeic average; spheric and exponenial models produced more imporan oscillaions around arihmeic weighs, while gaussian model produce violen oscillaions (Caselier, Laurenge, 993). Furher we presen examples of he kriging mehod applicaions o he differen se of daa. In Ch.4 and 5 we presen he case of he insrumens, sampled a equal inervals. Chaper 4 consider non-saionary examples of prices, while Chaper 5 concenraes on he MACD indicaors wih bounded pahs wihou disincive rends. Chaper 6 presens he examples of he applicaion of he kriging mehod o he case of he daa, sampled a uneven inervals. 4 A leas hree classical model considered above and in Caselier, Laurenge (993). 37

60 4 Kriging resuls: Non-saionary, evenly spaced imeseries daa This chaper analyses he applicaion of he kriging o four differen insrumens: () Bund; () DAX; (3) Bren; (4) X insrumen 5. The analysis of he Bund is presened in deails, while only rading oucomes are presened for oher insrumens. 4. Variogram analysis and opimal kriging weighs: Bund The Bund sample represens he quoes 6 for hree differen conracs 7 due March 8, June 8 and Sepember 8, 6, sampled a one-second frequency. The daa is sampled a very high frequency, herefore, even for Bund, an acively rading insrumen, he daa is no available a each poin of ime. In order o obain equally spaced daa, missed daa is inerpolaed a he levels of he las available daa. Example of he Bund price pah for he conrac due on March 8, 6 is given in Figure.8. Oher conracs can be find in Figures E, E in he Appendix E. All figures indicae he presence of rends in he daa. Moreover he rends are clearly non-linear. Figure.9 suppors he hypohesis of he non-saionariy wih he unbounded price variograms evaluaed for he Bund conracs. Linear variogram implies he price process could be modelled as an IRF-. We have fi he following linear model o he variogram for he March 8, 6 Bund conrac (h in seconds): γ 6 ( h) =.93 h The opimal weighs esimaed for n=36s under assumpion ha local covariance is A γ ( h) for some very large A > have similar srucure han he weighs considered in he Chaper 3 (Figure.7): λ = λ.5, λ = = λ. 36 = = The analysis of he variograms for he residuals is more complicaed. 5 Due o he confidenialiy reason we canno presen deailed descripion of he insrumen. 6 The Bund quoes, used for vaiogram and kriging calculaions, are in fac he index buil on he basis of differen Bund prices (quoes). Due o he confidenialiy reason we canno provide he formulae. 7 Bund is a fuures conrac. 38

61 3.5.5 quoes observaions(s) x 6 Figure.8. Bund (December 9, 5 - March 8, 6, frequency sec).5. March 8 June 8 Sep 8.5 Vario lag (in s) x 4 Figure.9. Variograms for differen Bund conracs due a differen daes in 6 (December 9, 5 - March 8, 6, March 9 June 8, 6, June9-Sepember 8, 6, frequency sec) 39

62 As have been discussed already in Chaper 3, in order o obain he esimaes of rue covariance, he rend should be eliminaed from he daa. Furher we provide he analysis of how differen EMAs (he rend esimaor) can affec he form of he variogram/covariance used in kriging applicaions. Moreover, we would like o analyse wheher covariance srucure of he residuals is sable over differen conracs. I is obvious ha differen lenghs of he EMAs, used for rend subracion, will have impac on he parameers of he variogram (a leas is sill). Effecive lengh of he EMA defines how close he EMA is approaching he price curve and how smooh i is; longer EMA lengh implies smaller disance beween wo curves and smooher EMA curve. As he resul, he variance of he residuals will increase wih he increase of he EMAs lengh. Figure. presens he variograms, esimaed for Sepember 8, 6 conrac, which correspond o seven differen EMA lenghs. As expeced, he sill of he variograms is an increasing funcion of he EMA lengh. Vario Variances: 'n=8':. 'n=36':. 'n=7':.4 'n=44':.7 'n=396':.9 'n=79':.39 'n=98':.84 n=8 n=36 n=7 n=44 n=396 n=79 n= lag (in s) x 4 Figure.. Bund (June9-Sepember 8, 6, frequency sec): Variograms for Bund (8 Sepember conrac) residuals, which corresponds o EMAs of differen lengh The impac of he EMA lenghs on he range of he variograms is hough unclear from he Figure.. Therefore, we propose o consider he variograms, normalized by heir respecive sill (see Figure.). From Figure. we can see he ranges of he variograms also depend on he lengh of EMA, used for he rend subracion: he longer is he EMA he larger is he variogram range. The analysis of he residuals over differen conracs shows wheher he covariance srucure is he same over he ime. Figures. and E3-E4 in he Appendix E show he examples of he variograms for he residuals for differen Bund conracs, obained by he subracion of he EMA of he following lenghs: n = 8 sec (3 minues), n = 7 sec ( hours), n 3 = 79sec ( days) respecively. We canno conclude from he observed variograms abou 4

63 residuals saionariy. All he variograms sabilized around some sills. However, hese sills are all well below he esimaed variances..4. Vario n=8 n=36 n=7 n=44 n=396 n=79 n= lag (in s) Figure.. Bund (June9-Sepember 8, 6, frequency sec): Normalized variograms for Bund residuals, which corresponds o EMAs of differen lengh 8 x March 8 June 8 Sep 8 Vario Variances: March 8=.9 June 8=. Sep 8= lag (in s) Figure.. Bund (December 9, 5 - March 8, 6, March 9 June 8, 6, June9-Sepember 8, 6, Sepember 9 December 8, 6, frequency sec): Variograms for Bund residuals (n(ema)=8 sec) for differen conracs 4

64 Sabilizaion of he variograms below variance implies persisen auocorrelaion (correlaion is no longer = a large lags). However, explanaions for his phenomena can lay also in he daa peculiariies such as non-consan volailiy caused by such evens as overnigh jump of he prices, high volailiy «afer 4:3 hours» ec. These evens can cause an overesimaion of he rue variance. Therefore, calculaion of he average variogram over he periods ha does no include hese effecs can be one of he soluions for his problem. For example, he esimaion of he average «daily» variogram can help o define wheher overnigh effec can be a cause for hese ypes of he variograms ha we have observed. Esimaion of he average variogram (and average variance) over all available (non-) days for hree differen conracs can be done according o he following formulas: N ( days) ( h) γ i i= Ε( γ ( h) ) = (I.4.) N N ( days) ( Bund ) vari i= Ε( var ( Bund )) = (I.4.) N For hese esimaions we ake only days wih complee series of prices. Our maximum lag is cerainly consrained by day (in realiy, by more lower value). Figure.3 shows he esimaed average variograms for hree Bund conracs. The variograms show ha daily variograms have sabilized around esimaed variance. New esimaed variance is lower han he variances esimaed over he 3 monhs sub-sample. This means ha he presence of he overnigh effecs can be he cause for variance overesimaion. The form of he average variogram is smooh and «model-like», which will faciliae significanly he variogram modeling. The variogram range is approximaely equal for all conracs, while he sill is differen. Therefore, we can expec he same weighs for he opimal MA forecas, bu differen variance of he esimaor for differen Bund conracs. As he resul of he analysis, we can see ha he range parameer of he residuals variogram is consan over ime (does no depend on he conrac), however is value depends direcly on he EMA lengh used for he residuals consrucion. The residuals variance is ime-dependen and also depends on he EMA lengh. In order o obain he kriging equaion sysem ha is non-singular, a model should be fied o he empiric covariance esimaes, which will guaranee posiive definieness of he variancecovariance marix. We choose o model he average daily variogram in Figure.3 for he March 8, 6 conrac. I means ha we use he EMA of he -hour (36 seconds) lengh as a h rend esimaor. Figure.4 proposes he fi of he exponenial model ( ) = γ h σ e a wih he parameers a = 9, σ =.. 4

65 .5 x -3 Vario March 8 June 8 Sep lag (in s) Figure.3. Bund (December 9, 5 - March 8, 6, March 9 June 8, 6, June9-Sepember 8, 6, frequency sec): Average variograms over one day for Bund residuals (n(ema)=36s ( hour)).6 x vario empiric (March 8, 6) heoreical (exponenial) lag (in sec) Figure.4. Bund (December 9, 5 - March 8, 6, frequency sec): Empirical average daily variogram for Bund residuals (n(ema)=36s ( hour)) and heoreical model h γ ( h) =. e 9 43

66 From Chaper 3 we know ha equally spaced samples will produce he opimal kriging weighs of he paricular form. In paricular, for he exponenial variogram model, hese weighs values can be even calculaed analyically according o he formula (I.3.4) (n=7s): λ = 9.9* i λ = λ.78, 7 = 5, i =,3,...,799 The opimal weighs are presened in Figure.5. The kriged moving average (KMA), which corresponds o hese weighs is given in Figure.6. As can be observed a his lengh KMA is very close o SMA of he same lengh, bu less smooh. More volaile naure of he KMA is demonsraed in Figure.7 for a shorer SMA and KMA. We can noice ha KMA oscillaes around SMA and has much less smooh naure. weighs value # of he weighs (#7 for he mos recen observaion) Figure.5. Bund (December 9, 5 - March 8, 6, frequency sec): Opimal weighs (mean h esimaor) calculaed for he heoreical exponenial variogram γ ( h) =. e 9 : window=7s. As we can see he weighs srucure is he same wheher we use residuals covariance or IRF- model variogram. As he resul we propose furher he analysis wihin saionary framework (residuals covariance model), as we can expec he rading oucomes o be approximaely he same. 44

67 price, MAs price.6 KMA (n=7) SMA (n=7) observaions (s) x 5 Figure.6. Bund (December 9, 5 - March 8, 6, frequency sec, observaions aprox. 7- ): Price, KMA and SMA (window lengh =7s).5..5 price, MAs..5 price KMA (n=8) SMA (n=8) observaions (s) x 4 Figure.7. Bund (December 9, 5 - March 8, 6, frequency sec, observaions aprox. 8- ): Price, KMA and SMA (window lengh =8s) As we can see, he opimal MA weighs, which ake ino accoun he auo-covariance of he insrumens value resuls in he MA, which is close by he lag o he SMA, bu has more volaile 45

68 srucure. The quesion is wheher rading sraegies based on KMA can bring higher oucomes han SMA. 4. Trading resuls: KMA versus SMA The following sub-chaper 4. analyzes rading resuls, obained for he simulaion of sraegies, based on KMA and SMA for such insrumens as Bund, DAX, Bren and X insrumen. We apply simple rading sraegy, based on he crossovers of he price and MA lines (in his case, SMA and KMA). The long posiion should be aken (and he shor posiion should be closed) when he price crosses he MA curve from below, which is a confirmaion of he upward rend; he shor posiion should be aken (and he long posiion should be closed) when he price crosses he MA curve from above, which is a confirmaion of he downward rend. Thus, he sraegy is defined as following: Trend-following sraegy (I.4.3). Trading coss are. Profis are defined in quoes unis 8.. Les define Ri = Pi MAi. 3. The iniial rading posiion Pos = ; rading oucome Π =. 4. The firs rade ( Pos i, i > ) is underaken a he firs crossovers of he MA and price lines, i.e. under following condiion: if ( R i R i ) and ( Ri < ) : Pos i =, P enry = Pi, Π i = if ( R i R i ) and ( Ri > ) : Pos i =, P enry = Pi, Π i = oherwise, Posi =, Π i = 5. Aferwards, a he new rading signals (curves crossovers) he following rades are execued: and R < : if ( R R ) ( ) exi (previously aken) posiion Shor ( Pos = ): Pos =, P exi = P enry posiion Long: Pos =, P enry = P cumulaive rading oucome: Π = Π + Pos ( P P ) if ( R R ) and ( R > ) : exi (previously aken) posiion Long ( Pos = ): Pos =, P exi = P enry posiion Shor: Pos =, P enry = P rading oucome for his operaion: Π = Π + Pos ( P P ) oherwise, Pos, Π = Π + Pos P P Pos ( ) = The resuls of he sraegy (I.4.3) simulaions for differen insrumens and differen lenghs of KMA and SMA are presened in Table.. Excep for he Bund case, we use differen samples for he variogram esimaion and rading simulaions. Each sample (DAX, Bren, X insrumen) is spli ino wo sub-samples of approximaely he same lengh; he firs sub-sample is used for he variogram esimaion, and second sub-sample - for he rade simulaions. 8 Insrumens values are usually quoed in icks, no in currency equivalens. 46

69 Noe ha he rading resuls can be compared only wihin paricular insrumen due o he difference in he lengh of he sample, is frequency and value of he quoes for differen insrumens: we canno compare Bund o DAX, bu only he DAX resuls for he sraegies, based on KMA and SMA. Table. The oucomes of he simulaed rading sraegies Opimal lengh Max profi (in quoes) Number of Insrumen Frequency (obs) rades KMA SMA KMA SMA KMA SMA Bund sec DAX 3 min Bren 3 min X insrumen hour Table. shows ha KMA are seadily more effecive a shor lenghs, while he opimal lengh of he SMA is varying from shor o long. In general a shor MA lenghs, KMA generaes fewer rades han he SMA (see Bren case). Besides he profi per rade was higher for he KMA for Bund and Bren insrumens. Furher we analyze how resuls of he sraegies, based on he differen MAs depend on heir lengh, as well as have a look a he P&L pahs for he opimal MA lenghs for KMA and SMA o see wheher hey exhibi monoone and posiive rend. 4.. Bund Figure.8 gives he end-of-period cumulaive oucomes for he rading sraegies based on he KMA and SMA of he differen lenghs. KMA does no show beer resuls han he SMA, hough i sill works beer a shor lenghs han in long. Similar oucomes for long SMAs and KMAs are explained by he fac ha a long lenghs boh curves almos coincides. Shorer KMA are more volaile han SMA and oscillaes wih larger ampliudes around SMA leading o he difference in he rading oucomes. SMA-based rading sraegy accumulaes fewer rades han KMA (see Figure.9). The KMA and SMA pahs, which correspond o heir respecive opimal lenghs, are given in Figure.. We can see ha boh pahs exhibi posiive rend. Moreover, afer 6 observaions heir behavior is synchronized, in fac he end-of-period difference in he oucomes for boh MA is caused by beer performance of he SMA periods a he beginning of he sample. From Figure.6 we can see ha he price paern up o he 6 observaions is characerized by he rendless period; so i seems like SMA due o heir more smooh naure perform beer during ime, when markes are no rending. 47

70 KMA SMA.5 end-of-period oucome window lengh Figure.8. Bund (December 9, 5 - March 8, 6, frequency sec, observaions -): Endof-period oucomes for he sraegies, based on KMA and SMA of he differen lenghs 3 KMA SMA end-of-period oal rades number window lengh Figure.9. Bund (December 9, 5 - March 8, 6, frequency sec, observaions -): Endof-period oal rade number for he sraegies, based on KMA and SMA of he differen lenghs 48

71 .5 P&L.5 KMA(n=74 obs) SMA(n=66 obs) observaions (in s) x 5 Figure.. Bund (December 9, 5 - March 8, 6, frequency sec, observaions -): Opimal P&L pahs for he sraegies, based on KMA (lengh=74 observaions) and SMA (lengh=66 observaions) 4.. DAX Shor descripion of he DAX sample, as well as variogram used in he kriging applicaions is given in Appendix F. Figure. presens he end-of-period cumulaive oucomes for he rading sraegies, based on he KMA and SMA of differen lenghs. I seems like KMA is more effecive han SMA a shor lenghs (beween approx. 3-6 observaions). A long lenghs SMA leads o higher oucomes, hough he resuls for boh MAs are comparable. Similar rading oucomes for he long SMAs and KMAs are explained by he fac ha wo curves almos coincide a hese lenghs. Shorer KMAs are more volaile han respecive SMAs: hey oscillae wih larger ampliudes around SMA leading o he difference in rading oucomes. Conrary o Bund case, he KMA accumulaes fewer rades han he SMA almos for all window lengh (see Figure.). I means ha even having more erraic naure, and herefore, higher probabiliy of sending false signals, KMA crosses price curve less frequenly han he SMA curve. The KMA and SMA pahs ha correspond o heir respecive opimal lenghs 9, are given in Figure.3. We can see ha boh pahs exhibi posiive rend. 9 We choose he KMA opimal lengh a he level ha does no generae he global P&L maximum, bu fells wihin he inerval of opimal KMA lenghs. 49

72 5 end-of-period oucome 5-5 KMA SMA window lengh 4 6 Figure.. DAX (3/7/3-7//6, frequency 3 minues, observaions 86-7): End-ofperiod oucomes for he sraegies, based on KMA and SMA of he differen lenghs KMA SMA end-of-period oal rades number window lengh Figure.. DAX (3/7/3-7//6, frequency 3 minues, observaions 86-7): End-ofperiod oal rades number for he sraegies, based on KMA and SMA of he differen lenghs 5

73 5 P&L 5 KMA(n=45 obs) SMA(n=8 obs) observaions (in 3 min) Figure.3. DAX (3/7/3-7//6, frequency 3 minues, observaions 86-7): Opimal P&L pahs for he sraegies, based on KMA (lengh=45 observaions) and SMA (lengh=8 observaions) 4..3 Bren Shor descripion of he Bren sample, as well as variogram used in he kriging applicaions are given in Appendix F. Figure.4 presens he end-of-period cumulaive oucomes for he rading sraegies, based on he KMA and SMA of differen lenghs. Again, he shorer lengh of he KMA leads o higher oucomes han he longer lengh, hough conrary o he DAX case no superioriy over SMA resuls are observed. As in he case of DAX insrumen, he KMA accumulaes fewer rades han he SMA for shor and medium lenghs, bu slighly higher number of rades for he long lengh (see Figure.5). For shorer lenghs KMA sends less false signals han SMA. The KMA and SMA pahs, which correspond o heir respecive opimal lenghs, are given in Figure.6. Conrary o he DAX case, KMA pah exhibi seeper rend han he opimal SMA pah ha is more random. 5

74 5 KMA SMA end-of-period oucome window lengh Figure.4. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): End-of-period oucomes for he sraegies, based on KMA and SMA of he differen lenghs 3 KMA SMA end-of-period oal rades number window lengh Figure.5. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): End-of-period oal rades number for he sraegies, based on KMA and SMA of he differen lenghs 5

75 6 4 P&L KMA(n=45 obs) SMA(n=49 obs) observaions (in 3 min) Figure.6. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): Opimal P&L pahs for he sraegies, based on KMA (lengh=45 observaions) and SMA (lengh=49 observaions) 4..4 X insrumen Shor descripion of he X insrumen sample as well as variogram, used in he kriging applicaions are given in Appendix G. Figure.7 presens he end-of-period cumulaive oucomes for he rading sraegies, based on he KMA and SMA of differen lenghs. Conrary o he oher insrumens, none of he MAs provides he profiable rading sraegy. One of he possible explanaions migh be ha he X insrumen has mean-revering naure in he long run, herefore, rend-following sraegies migh no work for his insrumen. As for he DAX and Bren cases, KMA generaes fewer rades han SMA (see Figure.8). 53

76 - end-of-period oucome KMA SMA window lengh Figure.7. X insrumen (frequency hour, observaions ): End-of-period oucomes for he sraegies, based on KMA and SMA of he differen lenghs end-of-period oal rades number KMA SMA 5 5 window lengh Figure.8. X insrumen (frequency hour, observaions ): End-of-period oal rades number for he sraegies, based on KMA and SMA of he differen lenghs 54

77 Comparing he performance of he KMA and SMA of he same lengh for he considered insrumens, we can conclude ha a shor lenghs KMA performs beer han SMA, producing fewer rades; a long lenghs he difference in he performance is less pronounced. The explanaion o his phenomenon lays in he behavior of boh curves. KMA is more volaile han he SMA and i oscillaes around he SMA curve. The volailiy and ampliude of he oscillaions is he indirec funcion of he KMA lengh: he longer he KMA he less i is volaile and i coincides more wih he SMA curve. KMA seems o perform beer during rending periods and worse during rendless periods han he SMA of he same lengh. The explanaion o his lays in he weigh srucure of he MAs. As markes are rending, KMA is more sensiive o he las large price changes, han SMA as i assigns larger weigh o he las available observaion 3. During he rendless periods such srucure makes he KMA more erraic han he SMA, causing more false signals, and hus, less profi. This migh be an explanaion why he rading resuls for he X insrumens are so poor. For rending price paerns KMA produces fewer rades han he SMA of he same lengh even having a more erraic naure. Taking ino accoun non-zero ransacion coss in real-life applicaions, fewer rades end up a lower rading coss. 5 Kriging resuls: Bounded, evenly spaced ime-series daa The previous Chaper 4 shows he resuls of he kriging mehod applicaion o he esimaion of he price mean for series ha are normally non-saionary due o he presence of rend. This chaper analyzes he case of he daa, which is bounded and has mean-revering naure. 5. Moving Average Convergence/Divergence indicaor (MACD) and rading sraegies Moving Average Convergence/Divergence indicaor (MACD) is a echnical indicaor, developed by G. Appel (Murphy, 999). Definiion.4 MACD is he difference beween wo exponenial moving averages of differen lengh: MACD ( α( n )) EMA ( α( )) = EMA, (I.5.),, n where EMA i, - exponenial moving average a he momen of ime ; α ( n i ) - parameer of he EMA, as a funcion of is effecive lengh n i. Figure.9 presens he example of he MACD indicaor for he effecive lenghs of n = 6 days esimaed for he sample of Bund (99-6) wih daily frequency. n = and 3 KMA opimal srucure also assigns more weigh o he firs observaion in he window, bu due o he presence of he rend his value, depending on he rend direcion, is much smaller or much bigger han he las available observaion; herefore is weighed impac is smaller on he mean value han for he las observaion in he window. 55

78 3 Bund quoes MACD & signal line MACD, KMA, SMA.5 5 MACD -.5 3,4, SMA observaions Figure.9. Bund (99-6, frequency=day), MACD ( n =, n = 6 days) Moving Average Convergence/Divergence indicaor (MACD) belongs o he group of rading oscillaors. As long as he choice of he EMAs effecive lenghs in (I.5.) implies he crossover of boh EMA curves, he MACD oscillaes around -level. The ampliude and frequency of he oscillaion depend on he choice of he n, n values (see Figure I in he Appendix I for he examples of he MACD indicaor calculaed for he Bund sample). From he saisical poin of view, he major qualiy of he MACD indicaor is is mean-revering naure; herefore i is more likely o be saionary han he price series. Differen rading rules are defined on he basis of he MACD values (Murphy, 999). One sraegy coincides wih he echnical rule, based on he crossovers of he wo MAs of differen lengh 3 : when MACD line crosses he zero-line above - he buy signal is generaed (poin # in Figure.9), when MACD line crosses he zero-line below - he sell signal is generaed (poin # in Figure.9). Anoher rule is based on he crossovers of he MACD line wih is signal line. Signal line is defined by MA (for example EMA), which is consruced on he basis of he MACD values. The crossing of he MACD line above he signal line is he buy signal (poin #3 in Figure.9); he crossing of he MACD line below he signal line generaes he sell signal (poin #4 in Figure.9). 3 The rading rule based on he wo MA of differen lengh saes he following: Buy signal: Shorer moving average rises above he longer moving average; Sell signal: Shorer moving average fall below he longer moving average. 56

79 A conrarian sraegy is consruced on he basis of he exreme MACD values, which sugges ha prices have gone oo far oo fas and herefore, are subjecs for some correcions: an overbough condiions are presen when he MACD is oo far above he zero line (poin #5 in Figure.9), while he oversold condiions are presen when he MACD is oo far away below he zero line. Finally, a rule is consruced on he convergence/divergence of he price and MACD rends. Negaive divergence akes place when he MACD line is well above he zero line and sars he negaive rend a he ime when prices exhibi posiive rend; in his case, he sell signal is generaed. The posiive divergence happens when he MACD line is well below he zero-line and sars exhibiing he posiive rend earlier han he price line does; hen, he buy signal is generaed. We choose for evaluaion and opimisaion he sraegy, based on he crossovers of MACD and signal lines. This will allow defining he opimal signal line as a kriged MA. 5. MACD sraegy: opimal signal line A sraegy, based on he crossovers of MACD and signal lines has a leas hree parameers o opimise: n,n - he lenghs of he EMAs and ns - he lengh of he MA chosen as a signal line. Taking ino accoun ha he goal of his sub-chaper is o find he opimal MA for he saionary daa, we choose o opimise only he value n S, while acceping some defaul values for he lenghs n, n involved in MACD calculaions. From he empirical variogram in Figure H (Appendix I) we have chosen defaul parameers n = and n = 6 for MACD calculaions, which represen some average variogram for he MACD indicaor for he Bund insrumen (see Figure.3). Figure.3 presens he empirical variogram for MACD(-6) and he combinaion of wo heoreical models fied o hese values. The model ha is fied o he daa is a sum of he h gaussian model ( ) h = γ h σ e a α and damped cosines model ( ) = a h γ h σ e cos : a h h h γ =.5 e.4 e cos. (I.5.) 8. ( h) The opimal weighs 3 for he KMA esimaes are presened in Figure.3 (window=5 observaions). Figure.3 represens he MACD and wo signal lines - KMA and SMA of he same lengh. We can see again ha KMA is more volaile han he SMA and oscillaes around i. 3 The kriging weighs ha correspond o he variogram model (I.5.). 57

80 .. Variogram empirical variogram heoreical model lag Figure.3. Bund (99-6, frequency = day): Empirical variogram of he MACD indicaor ( n = and n = 6 ) and heoreical model ( ) h h.4 7 = + 8 h γ h.5 e.4 e cos weighs value weighs #: #5 corresponds o he mos recen observaions Figure.3. Bund (99-6, frequency = day): Opimal weighs for he MA esimae for MACD indicaor ( n = and n = 6 ), variogram model 58

81 3 Bund quoes MACD & signal line MACD, KMA, SMA.5 MACD -.5 KMA SMA observaions Figure.3. Bund (99-6, frequency = day), MACD and is signal lines represened by KMA and SMA of he same lengh (5 observaions) Furher in Chaper 5.3 we compare he rading oucomes for he rading sraegies, based on he crossovers of MACD curve and he signal lines such as KMA, SMA and EMA of he same lengh. The analysis will be conduced for four insrumens 33 : () Bund; () DAX; (3) Bren; (4) X insrumen. The defaul parameers of he MACD indicaors are chosen a levels specific for each insrumen aken ino accoun is frequency. 5.3 Resuls of he rading sraegy, based on he MACD indicaor and is signal lines { } Suppose P i i> is an insrumen s prices and EMA,, EMA, are he exponenial MAs of he lengh n, n : EMAi, = α i P + ( α i ) EMA, α i =. + ni Then MACD indicaor is { MACD } : MACD = EMA, EMA,. i < i The rading sraegy based on MACD can be formulaed as following:. Trading coss are. Profis are defined in quoes unis 34. MACD for each momen we consruc he signal line in he form of MA of he lengh n S : { MA } S, i. i> The following are he ypes of he MA used as a signal line:. On he basis of he hisory of esimaed MACD indicaors { i} < i 33 Noe ha DAX, Bren and X insrumen samples are he same as in Chaper 4, while we consider new sample for Bund insrulen. 34 Insrumens values are usually quoed in icks, no in currency equivalens. 59

82 Exponenial moving average: EMAS, = α S MACD + ( α S ) EMA, α S = ; + n Kriged moving average: Simple moving average: ns KMA S, = λ imacd ns + i i= SMA n S P i+ i= S =,. ns 3. Les define R i = MACDi MAS, i. 4. The iniial rading posiion Pos ; rading oucome Π. = 5. The firs rade ( Pos i, i > ) is underaken a he firs crossovers of he MA and MACD lines, i.e. under following condiion: if ( R i R i ) and ( Ri < ) : Pos i =, P enry = Pi, Π i = if ( R i R i ) and ( Ri > ) : Pos i =, P enry = Pi, Π i = oherwise, Posi =, Π i = 6. Aferwards, for he new rading signals (curves crossovers) he following rades are execued: if ( R R ) and ( R < ) : exi (previously aken) posiion Shor ( Pos = ): Pos =, P = P = enry posiion Long: Pos =, P enry = P Π = Π + Pos cumulaive rading oucome: ( P P ) ( R ) ( R > ) if R and : exi (previously aken) posiion Long ( Pos ): Pos =, P = P enry posiion Shor: Pos =, P enry = P Π = Π = rading oucome for his operaion: + Pos ( P P ) Pos Pos Π = Π + Pos ( P P ) = oherwise,, This rading sraegy was applied o four differen daa samples of differen frequency. For DAX, Bren and X insrumen he same daa samples are used as in he Chaper 4. For Bund insrumen we chose he new sample of he daily frequency. For Bund case he variogram model (I.5.) esimaed on he whole sample, ha is also used for kriging applicaions. For he DAX, Bren and X insrumen we have divided each sample in wo sub-samples: he firs sub-sample is used for he variogram esimaions and he second sub-sample for he simulaion of he rading aciviy. The resuls of he applicaion of he rading sraegy, described above are summarized in Table.. The following general conclusions can be made:. The opimal signal line defined by KMA has much shorer lengh han he opimal SMA and EMA. (The only excepion is he X insrumen, bu hese resuls are no represenaive as opimal SMA and EMA in fac bring minimal losses no maximum profis).. KMA leads o higher absolue profis for all four insrumens. S exi exi 6

83 3. For Bund and Bren insrumens, he efficiency of rades (profis per rade) is he lowes for KMA. Table. The oucomes of he simulaed rading sraegies, based on MACD indicaor Insrumen Frequency Opimal lengh (obs) Max profi (in quoes) Number of rades KMA SMA EMA KMA SMA EMA KMA SMA EMA Bund day DAX 3 min Bren 3 min X insrumen hour Furher we consider he resuls of he sraegy simulaion for each insrumen in more deails Bund For he simulaion of he rading sraegy we use he sample of daily observaions for Bund (99-6) analyzed in Chapers 5. and 5.. MACD is defined as he difference of he EMAs of he lenghs n = and n = 6 ; he esimaed variogram follows model (I.5.). Figure.3 and.3 compare he profis and rade number for differen lengh and ypes of he signal line. The MACD sraegy leads o he posiive resuls for all ypes and lengh of he signal lines (Figure.3). The usage of he KMA as a signal line leads o he highes possible profis, alhough he rading oucomes are more volaile han for oher signal lines. The opimal lenghs of he KMA lay beween 5 observaions (days). EMA shows more consisen resuls: for longer EMAs he sraegy brings high and less volaile oucomes han he oher signal lines. Figure.3 shows ha excep for very shor lenghs - observaions, KMA leads o relaively lower rades number han he signal lines defined by SMA and EMA of he same lenghs. end-of-period oucome KMA 5 SMA EMA window lengh 3 5 Figure.3. Bund (99-6, frequency = day): End-of-period oucomes for he sraegies, based on MACD ( n =, n = 6 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. 6

84 35 3 KMA SMA EMA oal rades number window lengh Figure.3. Bund (99-6, frequency = day): End-of-period oal rades number for he sraegies, based on KMA, SMA and EMA of he differen lenghs As for he opimal P&L pahs (see Figure.3), all signal lines ypes leads o he pahs ha exhibis posiive rends P&L KMA (n=8 obs) SMA (n=37 obs) EMA(n=3 obs) observaions ( s) Figure.3. Bund (99-6, frequency = day): Opimal P&L pahs for he sraegies, based on KMA (lengh=8 observaions), SMA (lengh=37 observaions) and EMA (lengh=3 observaions) 6

85 5.3. DAX The descripion of he DAX daa sample is given in Appendix F. MACD indicaor was consruced as he difference of he EMAs of he lenghs n = 4 and n = 8 (see Figure E4). The MACD variogram model (see Figure F5 in he Appendix F) ha is used in kriging applicaions is: h h.3 ( ) h γ h = 3 e 3 e cos. 8 The rading oucomes generaed for differen lenghs of he signal line defined by KMA, SMA and EMA are presened in Figure.33. Conrary o Bund case, some signal lines lead o losses. The sraegy based on he KMA signal line generaes he highes possible profis; however, is oucomes are volaile. The opimal KMA lenghs belong o he inerval of low values -4 observaions; EMA opimal lenghs lay beween 8-5 observaions. The number of he oal rades for he KMA-based sraegy is higher for shor lenghs; a long lenghs i is similar o he number of rades generaed by oher ypes of he signal lines (see Figure.34). As for he monooniciy of he opimal P&L pahs in Figure.35, hey do no exhibi consanly he paerns wih posiive rend and are quie volaile for ally ypes of he signal line. end-of-period oucome KMA SMA EMA window lengh 3 5 Figure.33. DAX (3/7/3-7//6, frequency 3 minues, observaions 85-7): End-ofperiod oucomes for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. 63

86 end-of-period rades number KMA SMA EMA window lengh Figure.34. DAX (3/7/3-7//6, frequency 3 minues, observaions 85-7): End-ofperiod oal rades number for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. 5 KMA SMA P&L 5 EMA -5 KMA (n=36 obs) SMA (n=45 obs) EMA (n= obs) observaions x 4 Figure.35. DAX (3/7/3-7//6, frequency 3 minues, observaions 85-7): opimal P&L for he sraegies, based on MACD ( n = 4, n = 8 ) and opimal signal lines (KMA, SMA and EMA) 64

87 5.3.3 Bren Bren daa sample is briefly presened in Appendix G. The MACD indicaor was consruced as he difference of he EMAs of he lenghs n = 4, n = 8 (see Figure G4 in he Appendix G). The following variogram model (see also Figure G5) is used in he kriging applicaions: h h.3 ( ) h γ h =. e. e cos. 5 Trading profis/losses and rades numbers generaed for differen lenghs of he signal lines are presened in Figure.36 and Figure.37 respecively. As in he previous cases he KMA generaes he highes possible profis, while is oucomes are quie volaile (see Figure.36). The opimal lenghs of he KMA belong o he inerval of he low values beween -7 observaions. Again SMA and EMA perform beer as he signal line a he long disances. As for he oal number of rades he KMA generaes fewer rades han respecive EMA, excep for he shor lenghs up o approximaely 3 observaions (see Figure.37). Finally he opimal P&L pah for he KMA signal line exhibis he seepes posiive rend comparaively o he SMA or EMA signal lines (see Figure.38). end-of-period oucome KMA SMA EMA window lengh Figure.36. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): End-of-period oucomes for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. 65

88 end-of-period rades number KMA SMA EMA window lengh Figure.37. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): End-of-period oal rades number for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs KMA EMA 5 SMA P&L KMA(n=49) SMA(n=97) EMA(n=85) observaions Figure.38. Bren (7//4-7//6, frequency 3 minues, observaions 55-59): opimal P&L for he sraegies, based on MACD ( n = 4, n = 8 ) and opimal signal lines (KMA, SMA and EMA). 66

89 5.3.4 X insrumen The daa sample used for he analysis of he X insrumen is presened in Appendix H. The MACD indicaor was consruced as he difference beween EMAs of he following lenghs: n = 4, n = 8 (see Figure H4 in he Appendix H). The following model is fi o he empirical variogram esimaes (see Figure H5 in he appendix H) and used furher in he kriging applicaions: h h ( ) γ h =.7 e.3 e. Figure.39 shows ha only one sraegy based on he KMA of he lengh 37 observaions as he signal line generaes some posiive profis. All he oher MACD sraegies generae losses. Due o his fac we do no provide he opimal P&L pahs for his insrumen. As in he previous cases, KMA generaes higher rades number a shor lenghs and relaively lower rades number a long lenghs comparaively o he SMA and EMA. KMA SMA EMA end-of-period oucome window lengh Figure.39. X insrumen (frequency - hour, n = 4, n = 8, obs ): End-of-period oucomes for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. 67

90 end-of-period rades number KMA SMA EMA window lengh Figure.4. X insrumen (frequency - hour, n = 4, n = 8, obs ): End-of-period oal rades number for he sraegies, based on MACD ( n = 4, n = 8 ) and signal lines (KMA, SMA and EMA) of he differen lenghs. As he resul, we can see ha applicaion of he kriging approach o he parameer opimizaion of he rading sraegy, based on MACD indicaor leads o similar resuls as in Chaper 4. KMA also leads o he absolue maximum profis for he majoriy of he insrumens. The opimal lenghs of he KMA coincide wih he low values conrary o opimal EMA and SMA signal lines. KMA generaes higher rades number a shor lenghs, bu lower rades number a long lenghs relaively o he EMA and SMA of he same lengh. We can conclude ha KMA improves he MACD rading sraegy under zero-ransacion-coss hypohesis. Moreover i works beer a shor disances, which is unlikely for he majoriy of oher MAs ha generaes more false signals a hese lenghs. 6 Kriging resuls: unevenly spaced daa As have been shown in Chaper 3, for he equally spaced samples of he financial daa he opimal KMA, defined by he kriging mehod, has a specific weigh srucure; KMA is close in he lag o he SMA, calculaed on he same window, bu migh have higher volailiy. Moreover he opimal weighs are he same hroughou he ime, as he same disances (,,,..., n ) beween he momen of ime, where he MA is calculaed and observaions in he moving window of he lengh n ha precedes his momen are used o define hese weighs. However, frequenly he sampling of he financial daa is no done a equal disances, due o he fac ha he price/quoe is documened if he ransacion is underaken. Oher case of he unequal sampling appears when an insrumen s price is considered as a funcion of oher han 68

91 ime variable, for example volume. Such approach can be presened as a change of he coordinaes. In his case he price can be considered as a subordinaed process 35. The change of coordinaes helps o smooh he jumps in he price process, frequenly associaed wih he large changes in he raded volume. The incorporaion of he volume ino he process of decision-making migh improve he rading resuls. Blume, Easley, O Hara (994) demonsraed ha he raders who use informaion conained in marke saisics such as prices and volume do beer han he one ha do no. The oher aricles ha analyzed he imporance of volume for he price predicion are Lo, Wang (), Campbell, Grossman and Wang (99), Harris and Raviv (99), Wang (99). Taking ino accoun ha he kriging mehod accouns for he difference in he disance beween he poins hrough he covariance/variogram model, i can be used o consruc he opimal MA for he unevenly spaced daa. In his chaper we presen he examples of such mehod applicaions. In order o analyse he KMAs in more deails we have also simulaed rend-following sraegy, based on he crossovers of he price and MA curves. The same sraegy, based on he SMA curve, is considered as a benchmark for he resuls comparison. The crossovers of he price and MA lines (SMA and KMA) define he sraegy enry/exi signals. Les define variable R = P MA. Then he sraegy is formulaed in he following way. i i i Trend-following sraegy. Trading coss are.. The iniial rading posiion Pos = ; rading oucome Π =. 3. The firs rade ( Pos i, i > ) is underaken under following condiion: if ( R i R i ) and ( Ri < ) : Pos i =, P enry = Pi, Π i = if ( R i R i ) and ( Ri > ) : Pos i =, P enry = Pi, Π i = oherwise, Posi =, Π i = 4. Aferwards, if he rading signals are generaed he following rades are execued: if ( R R ) and ( R < ) : exi (previously aken) posiion Shor ( Pos = ): Pos =, P exi = P enry posiion Long: Pos =, P enry = P cumulaive rading oucome: Π = Π + Pos ( P P ) if ( R R ) and ( R > ) : exi (previously aken) posiion Long ( Pos = ): Pos =, P exi = P enry posiion Shor: Pos =, P enry = P rading oucome for his operaion: Π = Π + Pos ( P P ) oherwise, Pos, Π = Π + Pos P P Pos ( ) = As he resul Chaper 6 is organized in he following way. Chaper 6. presens he case of unevenly spaced ime-dependen daa caused by he missed observaions when no ransacion akes place. We show how o consruc KMA and compare is resuls wih he radiional SMA, 35 From he saisical poin of view he change of he ime coordinaes o he volume-based (or oher variable-based) axe inroduce he subordinaed processes for price. 69

92 esimaed on he same sample. We also analyse wheher he adjusmen of he sample o he regular spaced one by an inerpolaion of missed observaions can produce beer KMA. Chaper 6. presens he kriged volume weighed moving average (KVWMA) consruced for he irregular spaced sample due o he change of he coordinae from ime o volume. We also analyse he rading oucomes of he rend-following sraegy, based on his KVWMA. 6. Unevenly spaced ime dependen price series In his chaper we presen applicaion of he kriging mehod o he Bund daa sampled a second-frequency for he day of April 8, 6. The frequency is very high, herefore even for Bund ha is raded very acively, many gaps are presen in he daa. Some of hese gaps coun up o minues. Figure.4 presens he sample a real ime coordinae (in seconds); correspond o 9::, #3 o 9:5:, ec. price seconds ( corresponds o 9::) x 4 Figure.4. Bund (April 8, 6, frequency sec): quoes (unevenly spaced daa) In order o analyse how hese gaps migh impac he esimaion of he opimal MAs, we propose o consider wo samples: () irregular sample: raw daa a available ime poins of ime () regular sample: he sample obained by filling he gaps in he daa by he same values available in he previous momens of ime. Such approach is based on he assumpion ha he when here is no rading he price says a he level corresponding o he las ransacion. Figure.4 presens wo variograms esimaed on hese samples. As we can see, he adjusmen of sample o he regular sampling changes significanly he form of he variogram. 7

93 7 x -3 Vario (irregular ime) Vario (regular ime) 6 5 vario 4 3 variance= lag (s) Figure.4. Bund (April 8, 6, frequency sec): variograms for unevenly and evenly spaced price samples Two differen models were fi o he empirical variograms: () h irregular sample: γ ( h) =.7 e 3. () -6 regular sample : γ ( h) =. h. The weighs srucure corresponding o hese variogram models are given in Figures.43 and.45. Figure.43 presens he example of he weighs for he irregular sample. I should be sressed ha alhough he disribuion of he weighs is he same (he larges weighs are assigned o he firs and las observaion), he weighs will differ from window o window as hey are defined also by he disance from he esimaion poin o oher poin in he window, which are irregular. This observaion is suppor by he Figure.44 ha presens he firs weigh esimaed for each window. We can see ha while he firs weigh (and he las) values conained in he inerval beween.4 and.5, hey almos never consan. As for he regular sample, he same weigh srucure ha corresponds o he linear model preserves hrough ime: λ λ N.5, λ (see Figure.45). = < i< N The examples of he KMA and SMA, esimaed on he basis of irregular sample and KMA, esimaed on he basis of regular sample and re-sampled a he poins of irregular sample, are presened in Figure.46. Noe ha he effecive lenghs of he MAs esimaed on he regular and irregular samples are differen. 7

94 weighs value # of he weigh in he window (#8 correspond o mos recen observaion) Figure.43. Bund (April 8, 6, frequency sec): Example of he opimal weighs for he KMA for he daa sampled irregular (window beginning a 35 observaion).5.49 value of firs/las weigh # of he rolling window Figure.44. Bund (April 8, 6, frequency sec): Value of he firs/las weighs for he opimal KMA as a funcion of he window used for is esimaions 7

95 weighs value # of he weigh in he window (#6 correspond o mos recen observaions Figure.45. Bund (April 8, 6, frequency sec): The opimal weighs for he KMA for he daa resampled regularly price KMA (irregular) SMA (irregular) KMA (regular) price, MAs ime(sec) Figure.46. Bund (April 8, 6, frequency sec, obs. 6-35s): Price and differen MA ypes (window=8 observaions) 73

96 As we can see for his paricular lengh KMA esimaed on he irregular sample generae less false signals han he SMA and KMA esimaed on he regular sample. For example, beween 9-3 seconds he SMA, esimaed on he irregular sample and KMA, esimaed on he regular sample send false signals by crossing several imes he price curve, while he KMA esimaed on he irregular sample crosses i only once. Table. and Figure.47 summarize he maximum profis generaed by he rend-following sraegy, based on he differen MA ypes 36. Table. Opimal oucomes of he rend-following sraegy, based on he differen MA ypes Oucomes KMA (irregular) SMA (irregular) KMA (regular) MA lengh Cumulaive value Number of rades As we can see he highes oucome is achieved for he sraegy, based on he SMA. The KMA, esimaed for irregular samples generaes comparable profis a much shorer lenghs. A he same ime, we can noice ha he KMA calculaed on he adjused regular sample and resample for he irregular poins, generae more false signals han he KMA esimaed on he irregular sample; a he same ime i generaes some sable profis for differen MA lenghs cumulaive value KMA (irregular) SMA (irregular) KMA (regular) MA lengh Figure.47. Bund (April 8, 6, frequency sec): End-of-period cumulaive value of he sraegy for differen ypes of he MAs and daa samples 36 I should be noed ha he MA of he lengh considered in he analysis are oo shor and less likely o be used for he daa wih -second frequency in he real-life applicaions. We hough propose o consider hese values and daa as some general example. 74

97 The pahs generaed by he opimal MAs are presened in Figure.48. As we can see, he SMA does no generae consisenly large profis han he oher KMA. In fac he KMA, calculaed on he irregular sample generae he mos consisen profi pah. cumulaive value ofhe sraegy KMA (irregular,window=8 obs) SMA (irregular,window=3 obs) KMA (regular, window=6 obs) ime(sec) x 4 Figure.48. Bund (April 8, 6, frequency sec): Cumulaive value of he sraegy (pahs) for differen ypes of he opimal MAs As he resul we can see ha KMA, calculaed for he irregular samples generae consisen and he highes resuls a he shor lenghs of he MA, where radiional MAs do no work well. The KMA calculaed on he adjused regular sample generaes more false signals. 6. Unevenly spaced price series subordinaed o cumulaive volume Anoher case when daa is unevenly sampled appears when he coordinae is changed from ime o anoher variable, for example cumulaive volume. The rading lieraure proposes o consruc he MA ha incorporae informaion accumulaed in he rading volume. Volume-adjused moving average (VAMA) can be represened as following: VAMA = n i= w * i P n+ i *, w w ( ) * =, i V i { i } i> * P n { w i } i n where are prices; is MA lengh; and i { i } i> cumulaive volume a he h momen of ime V. are MA weighs as a funcion of he The idea behind VAMA is o use some addiional informaion abou he rading aciviy in he definiion of he MA. Tradiional ime-based (sampled) MA suppose ha all momens of imes are equal wih respec o he informaion ha hey bring. VAMAs rea he aciviy of rading as 75

98 he measure of he price imporance in he paricular momens of ime. For example, a sharp increase in rading aciviy migh indicae he change in he curren rend, herefore he price a his momen of ime have much higher imporance for rader han he prices when he marke is no raded acively. Some examples of such volume weighed MA exis. Richard W. Arms Jr ([], V.8:3) proposed o change he sysem of coordinaes from ime- o volume based. The new curve has prolonged price values for heavily raded momens of ime and shoren (or even negleced) he price values for lighly raded momens. The new VAMA can be calculaed on he basis of he new price curve. Arms ([], V.8:3) used graphical approach o he consrucion of VAMA, in paricular equivolume charing, which can be ranslaed ino programming algorihm. Suppose ha you have daily observaions of price (high, low) and volume for a paricular insrumen. According o he equivolume charing each price observaion is represened in he form of box wih high being he upper side of he box, low being he lower side of he box, and volume being he widh of he box. The mean price calculaed as sum of high and low divided by wo, is posed one or several imes in he box depending on is widh. The rading rule hen involves he new price curve and VAMA. As he resul of he consrucion, VAMA approaches price curve during he periods of heavy rading more rapidly han he ime-based MA. Wih his approach he days wih heavy rading have higher weighs and, hus, more imporance for he definiion of he MA. A our opinion he mehod presened above has several drawbacks. Alhough he VAMA akes ino accoun he rading volumes a each momen of ime, i canno incorporae he informaion abou possible price (auo-) correlaion a he volume poins. Besides, building he price-volume blocks of differen widh implies price inerpolaion for he non-available cumulaive volumes values; he inerpolaion of he price a he same level as previously available daa can be quesioned. Finally, while he MA calculaion procedure can be programmed, i involves many subjecive judgemens, among which he definiion of he volume frequency and he poins a which he MA is calculaed. Wih respec o his criique, he kriging mehod can be used o define he volume weighed moving average. In his case we use a variogram γ ( v) wih v as he cumulaive volume lags/incremens o define he opimal weighs for he kriged volume-weighed moving average (KVWMA). Comparing o he VAMA he mehod is more sraighforward. The KVWMA has even more advanages han he VAMA: i no only incorporaes he informaion abou he rading aciviy (volume), bu also abou he price auocorrelaion. Les consider he same Bund sample o presen he implemenaion of he mehod. Figure.49 presens he Bund quoes and rading volumes sampled a he frequency no higher han second April 8, 6. We can see from he volume curve ha he rading aciviy varies during he day; moreover, he average aciviy increases afer 4: for he European markes wih he opening of he US exchanges (beween 4-5 observaions in Figure.49). Figure.5 presens he same price series, bu ploed as a funcion of he volume. We can noice ha alhough some of he price jumps are reduced, he medium-erm rends are sill presen in he price pah. Taking ino accoun ha he variabiliy of volume can be significan, in order o calculae he empirical variogram γ () v we need o scale he cumulaive volume by dividing each value by some Δ. Oherwise he variogram calculaions migh be very ime consuming. We use he scaling h facor of for he Bund volume values (now he Bund volume is couned in unis). 76

99 The price variogram calculaed on he whole sample suppors he non-saionariy assumpion: he variogram breaks he variance of he sample (see Figure.5). We known already wha ype of he weighs he linear variogram provides λ λ N.5, λ = i N price volume observaions Figure.49. Bund (April 8, 6, frequency sec): quoes and volume cumulaive volume x 5 Figure.5. Bund (April 8, 6, frequency sec): Price as a funcion of he cumulaive volume quoes 77

100 .4.. variogram lag x 5 Figure.5. Bund (April 8, 6, frequency sec): Variogram of he price in he cumulaive volume coordinae, calculaed on he whole sample, lag is in volume unis. (variance=.8) We propose o consider he problem in he saionary conex. We can noice a Figure.49 some rend-less periods, for example, sub-samples (9; 4) or (45; 77). Therefore, we propose o make he following assumpions: he rendless periods of ime allow esimaing he noncorruped covariance srucure of he residuals, which is sable hroughou he whole sample. We choose sub-sample (45; 77) (see Figure.5) o esimae he empirical variogram. The following model is fi o he variogram in Figure.53: γ () v v 4 3 v v.5 e = v 4.5 e +.5, v 8 3, v < 8 78

101 price cumulaive volume x 5 Figure.5. Bund (April 8, 6, frequency sec): Price as a funcion of he cumulaive volume, observaions x -3 empirical model variogram lag (x ) Figure.53. Bund (April 8, 6, frequency sec): Variogram of he price in he cumulaive volume coordinae, calculaed on he sub-sample for observaions (variance=.) 79

102 The following figure.54 presens some examples of he opimal weighs for he lengh of he moving window: n =. The srucure of he opimal KVWMA is similar o he one obained for he equally spaced samples: higher weighs for firs and las observaions and relaively low weighs for oher observaions. Alhough some middle weighs are differen from. Some similariies wih he equally spaced daa can be explained by he following facors. One of he possible explanaions for such paern is ha he rading volumes wihin he day are lower han he rading volumes for daa of lower frequency (for example, daily or monhly daa). Besides, he scale parameer Δ = migh also smooh he volume discrepancy; hus, changing he scale facor can also change he srucure of he weighs. For he examples of weighs srucure differen from Figure.54 a he presence of he jump in volume see Appendix J weighs for obs. 599:5999 weighs for obs.599:6 weighs for obs.599:6 weighs for obs.5993:6 weighs value weighs #: # - he mos recen observaion Figure.54. Bund (April 8, 6, frequency sec): Some examples of he opimal weighs (window lengh= observaions) Figures presen he KVWMA and SMA for differen moving windows ( n = and n = ) for he Bund sub-samples [; 5] and [; 3] respecively. The following peculiariies of he kriged MA can be noiced:. The KVWMA is much more volaile han he SMA of he same lengh; i is oscillaing around he SMA curve.. The shorer KVWMA follows he price curve more closely han he longer one. 3. The behaviours of he KVWMA and SMA around price curve are differen, in paricular for he rending pars of he price curve. During he rending periods, SMA seems o send more false signals han he KVWMA (see Figure.56) for he rend following sraegy described above. We can noice ha for he decreasing rend beween approximaely -7 observaions, he SMA sends hree false signals (#,3,4) versus no false signals for he KVWMA. For he upward rend beween he observaions 7-5 he SMA sends four false signals (#6,7,8,9) versus wo false signals for he KVWMA (#3*, 4*). 8

103 4. During he rendless periods boh MAs seems brings bad resuls; his again suppor he hypohesis ha he rend-following sraegies work only during he rending periods price KVWMA SMA price, MAs observaions Figure.55. Bund (April 8, 6, frequency sec, observaions -3): Price, KVWMA and SMA (window lengh= observaions) 5.9 price, MAs 5.85 * *,4* * 6, ,9 5* ,4 5 price KVWMA SMA observaions Figure.56. Bund (April 8, 6, frequency sec, observaions -3): Price, KVWMA and SMA (window lengh= observaions) 8

104 The rend-following rading sraegy, applied o whole sample (; 77) generaes he following rading oucomes. Figures.57 and.58 summarize he end-of-period rading oucomes, simulaed for SMA and KVWMA of differen lenghs. end-of-period oucome(p&l) KVWMA SMA window lengh 4 5 Figure.57. Bund (April 8, 6, frequency sec, observaions -3): end-of-period oucomes (P&L) for KVWMA and SMA of differen lengh end-of-period oal rades number KVWMA SMA window lengh Figure.58. Bund (April 8, 6, frequency sec, observaions -3): end-of-period oal rades number for KVWMA and SMA of differen lengh 8

105 The highes possible cumulaive profis are generaed by he sraegy based on he SMA. A he same ime, KWVMA demonsraes higher profis a he window lenghs from he inerval [6; ] 37. The oucomes of he KVWMA based sraegy are less volaile. Figure.58 also shows ha he sraegy, based on he KVWMA generaes less ransacion han he sraegy, based on he SMA of he same lengh. Figure.59 presens he opimal P&L pahs, generaed for he KVWMA (lengh=9 observaions) and SMA (lengh= observaions). We can see ha SMA pah is more volaile han he KVWMA pah; i also generaes less efficien resuls (profis per rade) P&L.3.. KVWMA(n=9 obs) SMA(n= obs) observaions Figure.59. Bund (April 8, 6, frequency sec, observaions -3): opimal P&L pahs for he KVWMA (lengh=9observaions) and SMA (lengh= observaions) As he resul, in his chaper we have demonsraed how consruc he KVWMA and showed ha i can generae consisen profis a he lenghs where radiional MA does no work well. 7 Conclusions The main objecive of he par was o presen a saisical mehod for opimizaion of one of he mos popular rule of he echnical analysis moving average. Par one presened he kriging mehod, used in geosaisics for he mean esimaion and predicion of random variables. According o he mehod a predicor/esimaor is a weighed average of some neighborhood of daa, which akes ino accoun he correlaion of a random variable a his neighborhood. Such esimaor coincides wih he definiion of he MA, making his approach easy o use for he MA 37 Noe ha for he sraegy simulaions he window lengh was abulaed wih he sep of observaions. 83

106 opimizaion. The main advanage of he mehod is ha i can be applied o boh equally and unequally sampled daa. In realiy, he financial daa, sampled a high frequencies is irregular, as prices and oher informaion abou ransacion are recorded a he momen of ransacion. Anoher ransformaion, which makes he price daa irregular is he change of he coordinaes from ime o anoher variable, such as volume. As he resul, we have considered he wo cases of kriging mehod applicaion, wih respec o he sample used in hese applicaions: () equally sampled daa; () unequally sampled daa. The firs approach leads us o he definiion of he opimal kriged moving average (KMA), based on he insrumen prices or some indicaor values. We have seen ha for he equally spaced daa, he opimal weighs follow some specific srucure, similar o differen variogram models and window lenghs. In such opimal weigh srucure he larges weighs in absolue value has he firs and he las observaion in he window, wih relaively negligible weighs for all oher observaions. As he resul, KMA has approximaely he same lag as simple moving average, bu is less smooh han SMA. KMA oscillaes around he SMA curve. The volailiy and ampliude of he oscillaions is he indirec funcion of he KMA lengh: he longer he KMA he less i is volaile and i coincides more wih he SMA curve. Therefore, he rading sraegies based on KMA and SMA will ake differen posiions a shor window lenghs and he same posiion a long lenghs. As for he rading resuls, he rend-following sraegies, based on KMA generae higher profis a shor lenghs han SMA, and in some cases produce even fewer rades. A longer lenghs he difference in he performance is less pronounced. KMA seems o perform beer during he periods ha conain price jumps and worse during rendless periods; i also performs beer han SMA during he period wih jumps and worse han SMA during he rendless period. The explanaion o his lays in he weigh srucure of he MAs. As markes are jumping, KMA is more sensiive o he las large price changes han SMA as i assigns larger weigh o he las available observaion 38. During he rendless periods such srucure make he KMA more erraic han he SMA, causing more false signals, and hus, less profi. I should be noed ha hese conclusions should be considered wihin a conex of a paricular rading sraegy rendfollowing, based on he cross-overs of MA and price curves. The paricular, characerisic of he KMA is ha frequenly i produces fewer rades han he SMA even having a more erraic naure. Taking ino accoun non-zero ransacion coss in reallife applicaions his facor can increase even furher he discrepancy beween he oucomes for KMA and SMA (when KMA is more profiable). I also implies ha erraic naure of he KMA curve does no necessary leads o more false signals generaed by he rading sraegy. Finally, KMA seems perform worse for he daa sampled a higher frequency due o more volailiy presen in his daa; his migh explain poorer resuls for Bund case, compare o DAX and Bren daa. As for he unequally spaced daa we have considered wo examples of he applicaion of he kriging mehod. For he ime-dependen daa we have shown ha KMA, calculaed for he irregular samples generae consisen and he highes resuls a he shor lenghs of he MA, where radiional MAs do no work well. We also showed ha an adjusmen of sample o he 38 KMA opimal srucure also assigns more weigh o he firs observaion in he window, bu due o he presence of he rend his value, depending on he rend direcion, is much smaller or much bigger han he las available observaion; herefore is weighed impac is smaller on he mean value han for he las observaion in he window. 84

107 regular spaced one by he inerpolaion of he missed observaions does no produce opimal MA: he KMA calculaed on he adjused regular sample generaes more false signals. As for he unequally sampled daa due o he change of coordinae, he esimaion of he volume weighed MA lead o he following conclusion: kriging resuls largely depend on he variabiliy in he volume. If he variabiliy is small due o he high frequency of he daa he srucure of he weighs resembles he one for he evenly spaced daa se: he srucure of he weighs emphasized he firs and he las observaions. A he same ime he weighs are no consan in values and are changing wih ime. As he resul he KVWMA behaves similar o he KMA: i oscillaes around he SMA and is smoohness depends on he MA lengh. I also generaes good rading resuls (alhough no he highes possible, bu more sable). KVWMA also generaes fewer ransacions han he SMA of he same lengh. For he case, when he variabiliy of he volume is significan, he weighs srucure will be more ime dependen and volaile. In general, we can conclude ha kriging mehod allows obaining quie ineresing MA. A he framework of he rend-following sraegies considered in his par, opimal KMA or KVWMA has shor lengh, conrary o classic MAs, which are more profiable a long lenghs. I also generaes fewer ransacions han oher MAs a he medium lengh of he moving window. 85

108 9 Appendices I Appendix A Some MA opimizaion echniques A. Opimisaion echnique, based on he price series cyclic naure According o Achelis ( he lengh of he MA should fi he peak-o-peak cycle of a securiy price movemen. The ideal MA lengh should follow he formula: = C n l + op, (A.) where C l - is he lengh of he cycle. Achelis proposes he suiable MAs lenghs wih respec o he rends ha hey have o filer (see Table A.). Table A. Opimal lenghs of he moving averages Trend Moving Average lengh Very shor erm 5-3 days Shor erm 4-5 days Minor Inermediae 6-49 days Inermediae 5- days Long-erm - days Source: Achelis, S.B. Technical Analysis from A o Z, One of he principal criiques of he mehod lies in he necessiy o define parameer before he MA opimizaion. The analysis of he cycles ofen demands preliminary analysis of he marke naure. Each marke has is own characerisics, which migh define is cyclic naure; herefore, applicaion of he same lenghs parameer (see Table A.) can be quesioned. Besides, hese opimal lenghs for daily observaions are irrelevan for he inraday rader ha works on much higher daa frequencies. The oher imporan drawback of he mehod is is concenraion only on he MA s lengh, alhough weighs values have an impac on he form and posiioning of he MA curve. A. Moving average as a cener of graviy Ehlers ( proposed simple and didacic represenaion of he rade-off problem. He shows ha as long as a rader knows he lag she/he can olerae, she/he can calculae he opimal weighs of he MA. He proposes o consider MA as a cener of graviy (c.g.) of he window wih price series on is diagonal. The weighs of he prices are he weighs of he MA, while he lag is he disance beween his cener and is projecion on he righ verical side of he window. Picure A. demonsraes his approach in he simplified version for he case of he simple moving average (all weighs are equal). For SMA he cener of graviy lies in he middle of he diagonal; hus, he lag ( L ) of his MA is half of he window lengh ( n ): L = n. As we can see, he value of he lag L is deermined by he posiion of he cener of graviy on he price curve, which in is urn is deermined by he price weighs and size of he window. C l See Ehlers, J. "Signal analysis conceps", hp:// for more deail analysis. 86

109 c.g. L P n MA(n) Figure A.. Formaion of he simple moving average (Ehlers, [38]) The example of he exponenial moving average (EMA) can be used o demonsrae he applicaion of he P is a coninuous rend wih slope μ and has value Ehlers s mehod. Les assume ha price { ( )} > T ()a momen. EMA a momen is defined as following: () ( α) ( ) EMA() = αp + EMA. (A.) Figure A. presens he schemaic represenaion of he definiion of he opimal weigh α. T=rend T(-) T() μ L L=Lag EMA μ μ L T()- μ L=EMA() T()- μ - μ L=EMA(-) Figure A.. Calculaion of he opimal weighs for he exponenial moving average (Ehlers, [38]) According o he figure A. he values of he EMA a momens and can be defined as following: EMA T Lμ EMA = T Lμ μ. Then expression (A.) akes he form: () = (), ( ) ( ) T() Lμ = αt( ) + ( α) ( T( ) Lμ μ ) μ = αμ( + L), or We can derive ha 87

110 α = (A.) + L As he resul, Ehlers ges (A.) for he weigh of he EMA as a funcion of he MA lag regarding price curve. One of he principal criiques of he mehod is ha knowing only he lag of he MA is no sufficien o define he opimal MA parameers. Eiher one of he parameer (lengh or weighs) should be predefined or he smoohness crieria should be inroduced. From he box presenaion of he MA as a graviy cener in Figure A.3 i is obvious ha differen ses of he opimal weighs can generae he same lag L ha he rader can olerae. For example, he moving window n = 5 wih weighs ; ; ; ; and ;;;; will have he same cener of graviy and, hus, lag L. A he same ime, differen weighs se will generae differen smoohness of he MA curve; herefore, he smoohness crieria in addiion o lag can help o define he opimal weighs. μ L Picure A.3. Moving average as a cener of graviy n The oher imporan criique is ha he prices never form a sraigh line making he opimizaion procedure quie approximae. Finally, i is no eviden how o define he level of lag L ha he rader can accep. A3. Adapive MA The problem of lag choice is imporan as i defines how close a MA is o prices curve. This deermines he sensibiliy of he MA rule o false signals. One of he definiions of he false signals is relaed o he ransacion coss. Di Lorenzo (996) ried o develop adapive moving average ha akes ino accoun he ransacion coss and volailiy of he price daa. They assume ha he price is a linear rend, namely P( ) for window n: P + P P M ( n) = n n μ μ μ = n i = μ n n n i= n+ Disance beween price rend and MA: μ + μ = ( n ) n ( n ) = μ ( ) μ( n + ) n μ = μ. Consequenly, moving average a momen μ = n n i= ( i) = d = P M ( n) ( n ) μ μ( n ) = μ μ = (A3.) From equaion (A3.), Di Lorenzo (996) received he formula for rolling window: 88

111 d n = (A3.) μ + According o Di Lorenzo (996), he disance should be he funcion of he shor-erm volailiy of he price series and he ransacion coss. The expression (A3.) is received under assumpion ha prices follow some linear rend, which is quie simplified version of he realiy. Taking ino accoun ha he rend facor μ is generally unknown i should be esimaed firs. Finally, he rend is frequenly non-consan value, herefore, should be reesimaed from ime o ime. All hese criique makes he applicaion of he mehod on he rouine basis less likely. 89

112 Noaion: P - price series; { i } i { wi } i n - weighs of MA n - lengh of moving window (lengh of MA) Appendix B Examples of he MAs wih fixed weighs α - lengh parameer of he EMA: α =, wih n - he lengh of he SMA ha has he same lag. + n. Simple moving average (SMA): MA = n i= w n n n i P i = P i = i= n n i= P i. Exponenial moving average (EMA): EMA ( α ) EMA = αp + 3. Weighed moving average (WMA): MA = n i= w P i n+ i 3.. Linearly weighed moving average (LWMA): Widely used case: ( n + ) n i= MA = w P w j = f ( j) = a + bj n i i i w i = (Huchinson, Zhang, [55], V.:). n 3.. Triangular moving average (TMA) For his MA higher weighs are assigned o he middle period observaions: w < w <... < wn / = wn / + >... > wn > wn for n even w < w <... < w + / >... > w > w for n odd n n 3.3. Generally weighed moving average (GWMA) (Huchinson, Zhang, [55], V.:): n i= MA = w ( α ) P wih w (, n) n i i n j α α j = n (B.). α j Les consider he following cases for generally weighed moving average (, n,α ) α = w j =, i.e. GWMA (, n,) = SMA(, n) n j j () α =, w j () = =, i.e. GWMA (, n,) = LWMA(, n) n n( n + ) j (), ( ) j= i= GWMA : 9

113 j 6 j α j = =, i.e. GWMA(, n,) is a square-weighed MA n n( n + )( n + ) j (4) =, w ( ) i= (5) =.5, w (.5) j α = (no closed form), i.e. GWMA(, n,.5) is a square-rooweighed MA j n i= j Noe ha he weighs consruced for he GWMA according o he formulae (B.) are always higher for more recen observaions: as α increases larger weighs are assigned o he recen observaions. 9

114 Appendix C Examples of he MAs wih variable weighs C. Volume-adjused moving average Volume-adjused moving average (VAMA) can be represened as following: VAMA = n i= w * n i+ P, w * * = w ( ), i+ i V i { i } i> * P n { w i } i n where is a price series; - is MA lengh; and i { i } i> cumulaive volume a he h momen of ime V. - are MA weighs as a funcion of he The idea behind VAMA is o use some addiional informaion abou he rading aciviy a some poin. Tradiional ime-based (sampled) MA suppose ha all momens of imes are equal wih respec o he informaion hey brings. VAMAs rea he aciviy of rading as he measure of he price imporance in he paricular momens of ime: for example, he sharp increase in he rading aciviy migh indicae he change in he rend, herefore hese price observaions have much higher imporance for rader han he prices when he marke is no raded acively. * Differen approaches o define he funcion w can be used. Richard W. Arms Jr ([], V.8:3) proposes o change he sysem of coordinaes from ime- o volume based. The new curve will have prolonged price values for heavily raded momens of ime and shoren (or even negleced) he price values for lighly raded momens. The new VAMA can be calculaed on he basis of he new price curve. Arms ([], V.8:3) used graphical approach o consruc VAMA, in paricular equivolume charing, which can be ranslaed ino programming algorihm. Suppose ha you have daily observaions of price (high, low) and volume for a paricular insrumen. According o he equivolume charing each price observaions is represened in he form of box wih high being he upper side of he box, low being he lower side of he box, and volume being he widh of he box. The mean price calculaed as sum of high and low devided by wo, is posed one or several imes in he box depending on is widh. Then x-uni-volume MA of some pre-chosen weigh srucure (for example, SMA- or EMA-ype) is calculaed. The rading rule hen involves he new price curve and VAMA. Then VAMA approaches price curve during he periods of heavy rading more rapidly han he ime-based MA. Wih his approach he days wih heavy rading have higher weighs and, hus, more imporance for he definiion of he MA. A our opinion he mehod presened above has several drawbacks. Alhough he VAMA akes ino accoun he rading volumes a each momen of ime, i canno incorporae he informaion abou volume possible (auo-) correlaion. Besides, building he price-volume blocks of differen widh implies price inerpolaion for he non-available cumulaive volumes values; he inerpolaion of he price a he same level can be quesioned. Finally, while he MA calculaion procedure can be programmed, i involves many subjecive judgemens, among which he definiion of he volume frequency and he poins a which he MA is calculaed. C. Variable index dynamic average (VIDYA) Tradiional MA is based on he assumpion of consan daa volailiy: rend is filered by assuming ha he noise has consan volailiy. Tha is why drasic price movemens caused by he change in is volailiy would be misakenly reaed as rend enforcing or rend-breaking evens. Therefore, adaping a MA o he marke volailiy should improve is predicion power. Chande ([3], V.:3) proposes he following adapaion of EMA o he marke dynamic. He has subsiued a smoohing consan α in he formulae of he EMA by he following coefficien: α =, (C.) kv 9

115 where k = cons, V - dimensionless marke-relaed variable. V Chande ([8], V.:3) represens as a raio of wo volailiy measures: he sandard deviaion of he insrumen s closing price and he marke sandard deviaion defined as a reference measure. Such MA in rading lieraure is called variable index dynamic average (VIDYA ): VIDYA σ σ = n + n k P k VIDYA σ ref σ ref, (C.) where VIDYA, VIDYA - variable index dynamic average for he periods and -; k = cons, < k - smoohing coefficien when σ n = σ ref ; P - he las available closing price; σ, - rolling sandard deviaion of he analysed insrumen and marke (rolling) sandard deviaion n σ ref (reference value) (for ex. CAC4); he lengh of he periods over which he rolling sandard deviaions are calculaed, are n- and ref-periods; hey can be differen 3. As we can see from (C.) and (C.), if he analysed insrumen s volailiy is he same as he reference marke volailiy, hen α = k became a consan; if insrumen s volailiy increases during some period of ime Δ, hen α > α Δ and adaped MA approaches closer he price curve; oherwise, if insrumen s volailiy diminishes during some period of ime Δ, hen α < α Δ and adaped MA diverges from he price curve. The mahemaical exercise can prove his VIDYA s propery. ( ) ( ) σ n σ Suppose some insrumen Y and reference index has he same volailiy change, i.e. σ ref We can always find some α ha corresponds o VIDYA a some precise momen of ime T: EMA ( ema ) VIDYA T = EMA T Then he effecive lenghs of he EMA and VIDYA coincides, and α = α, or σ n k σ ref ( T ) = ( T ) n + vidya, (C.3) where n is he effecive lengh of EMA., which coincides wih VIDYA effecive lengh. Then ema ( ) ( ) n. σ ref I seems like auhor defines he coefficien as k =, where n is he lengh of he SMA which has he same n + lag as EMA wih smoohing coefficien k, when σ n = σ ref 3 In Chande s example ([8], V.:3) he reference sandard deviaion defines he long-erm volailiy, conrary o shor-erm insrumen s volailiy: n < ref 93

116 ( σ ) ref ( T ) kσ n ( T ) kσ ( T ) n =. (C.4) n The equaion (C.4) confirms he observaions made above: he higher is he insrumen s volailiy relaively o marke volailiy he smaller he effecive lengh of he VIDYA, i.e. he VIDYA converges owards he price curve; and vice versa. As he resul he VIDYA follows he price curve more closely when volailiy increases. Oher measures of relaive volailiy V exiss, such as Chande momenum oscillaor (CMO) 4: where S u, S d CMO ( S S ) u d = (C.3) S u + S d - sum (for n-days period) of he respecively upward/downward closing price change: S u S d = = + Δ i i= n+δ+, Δ i, i= n+δ+ Δ i = ( P i P i δ ) I( P( i) P( i δ ) ) ; where Δ + i = ( P() i P( i δ )) I( P( i) P( i δ ) > ) ; ( ) ( ) P () i - price a momen i; I () - indicaor funcion: I( V ( x) i) ( x), V i =, oherwise C3. Variable-lengh moving average (VLMA) Arringon ([6], V.:6) inroduced variable-lengh moving average. The idea behind his MA is o define is lengh as a funcion of relaive magniude of recen price incremens: if recen price changes are unusually large he lengh of he moving average is shorened and average auomaically becomes more sensiive o he emerging rends 5. Conversely, when he rend is sable he lengh of he MA is increasing. The usual price change is proposed o define by he means of he price changes disribuion: [ μ σ ; μ + σ ] R, (C3.) u R, ( ) where u R e - usual and exreme price incremens: R = P( ) P δ μ, σ - mean and sandard deviaion of he price incremens: R ~ L ( μ,σ ) We believe ha he hypohesis behind VLMA is ha he larger price movemens precede rend change. However, Arringon ([6], V.:6) does no explain in his aricle he assumpions on which his MA ress (for example, wha is he probabiliy disribuion of he price incremens). He also does no precise how o define he change in he MA lengh. Therefore we can see, ha alhough concepually he idea is ineresing, his VLMA should be beer defined. 4 Noe ha CMO is very similar o he oher echnical indicaor relaive srengh index (RSI): Su RSI =, Su + S d where S u, S d - sum (for n-days period) of he respecively upward/downward closing price change. 5 Arringon, G.R. The basics of moving averages Sock & Commodiies, V.:6, pp.75-78, Copyrigh Technical Analysis Inc. 94

117 Appendix D Le Some examples of he variogram γ ( h) and correlaion ( h) h - lag of he variogram (disance beween he observaions); γ () - variogram; C () - covariance; a - range parameer; σ - variance (sill); α - parameer. C models I. Saionary models. Sable model: < α γ ( h) = σ e C ( h) = σ e α h a α h a Paricular cases: () α = - exponenial model () α = - gaussian model. Spheric model: 3. Dumped cosines model: II. Non-saionary models γ C ( h) ( h) 3 3h h σ, h a = 3 a a σ, oherwise σ = 3 3h h +, h a a a, oherwise 3 h πh aα γ ( h) = σ e cos a C h a πh α σ a ( h) = e cos. Fracal model: < α < Paricular case: () α = - linear model γ ( h) = σ h α 95

118 Appendix E Shor descripion of he Bund insrumen observaions (s) x 6 Figure E. Bund (March 9 - June 8, 6, frequency sec) quoes observaions (s) x 6 Figure E. Bund (June 9 Sepember 8, 6, frequency sec) quoes 96

119 .5 x -3 March 8 June 8 Sep 8.5 Vario.5 Variance: March 8:.4 June 8:.4 Sep 8: lag (in s) Figure E3. Bund (December 9, 5 - March 8, 6, March 9 June 8, 6, June9-Sepember 8, 6, Sepember 9 December 8, 6, frequency sec): Variograms for Bund residuals (n(ema)=7 sec) for differen conracs.5. March 8 June 8 Sep 8.5 Vario..5 Variances: March 8=.4 June 8=.4 Sep 8= lag (in s) x 4 Figure E4. Bund (December 9, 5 - March 8, 6, March 9 June 8, 6, June9-Sepember 8, 6, Sepember 9 December 8, 6, frequency sec): Variograms for Bund residuals (n(ema)=79 sec) for differen conrac Shor descripion of he Bren insrumen 97

120 Appendix F Shor descripion of he DAX insrumen DAX index represens he socks marke; is hisoric quoes are given in Figure F. DAX daa corresponds o he ime inerval of 3/7/3-7//6 and he frequency of 3 minues observaions (in 3 min) x 4 Figure F. DAX quoes for period 3/7/3-7//6 (frequency 3 minues). quoes observaions x 4 Figure F. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions) residuals 98

121 As we can see from Figure F, he daa is posiively rending, and hus, non-saionary. Therefore, he EMA of he lengh of 5 observaions is exraced from he price daa o esimae he covariance of he residuals. As we can see from figure F, he obained residuals do no exhibi any rend. The esimaed variogram of he residuals for he sub-sample [;85] is presened in Figure F3. The following heoreical variogram model (he sum of exponenial and damped-cosines model) was fi o daa: h h ( ) πh = γ h e + 9 e cos variogram 5 5 empirical model 3 lag( lag=3 min) Figure F3. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions, observaions -85): empirical variogram of he residuals and is heoreical fi The MACD indicaor esimaed as he difference of he EMAs of he lenghs n = 4, n = 8 is presened in Figure F4, while heir variogram esimaed on he sub-sample [;85] is given in Figure F5. The following model was fir o he empirical variogram: γ ( h) = 3 e h e h.3 8 h cos 8 99

122 6 4 MACD observaions x 4 Figure F4. DAX (3/7/3-7//6, frequency 3 minues, n = 4, n = 8 ): MACD indicaor ( n = 4, n = 8 ) variogram 5 variance=46 5 empirical heoreical lag (in 3 min) Figure F5. MACD indicaor for DAX (3/7/3-7//6, frequency 3 minues, n = 4, n = 8, observaions -85): empirical variogram of he residuals and is heoreical fi

123 Appendix G Shor descripion of he Bren insrumen Bren is a fuures on crude oil. Therefore, we consider his insrumen as a represenaive of commodiy markes. The peculiariy of his insrumen is ha i is highly volaile. Our daa represens he period of 7//4-7//6 wih 3 min frequency (see Figure G) quoes observaions (in 3 min) Figure G. Bren quoes for period 7//4-7//6 (frequency 3 minues) residuals observaions Figure G. Bren residuals (7//4-7//6, frequency 3 minues, n(ema)= observaions)

124 The daa is rending, and hus, non-saionary. We eliminae he EMA of he lengh observaions o calculae residuals. The variogram of he ransformed residuals (see Figure G3) is evaluaed on he sub-sample [ ;54]: h γ ( h) = e empirical model variogram lag ( lag=3 min) Figure G3. Bren residuals (7//4-7//6, frequency 3 minues, n(ema)= observaions, observaions h :54). Variogram of he residuals and heoreical model fi o he daa γ ( h) = e 5 The MACD indicaor was consruced for he parameers n = 4, n = 8 (see Figure G4). The variogram esimaed on he firs half of he sample in presened in Figure G5. The following heoreical model is fi o MACD daa: ( h) =. e h 5 +. e h.3 5 h cos 5 γ.

125 .5.5 MACD observaions (in 3 min) Figure G4. Bren (7//4-7//6, frequency 3 minues). MACD indicaor, consruced for EMA lenghs n = 4, n = empirique model.5 variogram lag(in 3 min) Figure G5. MACD indicaor for Bren (7//4-7//6, frequency 3 minues, n = 4, n = 8, obs. :54): Variogram empirical and heoreical 3

126 Appendix H Shor descripion of he X insrumen Insrumen X represens an arificially creaed index, used by one bank for sraegy consrucions. Due o he confidenialiy reason we canno neiher presen is deail descripion, nor provide he informaion on is real quoes. Tha is why on Figure H, which presens he quoes pah during some period of ime, here are no icks on he Y-coordinae. The only informaion we can provide is ha daa frequency is hour and ha index has mean-revering naure in long run. quoes observaions (in hour) Figure H. X insrumen quoes (frequency hour) As we can see from Figure H, some rends are presened in he daa. We eliminae he EMA of he lengh of 5 observaions o obain residuals used for he covariance calculaions. As we can see from figure H, he residuals do no exhibi any rend. Empirical variogram of he residuals in he sub-sample [;37] is given in Figure H3. The following heoreical model is fi o empirical variogram: h γ ( h) = 5. e 8. The MACD indicaor was consruced as he difference beween EMAs of he following lenghs n = 4, n = 8 (see Figure H4). The empirical variogram of his indicaor esimaed on he subsample [; 37] is presened in Figure H5. The following model is fi o he daa: h h ( ) γ h =.7 e.3 e. 4

127 5 residuals observaions Figure H. X insrumen residuals (frequency hour, n(ema)=5 observaions) 6 5 variogram 4 3 empirical model 3 lag( lag= hour) Figure H3. X insrumen residuals (frequency - hour, n(ema)=5 observaions): Variogram of he ransformed residuals for sub-sample [:347] and heoreical model fi o he daa. 5

128 4 3 MACD observaions Figure H4. X insrumen (frequency - hour,): MACD indicaor ( n = 4, n = 8 ).4. empirical heoreical variogram lag (in hour) Figure H5. MACD indicaor for X insrumen (frequency - hour, n = 4, n = 8, obs. :37): Variogram empirical and heoreical (variance=). 6

129 Appendix I Examples of he MACD for he Bund daa The mean-revering naure of he MACD indicaor allows obaining bounded variograms for hese indicaors. Furher we consider how he choice of he EMAs lenghs has impac on he MACD indicaor and is variograms. Couples of EMAs lenghs, for which MACD and is variograms are calculaed, are given in Table I. Table I Effecive lenghs of he EMAs used for MACD calculaions Case # n n Figure I presens examples of he MACD indicaors calculaed for he EMA lenghs in Table I. As we can see MACD indicaors have similar forms. Some MACD curves have more smoohed form, bu oscillae wih higher ampliudes (case 4), some MACD curves are less smoohed bu oscillae wih lower ampliudes (case 5). As he MACD indicaor exhibi mean-revering paern, we migh expec he variogram o converge versus some level. However, aking ino accoun significan discrepancy beween he variances of differen MACD, we propose o consider he variograms normalized by heir respecive variances (see Figure I). Alhough he variograms curves do no coincide, he heoreic models used for heir modeling mos likely would be he same, which will differ only in parameers..5 MACD indicaors.5 MACD MACD-8-4 MACD--5 MACD-6-6 MACD observaions Figure I. Bund (99-6, frequency = day, observaions -5): MACD, consruced for differen EMAs 7

130 .4. Variogram.8 Variances:.6 Var(MACD-6-3)=.5 Var(MACD-8-4)=.7 Var(MACD--5)=.5.4 Var(MACD-6-6)=.5 MACD-6-3 Var(MACD--6)=. MACD-8-4. MACD--5 MACD-6-6 MACD lag Figure I. Bund (99-6, frequency = day): Normalized variograms for he differen MACD indicaors, consruced for differen EMA lenghs 8

131 Appendix J The examples of he weighs for he kriged volume weighed moving average in he case of large variabiliy in he cumulaive volume As have been discussed he presence of greaer variabiliy in he volume will have impac on he weigh srucure. We propose he following example o illusrae his hypohesis. Suppose he jump is presen in he cumulaive volume (see Figure J). The kriged weighs based on he observaions in he inerval [455;464] presens oally differen paern from wha we have observed before. The weighs are much more volaile. 5 cumulaive volume (in unis) oservaions Figure J. Bund (April 8, 6, frequency sec): cumulaive volume (scale facor=) a he observaions

132 .3.5. weighs value # of weighs: # corresponds o he mos recen observaions Figure J. Bund (April 8, 6, frequency sec): weighs of he kriged volume weighed MA for he price inerval [455;464].

133 Par II. An Alernaive o Bollinger bands: Daa ransformed bands Inroducion Trading bands are one of he mos frequenly used ools of echnical analysis. They are lines ploed around a measure of cenral endency, such as he moving average, shifed by some percenage up and down (upper and lower bands) (Bollinger, ). The schemaic represenaion of he concep is given in Figure.. Differen ypes of such bands are defined, such as envelopes, price channels (Bollinger, ; Chande, ; Murphy, 999). While he difference beween hem lies only in he way ha hey are defined and consruced; he sraegies based on hem are quie similar. Touching/breaching hese bands gives he rader informaion on he direcion of price movemens, or abou relaive price levels (wheher he insrumen is oversold or overbough), which are used as signals in sraegy consrucion. Upper band Cenral endency Lower band Figure.. Schemaic represenaion of rading bands According o Bollinger () rading bands were firs used more han 5 years ago. Some papers analyze he profiabiliy of he bands, as well as compare heir predicion power wih oher economic mehods (Williams, 6). Unforunaely, less work is dedicaed o he analysis of he concep from he heoreical and saisical poin of view. Laer we will provide a shor descripion of he bands wihou heir deep analysis and explanaions. An expanded overview of he differen bands is given in he book Bollinger on Bollinger bands (), o which he ineresed reader can refer for more informaion. In his paper we prefer o look a he concepual similariies and differences beween he mehods as we could no find his ype of overview in any oher papers. From our poin of view, he mehods differ in wo aspecs: See Bollinger () for more deail insigh ino hisory of he rading bands.

134 () he way ha hey are consruced, i.e. he indicaors used o consruc middle, upper and lower bands; () he concepual ideas, based on he daa saisical characerisics, ha are incorporaed in hese bands. The difference in he consrucion mehod is mainly jusified by he definiion of differen rading sraegies. For example, if he original Bollinger bands use simple moving average (SMA) as he middle line, some inerne rading sources propose o use exponenial moving average (EMA) as he middle line jusifying i by he possibiliy o generae higher rading profis. The problem wih such jusificaion is ha any obained profis are condiional on many rading facors such as sraegy iself, raded insrumen, is frequency, ec. Generalizaion of such resuls on larger samples of insrumens wihou heir preliminary esing is quie dangerous. Therefore, we found his approach more subjecive. Four groups of he concepual ideas can be disinguished: () consan volailiy; () non-consan volailiy; (3) disribuional characerisics of he price or residuals (Price-MA); (4) ohers. The fourh group includes bands which incorporae eiher echniques, based on he conceps o define he bandwidh oher han daa saisical characerisics (for example, dividends); or echniques ha were no described in he finance lieraure; or echniques ha are very subjecive in heir presenaion and usage. These among ohers include valuaion envelopes based on he dividend yield, ha were inroduced in 966 by he Invesmen Qualiy Trends newsleer; Le Doux s Twin-Line echnique; and envelopes defined by Hurs in 97 on he basis of (hand-made) cycles idenificaion. This group of echniques will no be discussed any furher in his paper. Our ineres lies wihin firs hree groups of bands. All he bands equally disanced from he middle line hroughou he ime are he represenaive of he firs group of sudies. The mos widespread example of such bands is a parallel shif of he MA up and down by some fixed percenage (Kaz, McCormick,, Lien, 6). The desire o incorporae he sochasic (non-consan) price volailiy led o he developmen of oher band ypes. This boosed he developmen of he second group of he sudies. Kelner s mehod was among he firs approaches, used in he 96s, which incorporaed a rolling esimae of he insrumen volailiy (Kelner, 96 in Bollinger ). In fac i used a -day MA, buil on he ypical price, as a cener line, and (moving) -day average range 3 as rading bands. Technical raders are aware ha range is one of he volailiy measuremens. The Bollinger bands, developed in early 98s by John Bollinger, became he mos popular represenaive of his concepual group. Bollinger used sandard deviaion o measure an insrumen s volailiy and o consruc bands around some SMA. We will discuss his mehod in more deail laer. Examples of he hird concepual groups include Bomar bands and he Donchian channel. In early 98s, asymmerical Bomar bands were proposed by Chaikin and Brogan (presened by Bollinger ), based on he following consrucion formula: upper/lower bands conain he 85% of daa above/below he -day MA for he pas 5 periods. As we can see, he developers should use he disribuional properies of he insrumen in order o define he confidence inervals ha conain he 85% of daa. We will discuss he problems linked o his mehod furher Typical price is he sum of he high, low and close prices, divided by 3. 3 Range is a difference beween high and low prices a given momens of ime.

135 in more general framework. Here we jus wan o cie Bollinger, who explained he main advanages and drawbacks of he mehod 4 : The major benefi conferred by Bomar Bands was ha analyss were no longer forced o provide heir own guesses abou wha he proper values for he bands were. Insead, hey were free o focus on decision making and le heir PCs se he bandwidh for hem. Unforunaely, Bomar Bands were exremely compuaionally inensive for heir ime, and o his day are no readily available beyond Insine s research and analyics (R&A) plaform. Thus, hey have no achieved he broad accepance hey deserve. John Bollinger,, Bollinger on Bollinger Bands, p.46. The firs conclusion is ha Chaikin and Brogan s sudy was he firs aemp o formulae a mahemaical opimisaion problem o find opimal bandwidh values. I was based on probabilisic conceps such as he disribuional characerisics of he random variable under sudy. I is clear, ha all he sudies and heir conclusions produced before Bomar bands were subjecive and daa sensiive (based on raders guesses or analysis and provided for concree insrumens, i.e. daa samples). The second conclusion is ha due o heir compuaional inensiy hese bands did no become very popular. Donchian s four-week rule is older (defined in earlier 96s), bu no less ineresing: buy when he four-week high is exceeded and sell when he four-week low was breached (Bollinger, ). This rule became a prooype for he Donchian channel - he concep ha ses he upper band a he n-period high and lower band a he n-period low. We classify hese band ypes in he hird group, as we can see in he basis of he rule he saisical concep of exreme values ha were eiher consciously applied or was empirically observed and used by he developers. Roughly speaking, he raders define he maximum/minimum (exreme) price values for some pas periods as a hreshold for he definiion of he overbough/oversold markes in some fuure periods. Again, we would like o cie Bollinger on a Donchian channel: This concep is rumored o be a he hear of one of he more 5 successful rading approaches in wide use oday, ha employed by he Turles 6. Bollinger J,, Bollinger on Bollinger Bands, p.4 Anoher unusual fac abou hese bands is heir sair -like form. This form, however, can be easily explained by he fac ha daa wihin some ime frames are locally saionary wih he same disribuional properies and hus, exreme values. Simply speaking, consider a subsample of n values on he inerval [ n +; ] o define he local maximum and minimum of he random daa. For locally saionary daa i is unlikely ha rolling he subsample forward by observaion (or a small Δ number of observaions) o he inerval [ n + ; + ] (or [ n + + Δ; + Δ] ) would significanly change he local maximum/minimum. Tha is why we can obain hese, le s call hem, Δ -seps of he sair -like bands. To be saisically correc, we should say ha he firs and second groups are jus special cases of he hird group in which he volailiy is one of he saisical characerisics ha are pars of more general disribuional analysis of he daa. In his paper, we propose an alernaive ype of rading bands, called daa-ransformed (DT) bands, which belong o he hird group of bands. DT bands are more general case han he Bollinger bands: 4 The auhor of his aricle uses underlined and bold fons. 5 Possibly should be mos insead of more ; probably Bollinger ex error. 6 Richard Dennis, a famous commodiy rader, augh a number of raders his proprieary echniques. Those raders are called Turles (Bollinger,, p.9) 3

136 () hey provide more rigorous heoreical inerpreaion (jusificaion); () he manner in which hey are consruced, allows for a mahemaical approach for bandwidh opimizaion ha is based on daa disribuional properies. Our goal is o produce a new heoreical concep and o show ha DT bands can be used as a new echnical indicaor o creae successful rading sraegies. For his he mehod mus be easy o apply 7 and ouperform Bollinger bands a leas in some cases. To illusrae is poenial, we give four numerical examples on differen ypes of he financial insrumens (bonds, socks, commodiies and arificially consruced porfolio indexes), a differen frequencies and ime periods. In hese examples we compare he DT bands wih he classical Bollinger bands and show ha in some cases DT bands give beer resuls. The empirical observaions of he DT bands sugges an ineresing hypohesis. We believe ha DT bands can help o deec Ellio waves and Suppor/Resisance levels, wo popular conceps in echnical analysis. This hypohesis will no be analyzed in deail as i goes beyond he scope of his paper. However, he sraegies based on hese signals are inroduced and hey show posiive (hough no he bes) oucomes in he majoriy of cases. The srucure of he par is he following. In he firs chaper we presen he classical Bollinger bands and he rading sraegies buil on hese bands, and discuss he assumpions on which he heory of Bollinger bands is based. The second chaper inroduces he new concep of he DT bands. The hird chaper proposes he rading sraegies and he framework for he comparisons of he DT and Bollinger bands. The fourh chaper presens four numerical examples. The conclusions are proposed a he end of he par. Classical Bollinger bands concep This par discusses he concep of Bollinger bands proposed in he early 98s and sill very popular among raders nowadays. The firs subchaper defines he bands and explains how hey are consruced (calculaed). The second subchaper presens several rading sraegies based on Bollinger bands. Finally, he hird subchaper proposes some plausible assumpions as he basis of he heory and as such plays an imporan role in our paper because here do no seem o be any heoreical papers on hese bands. This is raher surprising because his approach clearly inroduces he saisical conceps (volailiy measuremens) ino he rading mehods and provides a powerful insrumen o consruc profiable rading sraegies. Unforunaely Bollinger bands are no presened as rue heoreical concep: here is no framework, including assumpions; no cases are given o show when he mehod should work (a leas heoreically). Bollinger menioned hese assumpions indirecly in his works (books, web-sies) explaining in a simple manner when and how he approach should be used. Bu we believe ha hese assumpions should be formalized in more scienific manner. In paricular, as main criique of he echnical analysis is is subjeciviy and he absence of scienific jusificaion. This is no merely a heoreical exercise. We will show ha by quesioning hese assumpions we were able o produce a more general concep of DT bands. 7 In conras o he Bomar bands, DT bands are no compuaionally inensive. 4

137 . Bollinger bands definiion Bollinger bands are paricular examples of rading bands. Figure. shows an example of hese rading bands for Bund quoes. We can see he four key componens: () price (quoes), () he mid-line, (3) he upper band, and (4) he lower band. price, SMA, bands price SMA lower band upper band observaions Figure.. Bund quoes (3/7/3-7//6, frequency: 3 minues), SMA (lengh n=5 observaions) and Bollinger bands (Bollinger parameer k=.) In mos of he well-known sraegies, based on he Bollinger bands, he moving average (MA) represens he mid-line. Traders use differen ypes of moving averages such as a simple moving average (SMA) or an exponenial moving average (EMA). The upper and lower bands are obained by shifing he mid-line up and down by a value, which is a funcion of he price volailiy. These componens are represened mahemaically by he following formulas: () Price curves: { P }, () Mid-line: i > i m = n P i i= n+ I is an esimaor of he mean on he inerval [ n +, ] average (SMA) of lengh n a ime momen. (II..), which coincides wih simple moving (3) Upper line: b U = m + kσ (II..) 5

138 where m is he price mean esimaor on he inerval of lengh n ([ n +, ] ), or SMA of lengh n esimaed as in (II..) and where σ is an esimaor of he price variance on he same inerval n +, : of lengh n: [ ] ( P Ε( P) ) i i= n+ σ = (II..3) n k - Bollinger bands parameer, k = cons >. Ε in is pure saisical sense coincides wih SMA or mid-line for he Bollinger bands. This is one reason why Bollinger insiss on using his ype of he MA as he mid-line. Noe 8, however, ha he esimaor of he variance is no he weighed sum of he squared sum of he residuals R = P SMA : For saionary processes he expecaion ( P) i R Pi SMAi i= n+ i= n+ no σ =. n n i i ( P SMA ) i i= n+ σ =, (II..3 ) n ( ) (4) Lower line: b where m,σ, k, n are as in (II..). L = m kσ (II..4) From hese commens we can draw he following conclusions: The Bollinger rading band sraegy is adjused for price volailiy, as he bands definiion involves sandard deviaion, as a volailiy measuremen. Upper and lower bands a he same momen of ime are placed a equal disance wih respec o MA, i.e. hese bands are symmerical. However, his disance can be differen a differen ime momens, due o he changing naure of he price volailiy 9. The disance beween bands is narrowing wih decrease in he price volailiy and widen wih increase in he price volailiy. The usage of SMA as he middle line a he place of oher moving average is jusified by he fac ha SMA is he saisical mean of he price sub-sample he same value ha is used for he calculaions of he price sandard deviaion. In addiion, some research shows ha subsiuion of he SMA by more rapid MA does no produce higher sraegy oucomes (Bollinger, ). We have also shown in par ha opimal kriged moving average (KMA) on average coincides in lag wih he SMA; and KMA migh almos coincide wih he SMA in values a large lenghs of he mowing window. The ineres of inroducing EMA on he place of SMA migh be he recurren mehod of is calculaion (he curren EMA can be calculaed as a weighed sum of curren price and previous EMA). However, a his respec SMA can be also programmed as recurren formulae, bu i needs o save more daa a each ime momen han he EMA. 8 This observaion will be imporan furher when we will consider he daa-ransformed bands. 9 I is well-known fac ha hisoric price volailiy is no a consan value. 6

139 The choice of he parameers k, n should depend on he daa and opimizaion procedures. However, Bollinger () proposed he following defaul values for hese parameers: a. n = days (or monh) ; b. k = c. If n decreases o, hen k should decrease o.9, and if n increases o 5, hen k should increase o. (see Table.). Table. Defaul Bollinger bands parameers values Moving average lengh k.9 5. Source: Bollinger, J.. Bollinger on Bollinger Bands These defaul values were based on empirical findings ha a hese values one can ge conainmen beween 88 and 89 percen in mos markes. In he case he daa demands calculaion of he Bollinger bands for he periods longer han 5 or shorer han, Bollinger suggesed changing he daa sampling frequency in order o sick o he defaul values -5 periods. The defaul parameers are also proposed as a saring poin in opimizaion procedure of he Bollinger bands parameers search.. Bollinger bands and rading sraegies The classical rading sraegy based on Bollinger bands can be defined as a conrarian, i.e. he enry posiions are characerized by over- or under-pricing of he insrumens. The bands hemselves serve as he hresholds, which define wheher insrumen can be considered as misspriced. The simples Bollinger bands sraegy can be formulaed as following: - he long posiion is aken when prices approach (or cross) he lower band (Figure.3, poin ). An insrumen is under-priced and is price is expeced o reurn o is average, i.e. o increase. - he shor posiion is aken when prices approach (or cross) he upper band (Figure.3, poin 3). An insrumen is overpriced and is price is expeced o reurn o is average, i.e. o decrease. - he posiion can be closed when prices reurn o he average level (Figure.3, poins and 4). The middle line or MA represens his average price level. The probabilisic explanaion of he sraegy is presened in Figure.4. Le say he price has some densiy wih mean m, given by solid blue curve. The Bollinger bands m ± b define he areas of he exreme price values ha defines over- or under-priced insrumen (he areas marked by blue lines). Suppose curren price P has breached he upper Bollinger band m + b, indicaing ha he insrumen is overpriced. The rader ake shor posiion expecing ha he price will reurn o is mean value. Taking ino accoun he form of he densiy curve, he fuure price will reurn o more probable value wihin he inerval m ± b. Even if he fuure price value will be only P > m he ne worh of he posiion is posiive: W = P P >. Bollinger () has no precised he frequency of daa, bu we believe he alked abou daily (or monhly) frequency. Bollinger, J. Bollinger on Bollinger Bands, McGraw Hill,, p

140 price, SMA, bands price SMA lower band upper band observaions Figure.3. Bund quoes (3/7/3-7//6, frequency: 3 minues), SMA (lengh n=5 observaions) and Bollinger bands (k=.): Posiions, aken wihin conrarian sraegy: () enry-long; () exi-long; (3) enry-shor; (4) exi shor W W pdf m-b m+b m m* P P P3 Figure.4. Saisical explanaion of he Bollinger bands sraegy 8

141 Figure.4 also explains wha happens when he saisical characerisics of he variable has * changed, in paricular he mean has changed from m m. In such case price P will be misakenly defined as exreme value, and rader will ake he shor posiion expecing ha he price will fall o m. In realiy i is more probable ha he price will move up o is curren mean or even higher P 3. In such case he ne worh of he posiion will be negaive: W = P m * < or W 3 = P P3 <. Figure.4 also explains why he parameer k is abulaed for differen MA lenghs. The disribuion of he residuals, in paricular is dispersion, depends on he MA lengh used for heir calculaions. As he resul, he bands, which conain 88-89% of daa, depend on he MA lengh. There are some specific feaures of applying Bollinger bands, which are widely discussed in he rading lieraure and can be seen from Figure.3. Firs, during well-pronounced posiive (negaive) rends he price flucuaes mainly beween middle line and upper (lower) band. For example on Figure.3, prices vary beween (approximaely) -4 observaions for negaive rends and beween -5 observaions for posiive rends. The probabilisic explanaion is ha curren mean is changing, in paricular increasing (decreasing), hus higher (lower) price are mos probable. Second, Bollinger band sraegies, like all conrarian sraegies, send false signals during he upward or downward rends due o he misakes i makes in defining he rue insrumen value. For rending markes he rue value increases or decreases, hus, overvaluaion or undervaluaion saemens are wrong. For example, wha we believe o be over-valued, in realiy indicaes he increase in he rue value. Saisically i implies ha our curren bands parameers (esimaes) do no reflec he rue probabiliy disribuion, which is no consan. Generally speaking, applying conrarian sraegies in he rending markes leads o he losses. Due o hese false signals raders normally do no rely solely on he Bollinger band signs, bu consider hem in he combinaion wih oher echnical signals, which eiher confirm he overbough/oversold sae or predic he fuure price movemens, for example, rend-reversion. These indicaors include momenum, volume, senimen, open ineres, iner-marke daa, ec. (Bollinger,, Consequenly, Bollinger bands do no give he uncondiional signal o enry he posiion; hey give clues as o how he price migh behave in fuure bu hese clues normally need confirmaion. Confirmaion signals help o avoid hese rending pars of he price curve. We choose he momenum indicaor o confirm ha he marke is oversold or overbough. Momenum can be calculaed as follows: where Δ is he lag parameer. ( Δ) = P( ) P( Δ) M, (II..5) As can be seen from Figure.5 momenum is an oscillaor, which flucuaes around zero. (See Figures A-A in Appendix A for real-life examples). Local momenum maximum/minimum indicae ha he local upward/downward rend is slowing down (Figure.5, poins and 5), while he inersecion wih he level poins o he rend inversion (Figure.5, poins 3 and 6). The overbough/oversold markes are confirmed when momenum reaches some hreshold Noe ha he change in he rue mean will no be refleced sraigh away in he SMA (mid-line) and bands, as heir esimaes depends on he pas values. 9

142 levels while moving upward or downward 3. These (hresholds) values depend on he insrumen quoes and behavior. So hey have o be calculaed on a case-by-case basis. U s M L s M 3 6 M 4 Figure.5. Schemaic represenaion of a momenum indicaor 5 U L We can formulae he sraegy, based on momenum, as follows. Le s M, s M be he momenum hreshold values used o define he overbough and oversold signals. Then if momenum values are higher han he upper hreshold level, he insrumen is overbough (Figure.5, poin ); if momenum values are lower han he lower hreshold level, i is oversold (Figure.5, poin 4). Suppose for simpliciy ha he hresholds are symmeric U L s M = sm = sm. If M (, Δ ) > s M hen he overbough marke is confirmed and if Bollinger bands sraegy generaes he same signal ake shor posiion (S). By he symmery if M (, Δ ) < s M hen he oversold marke is confirmed and if Bollinger bands sraegy generaes he same signal ake long posiion (L). The usage of he momenum as confirmaion for miss-priced insrumens migh be confusing a firs sigh as his indicaor used mainly in he rend-following sraegy formulaion. There are several explanaions, why he usage of he rend-following indicaors in he conrarian sraegies migh be acceped. On one hand we can define he rend-less long-erm period as he combinaion of he local shor-erm upward and downward rends. Than slowing down of he rend implies he reversion of he rend and hus, reurn o he mean value. On he oher hand, he increasing/decreasing momenum wihin is hresholds inerval indicaes ha currenly he insrumen is in rending periods, allowing o avoid aking false posiion by he conrarian sraegy. Finally, he signal (sen by momenum) ha he curren rend is slowing down migh indicae is reversion ino rend-less periods; he conrarian sraegies produce he bes resuls for he price paern wihou any rend. Furher we show ha hese confirmed sraegies lead o higher oucomes han non-confirmed ones. 3 See Murphy (999) for more deails.

143 .3 Theoreical framework for Bollinger bands The main problem wih he Bollinger band approach is he absence of a precise heoreical framework, which could jusify heir usage in rading. Lieraure review indicaes hree groups of he research done in he field. Bollinger () and many Inerne resources explain how o calculae Bollinger bands and presens various rading sraegies based on indicaors or price paerns. The oher wo groups concenraes on he opimizaion of he bands parameers, such as scaling parameer k and MA lengh n ; he difference beween hem lies in he chosen opimizaion approach. Bollinger () provides he opimal couples of he parameers ( k, n) ha work for he majoriy of markes. These beliefs are based on he analysis of he prices and verificaion ha he defaul bands include a high proporion of prices (around 9%). Oher group of research (for example, Williams, 6) has defined parameers values (k, n) as being opimal, when hey guaranee posiive oucomes for he simulaed rading sraegies based on hese bands. Consequenly, we can see ha parameers opimizaion problem is based on a leas wo differen approaches. One approach is based on rading profi maximizaion, i.e. find k so as o obain he larges or sable posiive profi when using he rading sraegy. The main criicism of his approach is ha he opimal values are daa and sraegy ype dependen, because he opimizaion is based on pas hisoric daa and sraegy algorihm. According o he oher approach, he opimal parameer values guaranee ha he inerval beween he upper and lower bands conains some fixed percenage, K%, of price observaions. Normally he K lies beween 85-95%. This opimizaion approach is more appealing, as i explains why he sraegy should generae profis, from a saisical poin of view. For large values of K, he prices ouside he bands are exreme values ha have a low probabiliy of happening (he insrumen is overpriced/under-priced), hus, he fuure prices should probably move o he values close o he average one 4. Tha is, he insrumen will reurn o is average value. Under such problem definiion he sraegy characerisics, such as profi and number of rades, depend on he K% value. The ne profi per rade is he difference beween he (aained) exreme and he mean values, while he expeced profi is he weighed sum of all possible ne profis. The number of rades under his sraegy is an indirec funcion of K%: he larger K%, he less probable is he occurrence of he evens, when price is ouside he bands; and, hus, fewer rades are underaken. This conclusion is imporan for he choice of K% value. Boh opimizaion problems should be based on he number of saisical assumpions abou he daa ha can define he heoreical framework for he Bollinger bands applicaion. In paricular, he second opimizaion approach, as more saisically grounded, requires more assumpions abou daa, which, we define furher. Assumpion : Daa saionariy. Saionariy imposes he invariance of he join probabiliy densiy funcion and, herefore, all is momens under emporal (or spaial) shif (ranslaion), i.e. for any h probabiliy disribuion of he variables P ( ), P( ),..., P( n ) is he same as of he variables P( + h), P( + h),..., P( n + h). For he emporal daa i means ha all momens of he random variable does no depend on he ime a which he variables are observed, bu only on he disance beween hem. Simply speaking, informaion abou he process [is] he same no maer where i is obained 5. 4 The conclusion holds for he saionary process. 5 Anselin L. Variogram analysis, presenaion

144 However, even under saionariy assumpions join probabiliy disribuion is ofen difficul o esimae; herefore, he saisicians has inroduced second-order saionariy - he invariance of he firs and second momens under ranslaion (see Definiion.). Definiion. A random funcion { P ( ) } is saionary of order wo if is mean and covariance do no depend on ime, bu only on he disan beween he variable, i.e., h : Ε [ P ( ) ] = m (II..6) Cov P, P + h = Ε P P + h m = C h (II..7) [ () ( )] [ ( ) ( )] ( ) For Bollinger bands his means he following: m = Ε i= n+ ( P) = SMA = cons [( P Ε( P) ) ] n P i ( P m) i i n+ σ = Ε = n = cons (II..8) L m,σ P ~ ( ) In he case of he saionary daa, SMA can be used as a rue mean esimaor. Then saisically he prices ha breach he bands can be considered as he exreme values and applicaion of he conrarian sraegy will make sense. Therefore, if he price daa is saionary of order wo, hen:. Bollinger bands can be defined as (II..)-(II..4) in secion. and Bollinger sraegy has saisical jusificaion. Moreover, he bands will be a almos he same disance from MA hrough all he ime.. The same sraegy parameers (defined as opimal) can be applied hroughou he ime. The saionariy hypohesis can be weakening if we assume ha prices are only locally saionary. Local saionariy means ha expression (II..8) holds rue only wihin some inerval of ime and he values of saisical momens alhough consan wihin he inerval on which hey are esimaed, can be differen from one inerval o anoher: m l = Ε ( P) n P i i= n+ [( P Ε( P) ) ] m l = on he inerval [ u, ], cons ( P m ) i= n+ σ l = Ε =, l = cons n P ~ ( ) m l, l i l, where u > n σ on he inerval [ u, ] L σ, where L - disribuion law is he same on he inerval [ u, ], where u > n, where u > n We can imagine such paerns as a combinaion of some rending periods wih he rendless periods; or he presence of he jumps in he rendless price paerns. Then:

145 . Wihin saionary inervals he Bollinger bands can be defined as (II..)-(II..4) in secion. and Bollinger sraegy has saisical jusificaion.. The sraegy parameers need o be recalculaed locally, each ime he saionariy paern changes. In some cases financial daa can be considered rend saionary. A rend saionary variable is a variable ha is formed by he deerminisic rend and saionary process (Focardi, Fabozzi, 4): P = m + R, (II..9) where m - is some deerminisic funcion, which values we know or can calculae a each momen of ime ; R - saionary residuals. A rend-saionary process can be ransformed ino saionary by subracing known rend m. The main problem wih he rend saionary daa is ha normally he rend values m are unknown; hus, needs o be esimaed firs. The advanage of he Bollinger bands lies in heir flexibiliy. If in he case of he saionary daa SMA served as a mean esimaor, in he case of rend-saionary daa SMA can be considered as * esimaor of he rend m. Le say he lengh of he SMA n such as Ε( R ) =. In his case, he Bollinger bands can define he exreme values, as in his case: ( P ) = m + Ε( R ) = m SMA ( P ) σ ( ) Ε (II..) σ = (II..) R Alhough he saisical meaning of he bands is sill preserved, he applicaion of he conrarian sraegy should be under quesion as he mean of he price is rending. Therefore, even if he posiion is aken for he (locally) exreme price values, he posiion migh be liquidaed a he SMA, which is unfavorable from he poin of view of he enry price. Therefore, under he rend-saionary hypohesis, he Bollinger bands have saisical meaning, bu he applicaion of he conrarian sraegy is no profiable. Finally, if daa is non-saionary we can calculae he Bollinger bands, bu he rading sraegy canno be jusified saisically. For non-saionary daa he formulas for he mid-line and he bands will no longer be he esimaes of rue saisical characerisics; hus, he Bollinger bands sraegy loses is heoreical saisical sense and canno be considered correc. We canno apply he esimaed (opimal) sraegy parameers for oher inervals, as hese parameers vary over ime and he esimaes depend on pas daa. Therefore, only saionary or local saionary paerns can be used in he Bollinger bands sraegy applicaions. Assumpion : Symmerical probabiliy disribuion The form of he price probabiliy disribuion defines wheher he bands are equally disanced from MA or no. In he case of equally disanced upper and lower bands, he disribuion should be symmerical around he price mean. If he daa is skew, he bands ha saisfy K% are b U = m + k σ, = m σ, wih k k. b L k > 3

146 The examples of such non-symmerical bands are he Bomar bands, or Marc Chaiken innovaion (Bollinger, ). So we see ha he classical Bollinger bands should only be applied o daa, which has a symmerical probabiliy disribuion. Assumpion 3: Known daa probabiliy disribuion funcion Alhough daa saionariy and a symmerical probabiliy disribuion are sufficien assumpions o jusify using of he Bollinger bands, knowing he probabiliy disribuion significanly faciliaes he search for he opimal parameer k. Oherwise he esimaion of he opimal value k, which guaranees ha K% of daa is inside he bands, becomes complicae and compuer-inensive procedure. Mahemaically defined probabiliy densiy funcions make i much easier o calculae he bands, as no preliminary probabiliy disribuion funcion esimaion is needed. To summarize In order o apply Bollinger bands he daa should be saionary, a leas locally; i should have a symmeric disribuion if we wan equally disanced bands; oherwise we apply Bomar (no Bollinger) bands. Finally, knowing he daa disribuion faciliaes he opimal band calculaions, especially if he disribuion funcion is well-defined. Daa ransformed bands As have been discussed earlier, he analysis of he price (residuals) disribuion and inerval around mean ha conains K% of daa is saisically more consisen mehod o search for he opimal k-parameer in he Bollinger bands han rading sraegy simulaion. An advanage of his approach is ha by aking ino accoun he disribuion properies of he daa we incorporae some hidden informaion valuable for all daa samples. A he same ime we know ha in mos cases hisoric price (or residuals) daa have neiher symmerical nor well-defined known disribuion funcion. Thus, classical Bollinger bands migh no lead o he maximum rading profis. Menioned earlier he Bomar bands could be he imporan generalizaion of he Bollinger bands for he daa wih differen disribuional properies. Unforunaely, we could no find he bibliographical sources, which explain how his mehod is applied. We know, however, ha his mehod is compuaionally inensive and, hus no popular (Bollinger, ). We migh only guess ha hey eiher applied opimizaion procedure direcly o empirical hisogram or cumulaive disribuion funcion (cdf) 6 or firs fi some densiy funcion o he empirical hisogram and hen derived he soluion by he opimizaion of his funcion. Boh soluions needs addiional daa analysis and canno be held on he rouine basis. We propose differen approach o he opimal band definiion, which is simple and can be quickly implemened, i.e. i can be used on coninuous ime basis. 6 This opimisaion problem migh no have he single soluion and is complicae o implemen. 4

147 . Daa ransformed bands: algorihm descripion The consrucion of he daa ransformed bands (furher DT bands) involves he saisical mehod of daa ransformaion (anamorphosis) wih one disribuional characerisic ino he daa wih known (usually normal) disribuion. In he case of he known disribuion of he non-normal variable Z ( x), he ransformaion of he Z ( x) ino normal Y ( x) is performed by using he equaion: F ( z) G( y) Y = G ( F( Z )) where F ( z) is he known CDF of he Z ( x) ; ( y) If he disribuion ( z) =, (II..) G is he sandard normal CDF. F is no known, i can be esimaed empirically and hen he ransform funcion (II..) can be applied. Furher in he applicaions we use he Kaplan-Meier esimae of he empirical cumulaive disribuion funcion. The reader can be referred o Cox, Oakes (984). The ransformaion procedure is presened in Figure.6. We can see he ransformaion of he observaion of he lognormal random variable z * ino he respecive value y * of he sandard F z * = G y *. normal variable. As we can see: ( ) ( ) Lognormal variable Sandard normal variable cdf.5 cdf z* y* Figure.6. Transformaion of he lognormal daa (lef) ino sandard normal variable (righ) The equaion (II..) implies ha he following proposiion. holds rue. 5

148 Proposiion. Le Z some random variable wih CDF F ( z) and Y is he sandard normal variable wih CDF G ( y), such ha Y = G ( F( Z )). Then for any z and z, he observaions of variable Z and heir respecive ransformed values y and y he following equaion hold rue: K = Pr( z Z z ) = Pr( y Y y ). (II..) Proof: K = Pr( z Z z ) = F( z ) F( z ) = G( y ) G( y ) = Pr( y Y y ). The applicaion of he mehod is sraighforward. Firs, we ransform he empiric daa ino normal sandard variable according o some previously calibraed ransform funcion. The ransformed variable has symmerical normal disribuion, for which he opimal inervals around zero-mean ha conains K% of daa can be easily obained (upper and lower bands are equally disanced from he moving average). Aferwards hese (normal) bands are ransformed back using he calibraed ransformaion funcion. The obained bands, furher called daa-ransformed bands (DT bands), are he bands ha can be used wih he original empiric daa. Noe ha according o he proposiion. he bands will conain he same percenage of daa K%. The choice of he variable o which he ransformaion should be applied is imporan. The bands should be buil around he middle line, which in Bollinger case is represened by he SMA. We propose o keep he same middle line because i s an unbiased mean esimaor, bu also because we wan o keep he same basis for he comparison wih he Bollinger approach. In his case we can evaluae he effec of he subsiuion of he bands calculaed on he basis of Bollinger mehod by he DT bands 7. As he resul, a each momen of ime we have o consruc he bands around SMA. In order o esimae he bands values around SMA, we need o analyze he disribuion properies of he residuals X i = Pi SMA and o calibrae he ransform funcion according o his disribuion. On one hand such definiion of he residuals allows us o keep in line wih he Bollinger esimaion procedure: hese residuals used for he esimaion of he price volailiy. On he oher hand, we will see furher ha he usage of hese residuals will make our bands less sensiive o he insignifican flucuaions in he mid-line curve. The consrucion of he DT bands is made in he following seps:. Calibraion of he ransformaion funcion. The empirical daa X i = Pi SMA, i [ n +; ] (n is he lengh of SMA) wih CDF F ( x) is ransformed ino normal sandard random variable by ransform funcion ϕ : X = ϕ( X ), X ~ N (,), (II..3) such ha F( x) = G( ϕ( x) ), where G (.) is he CDF of he normal sandard variable.. Search for opimal k-parameer. The coefficien k defines he bands for he normal sandard variable ( σ = ), which corresponds o he inerval [ k;k] ha conains K% of ransformed normal daa. 3. Backward bands ransformaion. The bands for empirical daa are obained by he backward ransformaion of heir values by he indirec funcion, esimaed in (II..3): ~ ~ B U = ϕ ( k), B L = ϕ ( k) (II..4) ~ ~ 8 4. The bands are posiioned around he SMA as SMA + B U, and SMA + B L, and ogeher wih price and SMA curves define he enry/exi poins for he DT bands rading sraegy. 7 This is he reason why we do no subsiue SMA wih he kriged moving average analysed in par. 8 The lower band is creaed also as a sum of SMA and esimaed band, as lower band has a negaive sign. 6

149 The imporan assumpion under which he DT bands can be consruced on coninuous basis and be saisically jusified is daa local saionariy.. Daa ransformed bands: empirical observaions The example of he DT bands for Bund insrumen is presened in Figure.7. As we can see from he graph, he form of he DT bands differs from he classical Bollinger bands.. Conrary o he classical Bollinger bands ha are almos sricly decreasing or increasing he DT bands show he presence of sair -ype paerns (furher seps ).. During local rends, prices approach and, in many cases, cross boh Bollinger and DT bands, i.e. boh bands send false signals during rending markes. Moreover, he DT bands may send even more false signals han he Bollinger bands. price, bands price SMA DT upper band DT lower band Bollinger upper band Bollinger lower band observaions Figure.7. Bund quoes (3/7/3-7//6, frequency 3 minues, observaions -5): DT bands (k=.) and Bollinger bands (k=.). The specific sair -shape of he bands is worh of separae discussion. The bands are creaed as he sum of non-consan SMA and ransformed (residuals) bands. In order o achieve he consan sair-paerns on he price band, he residuals bands should change proporionally o he change in SMA, bu wih he opposie sign. The explanaion of he phenomena lies in he consrucion of he residuals X i = Pi SMA. In order o esimae he band a poin + we will need o calibrae he new ransform funcion on he sub-sample { X i}. Le say he n i + SMA increased by δ : SMA + = SMA + δ. Then alhough he sub-sample of prices { P i} n i + does no change significanly comparing o he sub-sample { Pi } used in he calculaions of n+ i band a poin, he new sub-sample { X i} will differ significanly due o he change in he n i + SMA used for heir calculaions. In paricular, all old values i n; will be decreased X, [ ] i 7

150 by δ. If he new price a momen + does no bring significan innovaion o he sample, we end up wih approximaely he same disribuion funcion, bu differen mean: he CDF curve of he non-ransformed residuals will be shifed o he lef by δ. This means ha he new band ha will correspond o he same ± k argumen for sandard normal disribuion will decrease by δ. Consequenly, he bands will move from sair o sair only if he presence of he new prices in he sample is so imporan ha i will change he form of he disribuional funcion. Due o he paricular form of he DT bands several conclusions can be made wih respec o he price movemens around DT bands. The following observaions were made from he large sample of Bund prices and were subsequenly confirmed for oher insrumens:. Jumps in he DT bands from one sep (sair) o anoher correspond o large price changes due o he explanaion given above. Le us define rend phases as pars of he price curve, which correspond o he sep in he DT band. We have observed ha he rend normally slows down afer a leas hree such phases, i.e. he probabiliy of rend reversing increases. The oal lengh of hese hree phases is likely o be imporan. If he seps are very shor more han 3 seps migh needed o complee he rend.. If prices largely cross he seps while he bands show increasing/decreasing paerns, he rend is likely o coninue. Obviously, several large price incremens would change local disribuion of he residuals and, hus, ake he bands o he new (sep) level. 3. We have observed also ha in some cases he prices rebound from a sep: exiss some small ε >, such ha: ~ P ( MA + BU ) ε (he price rebound from upper band sep) ~ P ( MA + BL ) ε (he price rebound from lower band sep) Then if he prices rebound from long (he same) horizonal par of he DT band (sep) he rend is likely o reverse..3 Daa ransformed bands: oher echnical rules definiion The empirical observaions made in he previous secion have poenially more applicaions han jus definiion of he DT bands sraegies. These observaions migh help o define wo popular bu very subjecive echnical rules: Ellio Waves and Suppor/Resisance paerns. Shor descripions of hese insrumens are presened in Boxes. and. respecively. 8

151 Box. Ellio Waves Ellio Waves are exremely popular in Technical Analysis, bu, a he same ime, are quie subjecive and complicaed o implemen. The firs reference o he Ellio waves appeared in 938, in The Wave Principle by Charles J. Collins. The idea was finalized in 946 by R.N. Ellio in Naure s Law The Secre of he Universe and was republished by A. Hamilon Bolon in 953 in Ellio Wave Supplemen o he Bank Credi Analys. The heory saes ha he marke follows a repeiive rhyhm of a paricular complee cycle paern, presened in Figure Box.. Wave a Wave 5 Wave b Wave c Wave 3 Wave 4 Wave Wave Figure Box.. The basic paern of Ellio Waves Each basic paern consiss of 3 upward waves and downward waves, followed by he hree-wave correcion phase. The upward waves (Wave, Wave 3 and Wave 5) are he impulse waves, and he downward waves (Wave and Wave 4) are correcive waves. The hree-wave correcion phase sars afer he 5-waves paern is compleed; his phase is idenified wih he Waves a, b, c. Ellio also idenified he degrees of he rend, or he ime-horizon of he rend, ranging from he very long-erm rend of he wo hundred years o he shor-erm rend of a few hours. However, no maer how long is he rend, i is represened by he same basic paern given in Figure Box.. In fac, each long-erm basic paern is spli ino he basic paerns of shorer rends, i.e. for example each Wave -5 can be represened by he similar, bu shorer 8-waves basic paerns, ec. There are many sraegies and paerns relaed o he Ellio Waves heory (for more deails see Murphy, 999). We will no presen hem here, as hey are ouside of scope of he paper. However, i is clear ha if we manage o idenify he 5 waves paern, he rend reversion (or correcion) of 3 waves can be prediced; and aferwards, he new reversion of he five-waves cycle can be expeced. The bigges problem is o idenify hese paerns in real price daa samples, as i is obvious ha he schemaic represenaion of he basic cycle from Figure Box. does no exis, due o he daa noise. Source: Murphy, John J. Technical Analysis of he Financial Markes, New York Insiue of Finance,

152 Box. The Suppor and Resisance Technical Rules Price paerns can be considered as a series of local peaks and roughs (local maximums and minimums). These peaks and roughs can be idenified wih resisance and suppor levels, wo well-known conceps of he echnical analysis. See Picure Box.. Resisance Resisance 3 4 Suppor Suppor Figure Box.. Schemaic represenaion of he suppor and resisance levels. In he Figure, poins & 3 represen local maximum or resisance levels (R & R3), while poins & 4 represen local minimum or suppor levels (S & S4). We now presen some of he echnical rading rules based on he Suppor/Resisance:. The upward rend is confirmed if each following suppor and resisance levels are sricly higher han he previous ones (R3>R, S4>S), and likewise, for downward rends: R3<R, S4<S.. The rend reversal could happen when a suppor (resisance) his he same level as (a leas) one of he previous suppor (resisance) levels: for example, S=S4 for change from downward o upward rend, and R3=R for change from upward o downward rend. As long as hese Suppor/Resisance levels are idenified he rading rules can be applied. Source: Murphy, John J. Technical Analysis of he Financial Markes, New York Insiue of Finance, 999. Taking ino accoun he framework of he Markov imes presened in he General Inroducion, neiher Elio waves nor Suppor-Resisance rading rules are Markov imes, which complicaes significanly heir implemenaion, i.e. in order o idenify he paerns we need o anicipae fuure. Bu we believe ha DT bands can help o define hese rules more easily and in a less subjecive way. In paricular, we believe ha firs five-wave par of he basic Ellio Waves paern corresponds o our observaion () in previous subchaper abou he rend slow down or inversing afer hree seps of DT bands. In paricular, hree seps coincides wih he waves, 3, and 5, while waves and 4 are no refleced in he band paerns as he prices innovaions, corresponding o hese waves, are no significan o change he disribuion properies of he residuals. Observaion () abou he price bouncing off he DT bands sep before rend reversion migh correspond o he Suppor/ Resisance rading rule abou he price bouncing from he same levels of he suppor/resisance (see rule #, Box ) 9. In fac, he same sep is serving he suppor/resisance level for differen prices. 9 In order o alk abou he consisency of he rule definiion, he price should bounce a leas wice from he same band. Furher, we consider only one bouncing as sufficien indicaor of he rend reversal, due o very low number of 3

153 As he resul, DT bands migh give us he possibiliy o define he Ellio and Suppor & Resisance rules wihou anicipaing he fuure, simply by using he saisical characerisics of he daa. The Bollinger () saes ha Bollinger bands can be used o define such echnical analysis paerns as W booms or M ops, head-and-shoulders. Though o our poin of view hese definiions are sill complicae o program, as hey are quie subjecive. Insead, he DT bands give sraigh crieria o define Ellio waves or Suppor/Resisance levels. However, he saemens ha paricular DT bands paerns can be used o define hese echnical indicaors need o be proven by some ess or experimens. For example, some paern recogniion echniques migh be used o idenify he Ellio or Suppor/Resisance paerns, which aferwards can be compared wih he DT bands resuls o show ha hese bands can recognize he same paerns. Taking ino accoun he scale of he analysis, his problem is worh of separae discussion. In addiion, his problem is ou of he scope of our paper as our objecive is he amelioraion of he rading bands no he definiion of he Ellio or Suppor/Resisance insrumens. A he same ime, wihou in-deph analysis, we canno sae ha we are using precisely Ellio waves heory and Suppor/Resisance levels o confirm he rend reversion, while consrucing confirmed DT bands sraegies. Thus, in res of he ex we will refer o hese condiions for price rend reversion as rading signals, generaed by DT bands, furher called Ellio and Suppor/Resisance signals:. Ellio signals: if DT bands form 3 consecuive seps of oal lengh Sl E, hen ake: shor posiion, if hese 3 seps (formed by upper DT band) moved upward and respecive residuals R = P SMA > (prices are above he SMA curve). long posiion, if hese 3 seps (formed by lower DT band) moved downward and respecive residuals R = P SMA < (prices are below he SMA curve).. Suppor/Resisance signals: if a leas one price observaions bounce from he same sep, hen ake: shor posiion, if respecive residuals R = P SMA > (he sep is formed by upper DT band) long posiion, if respecive residuals R = P SMA < (he sep is formed by lower DT band) Consequenly, he DT bands has he following advanages over he Bollinger bands:. DT bands provide much clearer heoreical saisical jusificaion of he bands usage.. DT bands can define he following echnical rading rules ha are no Markov imes, and make i possible o implemen hese rules as algorihms: a. Ellio waves heory b. Suppor/Resisance concep However, in order o jusify heir usage we need o show ha DT bands improve exising classical rading sraegies, based on Bollinger bands. The Chapers 3 and 4 presen he framework and he resuls of such analysis. rades, which is generaed by higher resricion on he parameer. We consider his signal only ogeher wih Elio signal as an addiional signal confirmaion. 3

154 3 Sraegies descripions As we have discussed above, he applicaion of he Bollinger bands sraegy is saisically jusified only in he cases of known symmerical disribuion. The DT bands in heir urn can be applied o more general daa case. The quesion is wheher he usage of DT bands a he place of he Bollinger bands improves he rading resuls. For his purpose we should back-es (simulae) Bollinger and DT bands rading sraegies on he hisoric daa and compare heir oucomes. The following chaper presens he sraegies and heir algorihms. I also discusses he choice of parameers values used for he rading simulaions, as well as he choice of he indicaors used for he analysis of he rading oucomes. 3. Sraegies The sraegies used in he analysis of DT bands are based on he radiional Bollinger bands sraegies (simple basic conrarian and confirmed conrarian), as well as confirmed sraegies based on he signals generaed solely by he DT bands. As he resul he following groups of he sraegies are considered in he paper:. Basic sraegy (or Benchmark sraegy), based on Bollinger bands DT bands. Confirmed sraegies (basic sraegy wih confirmaive signals):.. Confirmaion by he momenum indicaor of he oversold/overbough signals, generaed by: Bollinger bands DT bands.. Confirmaion by he Elio signal of he oversold/overbough signals generaed by: DT bands.3. Confirmaion by he Ellio and Suppor/Resisance signals of he oversold/overbough signals generaed by: DT bands Noe ha he las wo confirmed sraegies are consruced only for DT bands, as only his ype of bands generaes he Ellio and Suppor/resisance signals. The more deail presenaion of he sraegies algorihm is presened furher. Noaion used for he sraegies definiion can be found in Table.. 3

155 Table. Noaion for he sraegies parameers P n MA MA σ i= n+ ( P) = m n P i = Ε = [( P Ε( P) ) ] ( P m) i price a momen inerval lengh, on which he SMA and moving volailiy indicaor are calculaed SMA value a momen ; calculaed on he inerval n +;. [ ] price sandard deviaion a momen ; calculaed on he n +;. inerval [ ] i= n+ σ = Ε = n k, k parameers of he Bollinger and DT bands respecively, which B DT defines he disance beween he middle line and bands. l size of he sub-sample, used for he ransformaion of raw DT daa ino normal daa B ~ U, B ~ upper and lower DT bands, defined for he residuals L ~ ~ B U = ϕ ( k DT ), BL = ϕ ( k DT ) Ri = Pi m, m = Pi, i [ ldt +; ] ldt i= ldt + b U, b upper and lower DT bands, defined for price L ~ b U, = SMA + BU, ~ b L, = SMA + BL, Δ lag parameer used for momenum calculaion Δ M, price momenum a momen Δ M = P P Δ s momenum hreshold value, used for definiion of he rend M reversal and confirmaion of he overbough/oversold signals S number of he consecuive seps, defined by DT bands E Sl oal lengh of he consecuive seps, defined by DT bands E T olerance level o define he sep, i.e. he level a which wo E values can be considered equal: band observaions form a sep if bi bi k TE T olerance level o define price ouching he band : price SR ouches he band if Pi bi TSR q number of price observaions ha ouched he bands sep 33

156 . Benchmark (basic) sraegy A basic sraegy is a conrarian sraegy, where he bands define he hresholds for overbough/oversold insrumens. Breaching hese hresholds (upper and lower bands) defines enry poins for he sraegy (shor and long posiions respecively). The posiion is closed when he price reurns o is mean value, i.e. breaches MA curve... Basic sraegy, based on Bollinger bands Enry posiion: if P MA > kσ P MA < P MA > Exi posiion: if ( P MA )( P MA ) Long posiion (L): if ( ) Shor posiion (S): if ( ) <.. Basic sraegy, based on DT bands Enry posiion: ~ U ~ if ( P MA ) > B P MA < B Long posiion (L): if ( P MA ) < P MA > Exi posiion: if ( P MA )( P MA ) or ( ) L Shor posiion (S): if ( ) <. Confirmed sraegy wih he momenum confirmaion signal A confirmed sraegy is a conrarian sraegy, where he bands ogeher wih rend-reversal signals define he overbough/oversold insrumens. Breaching hese hresholds (upper and lower bands) ogeher wih he signal ha he rend will reverse is direcion defines enry poins for he sraegy (shor and long posiions respecively). The signal for rend reversal is he exceeding (in absolue erms Δ ) by he price momenum M some hreshold value s M. The posiion is closed when he price reurns o is mean value, i.e. breaches MA curve... Confirmed sraegy, based on Bollinger bands Enry posiion: Δ if P MA > kσ and M > sm Long posiion (L): if ( P MA ) < Shor posiion (S): if ( P MA ) > Exi posiion: P MA P MA if ( )( ) < In order o define wheher a rend reverses is direcion from increasing o decreasing or from decreasing o increasing i is imporan o know wha hreshold (upper or lower is breached). However, as long as we consider his P. signal in combinaion wih he band breaching signals, i is sufficien o know he sign of he residuals ( ) MA The breaching of he upper (lower) band from below (above) is possible only during increasing (decreasing) rend paern; hus, only he fac of breaching he hreshold by momenum is imporan o ake he righ posiion. 34

157 .. Confirmed sraegy, based on DT bands Enry posiion: Δ ~ U if M > s and ( P MA ) > B M L or ( P MA ) < B Long posiion (L): if ( P MA ) < Shor posiion (S): if ( P MA ) > Exi posiion: P MA if ( )( P MA ) < 3. Confirmed sraegy wih he Ellio confirmaion signal A confirmed sraegy is a conrarian sraegy, where he bands ogeher wih rend-reversal signals define he overbough/oversold insrumens. Breaching hese hresholds (upper and lower bands) ogeher wih he signal ha he rend will reverse is direcion defines enry poins for he sraegy (shor and long posiions respecively). The signal for rend reversal is he compleion of he firs five Ellio waves. The DT bands define hese waves as a cerain number of he consecuive seps S E (in he same direcion ) of he oal lengh Sl E. Therefore, before making any decision he sraegy should esimae how many seps ( STEPS ) and of wha lengh ( LGTH ) ha precede he momen of decision-making. A sep is formed if a leas wo consecuive price bands values coincide: bi bi k TE. The posiion is closed when he price reurns o is mean value, i.e. breaches MA curve. 3.. Confirmed sraegy, based on DT bands Enry posiion: ~ U ~ L if [ ( P MA ) > B or ( P MA ) < B ] and [ STEPS S E and LGTH SlE ] Long posiion (L): if ( P MA ) < Shor posiion (S): if ( P MA ) > Exi posiion: P MA P MA if ( )( ) < 4. Confirmed sraegy wih he Ellio and Suppor/Resisance confirmaion signals A confirmed sraegy is a conrarian sraegy, where he bands ogeher wih rend-reversal signals define he overbough/oversold insrumens. Breaching hese hresholds (upper and lower bands) ogeher wih he signal ha he rend will reverse is direcion defines enry poins for he sraegy (shor and long posiions respecively). The signal for rend reversal is he compleion of he firs five Ellio waves and rebounding of he price from he same Suppor/Resisance level. The DT bands define hese waves as a cerain number of he consecuive seps S E of he oal lengh Sl E. Therefore, before making any decision he sraegy should define how many seps ( STEPS ) and of wha lengh ( LGTH ) ha precede he momen of decision-making. A sep is formed if a leas wo price bands have he same value ( bi bi k TE ). The Suppor/Resisance level is defined by he sep from which q price observaions rebounded. The signal is observed if ~ Depending on wheher we consider increasing (prices are above MA curve) or decreasing (prices are below MA curve) price paerns, each consecuive sep should be higher or lower han he previous one. Oherwise, he Ellio wave paern is no confirmed. 35

158 he number of price observaions TOUCH ha ouched he same sep ( Pi bi TSR ) is larger han q. The posiion is closed when he price reurns o is mean value, i.e. breaches MA curve. 4.. Confirmed sraegy, based on DT bands Enry posiion: if ~ U ~ L [ ( P MA ) > B or ( P MA ) < B ]and[ STEPS S E and LGTH SlE ] and TOUCH q Long posiion (L): if ( P MA ) < Shor posiion (S): if ( P MA ) > Exi posiion: P MA P MA if ( )( ) < 3. Choice of he sraegy parameers As can be seen he sraegies presened in Ch.3. have an imporan number of parameers. For example, he simples basic sraegies can be represened as following (see Table. for noaion): for Bollinger basic sraegy: P & L = f( n, k B ) (II.3.) for DT basic sraegy: P & L = g ( n, l, k DT DT ) (II.3.) For he confirmed sraegies he number of parameers increases even more wih addiional parameers for each confirmaion indicaors. For example, he sraegies confirmed by he momenum (see Table. for noaion): for confirmed by momenum sraegy, based on Bollinger bands: P & L = f ( n, k B, Δ, s M ) (II.3.3) for confirmed by momenum sraegy, based on DT bands: P & L = g n, l, k, Δ, s (II.3.4) ( ) DT DT For he confirmed by Ellio and Suppor/Resisance signals sraegy, he parameers of he sraegies based on DT bands are presened as following (see Table. for noaion): P & L = g 3 n, l, k, Δ, s, S, Sl, T, T q (II.3.5) ( ) DT DT M E E E SR, The search of he opimal parameers can involve maximizaion of he P&L wih respec o all parameers. However, as we can see for some more advanced sraegy definiions he number of parameers goes o en, which makes he opimizaion procedure exremely complicaed. Taking ino accoun he fac, ha i is virually impossible o produce he same parameers, which will be opimal for all insrumens and markes, he procedure will be muliplied by he number of insrumens, for which he analysis should be performed. We propose less ambiious approach o simplify boh analysis and resuls presenaion, aking ino accoun ha our main objecive is o define wheher DT bands improve he rading oucomes comparing o he Bollinger bands sraegies. Therefore, we propose o predefine some of he parameers ha migh have less imporance. As he resul, we wan o show ha for some insrumens under oher equal condiions he DT bands sraegies work beer han he Bollinger classical echnical sraegies, presened in finance lieraure. Our goal is o presen he DT bands M 36

159 as sraegy building blocks and show ha hey are no a mere producs of he heoreical amelioraion of he exising bands, bu can produce some superior rading resuls. The parameers, which define he Bollinger/DT bands and are common for all esed sraegies need a very careful definiion. Ideally he sraegies should be o analyzed for differen lenghs of he SMA and k parameers. However, he main problem wih such approach is ha hese parameers are relaed o each oher. Scaling parameer k for he Bollinger bands depend on he disribuion of he insrumens residuals; i is chosen a he level o guaranee ha approximaely 9% of he raw daa is wihin he bands. A he same ime he lengh of he SMA used o define he residuals has direc impac on he daa disribuion. As he resul hese parameers have o be chosen in couples. Esimaion of he MA, volailiy and calibraion of he daa ransform funcion is made on he same sub-samples: l DT = n. The lengh of he MA is chosen a wo levels (medium and long), in order o analyze he impac of differen MA a he sraegy oucomes. The values of each ype of he MA lenghs will depend on he daa frequency: for frequency 3 minues - n = 5, n = 5; for frequency hour - n =, n = 5. We decided no o change he frequency of he daa, aking ino accoun ha nowadays raders work wih even higher frequency han 3 minues, while he MAs of he lenghs of - days (which coincide wih our n values) are quie common for shor-erm raders. The chosen SMA lengh also coincides wih he defaul lenghs ( n =, 5 observaions) according o he Bollinger bands concep. I should be noed however, ha he resuls for he shor MA lengh and inerval used for daa ransformaion ( n = ), should be analyzed wih cauious, as i is no saisically correc o derive some conclusions from such small samples. The longer SMA n = 5 (alhough no recommended by he Bollinger heory) was chosen for wo reasons. On one hand, comparison wih longer SMA helps o define wheher he defaul values are he opimal one, i.e. brings more profis han he oher parameers. On he oher hand, he longer SMA (and sample) allows more daa variabiliy o calibrae he ransform funcion (he same daa samples are used for boh calculaions). Low daa variabiliy will have impac on he choice of he value for parameer k : he larger k values are likely o be ransformed ino ± values for DT bands. For example, his is he case for he DT bands wih he parameers n = 5 and k =.. The simulaion of he rading aciviy is hen impossible. The choice of he Bollinger bands scaling parameers k is given in Table.3. Table.3 Choice of he k parameer value Lengh of SMA, observaions Bollinger bands parameer DT bands parameer As we saed already he k value implies ha 9% of daa lays wihin he bands [ k;k] (Bollinger, ). For he ransformed sandard normal variable his value ( k * =. 65 ) is he same independenly of he MA lenghs (he ransformed variable has always he same disribuion). For non-normal non-ransformed daa his value will depend on he variabiliy of he residuals: for As recommended by Bollinger (). 37

160 example, he shorer MA leads o less variable residuals, and hus smaller k values. Thus, he values for he k parameer for Bollinger bands are chosen a he defaul levels for he SMA lenghs n =, n = 5 (see Table.), while we keep his parameer for he DT bands a he same level k * =. 65 for boh SMA lenghs. As he resul, for he firs wo cases we will see wheher he k value ha guaranees he same share of daa wihin he bands gives he same rading resuls. For longer SMA ( n 3 = 5 ) we choose o es wheher he same k for Bollinger and DT bands gives comparable rading resuls for differen bands ypes. We keep he same value k 3 =. as no defaul values are defined for his SMA lengh. For he sraegies confirmed by he Ellio and Suppor/Resisance signals we should define wheher bands values in some close neighborhood creae a sep. For his purpose we define he olerance level a level T E =. ha allows considering wo bands values o be he same: bi bi k TE bi = bi k. The benchmark for oal seps number S E = 3 is chosen from empirical observaions a he level ha defines he firs five Ellio waves (see Ch..3 for explanaion). For he Suppor/Resisance signal wo parameers should be predefined. We choose o observe a leas one ouch of he Suppor/Resisance level o accep he signal of rend reversal, i.e. he number of ouches should be larger or equal o ( q 3 ). The choice of he olerance level T SR ha defines wheher he band was ouched ( P i bi TSR Pi = bi ) depends on he insrumen ick. The opimal T SR is found from he rade simulaion resuls as he one ha maximize he rading profis. As for he sraegies confirmed by momenum, he opimal momenum parameers (lag and hreshold) are defined by he maximum performance (surface), simulaed for differen lags ( Δ ) and hresholds ( s M ). These parameers are opimized on case-by-case basis. We have inroduced wo vecors: () vecor of momenum lags [5; 8; 5; 5; 8] and () vecor of heir hreshold levels. The choice of he laes depends on he ranges of he insrumen momenum values. Simulaion of he rading aciviies for differen combinaion of hese parameers creaes he surface (able) of he profi/losses (P&L). Taking ino accoun ha he surfaces for Bollinger and DT bands migh no coincide, we repor wo oucomes for DT bands: () global maximum for DT bands P&L surface; and () he P&L ha corresponds o momenum parameers, defined as opimal for he Bollinger P&L surface. Finally, we assume -ransacion coss and slippage for he simulaion framework. 3.3 Analysis of he oucomes For each sraegy ype he following rading oucomes are analyzed: () profi/losses (P&L); () oal number of rades; (3) profi/losses (P&L) per rade. We also consider he form of he cumulaive P&L pah (as a funcion of ime). Noe ha he sraegy, which ends up in large profis a he end of some (long) period of ime, bu has shown he losses during he period iself is less ineresing han he sraegy, which ends up a some 3 Our experience shows ha more sric consrain migh leads o he absence of rades. 38

161 lower profis, bu has shown sable or increasing profiabiliy during he whole period. This characerisic is especially imporan for he dynamic rading, such as he inra-day rading. For such rading i is more imporan o have he opporuniy o end up sraegy applicaion wih posiive oucome a any momen of ime raher han wai for some drasically larger profis during he long-erm periods. Therefore, we also analyze he P&L pahs ogeher wih he endof-period performance values. The increasing P&L paerns have higher prioriy for us han he absolue end-of-period performance values (under condiion ha he sraegy has posiive oucome). 4 Trading oucomes for differen insrumens The rading sraegies were simulaed for four differen insrumens ha represen differen markes: () Bund; () DAX; (3) Bren; (4) X insrumen. The resuls are summarized furher. 4. Bund Bund daa represens he price quoes for he period of 3/7/3-7//6 (see Figure.8). The quoes were sampled a he 3 minues frequency. quoes observaions (in 3 min) x 4 Figure.8. Bund quoes for period 3/7/3-7//6 (frequency 3 minues) Table.4 and.5 presen respecively he opimal (maximal) oucomes for differen sraegies applicaion and heir corresponding number of rades. Boh ables presen also he sub-opimal resuls for he confirmed by momenum sraegy, based on he DT bands, which corresponds o he momenum parameers opimal for he Bollinger bands sraegy. The comparison of hese absolue resuls shows ha he larges profis are produced by he sraegy confirmed by momenum and based on he DT bands for he SMA lengh of 5 observaions (988 ). 39

162 Moreover, his ype of DT sraegy shows superior resuls for boh SMA lenghs (5 and 5 observaions). The shorer SMA (5 observaions) works beer han he longer SMA (for he firs 3 ypes of he sraegies). Only he fourh DT bands sraegy, confirmed by Ellio and Suppor & Resisance signals works worse for SMA lengh of 5 observaions, and slighly improves for longer SMA. Unconfirmed sraegies (sraegy ), based on eiher Bollinger or DT bands, are non-profiable, confirming he fac ha breaching any ype of he bands by price curve does no produce unanimous rading signals. These signals need a confirmaion by oher echnical overbough/oversold indicaors. The unconfirmed basic sraegy, based on he DT bands, brings higher losses han he unconfirmed basic sraegy, based on he Bollinger bands. This means ha he DT bands by here own send more false signals han he Bollinger bands (more losses for more rades number); i was expeced in subchaper. from he visual observaions of he graphs. The analysis of he rades number (Table.5) leads o he similar conclusions. The confirmaion of he overbough/oversold signals reduces significanly he number of rades for boh Bollinger and DT bands (by rejecing he false signals). Wha is even more ineresing is ha he larges profi for DT bands is achieved for significanly lower number of rades han for Bollinger bands, meaning much higher efficiency of he DT sraegy per rade in comparison wih he confirmed Bollinger bands sraegy (44 /rade versus /rade). The efficiency of oher confirmed DT bands is lower han he efficiency of he confirmed Bollinger bands sraegy. Table.4 Bund (3/7/3-7//6, frequency 3 minues): Profi & Losses Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum (687*) 3 (-6*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy Table.5 Bund (3/7/3-7//6, frequency 3 minues): Number of rades Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum 386 4(538*) 8 5 (4*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy As we have discussed already, he end-of-period P&L does no gives he absolue answer, which sraegy is beer. The more imporan is a P&L pah. Boh basic sraegies produce negaive oucomes wih non-monoone P&L pahs (see Figures.9 and.). We can also noice ha boh sraegies ge big losses afer srong rend paern (compare he Figure.9,. and.8). 4

163 This jus proves he fac ha he uncondiional (non-confirmed) Bollinger or DT bands sraegies work well only on non-rending saionary markes (markes wih consan mean) Bollinger Bands P&L DT Bands observaions (in 3 min) x 4 Figure.9. Bund (3/7/3-7//6, frequency 3 minues): P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n=5 observaions - Bollinger bands P&L -4-6 DT bands observaions (in 3 min) x 4 Figure.. Bund quoes (3/7/3-7//6, frequency 3 minues): P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n =5 observaions 4

164 The pahs for he confirmed sraegies, based on he momenum indicaor (see Figures.-.) were simulaed for he following opimal momenum parameers: n(sma)=5: () Δ =5 (Lag), s M =.6 (hresholds) (for boh Bollinger and DT bands sraegies); () Δ =8, s M =. (for he DT bands sraegy). n(sma)=5: (3) Δ =8, s M =.3 (for boh Bollinger and DT bands sraegies); (4) Δ =8, s M =.7 (for he DT bands sraegy). These parameers were defined as opimal from he P&L surfaces, given in he Appendix A (Figures A-A and Tables A-A4). Noe ha he opimal momenum parameers for DT bands (() and (4)) are more sable over differen SMA lenghs han he opimal momenum parameers for Bollinger bands. For boh SMAs, DT bands sraegies, confirmed by momenum, produce higher in absolue values P&L wih he pah ha exhibi seeper posiive rend han he bes possible confirmed Bollinger band sraegy (see Figures. and.). I should be noed ha confirmed DT bands sraegies, simulaed for he same momenum parameers as Bollinger bands sraegies produce worse oucome. However, we suppose ha a raional rader will choose he bes possible rading sraegy. Thus, we can conclude ha confirmed by momenum DT bands sraegy is superior o confirmed Bollinger bands sraegy. 8 6 DT bands wih opimal parameers (lag=8,s=.) Bollinger bands (lag=5,s=.6) P&L 4 DT bands wih Bollinger parameers (lag=5,s=.6) observaions (in 3 min) x 4 Figure.. Bund (3/7/3-7//6, frequency 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n =5 observaions 4

165 DT bands wih opimal parameers (lag=8, s=.7) P&L 5 5 Bollinger bands (lag=8, s=.3) -5 - DT bands wih Bollinger parameers (lag=8, s=.3) observaions (in 3 min) x 4 Figure.. Bund (3/7/3-7//6, frequency 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n =5 observaions The hird and fourh ypes of confirmed sraegies are simulaed only for DT bands sraegies. For consisency heir resuls will be compared wih he bes Bollinger bands sraegy, confirmed by momenum (see Figures.3-.4). The rades were simulaed for he following DT bands sraegy parameers, derived from he ables A5-A7 in Appendix A: n(sma)=5: () Ellio sraegy: S E =3 (# seps), () Ellio & Suppor/Resisance : E Suppor/Resisance). n(sma)=5: () Ellio sraegy: S E =3, Sl E =; () Ellio & Suppor/Resisance : S E =3, Sl E =3 (oal lengh); S =3, Sl E =3, Sl E =, T SR =.4. T SR =. (olerance As we can see he parameers choice is quie consisen. Shorer MA leads o less dispersed residuals (price-ma), hus, he smaller is he olerance level o define wheher he price ouches he DT bands. The shorer MA implies he shorer local residuals rend; hus, he smaller is he oal seps lengh. For shorer SMA, confirmed by momenum Bollinger bands sraegy works beer han he DT bands sraegy, confirmed by oher echnical signals ( Ellio and Suppor/Resisance ). However, for longer SMA he DT bands sraegy confirmed by boh he Ellio and Suppor/Resisance signals brings higher resuls han he Bollinger bands. The P&L pahs for confirmed DT bands sraegy exhibi posiive rend for boh SMA lenghs, while he confirmed 43

166 Bollinger bands sraegy produce he pah wih significanly flaer rend of longer SMA (see Figures.3-.4). 8 Bollinger bands (lag=5, s=.6) 6 DT bands,"ellio" (lgh=3, seps 3) P&L 4 DT bands,"ellio &SP" (lgh=3, seps 3,Tsr=.) observaions (in 3 min) x 4 Figure.3. Bund (3/7/3-7//6, frequency 3 minues): P&L for he Bollinger sraegy, confirmed by momenum, he DT sraegy, confirmed by he "Ellio" signals and he DT sraegy, confirmed by he "Ellio and Suppor/Resisance" signals, SMA lengh n=5 observaions DT bands,"ellio &SP" (lgh=, seps 3,Tsr=.4) P&L 5 5 Bollinger bands (lag=8, s=.3) 5 DT bands,"ellio" (lgh=, seps 3) observaions (in 3 min) x 4 Figure.4. Bund (3/7/3-7//6, frequency 3 minues): P&L for he Bollinger sraegy, confirmed by momenum, he DT sraegy, confirmed by he "Ellio" signals and he DT sraegy, confirmed by he "Ellio and Suppor/Resisance" signals, SMA lengh n=5 observaions 44

167 From he Bund case we can derive he following conclusions:. Bollinger bands, wih preliminary chosen defaul parameer k, do no produce profis.. The signals sen by Bollinger bands and confirmed by momenum indicaor lead o he end-of-period profis and almos increasing pah of P&L. 3. The Bollinger bands sraegy, which incorporaes shorer SMA, produce higher profis hen he same sraegy incorporaing longer SMA. This shorer SMA coincides wih he defaul value defined in Table.. 4. The DT bands on heir own produce more false signals and, hus, larger losses han he Bollinger bands. 5. Confirmed by momenum DT bands sraegy produces he highes profis and he P&L pah wih posiive (rend) slope. The opimal momenum parameers for he DT bands sraegy are quie sable over boh SMA lenghs (he lag is he same, and hresholds are close in values); for he confirmed Bollinger bands sraegy he momenum parameers for differen SMA lenghs differ significanly. 6. Confirmed by Ellio and Suppor/Resisance signals DT bands sraegies produce posiive profis and in some cases larger oucomes han confirmed Bollinger bands sraegy. Their P&L pahs exhibi increasing paern. 7. The DT bands show more consisency in applicaion resuls in he erms of profi sabiliy over he ime and sraegies. Especially, his is eviden for longer SMA, where Bollinger bands sraegy works worse. 4. DAX DAX index represens one more ype of financial insrumens socks (see Figure.5) observaions (in 3 min) x 4 Figure.5. DAX quoes for period 3/7/3-7//6 (frequency 3 minues). quoes 45

168 DAX daa corresponds o he same inerval of ime (3/7/3-7//6) and frequency (3 minues) as Bund daa in previous example. As we can see from Figure.5, he DAX was posiively rending during his period. The end-of period oucomes for he analysed sraegies are given in Tables As we can see, confirmed by he momenum DT sraegy produces he highes possible oucome for he SMA lengh n=5 observaions. Confirmed Bollinger band sraegy works beer han DT bands sraegy for longer SMA, alhough is profis are sill lower han for he shor SMA. Unconfirmed (basic) sraegies, boh Bollinger and DT bands, produce very low profis and losses respecively. Table.6 DAX (3/7/3-7//6, frequency 3 minues): Profi & Losses Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum (385*) (75*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy Table.7 DAX (3/7/3-7//6, frequency 3 minues): Number of rades Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum (463*) 85 8(9*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy For he SMA lengh of 5 observaions he DT bands sraegy, confirmed by momenum, brings he highes profis, bu wih lower efficiency han for he Bollinger bands sraegy, confirmed by momenum (4 /rade versus 5 /rade). This means ha for DAX case he DT bands sraegy confirmed by momenum leads o more false signals han similar Bollinger bands sraegy, conrary o he case of Bund. A he same ime, he highes efficiency (55 /rade) is achieved for he DT bands sraegy, confirmed by Ellio signals (he SMA lengh is 5 observaions). Similar o he Bund case, we can see ha he number of rades for confirmed sraegies is significanly lower han for he non-confirmed sraegies due o he eliminaion of he number of false signals (Table.7). While basic (non-confirmed) sraegies generae negaive oucomes (see Figures.6 and.7), for he sraegies, confirmed by momenum, he siuaion is quie differen (see Figures.8 and.9). 46

169 .5 x Bollinger sraegy P&L DT bands observaions (in 3 min) x 4 Figure.6. DAX (3/7/3-7//6, frequency 3 minues): P&L for he basic sraegies, based on Bollinger and DT bands, SMA lengh n=5 observaions.5 x Bollinger bands P&L DT bands observaions (in 3 min) x 4 Figure.7. DAX (3/7/3-7//6, frequency 3 minues): P&L for he basic sraegies, based on Bollinger and DT bands, SMA lengh n=5 observaions 47

170 The pahs for he sraegies, confirmed by momenum, were simulaed for he following parameers, obained from he momenum values in Figures B-B and opimisaion Tables B- B4 in he Appendix B: n(sma)=5: () Δ =8 (lag), s M = (hreshold) (for boh Bollinger and DT bands sraegies); () Δ =8, s M =3 (for he DT bands sraegy). n(sma)=5: () Δ =8, s M =9 (for boh Bollinger and DT bands sraegies); () Δ =8, s M =3 (for he DT bands sraegy). As for he Bund case he opimal momenum parameers are more sable over SMA lenghs for he DT bands han for he Bollinger bands. As we can see, for he shorer SMA (5 observaions) he confirmed DT and Bollinger bands sraegies produce almos everywhere increasing P&L pahs (Figure.8). For longer SMA (5 observaions) he P&L pahs show quie unsable behaviour over ime even hough hey produce end-of-period profis (see Figure.9). Conrary o Bund case, he sraegies, confirmed by momenum, produce increasing pahs exclusively for he shor SMA lengh (5 observaions). 6 x DT bands wih opimal parameers (dela=8,s=3) Bollinger bands (dela=8, s=) P&L 3 DT bands wih Bollinger parameers (dela=8,s=) observaions (in 3 min) x 4 Figure.8. DAX (3/7/3-7//6, frequency 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n=5 observaions 48

171 P&L x Bollinger bands (dela=8, s=9) DT bands wih opimal parameers (dela=8, s=3) DT bands wih Bollinger parameers (dela=8, s=9) observaions (in 3 min) x 4 Figure.9. DAX (3/7/3-7//6, frequency 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n=5 observaions The pahs for he sraegies, confirmed by oher echnical indicaors were simulaed for he following parameers, derived as opimal from he ables B5-B7 in he Appendix B: n(sma)=5: () Ellio sraegy: S E =3 (#seps), () Ellio & Suppor/Resisance : E n(sma)=5: (3) Ellio sraegy: S E =3, Sl E =9; (4) Ellio & Suppor/Resisance : S E =3, Sl E =3 (oal lengh); S =3, Sl E =4, T SR =.45. Sl E =9, T SR =.5. All momenum sraegies lead o comparable profis, excep for one sraegy, based on he Ellio signal (shor SMA), which is non-profiable (he Figure is no presened). For longer SMA, he DT sraegy, confirmed by he Ellio signal leads o profis only slighly below he resuls of he DT bands sraegy, confirmed by momenum (see Figure.). Moreover, he P&L pah for his confirmed DT bands sraegy seems o be less volaile han for he confirmed Bollinger bands sraegy. Figures.-. presen he DT bands sraegy resuls, confirmed by boh Ellio and Suppor/Resisance signals for differen SMA lenghs. For he shorer SMA he P&L is increasing, bu end-of-period resul is much lower han for he confirmed Bollinger bands sraegy (Figure.). However, for longer SMA he P&L is increasing and ends up a he level only slighly lower han he confirmed Bollinger bands sraegy. In addiion he P&L is less volaile and almos never goes below zero conrary o he confirmed Bollinger band sraegy (see Figure.). 49

172 x 4.5 DT bands P&L Bollinger bands observaions (in 3 min) x 4 Figure.. DAX (3/7/3-7//6, frequency 3 minues): P&L for he Bollinger sraegy, confirmed by momenum and he DT sraegy, confirmed by he "Elio" signals, SMA lengh n=5 observaions 6 x 4 5 Bollinger bands 4 P&L 3 DT bands observaions (in 3 min) x 4 Figure.. DAX (3/7/3-7//6, frequency 3 minues): P&L for he Bollinger sraegy, confirmed by momenum and he DT sraegy, confirmed by he "Elio" and Suppor/Resisance signals, SMA lengh n=5 observaions. 5

173 x 4 DT Bands.5 P&L Bollinger Bands observaions (in 3 min) x 4 Figure.. DAX (3/7/3-7//6, frequency 3 minues): P&L for he Bollinger sraegy, confirmed by momenum and he DT sraegy, confirmed by he "Elio" and Suppor/Resisance signals, SMA lengh n=5 observaions. As we can see for he DAX case he conclusions are similar o he Bund oucomes:. Bollinger bands, wih preliminary chosen defaul parameer k, do no produce or produce very low profis.. The signals sen by he Bollinger bands, confirmed by momenum indicaor lead o he end-of-period profis, while he P&L pah is almos everywhere increasing only for he shor SMA (5 observaions). 3. The Bollinger bands sraegy ha incorporaes shorer SMA, produces higher profis hen he same sraegy incorporaing longer SMA. 4. DT bands on heir own produce more false signals and, hus, larger losses han he Bollinger bands. 5. Confirmed by momenum DT bands sraegy produces he highes profis and almos everywhere increasing P&L pah for he shor SMA. The opimal momenum parameers for DT bands sraegy are sable over boh SMA lenghs (he lag and hresholds are he same); for Bollinger bands only he lag parameer is he same. 6. Confirmed by Ellio and Suppor/Resisance signals he DT bands sraegy produces posiive profis (excep for he case of shor SMA lengh); for long SMA he resuls are comparable wih he oucomes obained for he confirmed Bollinger bands sraegy. Their P&L pahs are exhibi posiive rend, conrary o he Bollinger bands sraegies. 7. The DT bands show more consisency in applicaion in he erms of profi sabiliy over he ime and he sraegies. 5

174 4.3 Bren Bren is a fuures on crude oil. Therefore, we consider his insrumen as a represenaive of commodiy markes. The peculiariy of his insrumen is ha i is highly volaile. Our daa represens he period of 7//4-7//6 wih 3 minues frequency (see Figure.3) quoes observaions (in 3 min) Figure.3. Bren quoes for period 7//4-7//6 (frequency 3 minues). Table.8 Bren (7//4-7//6, frequency 3 minues): Profi & Losses Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum (-4*) 4 87(834*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy Table.9 Bren (7//4-7//6, frequency 3 minues): Number of rades Lengh of he SMA 5 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum (8*) 8 39(34*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy 5

175 Tables.8 and.9 represen he absolue end-of-period oucomes for differen sraegies, based on Bollinger and DT bands. As we can see unconfirmed sraegies brings losses for boh lengh of SMA; moreover, conrary o previous cases (Bund and DAX) he shorer SMA of 5 observaions produces even higher losses han he longer one. Neiher higher profis are deeced for shorer SMA for confirmed sraegies. As in he previous case-sudies, he DT bands sraegy (n(sma)=5 observaions), confirmed by momenum shows he highes resuls. The sraegies confirmed by he Ellio and Suppor/Resisance signals showed posiive profis for boh SMAs, bu hese resuls are considerably lower han hose, obained for he sraegies, confirmed by he momenum indicaors. The rades efficiency, calculaed for he Tables.8 and.9, is he highes for he sraegies, based on he DT bands. For shorer SMA, he difference beween he efficiency for he Bollinger and DT bands is very large ($39/rade for he DT bands sraegy, confirmed by momenum, versus 86 /rade for he Bollinger bands sraegy, confirmed by momenum) and he discrepancy decreases for he longer SMA ($48/rade for he respecive DT bands sraegy versus $443/rade for he respecive Bollinger bands sraegy). We should noe ha he DT bands sraegies, confirmed by oher (han momenum) echnical indicaors lead o very small number of rades (excep for he DT bands sraegy, confirmed by he Ellio signal, for n(sma)=5 observaions). Tha is why, furher we will presen he P&L pahs only for hree confirmed DT bands sraegies ha produce he number of rades higher han. As for P&L pahs, basic sraegies show non-monoone behavior for boh SMA lenghs (see Figures.4-.5). 5 DT bands P&L Bollinger bands observaions (in 3 min) Figure.4. Bren (7//4-7//6, frequency - 3 minues): P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n=5 observaions 53

176 4 DT bands P&L Bollinger bands observaions (in 3 min) Figure.5. Bren (7//4-7//6, frequency - 3 minues): P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n=5 observaions The sraegies, confirmed by momenum were simulaed for he following se of parameers, derived from Figures C-C and Tables C-C4 in he Appendix C: n(sma)=5: () Δ =9, s M =3 (hreshold) (for boh Bollinger and DT bands sraegies); () Δ =5, s M =3.6 (for he DT bands sraegy). n(sma)=5: () Δ =5, s M =3.6 (for boh Bollinger and DT bands sraegies); () Δ =5, s M =3.4 (for he DT bands sraegy). Again he parameers are more sable over differen SMA lenghs for he DT bands sraegy han for he Bollinger bands sraegy. Confirmed by momenum opimal sraegy, based on he DT bands, produces he P&L pahs wih seeper posiive slope han for he confirmed sraegies, based on he Bollinger bands (see Figures.6-.7). For boh SMAs, hese confirmed DT bands sraegies produce higher profis han he confirmed Bollinger bands sraegies. In paricular, for he shorer SMA, he confirmed sraegy based on he DT bands akes less false posiion, because he P&L drops less han for he Bollinger bands sraegy (see Figure.6). For he longer SMA, he pahs are almos parallel, while he P&L for he DT bands sraegy ges some boos in he middle of he observed period (see Figure.7). 54

177 P&L x DT bands wih opimal parameers (dela=5, s=3.6) DT bands wih Bollinger parameers (dela=9, s=3) Bollinger bands (dela=9,s=3) observaions (in 3 min) Figure.6. Bren (7//4-7//6, frequency - 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n=5 observaions.5 x 4 DT bands wih opimal parameers (dela=5,s=3.4).5 P&L.5 Bollinger bands (dela=5,s=3.6) DT bands wih Bollinger parameers (lag=5,s=3.6) observaions (in 3 min) Figure.7. Bren (7//4-7//6, frequency - 3 minues): P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n=5 observaions 55

178 As has been menioned already, only one DT bands sraegy, confirmed by Ellio signals (n(sma)=5 observaions), produces sufficien amoun of rades. The P&L pah for his sraegy is given in Figure.8. The pah was simulaed for he following parameers derived from Appendix C, Tables C5-C7: S =3 (seps #), Sl =6 observaions (oal lengh). E E 4 8 Bollinger bands 6 P&L 4 DT bands observaions (in 3 min) Figure 8. Bren (7//4-7//6, frequency - 3 minues): P&L for he Bollinger sraegy, confirmed by momenum and DT sraegy, confirmed by he Ellio signals, SMA lengh n=5 observaions As he resul, he Bren case leads o he similar conclusions as he previous Bund and DAX cases, wih one excepion (conclusion #3):. Bollinger bands, wih preliminary chosen defaul parameer k, do no produce profis.. The signals sen by he Bollinger bands, confirmed by momenum indicaor lead o he end-of-period profis, while he P&L pah is almos everywhere increasing only for long SMA (5 observaions). 3. Conrary o he previous cases, he SMA wih shor defaul lengh does no produce beer resuls han he longer SMA. 4. DT bands on heir own produce more false signals and, hus, larger losses han he Bollinger bands. 5. Confirmed by momenum he DT bands sraegy produces he highes profis and almos everywhere increasing P&L pah among all esing sraegies, including he confirmed Bollinger bands sraegy. The opimal momenum parameers for DT bands sraegy are sable over boh SMA lenghs (he lag is he same and he hresholds are almos equal); for he Bollinger bands boh momenum parameers are differen for differen SMA lenghs. 6. Confirmed (by he Ellio and Suppor/Resisance signals) DT bands sraegies produce posiive profis, bu smaller han confirmed Bollinger bands sraegy oucomes; hey also generae fewer rades han he oher sraegies. 56

179 4.4 X insrumen Insrumen X represens an arificially creaed index, used by one bank for sraegy consrucions 4. The daa frequency is hour. quoes observaions (in hour) Figure.9. X insrumen quoes (frequency hour) The sraegies end-of period oucomes are given in Tables.-.. As in previous cases, unconfirmed (basic) sraegies bring losses for shor SMA ( observaions), while for long SMA he basic Bollinger bands sraegy brings some small profis. The highes resuls are achieved for he Bollinger bands sraegy, confirmed by momenum (n(sma)=5). The DT bands sraegy, confirmed by momenum brings larger profis for he shorer SMA (n(sma)=) and comparable profis in he case of he longer SMA. Conrary o oher cases, he oher confirmed sraegies, based on he DT bands bring very poor resuls. Table. Insrumen X (frequency - hour): Profi & Losses Lengh of he SMA 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum 58 9 (38*) (48*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy 4 Due o he confidenialiy reasons we canno neiher presen he deail descripion, nor provide he informaion on he insrumen real quoes. Tha is why on Figure.9 ha presens he quoes pah during some period of ime, here are no icks on he Y-coordinae. 57

180 Table. Insrumen X (frequency hour): Number of rades Lengh of he SMA 5 Sraegies Bollinger DT bands Bollinger DT bands Sraegy : Basic Sraegy : Confirmed-Momenum 9 (33*) 5 79 (7*) Sraegy 3: Confirmed-"Ellio" Sraegy 4: Confirmed-"Ellio & Suppor/Resisance" * values ha correspond o he momenum parameers opimal for Confirmed-Momenum Bollinger bands sraegy Taking ino accoun he number of rades in Table., we can see ha rades efficiency is higher for he DT bands sraegy confirmed by momenum, han for he similar Bollinger bands sraegies. As for he P&L pahs, he Figure.3 shows decreasing pah for all basic sraegies consruced for SMA lengh of observaions. Non-monoone pahs for all basic sraegies for SMA lengh of 5 observaions are presened in Figure.3. Bollinger bands - P&L DT bands observaions (in hour) Figure.3. X insrumen: P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n= observaions 58

181 35 3 Bollinger bands 5 5 P&L DT bands observaions (in hour) Figure.3. X insrumen: P&L for he basic sraegies, based on he Bollinger and DT bands, SMA lengh n=5 observaions The pahs for he sraegies, confirmed by momenum, were simulaed for he following parameers, derived as opimal from he Appendix D, Figures D-D and Tables D-D4: n(sma)=: () Δ =, s M =8 (hreshold) (for boh Bollinger and DT bands sraegies) () Δ =, s M =5 (for he DT bands sraegy) n(sma)=5: () Δ =, s M =4 (for boh Bollinger and DT bands sraegies) () Δ =, s M =5 (for he DT bands sraegy) The sabiliy of he momenum parameers for he DT bands sraegies also holds for he case of he X insrumen. Confirmed by momenum sraegies produce posiive and almos everywhere increasing pahs for all SMA lenghs. The shorer SMA leads o higher profis for he DT bands sraegy, confirmed by momenum; however he difference is no ha large (see Figure.3). Similar siuaion is observed a Figure.33, he P&L for he DT bands sraegy for longer SMA is slighly lower a he end of he observed period han for he Bollinger bands sraegy. 59

182 4 3 DT bands wih opimal parameers (dela=, s=5) Bollinger bands (dela=,s=8) P&L - DT bands wih Bollinger parameers (dela=, s=8) observaions (in hour) Figure.3. X insrumen: P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n= observaions 6 5 Bollinger bands (dela=,s=4) P&L 4 3 DT bands wih Bollinger parameers (dela=,s=4) DT bands wih opimal parameers (dela=,s=5) observaions (in hour) Figure.33. X insrumen: P&L for he sraegies, confirmed by momenum and based on he Bollinger and DT bands, SMA lengh n=5 observaions 6

183 Conrary o oher examples, he DT bands sraegies, confirmed by he Ellio and Suppor/Resisance signals produce very poor resuls (see Table.); herefore, we do no provide he graphical represenaion of he P&L pahs. As he resul, he DT bands shows less ineresing resuls for his paricular insrumen:. Bollinger bands, wih preliminary chosen defaul parameer k, do no produce or produce very low profis.. The signals sen by he Bollinger bands and confirmed by momenum indicaor lead o he highes end-of-period profis, while he P&L pah is almos increasing for long SMA (5 observaions). 3. The Bollinger bands produce end-of period profis for he SMAs of boh lenghs. 4. DT bands on heir own produce more false signals and, hus, larger losses han he Bollinger bands. 5. Confirmed by momenum DT bands sraegies produce comparable profis and increasing P&L pah for longer SMA and higher profis wih increasing P&L pah for shorer SMA. A he same ime, rade efficiency is he highes for he DT bands sraegies. The opimal momenum parameers for he DT bands sraegy are sable over boh SMA lenghs (he lag is he same and he hresholds are almos equal); for Bollinger bands boh momenum parameers are differen for differen SMA lenghs. 6. The DT sraegies confirmed by he Ellio and Suppor/Resisance signals do no produce even comparable resuls. 5 Conclusions This paper presens he daa ransformed (DT) bands as he alernaive o he Bollinger bands. Alhough being he simple and comprehensive echnical analysis insrumen ha can be incorporaed ino a successful sraegy, Bollinger bands lacking he heoreical jusificaion or explanaion, why he bands should work and how o opimize is parameers, he lengh of he simple moving average and scaling coefficien. By providing he condiions under which he Bollinger bands can be saisically jusified, we showed ha usage of he radiional Bollinger mehod is limied o he paricular case of he saionary (a leas locally), symmeric and known disribuions. The easing of hese assumpions complicaes significanly he applicaions and calculaions of he opimal Bollinger bands. The DT bands under assumpion of local saionariy provide simple, bu powerful ransformaion of he Bollinger approach ino beer heoreical framework, which in addiion is easier o opimize. The DT bands, obained for he ransform funcion calibraed for he residuals Ri = Pi SMA, SMA = P i, i [ n +; ], has a n i specific sair-ype form. Such DT bands migh be helpful in he definiion of oher echnical rules he Ellio waves and Suppor/Resisance levels. The numerical examples show ha he DT bands are no only he insrumen ha is beer jusified heoreically, bu hey can also be a successful sraegy componen. Wih he excepion of he arificially consruced insrumen X, all oher insrumens shows more or less he same resul: confirmed DT bands sraegy usually produces higher profis, rade efficiency and increasing P&L pahs han confirmed (by momenum) sraegies, based on he Bollinger bands. Table. summarizes he obained resuls for he opimal sraegies. The hypohesis abou - ransacion coss, under which he sraegies were simulaed, is cerainly unrealisic. However, we can ge some esimaes of wha can happen in he real-life cases. For example, if we assume ha 6

184 he esimaes of he ransacion coss ( c ) for he Bund and Bren insrumens respecively are.35 /rade and $5/rade, he oucomes for Bund and Bren will remain posiive. Besides of he ransacion coss here is a noion of slippage, meaning ha frequenly he posiion canno be realized a he curren marke prices due o he difference in he demand and supply prices. Slippage is he difference beween he expeced execuion and acual price. While he ransacion fees are relaively fixed, he slippage varies significanly. As we can see from Table if he slippages esimaes are below he P/L per rade values he sraegies are profiable.in fac for cerain realisic esimaes of he slippage we can sae ha opimal sraegies are profiable for Bund, DAX and Bren for boh SMA lenghs and profiable for X insrumen for longer SMA. The P&L for he sraegies based on he DT bands are almos everywhere increasing, which guaranees posiive oucome a any momen of ime, i.e. he profiabiliy of our posiion does no depend on he momen ha we have decided o end up rading; moreover, he longer we say in game he higher could be he oucomes. The sraegy parameers, opimized for he DT bands (momenum parameers, oal seps lengh and olerance level) seems o be more sable across differen SMA lenghs han momenum parameers for he confirmed Bollinger bands sraegies; heir values are less dependen on he choice of he SMA lengh. The sraegy ha incorporaes he Ellio and Suppor/Resisance signals does no brings he highes resul. However, in many cases hese sraegies lead o he comparable oucomes. Several direcions of he fuure research can be figured ou. Some research should be devoed o he analysis of he relaionship beween he sraegy profiabiliy and he value of K% secion or k parameer of he DT bands. The daa ransformaion approach can be also applied o oher echnical sraegies ha are based on he exreme values. For example, in his work we have considered he symmerical hreshold for he momenum, used for he confirmaion of he signals sen by he bands. However, hese sraegies migh be amelioraed if he choice of he (opimal) momenum hresholds will be made on he basis of he daa ransformaion approach. For his purpose he momenum daa should be ransformed ino normal, he K% secion should be applied and he hresholds should be back-ransformed o he real daa. This migh lead o he asymmeric momenum hresholds. Finally anoher direcion should address in more deails he problem of he DT bands applicaion o he definiion of oher echnical indicaors, such as Ellio waves and Suppor/Resisance rading rules. 6

185 Table. Summary of he bes rading oucomes for differen sraegies Shor SMA Long SMA Bund DAX Bren X Bund DAX Bren X SMA lengh Winning sraegy Sraegy #: DT bands, confirmed by Momenum Sraegy #: DT bands, confirmed by Momenum Sraegy #: DT bands, confirmed by Momenum Sraegy #: DT bands, confirmed by Momenum Sraegy #4: DT bands, confirmed by Ellio and Suppor/ Resisance signals Sraegy #: Bollinger bands, confirmed by Momenum Sraegy #: DT bands, confirmed by Momenum Sraegy #: Bollinger bands, confirmed by Momenum P&L # of rades P&L/rade Monooniciy Monoone Monoone Monoone Monoone Monoone Non-monoone Monoone Non-monoone Tick Tick value.5.5 P&L/rade (in number of icks Sraegy profiable? yes yes yes No yes yes yes yes 63

186 6 Appendices II Appendix A Example : Bund.5 Momenum observaions (in 3 min) x 4 Figure A. Bund (3/7/3-7//6, frequency 3 minues): Momenum indicaor (lag Δ =5 observaions)..5.5 Momenum observaions (in 3 min) x 4 Figure A. Bund (3/7/3-7//6, frequency 3 minues): Momenum indicaor (lag Δ =5 observaions) 64

187 Table A Bund (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Momenum parameer momenum parameer (hreshold) (dela) Table A Bund (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=.65 Momenum parameer momenum parameer (hreshold) (dela) Table A3 Bund (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Momenum parameer momenum parameer (hreshold) (dela)

188 Table A4 Bund (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=. Momenum parameer momenum parameer (hreshold) (dela) Table A5 Bund (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio signal sraegy, based on DT bands (Seps # = 3) oal seps lengh parameer n(sma) Table A6 Bund (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

189 Table A7 Bund (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

190 Appendix B Example : DAX 3 Momenum observaions (in 3 min) x 4 Figure B. DAX (3/7/3-7//6, frequency 3 minues): Momenum value (lag Δ =5 observaions) Momenum observaions (in 3 min) x 4 Figure B. DAX (3/7/3-7//6, frequency 3 minues): Momenum value (lag Δ =5 observaions) 68

191 Table B DAX (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag (Momenum Threshold (momenum parameer) parameer) Table B DAX (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=.65 Lag (Momenum Threshold (momenum parameer) parameer) Table B3 DAX (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag Threshold (momenum parameer) (Momenum parameer)

192 Table B4 DAX (3/7/3-7//6, frequency 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag (Momenum Threshold (momenum parameer) parameer) Table B5 DAX (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio signal sraegy, based on DT bands (Seps # = 3) oal seps lengh parameer n(sma) Table B6 DAX (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

193 Table B7 DAX (3/7/3-7//6, frequency 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) Toal seps lengh parameer olerance SR

194 Appendix C Example 3: Bren 6 4 Momenum observaions (in 3 min) Figure C. Bren (7//4-7//6, frequency - 3 minues): Momenum values (lag Δ =5 obs) Momenum observaions (in 3 min) Figure C. Bren (7//4-7//6, frequency - 3 minues): Momenum values (lag Δ =5 obs) 7

195 Table C Bren (7//4-7//6, frequency - 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag Threshold (momenum parameer) (Momenum parameer) Table C Bren (7//4-7//6, frequency - 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=.65 Lag Threshold (momenum parameer) (Momenum parameer) Table C3 Bren (7//4-7//6, frequency - 3 minues): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag Threshold (momenum parameer) (Momenum parameer)

196 Table C4 Bren (7//4-7//6, frequency - 3 minues): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag Threshold (momenum parameer) (Momenum parameer) Table C5 Bren (7//4-7//6, frequency - 3 minues): End of period P&L for confirmed by Ellio signal sraegy, based on DT bands (Seps # = 3) oal seps lengh parameer n(sma) Table C6 Bren (7//4-7//6, frequency - 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

197 Table C7 Bren (7//4-7//6, frequency - 3 minues): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

198 Appendix D Example 3: Insrumen X Momenum observaions (in hour) Figure D. Insrumen X (frequency= hour): Momenum values (lag Δ = observaions) Momenum observaions (in hour) Figure D. Insrumen X (frequency= hour): Momenum values (lag Δ =5 observaions) 76

199 Table D Insrumen X (frequency hour): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=, k= Lag (Momenum Threshold (momenum parameer) parameer) Table D Insrumen X (frequency hour): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=, k=.65 Lag (Momenum Threshold (momenum parameer) parameer) Table D3 Insrumen X (frequency hour): End of period P&L values for confirmed Bollinger bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag (Momenum Threshold (momenum parameer) parameer)

200 Table D4 Insrumen X (frequency hour): End of period P&L values for confirmed DT bands sraegy for differen momenum parameers, n(sma)=5, k=. Lag (Momenum Threshold (momenum parameer) parameer) Table D5 Insrumen X (frequency hour): End of period P&L for confirmed by Ellio signal sraegy, based on DT bands (Seps # = 3) oal seps lengh parameer n(sma) Table D6 Insrumen X (frequency hour): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=, Seps #=3) oal seps lengh parameer olerance SR

201 Tabl e D7 Insrumen X (frequency hour): End of period P&L for confirmed by Ellio and Suppor/Resisance signal sraegy, based on DT bands (n(sma)=5, Seps #=3) oal seps lengh parameer olerance SR

202 Par III. Disjuncive kriging in finance: a new approach o consrucion and evaluaion of rading sraegies Inroducion The definiion of echnical rading sraegies frequenly involves signals, rigged by breaching predefined hresholds by price or oher indicaors. The examples of such sraegies were given in par II. In fac, almos all rading bands sraegies can be reformulaed in he forma of breach he hreshold definiion. { X } ( ( )) Le > be some price process. δ θ defines he hreshold level ha depends on some parameer (or vecor of parameers) θ. For example, for Bollinger bands his parameer is defined by price volailiy. Moving average can be represened as some funcion g of pas prices: MA = g( X, X,... ). Then rading bands are defined by MA (,,... = g X X ) as he middle line and MA ± δ ( θ () ) as he upper/lower bands. The bands rigger firs rading signal a momen τ, such ha: { [, + ) X MA ( θ )} τ = inf : δ. Le s consider a simple rend-following sraegy ha saes o buy ( u = ) he sock when is price reaches he upper band and sell ( u = ) if price reaches he lower band. The posiion is liquidaed a some fuure ime momen T. The hresholds (bands) in his sraegy serve as he confirmaion of he esablished upward or downward rends. The rading oucome under zeroransacion cos assumpion, is hen defined as: ( X ( T ) X ( τ )) π = u. If he bands are breached he rader would be always beer of aking he posiion earlier a ~ < τ : ( ( ) ( ~ X X )) ~ π = π + u τ. In he case of correc predicion of he rend he rader would generae larger profis: ~ π > π >. In he case when he rend, was prediced incorrecly he rader would generae smaller losses: π < ~ π <. We can see ha beer sraegy should no be solely based on he opimal (in he erms of parameers) rading bands, bu also involves he predicion ha he bands will be breached. This forecasing problem can be solved by differen approaches. The firs approach is o esimae (predic) he value of price ( + Δ) R * ( + Δ) and compare he forecas wih he respecive hreshold MA ( + Δ) ± * ( + Δ) X * or residuals (price-ma) * θ or θ *( + Δ). Noe ha as definiion of he hreshold someimes involves he informaion abou price, he hreshold esimaes can also depend on he price esimaes. Differen predicion mehod exiss in financial lieraure. Among hem are ARIMA (Box-Jenkins analysis), ARCH, 8

203 GARCH and sae space models. The references o he models can be found in Greene (7), Box, Jenkins, Reinsel (8), Harvey, Koopman, Shephard (4), Harvey (99). Anoher approach is o solve he opimal sopping problem for τ. Then if he opimal sopping * ime is below our invesmen ime horizon τ < T he respecive rading posiion can be aken. Finally, he hird approach is o esimae he indicaor funcion ha he hreshold is breached. Predicion of he indicaor funcion coincides wih he esimaion of he probabiliy ha he hreshold is breached a some poin of ime in fuure: PX ( Δ, θ X, X,...) or P ( θ R, R,...). R Δ, In he conex of rading band sraegy, frequenly we do no need o know he absolue fuure price values or precise momen of ime when he hreshold is reached. In order o consruc such sraegy, i is sufficienly o know ha he hreshold will be broken during some fuure inervals of ime. Thus, he esimaion of he probabiliy of breaching he hreshold can be applied direcly o he sraegy formulaion. A condiional probabiliy disribuion funcion is frequenly used in financial applicaions for he esimaion of sochasic model parameers by he maximum-likelihood or for he evaluaion of derivaives prices. Popular financial models such as Black and Scholes (973), Vasicek (977) rely on he closed-form expression for he disribuion funcion. However, for many oher financial models he closed-form expressions do no exis. Among ohers mehods ha are used for he esimaion of he parameers of sochasic processes are generalized mehod of momen (Hansen and Sheinkman, 995), non-parameric densiy maching (Air-Sahalia, 996). One of he approaches o he esimaion of he condiional probabiliy is o find a numeric soluion o he Kolmogorov parial differenial equaion (Lo, 988). Anoher approach is based on he Mone- Carlo simulaions of models pahs for more finely re-sampled daa se (Honoré, 997; Sana- Clara, 995). The problem wih hese approaches is ha we need o know or predefine a sochasic model of a process in advance. They also considered he Markov processes in heir applicaions. These approaches also do no provide he closed-form expression for probabiliy calculaions. Ai-Sahalia (999, ) proposes a mehod o produce an accurae approximaion of he unknown ransiion funcion for he Markov processes o calculae he maximum-likelihood funcion. This approach is based on he normalizaion of he diffusion X wih an unknown disribuion by a variance; he obained diffusion Y wih uni variance has he ransiion densiy ha can be represened as an expansion in he closed-form expression. Then he ransiion densiy PY can be ransform ino he ransiion densiy PX. Ai-Sahalia (999, ) uses he Hermie expansion of he P Y densiy around he Gauss densiy funcion. He shows ha his approach works well for he esimaion of he parameers of he classical ineres rae models. In he conex of he consrucion of rading algorihms, he absence of he closed-form soluions complicaes significanly he usage of he approach on he rouine basis, as well as increases he ime-inensiy of is implemenaion. In his par III we propose a mehod, based on he disjuncive kriging (DK) approach o esimae he condiional probabiliies The indicaor funcion I () is defined as following: I( V ( x) i) ( x), V i =, oherwise 8

204 P ( Z + Δ < zc Z Z,..., Z m, ). We do no limi our research o only Markov processes. The disjuncive kriging (DK) approach was developed by Maheron in 97s and has been widely used by he geosaicians since. The DK mehod, firs published in 976 in Maheron s paper A simple subsiue for condiional expecaion: he disjuncive kriging, allows producing an opimal esimaor for nonlinear funcions, such as a probabiliy or an indicaor funcion. I can be applied o boh normal and non-normal daa; and i can produce he esimaor for boh Markov and non-markov processes. The DK mehod allows esimaion he probabiliy of breaching he hreshold no only a some paricular poin of ime in fuure + Δ, bu also during some fuure inerval of ime [ + Δ ; + Δ ]. The laes has even more appeal for sraegy consrucion, as frequenly for raders i is sufficienly o know only ha a hreshold will be breached (sooner or laer) in near fuure. One of he firs applicaions of he DK mehod, performed by Orfeuil (977), was he evaluaion of he probabiliy ha cerain hresholds of he air polluion, defined as criical, would be breached during following hours. Furher he mehod was also applied o he mining indusry, geology, meeorology, environmenal sudies, ec. (see for example, Dai, Wei, Wang, 7; Emery, 6; Trianafilis, Odeh, Warr, Ahmed, 4; von Seiger, Webser, Schulin, Lehmann, 996). I should be noed ha probabiliy expansion by Hermie polynomials used by Ai-Sahalia (998, ) for he calculaion of he ransiion densiies is a paricular case of he DK mehod when he ransformed variable follows isofacorial Gaussian model. A he same ime, o our bes knowledge nobody ries o apply his mehod o he consrucion of rading sraegies. The objecive of his par III is o show ha his approach can be used in finance o improve rading sraegies ha incorporae differen hresholds. Firs, we presen he heory behind he DK mehod in Chaper. Chaper discusses peculiariies of financial daa and heir impac on he DK procedure; i also presens he way o adjus for hese peculiariies. Chaper 3 presens he examples of DK probabiliies. Chaper 4 provides he deep analysis of he inerval DK probabiliies, i.e. he probabiliies ha he hreshold will be breached during some fuure inerval of ime. Chaper 5 presens he oucomes of he rading sraegies ha incorporae he DK probabiliies. Finally, he resuls of he paper are summarized in he Conclusions. Theory Z( T ) Suppose ha we wan o predic some random variable (for example, price) or he value of linear funcion l( Z( T )) a some fuure poin of ime from he available daa Zα on some inerval of ime α : α [ ; n ]. As have been shown in Par I, we can use linear mehods, such as he kriging, o obain a linear predicor for Z ( ). However, if we wan o ge a predicion for some non-linear funcion η ( Z( T )) of Z( T ), which is he case of he probabiliy esimaion, i is no opimal o apply linear predicion mehods o he η ( Z( )), η( Z( )),..., η( Z( n )). Insead, a nonlinear predicion mehod, such as DK, should be used. This chaper briefly summarizes he DK mehod. For more deailed descripion ineresed reader can be referred o he works of Maheron (976), Cressie (99), Rivoirard (994), Chilès and Delfiner (999). Chaper is spli in four sub-chapers. Firs, we presen he general case of a η Z. Chaper. considers he paricular case of a non- predicor for a non-linear funcion ( ( )) 8

205 linear predicor η ( Y () ), wih he normally disribued argumen Y ( ). Chaper.3 shows how o apply DK mehod o a non-normal random variable. Chaper.4 presens he DK esimaor of he regularized variable, such as he proporion of observaions below a benchmark on some fuure inerval of ime. Finally, we consider he paricular case of he applicaion of he DK mehod o he variable wih an exponenial covariance model in Chaper.5.. Disjuncive kriging Z Le s assume ha a process is saionary wih known mean and finie variance. From he saisical poin of view, he bes predicor for η Z T is a condiional expecaion: ( Z( T )) = E η( Z( T )) Z( ) ( ( )) ( α ) = η( z) dft ( z Zα ) η *, (III..) where F - is he condiional disribuion of he from he inerval [ ] α ;. : α n Z a he momen of ime T on pas values Z α This condiional disribuion can be esimaed as following: F T ( z Z ) ( Z( T ) z Z ) = Ε[ I( Z( T ) z) Z ] α Pr (III..) α α where I() is an indicaor funcion, such ha: I ( V ( x) i) ( ) ( x), V i =, oherwise I is no appropriae o subsiue Pr Z( T ) z Z α by Pr ( Z( T ) z), as marginal disribuions of Z( T ) have more variabiliy han he condiional disribuion Z ( T ) Zα. The problem wih he opimal predicor (III..) lays in he necessiy o know he (n+)-dimensional disribuion of Z T Z ; i is complicae o esimae his disribuion from empirical daa. ( ) α The DK mehod allows avoiding he esimaion of he disribuion of Z ( T ) Zα. According o Maheron (976) he opimal predicor for η ( Z( T )) can be approximaed as a linear combinaions of measurable square-inegrable funcions{ f i : i =,..., n} : ( Z( T )) = f ( Z( n η * )) (III..3) i= i i According o he Hilber-space heory, opimal predicors saisfy he orhogonaliy propery for h j : j =,..., n : all measurable funcions { } Ε [{ η ( Z( T )) η *( Z( T ))} h ( Z( )] = As he resul, for he opimal funcions { f i i,..., n} : = ha guaranee he minimum meansquared predicion error, he following equaion holds: j j 83

206 n [ ( Z( T )) Z( )] = Ε f Z( ) i= [ ( ) Z( )] Εη (III..4) j i Conrary o (III..) he equaion (III..4) implies ha only knowledge of he bivariae Z Z, i < j Z T, Z j, i < j is a necessary condiion disribuions { ( ), ( )} n and { ( ) ( )} n i j for he calculaions of he predicor for a non-linear funcion. This is a less sric assumpion han he knowledge of he (n+)-dimensional disribuion. The soluion for he equaion (III..4) can be obained if he process ( ) Z can be represened as a paricular isofacorial model. For such model he cumulaive disribuion funcion (CDF) of he pair ( Z () s, Z() u ) saisfies: F ( dz dz ) = ( s u) χ ( z ) χ ( z ) F( dz ) F(,, } k = k k i k j ν dz, (III..5) where { χ : k =,,... are complee and orhonormal funcions wih respec o F, i.e. k ) χ k χ ( z) χ k ( z) F( dz) = χ k ( z) F( dz) = ( z) χ ( z) F( dz) =, k l =, k =,...,, k =,..., l, k =,,..., l =,,... (III..6) The properies lised above implies he following equaions: ν Le s define for i =,...,n : k Ε [ χ ( Z( s) ) χ ( Z( u) )] =, k l ν ( s u ) = ( s u ) = cov[ χ ( Z() s ) χ ( Z( u) )] = corr[ χ ( Z() s ) χ ( Z( u) )], k =,,... k k k l i ( Z( i )) k ( Z( i )) F( dz ) ( Z( T )) χ ( Z( T )) F( dz ) a χ, ik = f i b k = k T η. Compleeness and orhonormaliy of { : k =,,... } χ imply ha k f ( Z( )) a χ ( Z( )), i =,..., n i i = k = ( Z( T )) b χ ( Z( T ) = η k k ) k = As he resul, he disjuncive-kriging predicor of he funcion ( Z( T )) n ( Z( T )) = a ( Z( i= k= ik ik k k i i k k η is: η * χ )), (III..7) 84

207 where { } a ik are he soluion of he following sysem: k n ( T, j ) bk = ν k ( i, j ) ν a, k =,,...; j =,..., n (III..8) i= For k = he expression (III..8) is reduced o b =. The sysem (III..8) is runcaed a some value ik n a i i= he sysem (III..8) has nk equaions wih nk- unknowns. K b k k = [ ( )] k = K such ha var Z T =. As he resul, As have been shown in (III..5), he advanage of he isofacorial models is ha some of hem have polynomials facors, i.e. χ k are polynomials. The choice of he polynomials { χ k : k =,,... } depends on he form of he isofacorial model F u, s ( dzu, dzs ). Chilès, Delfiner (999) presens he following models and heir polynomials: I. Coninuous marginal disribuions: The Gaussian model wih Hermie polynomials; The gamma model wih Laguerre polynomials; The bea model wih Jacobi polynomials. II. Discree marginal disribuions: The binomial model wih Krawchouk polynomials; The negaive binomial model wih Meixner polynomials; The Poisson model wih Charlier polynomials; The discree Jacobi ype model wih discree Jacobi polynomials. Furher we consider he case of he Gaussian isofacorial model F ( dz, dz ) polynomials for a non-linear predicion of a normal variable. u, s u s wih he Hermie.. Disjuncive kriging: normal random process The gaussian isofacorial model is he mos frequenly used model in non-linear geosaisical applicaions. Le s consider variable Y ( ) ha is normally disribued wih -mean and uni variance. Is probabiliy densiy funcion g and cumulaive densiy funcion G are respecively: g () = e G π u ( u) = g( ) Maheron (976) shows ha if he funcions { : k =,,... } polynomials, he F, k d χ are represened by he Hermie in (III..5) is defined as bivariae normal disribuion wih sandard normal 85

208 marginal disribuions and correlaion coefficien ρ ; i.e. he couples { Y ( ) Y ( + h) } bivariae normal wih correlaion ρ = ρ( h) and he probabiliy densiy funcion: + u uρ g ( ) = ρ, u exp. π ρ ( ρ ) Hermie polynomials H k ( Y () ) of order k are defined by following formulae ( k ):, are assumed H k ( y) = k! g ( y) k d g dy ( y) k. (III..9) Hermie polynomials can be defined recursively by he following recurren relaionship: H H H H ( y) ( y) ( y) = = y = y k k + ( y) = yh ( y) H ( y) k + k k k + (III..) I can be shown Hermie polynomials of order k has he properies, defined by (III..6): cov Ε[ H k ( Y () )] = H k ( y) g( y) dy = var[ H k ( Y () )] = k ( Y () ), H p ( Y ( ) ) = Ε H k ( Y ) H p ( Y ( ) ) k [ H ( Y () ), H ( Y ( + h) )] = ( h) = ν [ H ] [ () ] =, p k [ ρ ] ( s u ) = ν ( h) cov k k k k (III..) Almos any funcion of Y () can be represened in he erms of Hermie polynomials: [ Y () ] = f + f H [ Y () ] + f H [ Y () ] + = f H [ Y () f ], (III..) wih he coefficiens of he expansion:... f k = f = Ε[ f ( Y ( ) )], [ f ( Y () ) H ( Y ( ) )] = f ( y) H ( y) g( y) = Ε dy k k k. Le s consider he example of he indicaor funcion ( Y ( ) ) his funcion are: k k I <. The expansion coefficiens of y c 86

209 f k ( yc ), k = ( y ) g( y ) yc G = I < ( ) ( ) = ( ) ( ) = y y H y g y dy H y g y dy c k k H, k c c k k As he resul, I = [ ] (III..3) k k () y G( yc ) + H k ( yc ) g( yc ) H k Y ( ) Y < c or I Y () y = IY () < y = G( yc ) H k ( yc ) g( yc ) H k [ Y ( ) ] c c k k The disjuncive kriging of he funcion of Y ( ) as in (III..): DK CK [ f [ Y () ] = f + f [ H [ Y ( ) ] + f [ H [ Y ( ) ] = f + f +... = K K [ H [ Y () ] + f [ H [ Y () ] +... f [ H [ Y () ] The cokriging (CK) of Hermie polynomials becomes a simple kriging (K) due o he orhogonaliy (independence) of he polynomials. As he resul he disjuncive kriging of he nonlinear funcion of he normally disribued variable becomes he simple kriging of he Hermie polynomials consruced for his normal variable. CK = k = k k K The kriging of he Hermie polynomial of he order k can be defined as following: K [ H ( Y () )] λ H ( Y ) where α are he experimenal poins: kriged coefficiens. k = kα k α, (III..4) = { } α α i, i,..., n = ; H ( Y ) H ( Y ( )) k α = k α ; α The kriged weighs λ k α are obained as he soluions of he following sysem: λ k β cov[ H k ( Yα ), H k ( Yβ )] = cov[ H k ( Yα ), H k ( Y ( ) )], i.e. β k [ ρ ] [ ρ ] k where ρ ρ( ) = cov( Y ( ), Y ( )) αβ α β λ k αβ = β α, (III..5) β =. α β λ k are he Cokriging is a mulivariae generalizaion of he kriging mehod. Cokriging is applied when for he predicion of some random process ( ) we can use no only pas hisoric daa on Z, bu also daa on oher random processes () p Z Z i, < i correlaed wih Z. As he resul he sysem of cokriging equaions involves no only he auocovariance marix of he process Z ( ), bu also he cross-covariance marix. More informaion abou he cokriging, as well as he cokriging equaions can be found in Chilès, Delfiner (999). 87

210 k Wih increase in n [ ( h) ] ρ and kriged esimaor ends o is mean ha is. Therefore, only some limied number of polynomials should be kriged and he number of Hermie polynomials can be runcaed a some level m. Taking ino accoun ha = var f Y f, he runcaed number m should saisfy he following expression: m ( f ) k var f [ Y ( ) ]. [ ()] ( ) The mos common choice of he covariance srucure for he Hermie polynomials is: T k ( h) = cov [ H ( Y ( ) ), H ( Y ( + h) )] = [ ρ( h) ] k k However, ohers models for covariance srucure are used. In pracice, four models are he mos popular (Chilès, Delfiner, 999):. The pure diffusive model: k Tk ( h) = ρ ( h), k. The mosaic model: T k ( h) = ρ( h), k 3. The barycenric model: (III..6) k Tk ( h) = βρ ( h) + ( β) ρ( h), k 4. The bea model: Γ ( ) ( β ) Γ( βρ( h) + k) T k h =, k Γ β + k Γ βρ h ( ) k ( ( )) Noe ha Gaussian model is a paricular case of he pure diffusive model. There are wo approaches o choose beween he models in (III..6) (Chilès, Delfiner, 999). One approach involves he analysis of he regression of χ k ( Y ( + h) ) on Y () or χ k ( Y ( ) ). However, he analysis should be performed for all values of k and h, which is complicaed o implemen in pracice. According o he second approach, he righ model can be chosen by he inspecion of he variogram of order, esimaed for he process Y ( ). Variogram of order is defined as following: [ ] ( h) = Ε Y ( + h) Y ( ) γ. Le s consider he variograms of firs and second 3 order, normalized by heir respecive sills C,C : ~ γ γ =, ~ γ γ = = ρ( h). C C k [ ] 3 A sandard semi-variogram: γ ( h) = Ε ( Y ( + h) Y ( ) ) 88

211 The following relaionship beween hese variograms holds for he respecive models in (III..6): ~ γ ( h) = β Γ Γ γ ( h) ~ γ ( h) ~ γ ( h) + ( β ) ~ γ ( h) ( β ) Γ βγ~ ( h) + Γ βγ~ ( h) ( β + ) ~ ( ) ( ) Ploing ~γ versus ~ γ helps o define wha covariance srucure for he Hermie polynomials should be chosen..3 Disjuncive kriging: non-normal random process We have presened he applicaion of he disjuncive kriging mehod o he bivariae sandard normal variable. However, i is rare ha in real-life applicaions he analyzed variables saisfy his hypohesis. Maheron (976) proposed he soluion o his problem: a non-normal random variable and is hresholds are ransformed ino normal variable and normal hresholds. Then he DK mehod is applied o his normal variable as described in Chaper. of par III. An anamorphosis (ransformaion) is he funcion Φ ( ) ha relaes a non-normal variable Z ( x) o a normal one Y ( x) : () = Φ( Y ( ) ) = Φ( ) Z z c y c. (III..7) This ransformaion is defined by he cumulaive probabiliy disribuions F ( z) = P[ Z( ) < z] and G ( y) = P[ Y ( ) < y] of he variables Z ( ) and Y ( ) respecively. The ransformaion associaes wih each value z such value y ha F ( z) = G( y). The graphical example is given in Par II, Figure.6. The ransformaion funcions are discussed in more deails furher in Chaper. The obained DK resuls for he normal variable will be valid for he respecive non-normal variable: ( Z < z ) = Pr( Y < y ) Pr. c.4 Disjuncive kriging: case of a regularized variable In Chapers.-.3 we have shown how o esimae probabiliy of breaching hreshold a some poin of ime in fuure. However, DK mehod also allows esimaing he proporion of poins, when he price is below (above) some hreshold during some inerval of ime in fuure. This proporion is called regularized indicaor and is defined as: c 89

212 where L is he inerval [ b] L b a I Z () < zd = I Z () < L b a z d, a;, on which he proporion is esimaed. The DK procedure should be applied o he normal regularized indicaor. Noe ha: L I Z () < zd = IY () < L L L y d, where z = Φ ( y) - he ransformaion funcion, defined in (III..7). Rivoirard (994) shows ha L L I Y () < yd = G( y) + H ( y) g( y) L k k L H k ( Y () ) d. (III..8) The kriged esimaor of H k ( Y () ) d b a b a can be obained from he sysem similar o (III..5): or β λ β kβ λ cov kβ b [ H k ( Yα ), H k ( Yβ )] = cov H k ( Yα ), H k Y ( ) b k [ ραβ ] = [ ρα ] b a a k b a a [ ( )] d (III..9) dx Then he DK esimaor is: b a b a H DK ( Y () ) d = G( y) + H ( y) g( y) λ H ( Y ) k k k α kα k α (III..) Furher for he sraegy consrucion we will use he DK esimaor of he regularized variable raher han he esimaor of he indicaor a one poin of ime..5 Disjuncive kriging: paricular case of he variable wih exponenial covariance model The financial random funcions ha can be defined as Markov processes, have he covariance ha fi an exponenial model. Suppose ha some sandard normal variable Y has he covariance ha follows he following exponenial model: h a ( h) = e ρ. 9

213 The empirical observaions are available a he poins α Y { } m i i x. Suppose we need an esimae of he probabiliy ( ) ( c y x Y < Pr ) a predicion horizon. x Le s consider wo cases of he predicion horizon : x Case : is a uni disance from he esimaion window x { } m i i x (see Figure 3.); Case : is a some larger disance from he esimaion window { } (see Figure 3.). x m i i x As have been shown in Chaper., he probabiliy esimae (III..3) involves he value of he kriging esimae of each Hermie polynomials of he order k. Therefore, he esimaion problem coincides wih he search of he opimal kriged weighs α λ k as he soluion of he sysems (III..5). Suppose he following covariance srucure for he Hermie polynomials of order : n ( ) A h a h n n e e h = = ρ, where n a A = Le s inroduce he following noaions: = m m C ρ ρ ρ ρ ρ ρ = m m B ρ ρ ρ ρ = m m b ρ ρ ρ = m m λ λ λ λ λ ρ λ λ = ~ Case. Uni predicion horizon (Figure 3.) m x x x x Figure 3.. Esimaion of he poin probabiliy a equally disanced poin The sysem of he DK equaions (III..5) akes he following form for he case presened in Figure 3.: ( ) ( ) = = n n n n x x x x α β β α α ρ ρ λ (III..) Taking ino accoun he marix noaion presened above, he sysem of equaions (III..) can be represened as: 9

214 or C ~ λ = b Cλ = B = ρb (III..) The sysem (III..) has a unique soluion: ρ ~ λ =... or λ = (III..3)... Case. Disan predicion horizon (Figure 3.) p xm x x x Figure 3.. Esimaion of he poin probabiliy a poin disan from he esimaed window In he case, when poin x is siuaed a some larger disance from he esimaion window, as in Figure 3., he disance o he ime horizon can be defined as following: x x = pδx, wih Δx a uni disance. Then he opimal weighs from he sysem (III..) are: p ρ λ = (III..4)... As we can see from he opimal soluions for he kriging weighs (III..3) and (III..4) only las (mos recen) observaion maers for he DK esimaion procedure in he case of he process wih an exponenial model. Peculiariies of he disjuncive kriging applicaions o financial daa According o Chaper wo ypes of he DK probabiliies can be defined and esimaed. The firs ype is he DK probabiliy, esimaed for he paricular fuure momens of ime h. We call hem furher poin DK probabiliies: p DK ( h, y ) = Pr( Y ( + h) y ) P, <. (III..) c The second ype is he regionalized DK probabiliies, called furher inerval DK probabiliies: c 9

215 P ( l, y ) = Pr( Y < y, i [ + ; l] ),. (III..) I DK c i c + Furher he DK resuls should be analyzed and validaed. Two approaches o he mehod validaion can be considered in he financial conex:. The obained DK probabiliies should be consisen; hey should be compared wih some alernaive probabiliy esimaes.. The mehod, incorporaed ino a rading sraegy, should improve he rading resuls. The mos rigorous evaluaion approach would be he comparison of he DK probabiliies wih respecive heoreically calculaed rue probabiliies of being below he designed hreshold; however, his is difficul o implemen for he processes wih unknown saisical characerisics. The bes possible probabiliy esimaes ha we can obain empirically are he frequencies or he share of cases when price is below some hreshold. Cerainly, his approach is applicable only for he evaluaion of he inerval DK probabiliies (III..), esimaed for regularized variables 4. For hese inerval probabiliies we can calculae he proporion of an insrumen price (residuals) values smaller han paricular hreshold on he sudied inerval. Furher in Chaper 3, 4 and 5 we will consider he examples of he poin probabiliies and will perform he analysis of he inerval probabiliies. In his Chaper we will concenrae on he peculiariies of he financial daa such as nonnormaliy and non-saionariy ha have direc impac on he DK applicaions. Daa nonnormaliy implies he necessiy of is ransformaion ino a sandard normal variable. Daa nonsaionariy implies ha neiher ransform funcion nor he esimaed covariance model can be assumed he same hroughou he ime. As he resul wo imporan quesions should be addressed:. Wha ransformaion funcion o use?. Wha daa sub-sample should be used o esimae he ransformaion funcion and how ofen he ransformaion should be adjused o he new daa?. Calibraion of he ransform funcion: kernel approach When hisorical daa Z ( x) has non-normal disribuion he following cases are possible:. The non-normal disribuion of he variable Z ( x) is known.. The non-normal disribuion of he variable Z ( x) is unknown. If disribuion funcion is known and well defined, hen he ransformaion ino a normal variable Y x can be done hrough equaion: ( ) F ( z) G( y) Y = G ( F( Z )) =, where F ( z) is he known CDF of he ( x) Z ; ( y) G is he sandard normal CDF. 4 I is impossible o calculae he same frequencies ha correspond o he poin probabiliies a he presence of sole price pah. 93

216 In some cases he exising well-defined saisical ransforms funcions ϕ () can be used o ransform daa ino normal. The mos popular exising funcion ha can be applied o he skewed disribuions is he Box-Cox ransform: ϕ ( z, λ) ln z, λ = λ = z, λ λ The oher well-known daa ransforms, developed o arge differen daa ypes (skewed nonsymmeric disribuions; symmeric disribuions, ec.), were proposed by Manly, John and Draper, Bickel and Doksum, Yeo and Johnson, ec. 5. However, he main drawback of hese welldefined ransforms is ha hey provide good resuls for unimodal disribuions. Therefore, when he empirical disribuions are mulimodal, he applicaion of he discussed ransformaions does no make daa normal. In he case when he disribuion funcion is unknown we have several approaches o is esimaion. The firs mehod is o fi a heoreical disribuion o he empirical hisogram (or CDF). Such mehods as maximum-likelihood (ML) are used o obain he esimaes of he parameers of he predefined disribuions. The main drawback of he mehod is ha a disribuion model should be chosen ad hoc, before he esimaion and opimizaion procedure. According o he second approach numerical esimaors of he empirical hisogram (or CDF) a observed poins can be used for he inerpolaion 6 of he poins, where he numerical esimaes are no available. Differen mehods exis o esimae/inerpolae numerically he empirical disribuion funcion. One of he mehods is o use he Kaplan-Meier esimaor of he empirical CDF, used in he previous par II o ransform daa ino normal variable. The drawback of he mehod is ha he CDF is esimaed very closely o he available empiric hisogram, making he ails of he densiy funcion converge drasically oward for he values ouside of he observed daa inerval. For he DK procedure i means ha if daa is ransformed locally, he hresholds ha were no reached during he analyzed period are ransformed ino infinie values, even if he values are quie close o he observed price/residuals 7. The infinie hresholds values imply or value for he DK probabiliies no maer how far are he hresholds from he observed daa; such probabiliy values are useless in he rading applicaions. The non-parameric mehods, such as kernel approach, allow beer (or managed 8 ) approximaion of he probabiliy densiy for non-observed exreme values. Saisicians apply he kernel smoohing mehod o he esimaion of he regressions or densiy probabiliy funcions. As a nonparameric mehod, i does no demand any predefined funcional represenaion and i allows uncovering some srucural characerisics ha parameric mehods canno reveal. As he resul, i can be fi o any empirical disribuion. For more informaion on kernel smoohing mehod he ineresed reader can be referred o Wand (995), Marinez, Marinez (), Sco (99). 5 See for more deails: Li, P. 5. Box-Cox Transformaions: An Overview, presenaion, hp:// See also Carroll, Rupper (988) for more informaion on daa ransforms. 6 Saisical/Mahemaical sofwares frequenly include he inerpolaion funcions in heir packages, making he procedure very easy. 7 For previous DT bands ransformaions his facor had less implicaions, as he bands were defined by he K% cus of he sandard normal disribuion, no by some prefixed value ha corresponded o he hisoric (raw) daa.. 8 The kernel fi can be managed by conrolling for he bandwidh parameer ha defines he closeness of he fi. 94

217 { } n Le he random variable Z i have a coninuous univariae densiy funcion ρ. Then a i kernel esimaor of he densiy funcion is: n z Z i ˆρ ( z, h) = K, (III..3) nh i= h where K u - is a kernel funcion, h - a bandwidh parameer. ( ) Expression (III..3) can be considered as a weighed average of all available daa poins wih he weighs defined by he Kernel funcion and bandwidh parameer. We can noice ha only disance beween one poin and all oher available daa poins (bu no heir relaive posiioning) has impac on he value of he weigh assigned o each of he poins. ( ) Kernel funcion K u can iself be considered as probabiliy densiy funcion ha is unimodal and symmeric wih respec o. The ypes of kernel include Gauss, uniform, riangular, Epanechnikov funcions. The bandwidh is a scaling parameer; i conrols he smoohness of he esimaed densiy curve. Small bandwidh parameer gives much closer fi for he esimaed densiy funcion, bu a he same ime probabiliy ouside esimaed inerval quickly converge oward (close fi). Larger bandwidh on he oher hand givers poorer fi. Therefore he choice of he bandwidh is combined wih he risk of over or under-smoohing. The mehods of he search of he opimal bandwidh parameer can be spli ino wo groups (Wand, 995): () quick and simple approaches ha does no guaranee he opimal bandwidh, bu which are less ime-consuming in inroducion; () high ech approaches ha are based on he opimizaion procedure of minimizing AMISE (asympoic mean inegraed squared error). We will no presen he mehods in more deails; he ineresed reader can be referred o Wand (995). Taking ino accoun he universaliy of he mehod wih respec o is applicaion o differen insrumens wih differen saisical characerisics, we choose he kernel mehod o esimae he empirical CDF, and as he resul, o calibrae he ransform funcion. The kernel weighs are defined by Gauss funcion ha assigns he highes weigh o he daa poin where he densiy funcion is esimaed, wih he decreasing weighs for more disan poins. The bandwidh parameer is defined by he defaul value proposed by he MATLAB procedure. In addiion, for DAX and Bund insrumens, we consider one more bandwidh value, larger han he defaul value ha allows more smooh densiy esimae in order o analyze he impac of his parameer on he DK resuls and rading oucomes. Le s consider he resuls of he kernel applicaion o he densiy funcion esimaion for he paricular case of he DAX insrumen. DAX index represens he socks marke; is hisoric quoes are given in Figure 3.3. DAX daa corresponds o he ime inerval of 3/7/3-7//6 and he frequency of 3 minues. 95

218 quoes observaions (in 3 min) x 4 Figure 3.3. DAX quoes for period 3/7/3-7//6 (frequency 3 minues). The sraegies, based on rading bands, can be ransformed ino he sraegies, based on he residuals, consruced as he difference beween a price and middle line. For example, signal riggered by breaching he upper/lower bands by price, can be subsiued by he signal riggered by breaching he hreshold, defined by he disance beween middle line and bands, by residuals series. Le s consider he rading bands sraegies wih he EMA as he esimae of he middle line. The residuals, obained by he subracion of he EMA (EMA lengh=5 observaions) from he DAX price daa are given in Figure 3.4. The residuals daa is no normal: he hypohesis ha he residuals represen a sandard normal variable can be rejeced under Kolmogorov-Smirnov es wih 5% significance level (KS(saisics)=.58 wih cu-off value of.). Thus, calibraion of he ransformaion funcion for residuals ransformaion is needed before applicaion of he DK mehod. 96

219 observaions x 4 Figure 3.4. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions) residuals.. DAX daa (:8 obs) bandwidh=7 bandwidh=5 bandwidh=5.8 Densiy Daa Figure 3.5. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions): empirical hisogram and CDF esimaes according o he kernel mehod for differen bandwidh parameers (sub-sample [;8]). The kernel esimaes of he CDF for differen bandwidh parameers, esimaed on he firs half of he sample are demonsraed in Figure 3.5 (defaul bandwidh=7). As we can see he smaller 97

220 he bandwidh parameer - he closer he esimaed curve follows he empirical hisogram and he hinner are he ails of he esimaed disribuion. The larger is he bandwidh parameer he hicker are he ails; a he same ime he esimaed mode of he disribuion is much lower han he empirical mode. Figure 3.6 shows he variograms of he normal variables, obained for differen ransform funcions ha are calibraed for hree bandwidh values. As we can see he bandwidh value has impac on he variogram parameers, in paricular on is sill. The explanaion for his phenomenon lays in he fac ha large bandwidh parameer hickens he ails of he densiy funcion; his resuls in he higher probabiliy values, aached o he (hisorically) observed exreme values. As he resul, he discrepancy (variance) of he normal daa diminishes..4. bandwidh=7 bandwidh=5 bandwidh=5 variogram lag Figure 3.6. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions, observaions -9): variograms of he normal variable, ransformed according o he kernel esimaor wih differen bandwidh values Furher in he DK applicaions we use wo bandwidhs values o calibrae he ransform funcion for DAX insrumen: b = 7 (defaul value) and b = 5. The following models (he sum of h h exponenial and damped-cosines model ( ) = a aα πh γ h σ + e σ e cos ) are fi a o hese wo variograms: b = 7 : ( ) h h πh = γ h.75 e +.5 e cos 45 b = 5 : ( ) h h πh = γ h.65 e +.5 e cos (see Figure 3.7 for he 45 variogram example) 98

221 .4. empirical model vario observaions Figure 3.7. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions, observaions -8): variograms of he normal variable, ransformed according o he kernel esimaor wih bandwidh b = 5 and is heoreical fi. Daa used for he esimaion of he CDF: locally adjused ransform funcion The oher peculiariy of he financial daa is is non-saionariy ha implies ha par of he parameers used in he DK applicaion procedure are no valid everywhere. Furher in Chaper 3 we provide he examples of he DK probabiliy esimaes, obained for he same CDF, ha was esimaed on he whole available sample, under he following assumpions: () daa is saionary; () he esimaed CDF represens rue disribuion funcion. We will show ha his framework implies poor probabiliy esimaes. Under he assumpion of local saionariy, he local esimaion of he densiy funcions is more appropriae. A he same ime he local esimaion of he CDF is frequenly based on significanly reduced sample size, making he esimaion resuls less consisen (especially if he esimaions are made for he hresholds ha were no reached during he mos recen period of ime). Therefore, in order o avoid bias and errors due o he small number of observaions in he esimaed inerval, we propose o esimae he CDF funcion on a longer sample and hen locally adjus is saisical momens, in paricular is variance. We assume ha he CDF (furher called basic CDF) and covariance ypes of models are he same hroughou some long period of ime, while some of is parameers, such as variance are only locally saionary 9. 9 Taking ino accoun ha furher he DK mehod is applied o he price residuals, we can assume ha is mean is consan and zero. 99

222 Suppose Z ~ ~ L (, ) is a random variable ha follows some arbirary disribuion law, wih ( ) ( ) mean and variance. Le s define anoher random variables Z, Z on he basis of Z ~ ha follows he same disribuion law wih -mean and differen variance: Z Z () ( ) ~ = σ Z, ~ = σ Z, ( ) Z ~ L (, σ ) ( ) Z ~ L (, σ ) Then Z () σ σ () = Z (III..4) Le s consider some ransformaion funcion ϕ, such ha: Z ( ) = ϕ( Y ) where Y is a random sandard normal variable: Y ~ N (,). Le s define he following CDFs for he variables Φ () ( ) ( ), ( ) Z, ( Z z) ( Z z) ( ) Z and Y : z = Pr < (III..5) ( ) ( ) ( ) Φ z = Pr < (III..6) F y = Pr Y < y (III..7) ( ) ( ) Taking ino accoun (III..4), (III..5) and (III..6), we can represen he CDF ( ) Φ as following: Φ ( ) ( ) ( ) u = Pr σ () () σ () ( Z < u) = Pr Z < u = Pr Z < u = Φ σ The following expression holds rue for he ransformaion funcions: Z ( ) () ( ( ) ) = ϕ Y Φ ( ) = F ϕ ( ) σ σ u σ Suppose ha he residuals Z are saionary in he erms of he disribuion law (i.e. he disribuion funcion is he same and does no depend on he momen of ime when i was esimaed) and has consan mean (equal o zero). A he same ime he variable Z is globally non-saionary in he erms of he variance (i.e. variance is ime dependen), bu can be considered consan wihin some (shor) inervals of ime (locally saionary). ( b Φ ) ( z) Le s define a basic CDF as he CDF ha is esimaed on he sub-sample long enough o capure he rue disribuion funcion. By analogy, le s define a basic volailiy σ b as he ( i sandard deviaion of he same sub-sample. Then a local CDF Φ ) ( u) of he residuals on some i- inerval wih consan local variance σ i can be defined as he following: () i ( b) σ b Φ ( u) = Φ u (III..8) σ i

223 We can derive he following conclusions from he expression (III..8): () i ( b. If σ i = σ b, hen Φ ( u) = Φ ) ( u), i.e. here is no need for local adjusmen. ( i. If σ i > σ b (residuals become more volaile), hen ) ( b ( ) ) σ b Φ u = Φ ( u ), where u = u < u. σ 3. If σ i < σ b (residuals become less volaile), hen Φ ( i ) ( b ( ) ) u = Φ ( u ) σ b, where u = u > u. σ The expression (III..8) is applied furher o obain he local esimaes of he CDF ha is used o calibrae he ransform funcion. According o his expression, he local CDF esimaors are obained as following: σ b. Each hisoric observaion in he esimaed window is adjused by he raio. σ i. This adjused hisoric daa is ransformed ino normal variable by he means of he basic CDF funcion. Figure 3.8 presens he examples of he volailiy raio σ i r i = for he DAX insrumen. Basic σ b volailiy σ b is esimaed as a sandard deviaion of he daa on he inerval [ ; N he oal lengh of he sample. Local volailiy i-h moving window of he lengh m : [ i m + ; i], i N / +. / ] i i, where N is σ i is represened by he sandard deviaion of he.5 window= window=5 window= volailiy raio window # Figure 3.8. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 observaions, observaions 9-58): Volailiy raio (rolling sandard deviaion (ih window)/ rolling sandard deviaion (basic)) for hisoric residuals for differen windows lenghs

224 As we can see he volailiy raio flucuaes around : a some poins local volailiy is lower han he basic volailiy, while a ohers - local volailiy is higher han he basic volailiy. Therefore, he adjusmen of he CDF for volailiy is jusified. Furher in Chaper 4 we will show ha conrary o he saionariy assumpion, local adjusmen of he CDF for he volailiy improves he DK probabiliy esimaes. Noe also ha we do no adjus he variogram parameers for local variance, as variogram is esimaed for already ransformed sandard normal variable ha always has uni variance by definiion. In he following applicaions we will esimae he basic CDF and variogram model only once. Though in real life rouine applicaion, he models should be esed for newly arrived daa and in he case of significan discrepancy he models should be re-esimaed. 3 Examples of he DK probabiliies under saionariy assumpion In his chaper we consider he examples of he DK probabiliies, obained under assumpion of daa saionariy. This implies he same daa ransform funcion and covariance model for he ransformed sandard normal variable. Chaper 3. presens he examples of he poin probabiliies for he DAX residuals, while Chaper 3. analyses he inerval probabiliies for he Bund residuals. The Bund/DAX residuals are obained by he exracion from price of he EMA of he one-week lengh (5 observaions). The ransform funcion is calibraed on he whole daa sample and covariance funcion is esimaed for he whole sample of he ransformed normal variable. Bund and DAX residuals are sill non-normal, bu can be considered saionary as heir semi-variograms converge oward residuals variance. Taking ino accoun ha he examples of he DK probabiliies in Chaper 3 are esimaed for he same sample on which he calibraion of he ransformaion funcion is done, his approach creaes he daa-snooping bias, when he obained resuls are validaed on he same sample, on which he parameers, used in he esimaion procedure, are esimaed. However, his allows us o analyze he resuls under he assumpion ha he obained variogram and CDF esimaes are he rue esimaes of he covariance and probabiliy disribuion. I should be noed ha he variogram of he ransformed h normal variable for Bund (Appendix C, Figure C3) fi he exponenial model γ ( h) = e 55, while for he DAX residuals he variogram is h h πh = γ h.75 e +.5 e cos (see Chaper.). 45 ( ) 3. Poin DK probabiliies: DAX case The examples of he poin probabiliies are provided for he sub-sample of he DAX quoes (see price pah in Figure 3.9). Appendix A explains he algorihm behind poin probabiliies calculaions. Our objecive is o show how he values of he poin DK probabiliy depend on he choice of he lengh of he window, ime horizon and hreshold level. Noe ha he hreshold is

225 fixed a he level of he las observaion in he window; his will allows us more inuiive inerpreaion of he resuls. 3 quoes observaions Figure 3.9. Transformed DAX residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): observaions 5-5 Figure 3. presens he esimaed poin probabiliies for differen hreshold level and ime horizon, esimaed on he differen windows, bu of he same lengh observaions: he DK probabiliy for he hreshold y c = -.36 is esimaed on he inerval [5; 599]; for he hreshold yc =.55 - on he inerval [65; 749]; and for he hreshold y c, 5 =. 9 - on he inerval [85; 949]. We can noice ha a long ime horizon he DK probabiliy values converge owards uncondiional normal probabiliy (CDF): F ( yc ) = P( Y ( h) < y c ). Moreover, his convergence achieved a he ime horizon ha is close o visual variogram range (where variogram convergence oward variance), confirming ha exisen auocorrelaion is aken ino accoun for DK probabiliy esimaion. Increasing or decreasing probabiliy a shor ime horizon depends on he price pah ha he insrumen follows in he esimaed window (see Picure 3.). For he inerval [5; 599], he negaive rend is well pronounced; hus, he probabiliy of being below he hreshold (las observed price) is higher han uncondiional normal CDF (i.e., we expec he price o decrease furher and breach he las price level). For he inerval [65; 749] a he presence of he posiive rend he probabiliy of saying below he designed hreshold (las available observaion) is less likely, herefore he probabiliy is lower han he uncondiional probabiliy for his hreshold. Finally, no rend is observed on he inerval [85; 949] making he probabiliy of being below he las observaions very close o he uncondiional value. 3

226 .8.7 poin DK proba hreshold=-.36 hreshold=.55 hreshold= ime horizon Figure 3.. Transformed DAX residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Probabiliy P ( Y ( + h) < yc ) for differen ime horizon (lags) h, for differen esimaion window of he lengh observaions and differen hreshold levels: y = -.36, y =. 55, y. 9. c c c, 5 = poin DK proba window=.6 window=5 window= ime horizon Figure 3.. Transformed DAX residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Probabiliy P ( Y ( + h) < yc ) for differen ime horizon (lags) h, for differen lengh of he esimaion window and he same hreshold level =.53. y c 4

227 The lengh of he window has no impac a very shor ime horizon, bu implies discrepancy a longer ime horizon. Figure 3. demonsraes he DK probabiliies esimaed for he same hreshold y c =.53 ha is defined by he same h observaion of he ransformed residuals (he end of all windows is a h observaion). Therefore, he longer window (5 observaions) conains he shorer windows ( and 5 observaions) and daa overlaps. For he ime horizon close o - observaions hree curves coincide, while for he longer ime horizon he speed of he convergence versus uncondiional value depends on he window lenghs (see Figure 3.). The shorer ( observaions) and medium (5 observaions) windows conain a negaive rend; herefore, convergence versus uncondiional level is less rapid han for longer window, where no global rend can be defined. Alhough he poin DK probabiliies canno be back-esed, he resuls of his paricular es follow common sense: () a disan ime horizon he probabiliy of being below some hreshold converges versus he uncondiional normal probabiliy; () he poin DK probabiliy value a low ime horizon as well as he speed of is convergence versus uncondiional level depend on he insrumen pah in he esimaed window; (3) he impac of he pas observaions (esimaed window) disappears a he ime horizon longer han he variogram range. 3. Inerval DK probabiliies for global ransformaion funcion: Bund case Conrary o he poin probabiliies, he analysis of he inerval DK probabiliies can be enriched by he back esing resuls: he DK probabiliies can be compared wih he empirical frequencies P emp ( Y[ + ; + l ] < yc ), esimaed on he same inervals. The ransformaion funcion as well as he covariance funcion is esimaed on he whole available sample of he Bund residuals; hey assumed o be he same for he whole ime. In order o analyse he impac of he differen facors, he inerval DK probabiliies are calculaed for he following parameers values: Lengh of he window: 54 observaions (he window is rolled by he sep observaions) ; Lengh of he inervals: 3, and 5 observaions. Thresholds values: 3 : 3. y c [ ] Taking ino accoun he large number of es parameers (inerval lengh, hreshold value) he volume of he analysis resuls is considerable. Many of hese resuls are very similar. Therefore, we presen here only some of he resuls ha allow us o derive main conclusions abou DK applicaions oucomes under saionariy assumpion. Figure 3. and B-B3 presens he examples of he inerval DK probabiliies and respecive empirical frequencies, esimaed for he following parameers: window=54 observaions, inerval= observaions and hreshold=.9. The X-coordinae represens he number of he 54-observaions window, on which he DK probabiliy was calculaed for he -observaions We do no analyse he impac of he window lengh on he inerval DK resuls. The covariance model used in he DK procedure is exponenial; hus, only las observaion in he window defines he DK probabiliy. As he resul he window lengh does no have impac on he DK probabiliy values. Oher resuls can be provided on he reques. 5

228 inerval ha follows he window, i.e. window # corresponds o he window [;54] and he inerval [55;54], window # corresponds o he window [;64] and he inerval [65;64], ec. As we can see from Figure 3. he DK probabiliy and empirical frequency resembles some random variable: hey oscillae around.97-levels wih differen ampliudes. The empirical frequencies curves deserve separae discussion as heir forms migh raise several quesions. As we can see from Figure 3., he empirical frequencies ofen ake he value of (or for oher hresholds), increasing he dispersion of his variable. The fac ha % of all observaions in he esimaed inerval is found above or below he hreshold value poins a some daa peculiariy, i.e. he residuals are moving around some local mean values differen from and oscillae around in jumps. I means ha we can always find some shor inervals of daa during which he daa is below or above some chosen hreshold level. The inervals wih such behavior can be found in he Figures B-B3 in Appendix B. DK proba empirical frequencies window # window # Figure 3.. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 obs): DK probabiliy P DK ( Y[ + ; + l ] < yc ) and respecive empirical frequencies, hreshold value y c =.9, window lengh=54 obs, inerval= obs. I is clear ha he direc comparison of he DK probabiliies wih he frequencies curves in Figure 3. is very difficul. Therefore, we propose o consider DK curves as some random variable and use is firs wo momens (mean and sandard deviaion) o summarize he resuls over all moving windows. Figures 3.3 and B4-B5 in he Appendix B represen he analysis and comparison of he DK probabiliies over mean and sandard deviaion. Le F ( yc ) = P( Y ( h) < yc ) represens he CDF of he random variable. Figure 3.3 represens he average probabiliy P a ( Y( h) < y c ), or an average CDF of he variable Y ( h). The average CDF in Figure 3.3 resembles he normal CDF; his hypohesis is confirmed by he scaer poin graph B7 in Appendix B. 6

229 mean of he DK probabiliy hresholds value Figure 3.3. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 obs): Mean of ( Y y ) P < he DK probabiliies DK [ + ; + l ] c for differen hresholds ( ); window lengh= 54 observaions, inerval = observaions..3 y c s. deviaion of he DK probabiliy hresholds value Figure 3.4. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): ( Y y ) P < Sandard deviaion of he DK probabiliies DK [ + ; + l ] c for differen hresholds ( ); window lengh=54 observaions, inerval lengh= observaions. y c 7

230 The measure of he probabiliy dispersion for differen hreshold values is represened in Figure 3.4. The following conclusions can be derived abou DK probabiliy dispersion:. The dispersion (sandard deviaion) is no very large, meaning ha DK probabiliies say quie close o is mean value- uncondiional normal probabiliy.. The value of dispersion depends on he hreshold value, achieving is peak a he - hreshold value and diminishes oward wih he increasing of he hreshold value in absolue erms. I means ha oscillaions of he DK probabiliy are he highes for - hreshold. This anomaly can be easily explained: as zero is he mean of he ransformed variable, his value is crossed more ofen han he oher hreshold levels. Thus, we can conclude ha DK probabiliies oscillae around uncondiional normal probabiliy. Figures B-B (Appendix B) analyze wheher he lenghs of he inervals have impac on he inerval DK probabiliies. The following conclusions can be derived:. The mean value of he inerval DK probabiliies does no depend on he lengh of he inerval for which he esimaions are performed (see Figures B4 in he Appendix B).. The sandard deviaion of he inerval DK probabiliies depends on he inerval lengh for which he esimaion is conduced: he sandard deviaion of he DK probabiliy decreases wih increase of he inerval lengh (see Figure B5 in he Appendix B), i.e. increasing of he inerval for which he DK probabiliies are esimaed increases he error of he esimaions. The comparison of he DK probabiliy wih he empirical frequencies, esimaed for he same inervals, is one of he mos imporan pars of he analysis. If boh eniies coincide, he DK probabiliy can be acceped as he esimaor of he rue probabiliy and used furher for successful sraegy consrucions. Figures compare he main saisical characerisics of DK probabiliies and empirical frequencies. Figure 3.5 presens mean value of he DK probabiliy versus mean value of he empirical frequencies. We can conclude ha on average he values of DK probabiliy and frequencies coincide. A he same ime dispersion of he DK probabiliy and frequencies differs significanly (see figure 3.6): he sandard deviaion of he empirical frequencies exceeds he same characerisics of he DK probabiliy almos imes. I means ha if we assume ha he empirical frequency is he esimae of he rue probabiliy, he usage of he DK probabiliies as hese esimaors will lead o he significan errors. Increasing of he inerval lengh decreases he sandard deviaion of he empirical frequencies, as he more daa we use o esimae he rue probabiliies he beer he esimaes are (see Figure B6 in Appendix B). However, i is sill larger han he DK probabiliy dispersion. This resul was predicable from figure 3.: he empirical frequencies ofen ake he value of or ha increases he dispersion of he variable. 8

231 mean of empirical frequencies mean of DK proba Figure 3.5. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Mean of he DK probabiliy versus mean of he empirical frequencies for he same hresholds: window lengh=54 observaions, inerval lengh= observaions. s. deviaion of he DK probabiliy, empirical frequencies s.dev of DK proba s. dev. of frequencies hresholds value Figure 3.6. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Sandard deviaion of he DK probabiliies and empirical frequencies for differen hresholds ( ): window lengh=54 observaions, inerval lengh= observaions y c 9

232 The conclusions abou average values of he DK probabiliies and heir dispersion are ineresing from he heoreical poin of view. However, hey are no very useful for sraegy consrucions. The fac ha here is a large discrepancy beween DK probabiliies and empirical frequencies, makes i impossible o use he DK esimaes as he rue probabiliy esimaors. In addiion i is also obvious ha here is no only he discrepancy in values ha is imporan, bu also he discrepancy in he movemens of he DK probabiliies and frequencies, i.e. peaks or jumps in he empirical frequencies do no coincide (or even follow) wih he peaks in DK probabiliies (see Figures B-B3 in he Appendix B). Tha is why he obained resuls need he improvemen in boh senses: he value of DK probabiliy and is relaive movemens wih respec o he previous values. The following approaches migh improve he discrepancy beween DK probabiliy and empirical frequencies:. The lengh of EMA, used o esimae he residuals, should be shorening. The shorening of EMA can help o change he behavior of he residuals. The shorer he EMA, he closer i is o he price curve, herefore he residuals should oscillae more around. This migh help o avoid he presence of he residuals clusers around some local mean differen from zero (he peculiariy observed above).. The daa ransformaion ino normal should be made locally. On one hand, he usage of he large daa samples gives more informaion abou random variable. On he oher, i smoohes he daa oo much and eliminaes he local rends. Afer he DK analysis we found ou ha he firs approach does no provide saisfying resuls: shorening of he EMA for he residuals exracions does no improve he discrepancy beween DK probabiliies and he empirical frequencies. Alhough he mehod diminishes he variabiliy of he empirical frequencies, i is also diminishes he sandard deviaion of he DK probabiliies; hus, he discrepancy beween wo values does no improve. The local ransformaion of boh daa and hresholds allows aking ino accoun local rends or heir absence. The jusificaion of such approach is based on he hypohesis ha in a close neighborhood he daa behaves according o he same rules (local saionariy), however, hese rules differs for he disan inervals (non-saionariy). Thus, aking ino accoun only local informaion can improve he shor-erm daa forecas. We can see ha he local ransformaion will have impac on he DK probabiliies hough he analysis of he hresholds, ransformed once on he basis of he whole daa sample and he hresholds, ransformed locally on he basis of he coninuously rolling sub-samples of he lengh 3 m. The Figures show he hisoric hreshold z C =. 5 ransformed ino normal variable on he rolling sample of he lenghs m = 5 and m = respecively. Each poin of he blue curve on he figures corresponds o he number of he sub-sample, used for hreshold ransformaion: # corresponds o daa { Z i}, #k o daa{ Z } i m i, ec. The red do line k i k + m represens he value of he hreshold y C =. 56, obained for he ransform funcion, calibraed on he whole available daa se (7 observaions). Due o he large volume of he analysis he resuls are no presened in he paper. These resuls can be obained from he auhor a reques. 3 For he local ransformaion of a hreshold, he empirical CDF, used o calibrae he hreshold ransformaion funcion is esimaed on he rolling sub-sample.

233 3.5 ransformed hreshold # of window x 4 Figure 3.7. Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): The hreshold zc =.5 ransformed on he rolling DCI(hreshold) of lengh m = 5 observaions (blue line) and whole sample DCI(hreshold) (red line) ransformed hreshold # of window Figure 3.8. Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): The hreshold zc =.5 ransformed on he rolling DCI of lengh m = observaions (blue line) and whole sample DCI(hreshold) (black line) x 4

234 We can see from figures how he values of he ransformed hresholds are daa-sensiive, i.e. hey depend on he sub-sample, on which he ransform funcion is calibraed. Noe ha he gaps in he blue curve imply infinie values of he ransformed hresholds. This happens if a sub-sample, used for he calibraion of he ransform funcion does no conain any observaions of value 4 z C =.5. As we can see he ransformed values varies a lo. This variabiliy depends on he lengh of he sample used for daa ransformaion: longer samples correspond o less variable hresholds values (compare Figures 3.7 and 3.8). The following Chaper 4 provides he analysis of he inerval DK probabiliies, esimaed under assumpion abou local saionariy. 4 Examples of he DK probabiliies under local saionariy assumpion As have been shown in Chaper 3, he esimaes of he DK inerval probabiliies (for he same ransform funcion) coincide on average wih he empirical frequencies. However, hese resuls canno be used in he sraegy consrucion. We have also shown ha local ransformaion of he hreshold brings he resuls significanly differen han he hreshold value ha is obained for he ransformaion funcion, calibraed on he whole sample. Therefore, in his chaper we propose o analyze he inerval DK probabiliy under he assumpion of local saionariy. The DK esimaes in his Chaper 4 are obained for he local ransform funcion, adjused o he local volailiy as discussed in Chaper.. Chaper 4. explains he framework for he esimaion of he inerval DK probabiliies. Chaper 4. compares he inerval DK probabiliies wih he empirical frequencies, as well as analyzes he impac of he parameers choice (inerval, window lenghs, ec.) on he DK esimaes. 4. Framework for he esimaion and analysis of he inerval DK probabiliies Le s define he general framework for he analysis. Le is available sample of he hisoric daa () () for he i-h insrumen. Le and represen he firs and second halves of he sample S i S /, i S /, i (). Sub-sample S /, i (firs half of he original sample) is used o esimae he models furher used for DK applicaion procedure: () basic CDF and () variogram model for he normal () variable, ransformed from he hisoric daa according o he basic CDF. The inerval DK S /, i () probabiliies are esimaed for he second sub-sample S /, i. Such approach allows avoiding he menioned daa-snooping bias. This approach also enables us o apply he mehod on he rouine basis, as i does no anicipae he fuure. I The inerval DK probabiliy P ( l, y ) = Pr( Y < y, i [ + ; l] ) DK, c i c + is esimaed for he inerval of lengh l and hreshold yc. Threshold yc is obained by he ransformaion of he hisoric hreshold z c. We choose some vecor of consan values (fixed for he whole ime period) for he hreshold zc : z c = cons. Such hreshold choice coincides wih he formulaion of he rading bands sraegies wih he MA as he middle line and parallel shif of he MA up/down for he bands values (see Par II for more deails). The algorihm for hese sraegies 4 In his case he empirical probabiliy of observing his hreshold value is equal o ; hus, he ransformed variable akes infinie values. S i

235 can be reduced o he comparison of he residuals (difference beween he price and MA) wih a consan hreshold. The analysis of he predicion qualiy of he inerval DK probabiliies is made hough he comparison of he DK probabiliies wih he respecive empirical frequencies, as well as hough he analysis of he impac he parameers choice, such as lengh of he esimaed window (n) and inerval lengh (l), has on he DK values. Due o he large number of differen parameers and sub-samples, used in he DK applicaions we propose he following vocabulary of erms for he resuls descripion:. Disribuion calibraion inerval (DCI) is a sub-sample of hisoric daa, used for he esimaion (calibraion) of an empirical disribuion funcion or adjusmen of a basic CDF (as discussed in chaper.).. Window is a sub-sample of normal (ransformed) daa, used in he DK procedure for kriging of he Hermie polynomials. 3. Inerval is a sub-sample of hisoric daa, used for empirical probabiliy (frequency) calculaions. I 4. DK probabiliies are probabiliies PDK (, l, yc ) = Pr( Yi < yc, i [ + ; + l] ), esimaed according o he DK procedure. 5. Z-frequencies are empirical frequencies hisoric daa (residuals). P Z emp = l + i= + I ( Z < z ) l i c calculaed on he basis of he Le s use he schemaic represenaion of hese erms in Figure 3.9. I explains how he subsamples are siuaed a he ime coordinae wih respec o some curren momen of ime. -m+ window inerval -n+ +l DCI Figure 3.9. Schemaic represenaion of he sub-samples in he ime space This figure 3.9 also represens he way he DK calculaions are performed. Suppose we wan o esimae DK inerval probabiliies a momen of ime and compare hem wih real-life Z, where << N. Then we use frequencies using available hisoric daa sample { i} <i< N { Z i } m i b, he sub-sample DCI of he lengh, o adjus he basic ransformaion funcion m + ϕ for volailiy (see Chaper. for more deails) and ransform hisoric daa Z and hreshold 3

236 z ino normal variable Y : c { i } Y = ϕ ( Z ) m+ i and yc ϕ ( zc ) =. For he DK calculaions we use he hreshold value y c and par of he normal daa Y [ n+ ; ] Y[ m+ ; ] as he window of he I lengh n. We esimae he DK inerval probabiliies PDK (, l, yc ) = Pr( Yi < yc, i [ + ; + l] ) for he inerval [ + ; + l] of he lengh l. We calculae he Z-frequencies for he daa { Z i } o + i + l I compare wih he DK probabiliies P, l, y. DK ( ) c The DK analysis is performed according o he following algorihm:. Basic (non-adjused) ransformaion funcion ϕ b for he ransformaion of he hisoric daa Z and is hresholds z ino normal variable Y and is calibraed on he firs half c { Z i } i / () of he sample S =, where N is he lengh of he sample. / N. Variogram parameers are esimaed once, on he basis of he sample of { } : y c Y i i N / Y i b ( Z ), i [ ; N / ] = ϕ. i 3. The local ransformaion funcion ϕ is he basic ransformaion funcion ϕ b, adjused for he local volailiy σ according o he following procedure (see Chaper.). Le () Φ ( Z ) be he local CDF funcion ha we wan o obain (adjus) a he momen for he hisoric daa/hreshold ransformaion: ( ) ( ( ) ) Y Φ ( Z ) = F ( Y ) Z = ϕ ϕ ( ) ( ) b We have esimaed already he basic CDF Φ Z a he sep #. The adjusmen of he basic CDF coincides wih he adjusmen of he hisoric daa (hreshold) ha we wan o ransform: Φ () ( b) ( ) u = Φ σ b u, σ ( ) ( b ) ( Z ) = F ( Y ) Φ ϕ. b The volailiy raio is defined by he following values: σ σ b = Ε[ ( Z i Z b ) ], i [ ; N / ] ( Z Z ), i m + [ i l ] [ ] = Ε ; m N Z b Ε[ Z i ], i [ ; N / ] () S Z Ε[ Z ], i [ m + ] where is DCI lengh, - sample lengh, half of he sample, / l i ; = - mean of he firs = - local mean of he sub-sample. 4. Sub-sample Z and hreshold z are ransformed ino normal Y and y according DCI o he local ransformaion funcion ϕ. c DCI c 4

237 ( ) 5. For he esimaion of he DK probabiliies P DK, Y[ + ; + l ] < yc a ime > N / we use normal variable sub-sample Yα Y DCI : Yα = { Yi }, i [ m n +; m], where n - lengh of he window, used for DK procedure. Noe ha < n m. ( ) 6. DK probabiliies a momen i P DK, i Y[ i+ ; i+ l ] < yc are calculaed for he sample i [ N / + ; N l], where l is he lengh of he inerval, for which he esimaion procedure is performed. ( ) 7. Respecive empirical probabiliies a momen i P emp, i Z [ i+ ; i+ l ] < zc are calculaed for Z i, hisoric daa { } : [ N / + N ] P l + I( Z i < zc ) Z i= + emp, =. 8. The end of he esimaion window is moved on he ime line wih he sep of observaions 5 : k = k +. Several parameers inervene direcly or indirecly in he DK calculaions. The following parameers used direcly in he DK calculaions: () he lengh of he moving window of normal variable, used for Hermie kriging; () he lengh of he inerval, for which he probabiliies are esimaed. The oher parameers can inervene indirecly hrough esimaion of he CDF, used in ransformaion: () he lengh of he DCI inerval on which he adjusmen of he basic CDF is made; () he bandwidh value used in he kernel procedure for CDF esimaion. In order o analyze he impac of hese parameers on he DK resuls we choose he lengh of he DCI, window and inervals a he levels ha represen he shor, medium and long windows and inervals. As his classificaion is quie general and subjecive, we pu he following characerisics in he base of he differeniaion beween shor, medium and long. The choice of he DCI covers sub-samples from 5 o 5 observaions. For he windows he lengh is defined by he range parameer of he variogram, as i represens how long he pas observaions has impac on he fuure predicion: long lengh is represened by he disance, a which he variogram (graphically) converges oward he variance level; medium and shor lenghs represen approximaely half and forh par of he long window. For he inerval lengh, he shor inerval should sill conain sufficienly observaions o calculae he empirical frequencies (i.e. a leas observaions). Furher, Chaper 4. presens he examples of he DK probabiliies for he DAX residuals afer exracion of he EMA of he one-week lengh (5 observaions). 4. Inerval DK probabiliies: DAX insrumen The saisical comparison of he DK probabiliies wih he empirical frequencies involves esimaion of some indexes or saisics ess ha allow judging abou he predicive properies of he DK mehod. However, as we will see furher due o he specific naure of he esimaes he classical indexes, such as mean squared error, are meaningless or complicae o calculae. 5 The abulaion sep of observaions is used in order o reduce he calculaions ime due o he lengh of he samples. l 5

238 Therefore, we use graphical represenaions of he DK probabiliies and Z-frequencies o compare hem. The numerical evaluaion of he DK predicion power is subsiued by he analysis of he rading oucomes of he sraegies ha incorporae he DK probabiliies (see Chaper 5 for he analysis). Noe ha some of he ransformed hresholds values presened furher ake he infinie values ha graphically are refleced in he gaps in he DK probabiliies curves. Such siuaion is observed for he hisoric hresholds ha afer adjusmen for he volailiy raio were never aained in he firs half of he sample. For example, le s consider he non-volaile periods when he local volailiy of rolling sub-samples is wice as small as basic volailiy. Then he hreshold of 7 is adjused o he 4 level ha was never aained on he firs half of he sub-sample (see Figure 3.4). Such siuaion is more likely for he hresholds more disan from. We will observe hese gaps furher in all figures, as hey represen he paricular examples of he DK probabiliies for he hisoric hreshold = 7. z c The following subchapers analyse he impac of he window, inerval, DCI lenghs and bandwidh parameer on he inerval DK probabiliies. 4.. Impac of he window lengh The experimen is organized as following:. Variable parameer window lengh: n = 3 observaions; n = 6 observaions; n 3 = 5 observaions.. Fixed parameers:. DCI: m = observaions ( m = 5 for he n 3 = 5 ). Inerval: l = observaions z 5 : :6 3. Hisoric hresholds (non-ransformed): [ ] The examples of he DK probabiliies and respecive empirical probabiliies for he hreshold z c = 7 are given in Figures The X-coordinae represens he number of he inerval, for which he esimaions was done, for example, # - + ; + l + ; + + l ; ec. 6 c [ ]; # - [ ] The following peculiariies are observed: () some discrepancy in he absolue erms beween DK probabiliies and frequencies; () he presence of he gaps in he DK curves (explained above). The DK probabiliies in some cases lag he empirical frequencies curves, bu his lag is very small. Moreover, he periods when boh curves are differen from one ofen coincides leading us o he conclusion ha he predicion power (a leas iming) of he DK mehod is quie good. As we can see, he window size does no have significan impac on he DK probabiliies. Large window produces slighly differen resuls, bu he difference is no very large. 6 Noe ha he momen of ime is abulaed wih sep (see Ch.4.) 6

239 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)= obs, inerval= obs, window=3 obs, hreshold(z)=7 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)= obs, inerval= obs, window=6 obs, hreshold (z)=7 7

240 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)= obs, inerval= obs, window=5 obs, hreshold (z)=7 4.. Impac of he inerval lengh The experimen is organized as following:. Variable parameer inerval lengh: l = observaions; l = 5 observaions; l 3 = 5 observaions.. Fixed parameers: DCI: m = observaions Window: n = 6 observaions Thresholds (non-ransformed): z [ 5 : :6] As in he previous case he DK probabiliies show some good predicion power in he erm of iming. If we compare figures 3., 3.3 and 3.4 we can see ha increase of he inerval improve he predicion power of he DK mehod in boh iming and absolue erms, alhough o cerain poin: increase of he inerval o 5 observaions decreases he predicion power due o he significan smoohing of he empirical frequencies curve. c 8

241 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.3. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)= obs, inerval=5 obs, window=6 obs, hreshold (z)=7 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.4. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)= obs, inerval=5 obs, window=6 obs, hreshold (z)=7 9

242 4..3 Impac of he DCI lengh The experimen is organized as following:. Variable parameer DCI inerval lengh: m = 6 observaions; m = observaions; m 3 = 5 observaions ; m = 5 4 observaions.. Fixed parameers: Inerval: l = observaions; Window: n = 6 observaions; Thresholds (non-ransformed): z [ 5 : :6] Conrary o he previous wo cases, we can noice ha he predicion power of DK probabiliies depends on he DCI lengh. Figures 3. and show ha he shor DCI inervals (6 and observaions) allow obaining he bes DK esimaes in erms of iming and values. For he longer DCI of 5 and 5 observaions he lag and discrepancy (in absolue erms) beween wo curves increase. Therefore, furher for rading simulaions we are considering only shor DCI inervals of 6, and 5 observaions. c probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.5. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)=6 obs, inerval= obs, window=6 obs, hreshold (z)=7

243 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.6. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)=5 obs, inerval= obs, window=6 obs, hreshold (z)=7 probabliy (DK, empirical) DK proba Z-frequency inerval # Figure 3.7. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies and empirical Z-frequencies: DCI (calibraion)=5 obs, inerval= obs, window=6 obs, hreshold (z)=7

244 4..4 Impac of he bandwidh parameer The experimen is organized as following:. Variable parameers bandwidh parameer: b = 7 b = 5. Fixed parameers: DCI inerval lengh: m = observaions; Inerval: l = observaions; Window: n = 6 observaions; z 5 : :6 Thresholds (non-ransformed): [ ] We propose o consider he scaer poin beween he DK probabiliies, calculaed for he ransform funcions ha are based on he esimaes of he empirical CDF wih differen bandwidh parameers: b = 7 and b = 5. As we can see from Figure 3.8 he bandwidh parameer does no have a very big impac on he DK probabiliy values: he scaer poin graph form a line. We will see he confirmaion of his hypohesis in Chaper 5; he oucomes of he rading sraegies, based on DK probabiliies for DAX (Ch. 5.3.) and Bund (Ch. 5.3.) do no differ significanly for differen bandwidhs values. This is why for oher insrumens we will repor only he rading oucomes for bandwidh value, chosen auomaically by MATLAB procedure o fi he disribuion curve. c DK proba (bandwidh=5) DK proba (bandwidh=7) Figure 3.8. DAX residuals (3/7/3-7//6, frequency 3 minues, n(ema)=5 obs): DK probabiliies for bandwidh=7 versus DK probabiliies for bandwidh=5: DCI (calibraion)= obs, inerval= obs, window=6 obs, hreshold (z)=7

245 The following conclusions summarize he resuls of he analysis:. The predicion power of he DK mehod is quie good. The DK probabiliies in some cases lag he empirical frequencies curves, bu his lag is very small. Moreover, he periods when boh curves are differen from ofen coincides. A he same ime some discrepancy exiss in he absolue values of he DK probabiliies and empirical frequencies. These discrepancies as well as he gaps in he DK probabiliies curve make he classical numerical measure of he predicion power of he DK mehod impossible.. Neiher window lengh nor bandwidh parameer has impac on he values of DK probabiliies. 3. Inerval lengh has an impac on he predicion power of he DK probabiliies. A low values increase of he inerval lengh improves he predicion power, while a high levels increase in he inerval lengh diminishes he predicion power. 4. DCI lengh has an impac on he predicion power of he DK probabiliies. The shorer is he inerval he beer is he predicion properies of he DK mehod. I is likely ha he long DCI samples smooh oo much he local volailiy esimaes. Someone migh argue ha he discrepancies in he values of he DK probabiliies and frequencies should lead o he conclusion abou low predicion power of he mehod. However, for he consrucion of he rading sraegies he mos imporance has he iming of he predicion. A he same ime bad approximaion of he probabiliy value can be avoided by is comparison wih he (probabiliy) hreshold. For example, he probabiliy can be considered significan (ha leads o aking a posiion) if i is larger han some hreshold value. Chaper 5 proposes he example of a rading sraegy ha incorporaes he DK probabiliies, as well as he benchmark sraegy o evaluae wheher he inroducion of he DK mehod improves he rading resuls. 5 Applicaion of he DK mehod o sraegies consrucion We have shown in chaper 4 ha he DK mehod produces probabiliies ha are comparable wih empirical frequencies. In his chaper we presen he example of he sraegies ha incorporae DK probabiliies, as well as analyse heir oucomes. We compare heir resuls wih some benchmark sraegy, consruced on he basis of he well-known hypohesis in he finance random walk. The sub-chaper 5. define a DK rading sraegy he sraegy consruced on he basis of he DK probabiliies; while sub-chaper 5. presens a Benchmark sraegy he sraegy, he resuls of which will be used as he comparison benchmark. The rading resuls for four differen insrumens are analysed in sub-chaper 5.3. Furher we use he following noaion o define he sraegies: { PRICE i } - price series; i> { Z i } i> - price residuals: Z i = PRICEi EMAi ; Pos () - posiion aken a momen PRICE - price a which he posiion is aken; enry PRICE exi - price a which he posiion is liquidaed; Π = Pos { } Π j> j - profi per ransacion: j j ( PRICEexi PRICEenry ) 3

246 5. DK rading sraegy Consider rading band sraegy 7 wih EMA as he middle line, and upper/lower bands represened by he shif of EMA up/down by r unis. Le s consider rend-following sraegy, i.e. he objecive is o buy a low price and sell a high. Then if he price a curren momen is wihin he bands () ake long posiion L a momen if probabiliy of breaching upper band during he following inerval of lengh l is higher han probabiliy of breaching lower band; or () ake shor posiion S, if he probabiliy of breaching lower band is higher han he probabiliy of breaching upper band; oherwise (3) ake no posiion, if probabiliies of breaching any of he bands (upper or lower) is. Exi (non-zero) posiion (L or S): () when he price breaches upper/lower bands or () a he end of period +l. We suppose ha enry and exi prices are equal o he prices (quoes) a he momen of enry/exi (no slippage or ransacion coss). DK rading sraegy (algorihm): If Z r ( ) ( [ ] ) Enry: if PDK Z [ + ; + l ] < r < PDK Z + ; + l < r Then Pos = ; PRICE enry = PRICE() Oherwise P () ( Z < ) if DK [ ; + l ] + r > Pos ; PRICE enry = PRICE() Then () = Exi: if Z v r, v [ + ; + l] Oherwise Pos ( ) = Then ( ν ) = Pos ; PRICE exi = PRICE(v) Pos ; PRICE exi = PRICE( + l) Oherwise ( + l) = 5. Random-walk (or benchmark) sraegy Consider rading band sraegy as in Ch.5.. The rend-following sraegy is formulaed in he following way: ake long/shor posiion L a momen wih probabiliy.5; exi posiion (L or S) when he price breaches upper/lower bands or a he end of period +l. RW rading sraegy (algorihm) : If Z r Enry: Draw uniformly disribued variable U on he inerval [ ;] If U >. 5 Then Pos = ; PRICE enry = PRICE() () Oherwise () = Exi: if Z v r, v [ + ; + l] Pos ; PRICE enry = PRICE() 7 See General Inroducion and Par II for rading bands definiions. 4

247 Then ( ν ) = Pos ; PRICE exi = PRICE(v) Pos ; PRICE exi = PRICE( + l) Oherwise ( + l) = 5.3 Sraegies oucomes Boh sraegies as in 5. and 5. are applied o four differen insrumens: () DAX; () Bund; (3) Bren; (4) X insrumen 8. Descripion of he DAX insrumen is given previously in Chapers. - 4, while shor review of oher insrumens can be found in he Appendices C, D and E respecively. Their main saisical characerisics are presened in Table 3.. Table 3. Saisics for he insrumens residuals Saisics for residuals DAX Bund Bren X insrumen Mean Sandard deviaion S.deviaion / Mean Max Min As in chaper 4 we divide each sample in wo sub-samples. Firs par is used for he evaluaion of he basic CDF and volailiy, as well as he variogram model for he ransformed ino normal hisoric daa. The second par is used for he DK probabiliy esimaions and rade simulaions. All rade simulaions for each insrumen sar a he same poin, making he oucomes for differen parameers values, used in he simulaions, comparable. As in previous applicaions, we calculae DK probabiliies and make rading decision each h observaion (, +, +,... ); a he same ime we use all available daa before (,,..., DCI ) and afer (, +,..., + l ) ime momen for CDF calibraion and DK probabiliies calculaions, as well as ransacion simulaions. For each poin, +, +,... he profi series { Π j } j> are generaed for boh sraegies. In order o compare he oucomes for boh sraegies we define he cumulaive value of sraegy (III.5.) and average profi per ransacion as in (III.5.): where J is a number of rades. V V j a = j u= J Π u (III.5.) Π u u= = (III.5.) J Noe ha alhough hese values represen P&L prooypes, hey canno be reaed as such. Our prices are defined in quoes/icks (no in moneary unis); herefore, our ransacions values are calculaed in hese quoe unis. Zero-ransacion coss assumpion is applied hough all simulaions of rading sraegies. This hypohesis will no invalidae our conclusions, as he principal goal of he analysis is o see if DK sraegy gives beer resuls han random-walk sraegy. 8 Due o confidenialiy reason we canno presen he insrumen in deails. 5

248 5.3. DAX Figure 3.9 presens he (end-of period) sum of he payoffs simulaed for differen DCI lenghs. As we can see he DK sraegy brings posiive oucomes for he hresholds inerval of medium values r [ 5;8). The random walk sraegy generaes low and volaile oucomes ha flucuae around 9 (see Figure 3.9). Global maximum for all oucomes is reached for he DK sraegy a hreshold level of r = 6. The value of he DCI has an impac on he oal oucome: he highes oucome is generaed for he DCI lengh of 5 observaions. 6 cumulaive value of he sraegy DK sraegy (DCI=6obs) -6 DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) (absolue) hreshold Figure 3.9. DAX (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value (a he end of period) of he rading band sraegy for differen hresholds, based on DK and random walk (RW) approaches: n(ema)=5 observaions, DK parameers - window lengh=6 observaions, inerval lengh= observaions, DCI lengh=6, and 5 observaions Figure 3.3 presens he pahs of he DK and RW sraegies for differen DCI parameer values wih he highes cumulaive oucomes ( r = 6 ). DK pah is more volaile han for he RW sraegy, however i sill exhibis posiive general rend. The chosen pah for he random walk sraegy accumulaes losses. We can noice he drop in he sraegy value (around 6 observaions) afer some relaively posiive rend; i is possible ha recalculaion of some of he parameers (for example, variogram model, can be used o improve he predicion power of DK mehod. 9 Figure 3.9 presens only one case for he RW sraegy for DCI=6 observaions in order o no overload he graph. However, for oher DCI lenghs he form of he RW oucomes curve is similar: i is volaile around and generaes lower profis han he DK sraegies. 6

249 cumulaive value of he sraegy DK sraegy (DCI=6obs) DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) inerval # Figure 3.3. DAX (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value of he DK rading band sraegy for he hreshold r=6 and differen DCI parameer value (DCI=6, and 5 observaions): DK parameers - window=6 obs, inerval= obs Densiy x -3 DK sraegy DK sraegy (bandwidh=8) RW sraegy RW sraegy (bandwidh=4) Daa Figure 3.3. DAX (3/7/3-7//6, frequency 3 min, observaions 9-56). Disribuion of he payoffs for aken posiions wihin DK and RW rading band sraegies (kernel fi): n(ema)=5 observaions; hreshold r=6; DK parameers - window=6 obs, inerval= obs, DCI lengh= 5 observaions; RW parameers - window=6 obs, inerval= obs, DCI lengh= observaions 7

250 Figure 3.3 presens he disribuion of he profis per rade for DK and RW sraegies (hreshold r = 6 ). As we can see he disribuion of he payoffs for RW sraegy is almos symmerical around. The DK sraegy in is urn allows diminish he number of he rades wih medium losses and increase he number of rades wih he medium profis. The mean of he profi per rade for he DK sraegy is posiive and larger han for he RW sraegy (7 unis versus -.4 unis); he median for he DK sraegy is posiive (6 unis). More saisics for he sraegies oucomes are given in Table 3., in Ch.6. Figure 3.3 presens he rading oucomes for he DK sraegy for differen ransform funcions, calibraed for differen bandwidh parameers. I shows ha he choice of he bandwidh parameer does no really maer for he oucome of he rading sraegy: bandwidh=5 ha corresponds o hicker ails of he disribuion produces only slighly higher oucomes. As he resul we see ha he DK sraegy provides beer rading oucomes han he corresponding RW sraegy. 6 4 bandwidh=5 bandwidh=7 cumulaive value of he sraegy (absolue) hresholds Figure 3.3. DAX (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value of he DK rading band sraegy for differen bandwidh parameer: n(ema)=5 observaions, DK parameers - window=6 obs, inerval= obs, DCI lengh= 5 observaions, kernel bandwidh=5 and Bund Shor descripion of he Bund insrumen, as well as he variogram model, used in he DK procedure is presened in Appendix C. Figure 3.33 presens he end-of-period value of each sraegy for differen hreshold values. The r.;.5. For random-walk DK sraegy produces posiive oucomes for he hresholds [ ) 8

251 sraegy is end-of-period oucomes have random naure hey flucuae around. Insead he DK sraegy guaranees sable profis for he whole inerval of low hresholds values. We can see ha he choice of DCI lenghs has impac on he final sraegy oucomes, paricularly in he erms of he global maximum: as in DAX case, he DCI lengh of 5 observaions provides he highes cumulaive profis for r =.4. cumulaive value of he sraegy DK sraegy (DCI=6obs) DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) (absolue) hreshold Figure Bund (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value (a he end of period) of he rading band sraegy for differen hresholds, based on DK and random walk (RW) approaches: n(ema)=5 observaions, DK parameers - window lengh=55 observaions, inerval lengh= observaions, DCI lenghs=6, and 5 observaions, kernel bandwidh=.5 Figure 3.34 presens he pahs of he DK sraegy wih he highes cumulaive value for hreshold r =.4. As we can see all DK pahs are quie monoone and posiive afer inerval #3. Figure 3.35 presens he disribuion of he profis per rade for DK and RW sraegies (hreshold r =.4 ). As we can see he disribuions of he payoffs for he RW sraegy is almos symmerical around, while he DK sraegy allows diminishing he number of rades wih medium losses and increasing he number of rades wih medium profis. The mean of he profi per rade for he DK sraegy is posiive and larger han for he RW sraegy (.5 unis versus unis); he median for he DK sraegy is posiive (.6 unis). More saisics for he sraegies oucomes are given in Table 3., in Ch.6. Figure 3.33 presens only one pah for he RW sraegy for DCI=6 observaions in order o avoid overloading of he graph. Oher RW pahs for oher DCI parameers have he same random oucomes ha are lower han he respecive global maximums of profi for he DK sraegies. 9

252 cumulaive value of he sraegy DK sraegy (DCI=6obs) DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) inerval # Figure Bund (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value of he DK rading band sraegy: n(ema)=5 observaions, hreshold r=.4, DK parameers - window lengh=55 observaions, inerval lengh= observaions, DCI lengh=6, and 5 observaions, kernel bandwidh=.5. DK sraegy DK sraegy (bandwidh=.) RW sraegy RW sraegy (bandwidh=.4) Densiy Daa Figure Bund (3/7/3-7//6, frequency 3 min, observaions 9-56). Disribuion of he payoffs for aken posiions wihin DK and RW rading band sraegies (kernel fi): n(ema)=5 observaions; hreshold r=.4; DK parameers - window=55 obs, inerval= obs, DCI lengh= 5 observaions; RW parameers - window=55 obs, inerval= obs, DCI lengh= observaions. 3

253 Figure 3.36 presens he cumulaive oucomes for he DK sraegies for differen bandwidh parameers, used o define he ransformaion funcion. The figure suppors he conclusion ha he value of he bandwidh parameer does no have significan impac on he oucome. 4 3 bandwidh=.5 bandwidh=.5 cumulaive value of he sraegy (absolue) hresholds Figure Bund (3/7/3-7//6, frequency 3 min, observaions 9-56). Cumulaive value of he DK rading band sraegy for differen bandwidh parameer: n(ema)=5 observaions, DK parameers - window lengh=55 observaions, inerval lengh= observaions, DCI lengh= 5 observaions, kernel bandwidh=.5 and.5. The resuls of he rading sraegies suppor he conclusion ha he DK probabiliy can improve he rading resuls for Bund comparing o he RW sraegy Bren Shor descripion of he Bren insrumen, as well as variogram model, used in he DK procedure is presened in Appendix D. Figure 3.37 presens he end-of-period values of each sraegy for differen hreshold values. Conrary o Bund and DAX cases, he posiive oucomes of he DK sraegy have less sable characer: hey correspond o several poins and one inerval of he hresholds. The highes oucomes are reached for he hresholds inerval r [.4;3.5]. Conrary o he Bund and DAX cases wih he relaively low levels of he opima l hresholds, for Bren he highes profis are reached for he hresholds of much higher values. One of he possible explanaion is he insrumen volailiy: Bren shows much higher volailiy han Bund and DAX, herefore he bands ha are o close o he EMA (small hresholds values) produce more false signals and canno be opimal (see Table 3.). 3

254 cumulaive value of he sraegy DK sraegy (DCI=6obs) DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) (absolue) hreshold Figure Bren (7//4-7//6, frequency 3 minues, observaions ). Cumulaive value (a he end of period) of he rading band sraegy for differen hresholds, based on DK and random walk (RW) approaches: n(ema)= observaions, DK parameers - window lengh=5 observaions, inerval lengh= observaions, DCI lengh= observaions cumulaive value of he sraegy DK sraegy (DCI=6obs) DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) inerval # Figure Bren (7//4-7//6, frequency 3 min, observaions ). Cumulaive value of he DK rading band sraegy for he hreshold r =. 5 ha produces he highes oucomes: n(ema)= obs, DK parameers - window=5 obs, inerval= obs, DCI lengh= obs 3

255 The pahs of he cumulaive values for he opimal hreshold for r =. 5 ha provides he maximum possible oucomes for DK sraegy are given in Figure We can noice ha he opimal pah for he DCI lengh of 5 observaions is increasing and almos monoone; for oher DCI lenghs he oucomes goes well below zero. A he same ime he pah of he randomwalk (RW) sraegy ha corresponds o one of he DCI values, exhibis random paerns and ends up in negaive values. Figure 3.39 presens he disribuion of he profis per rade for he DK and RW sraegies (hreshold r =.5 ). As we can see he payoffs for RW sraegy are close o symmerical around, while he DK sraegy exhibi more posiive han negaive profis. The mean of he profi per rade for he DK sraegy is posiive and larger han for he RW sraegy (.89 unis versus -. unis); he median for he DK sraegy is posiive (.5 unis). More saisics for he sraegies oucomes are given in Table 3., in Ch.6.. DK sraegy DK sraegy (bandwidh=.78) RW sraegy RW sraegy (bandwidh=.84).5 Densiy Daa Figure Bren (7//4-7//6, frequency 3 min, observaions ). Disribuion of he payoffs for aken posiions wihin DK and RW rading band sraegies (kernel fi): n(ema)= observaions; hreshold r =.5 ; DK parameers - window=5 obs, inerval= obs, DCI lengh= 5 observaions, RW parameers - window=5 obs, inerval= obs, DCI lengh= observaions The resuls of he rading sraegies suppor he conclusion ha he DK probabiliy can improve he rading resuls for Bren comparing o he RW sraegy X insrumen Shor descripion of he X insrumen, as well as he variogram model, used in he DK procedure are presened in Appendix E. Conrary o previous insrumens, he DK sraegy does no give good resuls (see Figure 3.4). The random walk sraegy as usual produce some random oucomes, wihou some consisency. 33

256 cumulaive value of he sraegy DK sraegy (DCI=6obs) - DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) (absolue) hreshold Figure 3.4. X insrumen (frequency hour, observaions ). Cumulaive value (a he end of period) of he rading band sraegy for differen hresholds, based on DK and random walk (RW) approaches: n(ema)=5 observaions, DK parameers - window lengh=4 observaions, inerval lengh= observaions, DCI lengh= observaions 5 cumulaive value of he sraegy 5-5 DK sraegy (DCI=6obs) - DK sraegy (DCI=obs) DK sraegy (DCI=5obs) RW sraegy (DCI=obs) -5 5 inerval # 5 5 Figure 3.4. X insrumen (frequency hour, observaions ). Cumulaive value (a he end of period) of he DK rading band sraegy: n(ema)=5 obs, hresholds r=.8, DK parameers - window =4 obs, inerval = obs 34

257 The highes profis are generaed for he RW sraegy, while DK sraegy does no produce posiive resuls a any hresholds levels. Change in he DCI lengh allows obaining some low posiive oucomes for DK sraegy very shor hresholds inervals. However, hey seem o exhibi raher random naure; herefore, we do no presen hese resuls. Figure 3.4 presens some of he pahs of he cumulaive values of he DK sraegy; hey exhibi a random paerns and ends up in negaive values. We can see ha for he X insrumen he DK mehod does no improve rading resuls. 6 Conclusions This par discusses he disjuncive kriging (DK), an approach ha is used o esimae he probabiliy of breaching some hreshold level. If prediced correcly, hese esimaes can improve he rading sraegies, based on he rading bands. However, he applicaion of he DK mehod o he financial daa needs some adjusmen due o he daa peculiariies. One of hese peculiariies is daa non-saionariy; as he resul he ransform funcion, used in he DK procedure, should be recalibraed locally. We proposed he mehod of adjusing ransformaion funcion for local non-saionary volailiy under assumpion ha he daa follows he same disribuion law. One of he direcions for fuure research in he field can be he adjusmen of he CDF no only for he volailiy, bu also for he change in local mean or skewness. Two differen probabiliies ypes were considered in he par III: poin and inerval. The poin DK probabiliy P DK () = Pr ( Y ( + h) < yc ) shows convergence versus he uncondiional normal CDF F( y c ). Disance (h) a which his convergence akes place coincides wih he range parameer of he variogram of he normal variable Y. The values of he DK probabiliies a disances smaller han h depend on he pah of he Y process before. However, he poin probabiliies canno be back-esed. Besides for he rading applicaion i is ofen more imporan o know ha he hreshold will be breached in (near) fuure, hen ha he hreshold will be breached a he precise momen of ime. ( ) The inerval DK probabiliy P DK ( ) = Pr Y[ + ; + l ] < yc - he probabiliy of breaching hreshold during some inerval of ime in fuure - were analysed in deph, as hey can be back-esed by he respecive empirical frequencies calculaed on he inerval l. We have found ha he values of he inerval DK probabiliies depend on he lengh of he disribuion calibraion inerval (DCI), used for he adjusmen of he ransform funcion, and he lengh of he inerval, for which he DK probabiliies are calculaed. The bes probabiliy esimaors correspond o he medium DCI inervals and shor and medium inerval lenghs. The inerval probabiliies do no depend on he window lengh (even for he insrumens wih he non-exponenial variogram model) or bandwidh parameer, used in he kernel procedure for he esimaion of he basic CDF funcion. Comparing o he empirical frequencies, he probabiliy esimaors exhibi almos no lag, while some over- or under-esimaion in absolue values akes place. The predicion power of he DK mehod was evaluaed indirecly hrough he oucomes of he rading sraegies, based on hem. We have consruced wo sraegies: () DK sraegy, where decision abou enry posiion was made on he basis of DK probabiliies; and () Random-walk sraegy, where decision abou enry posiion was made randomly. Our analysis shows ha he DK sraegy produces posiive oucomes for he coninuous inerval of hresholds wih almos monoone and increasing pahs of he sraegy value. For some insrumens DK sraegy 35

258 produces he highes possible oucomes (DAX, Bund, Bren). The only insrumen for which DK does no produce any valid resuls is he X insrumen. The RW sraegy produces for all insrumens he profis ha have some random naure. The disribuions of he profis per rade for he opimal DK sraegy shows consisenly more profiable and less non-profiable rades comparing wih he resuls of he RW sraegy. Main saisical characerisics for hese disribuions are summarized in Table 3.. Table 3. Saisical characerisics of profis per rade for he DK opimal sraegy and respecive RW sraegy DAX Bund Bren DK sraegy RW sraegy DK sraegy RW sraegy DK sraegy RW sraegy Mean Median Sandard Deviaion Table 3.3 summarizes he oucomes of DK and RW sraegies for all insrumens. We can see ha he applicaion of he DK mehod o sraegies consrucion can improve rading resuls. 36

259 Table 3.3 Resuls of he applicaion of he DK and RW sraegies o differen insrumens opimal DK sraegy opimal RW sraegy Insrumen Variogram ranges/ frequency Opimal sraegy parameers Threshold value Cumulaive value* Mean value* Monooniciy Par of he inerval? Threshold value Cumulaive value* Mean value* Monooniciy Par of he inerval? Bund n(ema)=5 DK parameers: a=55 window=55 freq=3 min inerval= DCI=5 r= Yes Yes [.;.5] r= N/A No DAX n(ema)=5 a()=5, DK parameers: a()=45 window=6 freq=3 min inerval= DCI =5 r= Yes Yes (5; 8] r= 3.4 N/A No Bren n(ema)= DK parameers: a=5 window=5 freq=3 min inerval= DCI =5 r= Yes** Yes [.4; 3.3] r= N/A No X insrumen a()=4, a()= freq=3 min n(ema)=5 DK parameers: window=4 inerval= DCI =5 r= N/A N/A r= N/A N/A * - in quoes unis ** - afer #3 observaion 37

260 Appendices III Appendix A Esimaion of he poin probabiliies for DAX The DK analysis of he poin probabiliies (.) is performed according o he following algorihm:. The whole available sample is used o esimae empirical CDF. The poin DK probabiliies are S i S i esimaed for he same sample. This approach creaes he dilemma, when he obained resuls are validaed on he same sample, on which he parameers, used in he esimaion procedure, are esimaed. However, his allows us o analyse he resuls under he assumpion ha he obained variogram and CDF esimaes are he rue esimaes of he covariance and probabiliy disribuion.. The rolling windows of n-lengh are used as sub-samples for DK esimaion procedure: { Yi }, >, where Y is he normal sandard variable from he analysed daa sample S : n+ i Y Yi S Y. p 3. The DK probabiliy P DK (, h, yc ) = Pr( Y ( + h) < yc ) is esimaed for paricular hresholds value y c = Y (), i.e. hreshold is equal o he las available observaion in he window. This choice of he hreshold value faciliaes common sense validaions of he obained probabiliies. 38

261 Appendix B Esimaion of he inerval probabiliies for Bund: global ransformaion funcion The following algorihm lies behind he esimaion of he inerval probabiliy wih sole ransformaion funcion:. The whole available sample is used o esimae he empirical CDF. The inerval DK S i probabiliies are esimaed for he same sample. This approach creaes he dilemma, when he obained resuls are validaed on he same sample, on which he parameers, used in he esimaion procedure, are esimaed. However, his allows us o analyse he resuls under he assumpion ha he obained variogram and CDF esimaes are he rue esimaes of he covariance and probabiliy disribuion.. The rolling windows of n-lengh are used as sub-samples for DK esimaion procedure: { Yi }, >, where Y is he normal sandard variable from he analysed daa sample S : n+ i Y Yi S Y. p P, h, y = Pr Y ( + h) < y is esimaed for he vecor of hresholds. 3. The DK probabiliy ( ) ( ) DK c The variogram of he ransformed normal variable Y, used in he DK procedure is he same as in Figure C3. S i c 4 Bund ransformed residuals: observaions : residuals DK proba observaions Proba DK & empirical frequency: window=54, inerval=, Th= window #.8.6 freq.4 Figure B. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions, observaions :), DK probabiliies and empirical frequencies (hreshold value y c =, window lengh=54 observaions, inerval lengh= observaions). 39

262 Bund ransformed residuals: observaions : residuals observaions x 4 DK proba and empirical frequencies: window=54, inerval=, Th=.95 DK proba freq window # Figure B. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions, observaions -) and esimaed DK probabiliies and empirical frequencies (hreshold value y c =, window lengh=54 observaions, inerval lengh= observaions) 4 Bund ransformed residuals: observaions 75- residuals observaions x 4 DK proba and empirical frequencies: window=54, inerval=, Th= DK proba freq window # Figure B3. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions, observaions 75-) and esimaed DK probabiliies and empirical frequencies (hreshold value y c =, window lengh=54 observaions, inerval lengh= observaions) 4

263 DK probabiliy inerval=3. inerval= inerval= hresholds value Figure B4. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): ( Y y ) P < Mean of he DK probabiliies DK [ + ; + l ] c for differen hresholds ( ) and inerval lenghs: window lengh=54 observaions, inerval lengh=3, and 5 observaions. y c inerval=3 inerval= inerval=5 sandard deviaion of DK probabiliy hresholds value Figure B5. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): ( Y y ) P < Sandard deviaion of he DK probabiliies DK [ + ; + l ] c for differen hresholds ( c ) and inerval lenghs: window lengh=54 observaions, inerval lengh=3, and 5 observaions. y 4

264 sandard deviaion of he empirical frequencies inerval=3 inerval= inerval= hresholds value Picure B6. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Sandard deviaion of he empirical frequencies for differen hresholds ( lengh=3, and 5 observaions. normal heoreical proba y c ) and inerval lenghs: inerval proba mean Figure B7. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): Mean of he DK probabiliy versus normal heoreical probabiliy for he same hresholds: window lengh=54 observaions, inerval lengh= observaions. 4

265 Appendix C Shor descripion of he Bund insrumen Bund insrumen (fuures on Bund) represen he bond marke. The daa covers he period 3/7/3-7//6 wih frequency 3 min (see Figure C). We have eliminaed he EMA of he lengh 5 observaions o obain residuals in Figure C. quoes observaions x 4 Figure C. Bund quoes (3/7/3-7//6, frequency 3 min).5.5 residuals observaions (freq=3 min) x 4 Figure C. Bund (3/7/3-7//6, frequency 3 min): Residuals afer exracing he EMA of he lengh n(ema)=5 observaions 43

266 As we can see from figure C, he residuals do no exhibi any rend. A he same ime daa is no normal (see Figure C3). Thus, residuals ransformaion ino normal variable is needed before applicaion of he DK mehod..4. empirical kernel: b=.5 kernel: b=.5 Densiy Daa Figure C3. Bund (3/7/3-7//6, frequency 3 min, observaions :8):empirical hisogram and kernel fis for differen bandwidh parameers value [ ] The variogram of he ransformed residuals (see Figure C4) is evaluaed on he sub-sample ;8 : h γ ( h) = e

267 .4. empirical model variogram lag Figure C4. Transformed Bund residuals (3/7/3-7//6, frequency 3 min, n(ema)=5 observaions): h Empirical (solid line) and esimaed exponenial model (do line): γ ( h) = e

268 Appendix D Shor descripion of he Bren insrumen Bren is a fuures on crude oil. Therefore, we consider his insrumen as a represenaive of commodiy markes. The peculiariy of his insrumen is ha i is highly volaile. Our daa represens he period of 7//4-7//6 wih 3 min frequency (see Figure D) quoes observaions (in 3 min) Figure D. Bren quoes for period 7//4-7//6 (frequency 3 minues) residuals observaions Figure D. Bren residuals (7//4-7//6, frequency 3 minues, n(ema)= observaions) 46

269 The residuals in Figure D are obained by eliminaing he EMA of he lengh observaions. Residuals daa is no normal (see Figure D3). Thus, residuals ransformaion ino normal variable is needed before applicaion of he DK mehod..3 empirical kernel fi (bandwidh=.33) kernel fi (bandwidh=.8).5 Densiy Daa Figure D3. Bren residuals (7//4-7//6, frequency 3 minues, n(ema)= observaions): Hisogram and kernel fis for differen bandwidh values The variogram of he ransformed residuals (see Figure C4) is evaluaed on he sub-sample [ ;459] : h γ ( h) = e 5. 47

270 .4. empirical model variogram lag Figure D4. Bren residuals (7//4-7//6, frequency 3 minues, n(ema)= observaions). Variogram of h he ransformed residuals for sub-sample [;459], as well as heoreical model fi o he daa γ ( h) = e 5 48

271 Appendix E Shor descripion of he X insrumen Insrumen X represens an arificially creaed index, used by one bank for sraegy consrucions. Due o he confidenialiy reason we canno neiher presen is deail descripion, nor provide he informaion on is real quoes. Tha is why on Figure E ha presens he quoes pah during some period of ime, here are no icks on he Y-coordinae. The only informaion we can provide is ha daa frequency is hour. quoes observaions (in hour) Figure E. X insrumen quoes (frequency hour) We eliminae he EMA of he lengh of 5 observaions o obain residuals in figure E. They are no normal (see Figure E3). Thus, residuals ransformaion ino normal variable is needed before he applicaion of he DK mehod. 49

272 5 residuals observaions Figure E. X insrumen residuals (frequency hour, n(ema)=5 observaions). empirical kernel fi (bandwidh=.44) kernel fi (bandwidh=.5).5 Densiy Daa Figure E3. X insrumen residuals (frequency hour, n(ema)=5 observaions): Hisogram and kernel fis for differen bandwidh parameer values 5

273 The following vaiogram model, esimaed on he sub-sample # [ ;346] is used for furher DK calculaions (see Figure E): 3 h ( ) 3 h h γ h = e empirical heoreical variogram lag Figure E4. X insrumen residuals (frequency - hour, n(ema)=5 observaions): Variogram of he ransformed residuals for sub-sample [:347], creaed on he base of he hisoric daa, as well as heoreical model fi o he daa. 5

274 General conclusions In his hesis we have proposed several approaches o improve and opimize rading bands sraegies. Pars I and par II concenraed on he opimizaion of he rading bands componens: middle line (in he form of he moving average) and bands. Par III was dedicaed o he improving of he process of he decision-making. Three pars of he hesis concenraed on slighly differen rading bands. While firs par produced he opimal middle line in he form of he kriged moving average (KMA), he second and he hird pars used respecively simple moving average (SMA) and exponenial moving average (EMA) as middle line componens. Besides, while par II considered he rading bands defined by he daa saisical characerisics (for example, variance), par III analyzed he rading sraegies for he bands creaed by he parallel shif of heir middle line. As for he firs wo pars, we waned o avoid mixing he effecs from he improvemens of he rading bands componens. Therefore, we did no inroduce he opimal middle line (KMA) in he definiion of he opimal bands values (DT bands). As for he hird par, our sraegy choice was explained by he huge populariy of his ype of he bands in he rading applicaions. A he same ime fuure research can address hese quesions and adjus he analyzed sraegies. In par I we considered he kriging mehod for he opimizaion of moving average (MA) weighs. The kriging mehod is based on he saisical characerisics of daa such as covariance (auocovariance) funcion. This allows obaining opimal esimaes ha depend on he saisical characerisics of he daa raher han on he hisorical daa iself as in he case of he simulaion sudies. We have shown he examples of he mehod applicaion o boh equally and unequally sampled daa. For he equally spaced daa under assumpion of he locally consan mean (wihin he moving window), he opimal weighs follow some specific srucure for cerain covariance models sable over differen window lenghs: he larges weighs in absolue value has he firs and he las observaion in he window, wih smaller or relaively negligible weighs for all oher observaions. As he resul, KMA oscillaes around he SMA curve. The volailiy and ampliude of he oscillaions is an indirec funcion of he KMA lengh: he longer he KMA he less i is volaile and i coincides more wih he SMA curve. Therefore, rend-following sraegies, based on KMA and SMA will ake differen posiions a shor window lenghs and he same posiion a long lenghs. We also saw ha for he paricular rend-following sraegies, shor KMA produces higher rading oucomes han he radiional MAs. I also seemed ha he erraic naure of KMA curve did no necessary lead o more false signals, generaed by hese rading sraegies. As for he unequally spaced daa we showed ha he adjusmen of nonregular sample o he regularly spaced one and calculaion on hem of he opimal MA, migh lead o less effecive rading sraegies. Kriging resuls on such samples largely depend on he variabiliy in he dependen variable. Par II presened he daa ransformed (DT) bands as he alernaive o he Bollinger bands. Tradiional Bollinger mehod is saisically jusified for he case of saionary, symmeric and known disribuions. Deparure from hese assumpions complicaes significanly applicaions and calculaions of hese bands. The DT bands under assumpion of local saionariy provide simple, bu powerful ransformaion of he Bollinger approach ino beer heoreical framework, which in addiion is easier o opimize. The mehod is based on he ransformaion of raw daa ino normal random variable, for which he inerval ha conains some predefined percenage of daa is known; hen his inerval is back-ransformed ino he bands for he raw daa, which now conain he same percenage of daa. We have considered paricular residuals R = P SMA, i i 5

275 SMA = P i, i [ n + ; ] o calibrae ransform funcion. Our main objecive was o say n i in he line wih he Bollinger bands heory ha uses hese residuals for he calculaion of he daa sandard deviaion. However, his allowed us o obain he bands, which are less dependen on he movemens of he moving average. As he resul, a specific sair-ype form emerged for he DT bands, which change he level only when he large price innovaions happen. The numerical examples showed ha he DT bands are no only he insrumens ha are beer jusified heoreically, bu also can be a successful sraegy componen: in majoriy of cases confirmed DT bands sraegies produced higher profis, rade efficiency and seeper P&L pahs han confirmed (by momenum) sraegies, based on he classical Bollinger bands. Moreover, he DT bands sraegy was sill profiable under non-zero ransacion coss and slippage. Finally, he DT bands migh be helpful in he definiion of oher echnical rules he Ellio waves and Suppor/Resisance levels. Par III addressed he problem of he probabiliy esimaion of breaching some hreshold levels. Anoher geosaisical approach, disjuncive kriging (DK) was used for his probabiliy esimaion. The applicaion of he DK mehod o he financial daa hough needs some adjusmen due o he daa peculiariies. The main problem is is non-saionariy ha demands local re-esimaion of he DK parameers, in paricular he ransform funcion. We proposed he mehod o adjus he CDF for local volailiy under assumpion ha he daa followed he same disribuional law, parameers of which are ime dependen. The predicion power of he DK mehod was evaluaed indirecly hrough he oucomes of he rading sraegies, based on hem. We consruced wo sraegies: () DK sraegy, where decision abou posiion enry was made on he basis of DK probabiliies; and () Random-walk sraegy, where decision abou posiion enry was made randomly. Our analysis showed ha he DK sraegy produced posiive oucomes for he coninuous inerval of hresholds wih increasing sraegy value over ime. The RW sraegy produced for all insrumens he profis ha had some random naure. Fuure research The resuls of he applicaion of geosaisical and saisical mehods o he improvemen of he echnical analysis echniques indicae he field for fuure research. In his work, we have concenraed on he echnical analysis applied o one insrumen. The fuure research should consider he rading of a porfolio of insrumens. In paricular, anoher geosaisical mulivariae mehods, such as cokrging can be used o he esimaion of he porfolio mean and predicion of is value. In order o separae he effecs of he improved middle line (by he inroducion of he KMA) and improved bands (by he inroducion of he DT bands), we have no considered he DT bands ha incorporae he KMA as he middle line. We have also seen ha a long MA lenghs he DT bands, based on SMA and DT bands, based on KMA would be he same. However, i would be ineresing o analyse he DT bands sraegy, based on he shor KMA. As for he mehods ha have been considered in he hesis more aenion should be devoed o he pracical side of he applicaion of he kriging mehod o subordinaed random variables. In paricular, he search of he opimal scaling facor ha has impac on boh he opimal weighs srucure as well as ime-inensiy of he calculaion procedure should be sudied in more deails. 53

276 Besides, cokriging of he principal random variable and he variable, o which he prinicipal process is subordinaed, can be considered as an alernaive approach o he creaion of a weighed MA. The DT bands approach indicaes he following direcions of he research. The adjusmen of he ransform funcion o he local volailiy, performed in he par III for he DK mehod, can be applied o he definiion of he DT bands. The analysis of he relaionship beween he sraegy profiabiliy and he value of K% secion or k parameer of he DT bands will allow o creae more successful rading sraegies, based on hese bands. The DT approach creaes he new opporuniies o he improving of oher echnical analysis echniques. For example, he momenum based rading sraegy migh be amelioraed by he definiion of opimum momenum hresholds on he basis of he daa ransformaion approach. For his purpose momenum daa should be ransformed ino normal, he K% secion should be applied and he hresholds should be back-ransformed o he real daa. This migh lead o he asymmeric momenum hresholds. The oher example is he applicaion of he DT bands o he definiion of oher echnical indicaors, such as Ellio waves and Suppor/Resisance rading rules. Finally, he applicaion of he DK mehod o he financial daa can be improved furher by he addressing he problem of adjusmen of he CDF, used for daa ransformaion, for he change in local mean or skewness. 54

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