Financial Market Microstructure and Trading Algorithms

Transcription

1 Financial Marke Microsrucure and Trading Algorihms M.Sc. in Economics and Business Adminisraion Specializaion in Applied Economics and Finance Deparmen of Finance Copenhagen Business School 2009 Jens Vallø Chrisiansen Submied January 9, 2009 Advisor: Marin Richer

2 1 SUMMARY The use of compuer echnology has been an imporan par of financial markes for decades. For some ime, banks, hedge funds and oher sophisicaed marke paricipans have used compuer programs, known as rading algorihms, o rade direcly in he marke. As elecronic rading has become more widespread, compuer based access o he markes has become more broadly available as well. Many banks and brokerages offer heir cliens, including privae invesors, access o he financial markes by means of advanced compuer sysems ha roue orders o he opimal price. The consequence has been increased rading volume, beer liquidiy, and igher spreads. On major sock exchanges such as NASDAQ in he Unied Saes, rading algorihms now represen he majoriy of daily volume. This means ha he majoriy of rading akes par wihou direc conac beween human raders. There are wo main ypes of rading algorihms, hose ha are used for opimal execuion, i.e. obaining he bes possible price for an order, and hose used for speculaion. This paper describes boh from a heoreical perspecive, and shows how wo ypes of speculaive algorihms can be designed. The firs is a sraegy ha uses exponenial moving averages o capure price momenum. The second is a marke neural relaive value sraegy ha rades individual socks agains each oher known as pairs rading. Boh are esed using empirical daa and he resuls are encouraging. Despie he widespread use of algorihms in he markes, evidence remains of posiive excess reurns. In paricular, he resuls of Gaev e al (2006) based on pairs rading are confirmed using more recen daa from he London Sock Exchange. The idea of univariae pairs rading is exended o a mulivariae framework in wo ways. The second is based on sae space mehods. The resuls show ha for he daa sample used, higher ransacion coss ouweigh any benefis from his exension. The heoreical foundaion of rading algorihms is marke microsrucure heory. This heory deals wih he dynamics of rading and he ineracion ha akes place beween marke paricipans. Among he imporan issues are he exisence of asymmeric informaion and he adjusmen of marke prices o new informaion, eiher privae or public. The mehodology of Hasbrouck (1991) is used o analyze he informaion conen of high frequency ransacion daa, also from he London Page 2 AEF Thesis

3 Sock Exchange. The resuls obained show ha he main conclusions in Hasbrouck s paper remain valid. The concep of opimal execuion is given a heoreical reamen based on Almgren and Chriss (2001) and McCulloch (2007). The firs paper shows ha he problem of minimizing implemenaion shorfall can be expressed as a quadraic opimizaion problem using a simple uiliy funcion. The second paper shows how he inra day volume weighed average price can be used as a benchmark for opimal execuion. AEF Thesis Page 3

4 2 CONTENTS 1 Summary Inroducion Overview Lieraure Noaion Background Par I: Marke microsrucure heory Inroducion and overview Liquidiy pools and aggregaors The informaion conen of sock rades Bins and ransacion ime Model specificaion Vecor auoregression A simple microsrucure model Background Par II: Trading algorihms Overview Opimal execuion algorihms Opimal execuion wih a quadraic uiliy funcion Exensions o he opimal execuion model: Drif Exensions o he opimal execuion model: Serial correlaion Sub conclusion Opimal VWAP Trading Speculaive algorihms Overview Pairs rading Coinegraion Pair selecion and rading signals Mulivariae pairs rading Background Par III: Sae space models Inroducion The linear Gaussian sae space model An example of a srucural model Page 4 AEF Thesis

5 6.4 The Kalman filer The Kalman smooher and Disurbance smoohing Maximum likelihood esimaion Sandard error of maximum likelihood esimaes The expecaion maximizaion algorihm Analysis Par I: High frequency equiy daa Daa descripion Descripion of a represenaive sock: Anglo American PLC Dealing wih ransacion ime Empirical analysis of ick daa Analysis Par II: Speculaive algorihms Momenum sraegies using moving averages EMA momenum sraegy wih differen sampling frequencies Univariae pairs rading Mulivariae pairs rading Conclusion Sources Appendix Even ypes in he FTSE 100 ick daa Volume and open ineres of Bund and Euribor fuures Addiional co inegraion ess Derivaion of Kalman filer recursions and illusraion Empirical analysis of ick daa Derivaion of he exponenial moving average EMA momenum sraegies Univariae pairs rading Mulivariae pairs rading AEF Thesis Page 5

6 3. Inroducion 3 INTRODUCTION 3.1 Overview Over he las few decades algorihmic rading has become an imporan par of modern financial markes. As he use of compuer echnology has become more broad based, invesors are demanding faser, cheaper, more reliable, and more inelligen access o financial markes. Banks and hedge funds are aking advanage of his rend and have begun an arms race owards creaing he bes elecronic rading sysems and algorihms. Algorihmic rading represens an ever growing share of rading volume, and in some markes he majoriy (KIM, K., 2007). The heoreical foundaion for algorihmic rading is found primarily in he fields of financial economerics and marke microsrucure. The sudy of he ime series properies of securiy prices is among he mos pervasive subjecs in he financial lieraure, and has grown rapidly in andem wih cheaper access o compuing power. The field is characerized by he vas amoun of daa available o researchers in he form of daabases of hisorical ransacion daa. The saisical heory needed o analyze such daa is differen from he daases known from convenional economics wih less frequen observaions, and is ofen much more compuaionally inensive. Tradiional mehods in exploraory daa analysis such as vecor auoregression have been joined by new mehods such as auoregressive condiional duraion models, o ake ino accoun he nonsynchronous naure of highfrequency ransacion daa (ENGLE, R. and Russell, J., 1998). The primary ambiion of his paper is o provide a brief inroducion o an exensive subjec. The conen has been seleced wih he aim of covering core areas of he heory while mainaining a coheren whole. The papers and models ha will be covered are paricularly well suied o empirical esing as opposed o much heory in he microsrucure lieraure. The secondary aim of he paper is o deermine wheher i is possible o creae speculaive rading algorihms ha earn posiive excess reurns. The analysis will focus on wo main areas, he microsrucure of financial markes and rading algorihms. The wo are conneced in he sense ha microsrucure heory provides he heoreical basis for he developmen of rading algorihms. The disincion beween microsrucure heory and financial economerics is ofen blurry, and elemens from each field will be used as deemed Page 6 AEF Thesis

7 3. Inroducion appropriae. The srucure of he paper is hus divided ino wo main pars, a par ha presens heoreical background, and an empirical analysis using marke daa. In he firs par, secions 4 6 presen he background for he empirical analysis. Secion 4 presens a mehod for he analysis of high frequency equiy ransacion daa. Secion 5 covers rading algorihms, boh for opimal execuion and speculaion. Secion 6 oulines he general heory of sae space models which can be used o design speculaive algorihms. In he second par, secions 7 & 8 apply he heory o marke daa. Secion 7 carries ou an empirical analysis of equiy ick daa based on he mehodology of secion 4. Secion 8 proceeds o es wo kinds of speculaive algorihms, one based on securiy price momenum, he oher based on he relaive value of securiies. Secion 9 concludes. 3.2 Lieraure The academic lieraure on he subjec of marke microsrucure is vas. In his paper he main sources used were Hasbrouck (1991) and Hasbrouck (2007). For opimal execuion he main sources were Almgren & Chriss (2001) and McCulloch & Kazakov (2007). Academic work on speculaive rading algorihms is scarce, and pairs rading in paricular, bu Gaev e al (2006) gives a useful overview. The empirical analysis was done using MATLAB 1, and o his end Kassam (2008) was a grea help. Shumway & Soffer (2006) was he source for general heory of ime series analysis, and Campbell e al (2006) for financial economerics, including a chaper on marke microsrucure. Durbin & Koopman (2001) was he main reference for sae space models. 1 The auhor may be conaced a [email protected] for he MATLAB code used in he empirical analysis. AEF Thesis Page 7

8 3. Inroducion 3.3 Noaion The following noaion has been used hroughou he paper unless oherwise specified: denoes a paricular poin in ime while denoes a ime incremen Δ for 1,2, ~, denoes a normally disribued random variable wih mean and variance. ~, denoes a random variable which is disribued wih independen and idenical incremens from a normal disribuion wih mean and variance. denoes he ideniy marix. Page 8 AEF Thesis

9 4. Background Par I: Marke microsrucure heory 4 BACKGROUND PART I: MARKET MICROSTRUCTURE THEORY 4.1 Inroducion and overview This secion will presen selec pars of he heory of marke microsrucure, and prepare he reader for he empirical analysis in secion 7. Wha follows is a brief descripion of he dynamics of a modern securiies marke. The main marke mechanism in modern elecronic markes is he limi order book. The limi order book consiss of a lis of buy and sell orders a differen prices and for differen quaniies. An example could be an order o buy 100 shares a $30.10 or less or sell 300 shares a $30.50 or more in 100 share incremens. By consolidaing all such orders in a cenral sysem, as for example a sock exchange, i is possible o know he bes bid (buy order) and bes offer (sell order) a any given ime. The difference beween he bid and he ask is known as he spread. Trades ake place when a rader is willing o cross he spread, ha is, o buy a he offer or sell a he bid of someone else. The marke is ypically anonymous, and rades may be made based only on he price and quaniy being bough or sold. This is known as a coninuous aucion. An acive limi order ha hasn ye been execued is known as a quoe. Once an order is execued, i.e. a ransacion akes place a a quoed price (eiher a he bid or ask), he quoe disappears and is replaced by he nex bes available bid or ask. Noe ha a quoe may be wihdrawn before i is ever execued. The hisorical observaions of ineres o marke paricipans are hus boh hisorical quoes and hisorical rades. These are also he quaniies ha will be used for empirical analysis. The effeciveness of coninuous aucion marke depends on he amoun of acive marke paricipans, and he amoun of a given securiy hey are willing o rade a any given ime. This evasive concep is known as marke liquidiy. One of he main challenges of marke microsrucure heory is o define and quanify i. In a well funcioning marke here are many paricipans ha rade significan amouns of a securiy wih each oher coninuously. The price of he securiy being quoed a any given poin in ime will hus reflec wha many paricipans believe i should be oherwise he orders would be filled, and he marke would move up or down in he order book. This process is known as price discovery. See Hasbrouck (2007) for furher deails. AEF Thesis Page 9

10 4. Background Par I: Marke microsrucure heory The following secion will ouch briefly upon some of he main insiuional feaures of modern elecronic markes. 4.2 Liquidiy pools and aggregaors Modern markes are characerized more by heir fragmenaion han heir consolidaion. This is despie of he progress of echnology and elecronic rading in paricular. The main reason is ha marke paricipans consanly seek cheaper and beer venues for heir rading. A good example is he recen success of mulilaeral rading faciliies (MTFs) such as Chi X and Turquoise in Europe. MTFs are hybrid rading venues in he sense ha hey connec marke paricipans o decenralized liquidiy pools. The erm liquidiy pool is used o describe he exisence of a marke in a securiy ouside a cenral exchange. Liquidiy pools are ypically operaed by large banks, and are inended o lower he cos of rading by moving i off he cenral exchanges (TURQUOISE, 2008). The obvious implicaion of his is marke fragmenaion, and his is he gap ha MTFs bridge by connecing he various liquidiy pools a low cos. Marke paricipans communicae across he MTFs, cenralized exchanges and privae dark pools using common communicaion proocols. The mos popular is called he Financial Informaion exchange (FIX) (WIKIPEDIA, 2008a). Marke paricipans naurally wan o access as many differen liquidiy pools as possible, o obain he bes possible price available. To his end hey use aggregaors, compuer sysems designed o roue orders o he bes possible price, wherever his price may be quoed. Aggregaors are ypically provided by invesmen banks, for example Deusche Bank s auobahn sysem (DB, 2008). The imporance of price aggregaors has been furher increased by he need o provide cliens wih bes execuion in accordance wih securiies regulaion such as MiFID in Europe (Markes in Financial Insrumens Direcive) (WIKIPEDIA, 2008b). 4.3 The informaion conen of sock rades This secion will provide he heoreical background for he subsequen empirical analysis of ick daa from he London Sock Exchange. The heoreical framework is ha of Hasbrouck (1991) and involves he use of vecor auoregression o exrac he informaion conen of sock rades. Page 10 AEF Thesis

11 4. Background Par I: Marke microsrucure heory Bins and ransacion ime A common way o analyze sock daa is o observe he daily closing prices of a given sock for an arbirary number of rading days. The frequency of price observaion is merely a maer of scaling, however, and his paper will analyze sock daa from a more deailed perspecive, namely a he ick level. The highes possible frequency of observaion for a given sock is he observaion of every single rade even. This may be combined wih he observaion of every single quoe even o give a deailed view of he inra day rading process. Large company socks rade very frequenly 2 however, so rade evens are ypically grouped in fixed ime inervals known as bins. Daily price observaions for a given sock may be seen as he creaion of one day bins of ransacion daa. Each bin has an opening price and a closing price. The price may in his case be he bid, he ask, he quoe midpoin (he average of he bid and he ask) or he las raded price. Typically he las raded price is used for daily observaions. Possible bin sizes range from a year or more o a minue or less. Bins are paricularly useful for graphing price daa. Figure 4 1 shows a candle graph of Anglo American PLC (AAL.LN) wih hourly bins. The green candles indicae ha AAL.LN closed a a higher price han he opening price of a given one hour period. The red candles indicae he opposie and he black whiskers a he op or boom of a candle indicae he range in which he sock raded during he ime inerval. 2 As an example, on March 3, 2008 beween 8:00 and 16:30, Anglo American PLC (AAL.LN) lised on London Sock Exchange experienced 51,412 evens of which 9,941 were rade evens and 41,471 were quoe evens. AEF Thesis Page 11

12 4. Background Par I: Marke microsrucure heory Figure 4 1: AAL.LN hourly bins March 3 12, Source: E*TRADE. When every rade and quoe even of a sock is observed, each even is a new poin in wha is known as ransacion ime. Clock ime incremens in ransacion ime can vary from even o even. On March 3, 2008, he average ransacion ime incremen of AAL.LN was seconds wih a sandard deviaion of , a minimum of 0 and a maximum of Model specificaion Following he analysis in Hasbrouck (1991), he primary price variable of ineres is he quoe midpoin. A ime he bes bid and ask quoe in he marke are denoed by and respecively, and ransacions are characerized by heir signed volume (purchases are posiive, sales are negaive). Define he value of he securiy a some convenien erminal ime in he disan fuure as, and le be he public informaion se a ime. Then he symmery assumpion is: /2 /2 0 (4.1) I.e. he quoe midpoin a ime, /2, conains all available public informaion of he fuure value of he securiy,. The informaion inferred from he ime rade ( ) can hen be summarized as he subsequen change in he quoe midpoin: Page 12 AEF Thesis

13 4. Background Par I: Marke microsrucure heory r (4.2) Convenienly, due o he symmery assumpion in (4.1), he informaion impac of is no affeced by he ransacion cos based componen of he spread. This specificaion is characerized by a rade impac ha is fully conemporaneous and can be wrien as,. In realiy, quoe revisions are likely o show a lagged response o rade innovaions for various reasons relaed o he marke microsrucure. Hasbrouck menions hreshold effecs due o price discreeness, invenory conrol effecs, and lagged adjusmen o informaion (HASBROUCK, J., 1991). Threshold effecs are of psychological naure, marke paricipans respond o some price levels differen han oher because of heir numerical value (round numbers or hisorical highs and lows). For example, a sock ha approaches he GBp 1,000 mark for he firs ime is likely o moivae a differen kind of behavior from marke paricipans han he behavior seen when he sock was range bound beween GBp 920 and GBp Invenory conrol effecs are probably less prominen in oday s largely elecronic markes compared o he marke on he NYSE in 1989 ha Hasbrouck was invesigaing. Lagged adjusmen o informaion is likely o remain an issue, bu has possibly diminished since 1989 due o he presence of more marke paricipans, he emergence of compuerized rading algorihms, and more efficien rading sysems in general (HASBROUCK, J., 2007) Vecor auoregression A more flexible srucure ha allows he curren quoe revision o be affeced by pas quoe revisions and rades, can be made using vecor auoregression. A vecor auoregressive model of order, VAR(), is wrien as v αγ v w (4.3) where each γ is a ransiion marix ha expresses he dependence of v on v j. The vecor whie noise process w is assumed o be mulivariae normal wih mean zero and covariance marix 3 GBp is an abbreviaion for one penny, which is 1/100 of a Briish pound serling, he price uni used for socks in Grea Briain. AEF Thesis Page 13

14 4. Background Par I: Marke microsrucure heory w Σ. As an example, a bivariae VAR(1) model, i.e. 2, consiss of he following wo equaions: v α γ v, γ v, w (4.4) v α γ v, γ v, w (4.5) (SHUMWAY, R. H. and Soffer, D. S., 2006). The relaionship beween quoe revisions and rades may be modeled using he srucure in (4.4). We ge a r a r b x b x, (4.6) where, is a disurbance erm. The quoe revision a ime is expressed as a funcion of pas quoe revisions and pas rades. This implies ha here is serial correlaion in he quoe revisions. Since he symmery assumpion in (4.1) is incompaible wih such serial correlaion in he quoe revisions, i is replaced by a weaker assumpion: As st, /2 0 (4.7) For some fuure ime s,, condiional on he informaion se a ime. This allows any deviaion in he quoe midpoin from he efficien price o be ransien. To allow for causaliy running from quoes o rades, rades may be modeled in a similar fashion: x c r c r d x d x, (4.8) The innovaion,,, capures he unanicipaed componen of he rade relaive o an expecaion formed from linear projecion on he rade and quoe revision hisory. Joinly equaions (4.6) and (4.8) comprise a bivariae vecor auoregressive sysem. I is assumed ha he error erms have zero mean and are joinly and serially uncorrelaed: E, E, 0, E,, E,, E,, 0, for. (4.9) The expeced cumulaive quoe revisions hrough sep m in response o he rade innovaion, may be wrien as α, Er,. (4.10) Page 14 AEF Thesis

15 4. Background Par I: Marke microsrucure heory By (3.5) as m increases, α, E 2, 0 2, (4.11) Tha is, he expeced cumulaive quoe revision converges o he revision in he efficien price. For his reason α, can be inerpreed as he informaion revealed by he rade innovaion, and consiues he underlying consruc of Hasbrouck s framework. A way of seeing why i is imporan o include lagged rades and quoe revisions in he model is o consider an alernaive o he vecor auoregressive seup. This could be a simpler model ha assumes he complee absence of any ransien effecs in he price discovery process, including liquidiy effecs. This can be wrien as, where he " " symbol denoes ha he model is incorrecly specified. The regression coefficien, / is likely o overesimae he immediae effec of a rade on he quoe revision due o invenory and liquidiy consideraions. Insead of capuring he lagged adjusmen of he efficien price o he rade innovaion, his oversimplified model will embed all shor erm effecs in he regression coefficien. Hasbrouck (1991) shows ha anoher alernaive specificaion of he model, which does no include lagged versions of he dependen variable, will also be inferior o he full specificaion. An imporan feaure of he VAR model of equaions (4.6) and (4.8) is he implicaion ha all public informaion is capured by he innovaion, and all privae informaion (plus an uncorrelaed liquidiy componen) wih he rade innovaion,. The raionale for he firs implicaion is ha all public informaion is immediaely refleced in he quoes posed by marke makers (oherwise he marke makers would be exposed o arbirage). The second implicaion is due o he fac ha rade innovaions reflec informaion ha was no already conained in he hisory of rades and quoe revisions, and mus herefore be based on exernal (or privae) informaion or liquidiy rading. In oher words, public informaion is no useful in predicing he rade informaion. Leing be he public informaion immediaely subsequen o he ime quoe revision, Ev, 0, for 0. (4.12) AEF Thesis Page 15

16 4. Background Par I: Marke microsrucure heory A simple microsrucure model The VAR seup as i has been presened so far is an economeric represenaion of a simple microsrucure model. The following descripion is also adaped from (HASBROUCK, J., 1991). The model exhibis boh asymmeric informaion and invenory conrol behavior. Le be he efficien sock price, he expeced value of he sock condiional on all public informaion. The dynamics of are given by m zv, v, (4.13) where v, and v, are muually and serially uncorrelaed disurbance erms, and inerpreed in he same way as above. The coefficien reflecs he privae informaion conveyed by he rade innovaion v,. The quoe midpoin price has dynamics q m aq m bx (4.14) where x is again he signed rade a ime and and are adjusmen coefficiens wih 01 and 0. Equaion (5.14) has an invenory conrol inerpreaion. Say ha a ime 0, q m. If x 0, i.e. an agen purchases from he marke maker a he exising quoe, he marke maker will reac by raising his bid o elici sales. If he spread ( ) remains consan, his implies ha q rises. The case of 1 is associaed wih imperfec invenory conrol: compeiion from public limi order raders, for example, forces q o move closer o m wih he passage of ime. The final equaion in he model describes he evoluion of rades: x cq m v, (4.15) where 0 defines a downward sloping demand schedule, i.e. when he quoe midpoin rises above he perceived efficien price (by a magniude greaer han half he spread), marke paricipans reac by selling. From an economeric viewpoin he efficien price m is unobservable, so an empirical model mus involve only x and q. This forms he basis for he VAR framework. Secion 7.3 presens an empirical analysis of hisorical ransacion daa from London Sock Exchange. Page 16 AEF Thesis

17 5. Background Par II: Trading algorihms 5 BACKGROUND PART II: TRADING ALGORITHMS 5.1 Overview Algorihmic rading volume has increased dramaically in he pas several years. The NYSE repors ha in 2000, 22% of all rading was execued via rading algorihms, up from 11.6% in In 2004, ha number had increased o 50.6% (KIM, K., 2007). There are several reasons for he emergence and relaive success of rading algorihms. One reason is ha much rading in he financial markes is done based on discreionary human decision making wihou consisen adherence o specific rules or sysems. Broadly speaking a rading rule is a se of insrucions a rader follows ha depend on he marke price of one or more financial insrumens. A rading rule akes marke prices as inpu and gives orders o buy or sell a a given poin in ime as oupu. Mos raders do use rules and sysems ha hey believe have worked in he pas, bu hey are hard o repea consisenly, and raders will be emped o deviae from heir rules once i appears ha hey may no be working anymore. Such small deviaions in rading paerns may do a criical amoun of damage o an oherwise successful rading sraegy. Trades execued by rading algorihms, on he oher hand, are based on rules ha may be formulaed mahemaically or in compuer code, and are herefore possible o repea wih perfec consisency. We may disinguish beween wo main ypes of rading algorihms: opimal execuion algorihms and speculaive algorihms. Opimal execuion algorihms seek o execue orders in he marke a he lowes possible cos, whereas speculaive algorihms ake marke risks in he hope of earning a profi. As depiced in Figure 5 1, he process of developing and hen implemening rading algorihms sars wih financial modeling in a suiable saisical sofware environmen such as Excel or MATLAB. Based on hisorical daa he algorihm is designed and back esed using hisorical price daa o he poin a which i has aracive ou of sample characerisics. In he case of speculaive algorihms his could be a high risk adjused reurn or low correlaion wih he reurn from invesing in he broader marke. Developers of opimal execuion algorihms will insead focus on achieving fas and cos efficien execuion. The implemenaion hen proceeds by bridging he developmen environmen wih rading infrasrucure ha may execue orders generaed by he algorihm. The algorihm now akes as inpu real ime daa from he financial markes. AEF Thesis Page 17

18 5. Background Par II: Trading algorihms Financial modelling Live implemenaion Hisorical daa Saisical sofware Trading sysem Real ime daa Analysis & design Order execuion Back esing Risk managemen Refinemen P&L Figure 5 1: Trading algorihm developmen and implemenaion. The implemenaion of boh opimal execuion and speculaive algorihms requires ha he algorihm can be programmed in a compuer language, so he rading process can be fully auomaed. If he process of geing orders o he marke a any sage requires human inervenion, he reacion speed and accuracy of he algorihm will fall, and hus many of he appealing characerisics of rading algorihms will be impaired. On he oher hand, he flexibiliy of he algorihm and is abiliy o adap o a changing marke environmen will only be as good as he compuer code i is based on an obvious disadvanage compared o human raders. The speed wih which a rading algorihm reacs o incoming real ime marke daa, processes he daa and reacs by issuing new marke orders or by waiing, will depend upon he compuer infrasrucure and programming language used. High level programming languages such as MATLAB are suiable for he financial modeling of rading algorihms, bu faser languages are ypically used o build rading sysems. C++, C# and Erlang are examples of low level programming languages. The laer is a concurren language ha faciliaes simulaneous execuion of several ineracing compuaional asks, and is paricularly fas (WIKIPEDIA, 2008c). Such characerisics may improve he performance of he rading algorihm and decrease he ime o marke which is a criical facor for boh opimal execuion and speculaive algorihms. The faser Page 18 AEF Thesis

19 5. Background Par II: Trading algorihms he algorihm can reac o incoming real ime marke daa, he more likely i is o obain he liquidiy i seeks, i.e. successfully execue scheduled rades before oher marke paricipans Opimal execuion algorihms The aim of opimal execuion algorihms is o minimize he ransacion coss involved in execuing large orders, which is also known as execuion coss. The lieraure on opimal execuion usually focuses on he equiy markes, as hey are he mos ransparen and mos horoughly researched markes (many of he resuls found can be direcly applied in oher markes). The benchmark of execuion coss is ypically he arrival price, which is he average of he bid and ask price in he marke when execuion of an order begins (eiher a buy or sell order). The difference beween he arrival price and he average price obained for he order is known as he implemenaion shorfall. If he aim is o liquidae a given posiion, he implemenaion shorfall is he difference beween he marke value of he posiion a he beginning of liquidaion, and he amoun of cash obained a he end of liquidaion. The reason ha implemenaion shorfall is differen from zero, is he limiaions imposed by marke liquidiy (or deph) and bid ask spreads. A any given poin in ime here are buyers and sellers available in he marke for a specific number of shares which may be much less han he size of he order o be execued. Once execuion begins, he process of rading he order is likely o move he marke price agains he execuion rader. When he aim is o buy a number of shares, he ask price will go up, and when he aim is o sell he bid will fall. This is known as marke impac, and can be eiher emporary or permanen. An example of emporary marke impac is when he bid ask spread widens is response o a large rade. Typically he bid ask will rever o is previous more narrow level once marke paricipans have reaced o he change in price. Permanen impac is a change in he efficien price ha will no immediaely readjus o is previous level. As described in secion 4.3.2, he efficien price can be changed by he ac of rading alone if such an ac is assumed o convey privae informaion o he marke, or if i is a lagged response o informaion already made public. 4 Time o marke is a significan issue for hedge funds using algorihmic rading sraegies. Some of hose hedge funds are known o have placed heir servers close o he New York Sock Exchange and oher sraegic venues o decrease he laency of daa ransfer (TEITELBAUM, R., 2007). AEF Thesis Page 19

20 5. Background Par II: Trading algorihms Opimal execuion wih a quadraic uiliy funcion The implemenaion shorfall problem may be solved by creaing an objecive funcion for he execuion rader ha akes ino accoun risk aversion. Almgren & Chriss (2001) define such an objecive funcion, and show ha i may be minimized wih respec o a quadraic uiliy funcion or a value a risk (VaR) meaure. They show ha here are wo exremes in he approach o execuing a buy or sell order. One is o execue he enire order immediaely, and he oher is o execue i a evenly spaced inervals hroughou he rading horizon. The rading horizon places an upper limi on he amoun of ime he order execuion may ake. In beween he wo exremes here exiss an efficien fronier in he space of ime dependen liquidaion sraegies. Tha is, for a given level of posiive risk aversion, here is a single rading sraegy ha dominaes all oher possible sraegies. The derivaion of he resuls ha follow may seem dauning, bu ress on a simple quadraic minimizaion problem. 5 Using linear rade impac funcions faciliaes he derivaion of explici soluions o he minimizaion problem. Following Almgren & Chriss (2001), suppose ha we hold a block of securiies ha we wish o liquidae before ime. We divide ino ime inervals /, and define he discree imes 0,, given by, for 0,,. We define a rading rajecory o be a lis,,, where is he number of unis ha we plan o hold a ime. Our iniial holding is, and liquidaion a ime requires 0. We may equivalenly define a sraegy by he rade lis,, where is he number of unis ha we sell beween imes 1 and. Clearly, and are relaed by, 0,,. A rading sraegy can hen be defined as a rule for deermining in erms of informaion available a ime 1. An imporan poin is ha he opimal sraegy is he same a all imes for 0,, if prices are serially uncorrelaed (see Almgren & Chriss (2001) for proof). Now suppose ha he iniial value of our securiy is, so he iniial marke value of our posiion is. The securiy s price evolves according o wo exogenous facors: volailiy and drif, and one 5 The approach is similar o he derivaion of he Capial Asse Pricing Model, alhough he CAPM is a maximizaion problem (CAMPBELL, J. Y. e al., 1997). Page 20 AEF Thesis

21 5. Background Par II: Trading algorihms endogenous facor: marke impac. These characerisics of price movemen may be summarized as a discree arihmeic random walk S S σ τξ τgn /τ, for 1,,. (5.1) Here is he volailiy of he asse, measured in sandard deviaions per year, ξ are iid normal random variaes, and he permanen impac gv is a funcion of he average rae of rading vn /τ. In equaion (5.1) here is no drif erm which we inerpre as he assumpion ha we have no informaion abou he direcion of fuure price movemens. Noe ha ypically a coninuous geomeric random walk of he kind (5.2) is used o model sock prices, where dz denoes a Wiener process (HULL, J. C., 2006). (5.2) may be approximaed in discree ime by Δ Δz Δz Δ ~ 0,1 (5.3) Equaion (5.2) models changes in he sock price whereas (5.1) models he level of he sock price. For he purpose of modeling sock prices inra day, (5.1) is a useful approximaion of (5.3) and leads o racable resuls. Reurning o he model in equaion (5.1), we define emporary marke impac as a change in caused by rading a he average rae, bu we do no include his direcly in he process of he efficien price. Raher i is added separaely o he objecive funcion. I can be expressed as n /τ. We now define he capure of a rajecory o be he full rading revenue upon compleion of all rades, N n N σ τξ τgn /τ x N n hn /τ (5.4) The firs erm on he righ hand side is he iniial marke value of our posiion. The second erm is he effec of price volailiy minus he change in price as a consequence of he permanen impac of rading, and he hird erm is he fall in value caused by he emporary impac of rading. The oal cos of rading can be expressed as N n and is he implemenaion shorfall. AEF Thesis Page 21

22 5. Background Par II: Trading algorihms Prior o rading he implemenaion shorfall is a random variable wih expecaion and variance. We readily obain N τgn /τ x N n hn /τ (5.5) σ N τx (5.6) The objecive funcion may hen be expressed as Ux Ex λvx (5.7) where is a Lagrange muliplier ha may be inerpreed as a risk aversion parameer. The objecive of he analysis is o minimize he objecive funcion in (5.7) for a given risk aversion parameer, hus minimizing he expeced shorfall while aking ino accoun he uncerainy of execuion. Using a linear impac funcion gv γv he permanen impac erm becomes N and he emporary impac erm becomes τgn /τ x 1 2 γx 1 2 γ N n hn /τ sgnn η τ n Where sgn is he sign funcion. Wih linear impac equaion (5.5) becomes 1 N 2 γx n η τ N n (5.8) in which. Almgren & Chriss (2001) show ha we may consruc efficien sraegies by solving he consrained opimizaion problem min x:vxv Ex for a given maximum level of variance V. This corresponds o solving he unconsrained opimizaion problem minex λvx (5.9) where, as already menioned, he risk aversion parameer is a Lagrange muliplier. The global minimum of (5.9), can be found by differeniaing (5.7) wih respec o each yielding Page 22 AEF Thesis

23 5. Background Par II: Trading algorihms 2λσ (5.10) where / is he parial derivaive of (3.15) for 1,,1. Seing (3.18) equal o zero and simplifying we obain 1 τ 2 κ κ 1 2 (5.11) The opimal rajecory is now expressed as a linear difference equaion and may be solved using he hyperbolic sine and cosine funcions giving he expressions X 0,, (5.12) cosh κ T 1 j X 1,, (5.13) 2 where is he associaed rade lis. See Almgren & Chriss (2001) for deails on he derivaion. The exposiion here shows ha i is possible o find a closed form expression for he opimal rading rajecory using linear impac funcions. The resul is depiced in Figure 5 2 below as he efficien fronier of ime dependen liquidaion sraegies. Figure 5 3 shows he opimal rading rajecories for he hree poins A, B and C in Figure 5 2. AEF Thesis Page 23

24 5. Background Par II: Trading algorihms Figure 5 2: The efficien fronier of ime dependen liquidaion sraegies. (ALMGREN, R. and Chriss, N., 2001) Figure 5 3: Three differen rade rajecories for differen values of. (A) 0, (B), (C) 0. (ALMGREN, R. and Chriss, N., 2001) The poin B is he naïve minimum variance sraegy ha corresponds o a risk aversion parameer value of 0. The sraegy disregards he role of he variance of and rades a a consan rae hroughou he rading period. The poin A is an example of an opimal sraegy for a rader wih a posiive risk aversion coefficien. For a relaively small (firs order) increase in expeced loss, a relaively large (second order) reducion in loss variance is obained. Poin C illusraes he opimal sraegy of a rader who likes risk, and herefore has a negaive risk aversion coefficien. I is clear ha all risk averse raders will have convex rading rajecories. Almgren & Chriss (2001) proceeds o show ha equivalen resuls can be obained by minimizing he liquidiy adjused value a risk (L VaR). This is he maximum amoun an execuion rader is willing o lose, wih a given saisical confidence over he rading period. The L VaR objecive funcion may be wrien as Var x Ex λ Vx (5.14) where he confidence inerval is deermined by he number of sandard deviaions λ v from he mean by he inverse cumulaive normal disribuion funcion (we call λ from (5.7) λ o disinguish beween he wo). is he probabiliy wih which he sraegy will no use more han Var p x of is marke value in rading. In oher words, he implemenaion shorfall will no exceed Var p x a fracion of Page 24 AEF Thesis

25 5. Background Par II: Trading algorihms he ime. Because Var x is a complicaed nonlinear funcion of he, we canno obain an explici minimizing soluion such as (5.12) (5.13). Bu once he efficien fronier has been calculaed using (5.12) (5.13) i is easy o find he value of λ corresponding o a given value of λ Exensions o he model in (5.8) can be made by expanding he informaion se of he rader. This can be done by including a drif erm in price process (5.1) or by assuming ha he error erm in (5.1) is serially correlaed Exensions o he opimal execuion model: Drif A drif erm may be added o (5.1) o give The opimaliy condiion (5.11) becomes S S σ τξ αττgn /τ, for 1,,. (5.15) 1 τ 2 κ (5.16) in which he new parameer /2 is he opimal level of securiy holding for a imedependen opimizaion problem. The opimal rading rajecory and corresponding rade lis become sinh 1 sinh (5.17) cosh κ T 1 j X cosh κ j 1 2 cosh κ T 1 j 2 (5.18) 0,,. (5.17) is he sum of wo disinc rajecories: he zero drif soluion in (5.12) plus a correcion which profis by capuring a piece of he predicable drif componen by holding a saic posiion,, in he sock (ALMGREN, R. and Chriss, N., 2001). The difference beween he soluion in (5.12) and he one in (5.17) can be seen in a highly liquid marke when 1. For a risk averse rader, he opimal rajecory in such marke condiions approaches sraegy B in Figure 5 3, as he imporance of he risk aversion parameer diminishes. In he case of (5.17), however, he opimal AEF Thesis Page 25

26 5. Background Par II: Trading algorihms rajecory approaches he opimal saic porfolio holding. Near he end of he rading period his final holding is also sold o saisfy 0 a. Define as he opimal soluion using (5.12) and when using (5.17). The gain from he drifenhanced sraegy can hen be expressed as (5.19) 2 Since anhx / is a posiive decreasing funcion, his quaniy is posiive and bounded above by. Almgren & Chriss (2001) show ha for any realisic values of he parameers, his quaniy is negligible compared o he impac coss incurred in liquidaing an insiuional size porfolio over a shor period of ime Exensions o he opimal execuion model: Serial correlaion When, 0,,, are serially correlaed he opimal sraegy becomes dynamic, ha is, he bes possible sraegy a ime 0, is no longer he same as he opimal sraegy a ime 0. If we denoe he period o period correlaion of by, we may express he maximum per period gain as /4. As in he case of he drif enhanced rajecory, he gains are negligible when using realisic values for he parameers. Only in he case of an exremely liquid sock wih exremely high serial correlaion will he gains be significan for insiuional rading (ALMGREN, R. and Chriss, N., 2001). Bu in realiy hose wo characerisics are muually exclusive Sub conclusion The opimal execuion model of Almgren and Chriss (2001) gives a clear undersanding of he radeoff beween execuion uncerainy and marke impac. I shows ha a uiliy maximizing risk averse rader who has one day o execue an order, will divide he order over he enire day o minimize marke impac and make opimal use of liquidiy pockes during he day. Given he assumpion of no serial correlaion in prices, he opimal sraegy is saic and herefore unchanged hroughou he rading period. Page 26 AEF Thesis

27 5. Background Par II: Trading algorihms The cenral feaure of he model is he creaion of an efficien fronier of ime dependen execuion sraegies. The fronier is depiced in a wo dimensional plane whose axes are he expecaion of oal cos and is variance. Each poin on he fronier corresponds o he opimal sraegy of a rader wih a given level of risk aversion. I is ineresing o noe ha aking ino accoun possible serial correlaion or drif in prices, does no improve he performance of he sraegy o a significan exen Opimal VWAP Trading Anoher approach o he opimal execuion problem is o use he volume weighed average price as an execuion benchmark. The volume weighed average price is defined as where is he raded price of rade, and is he raded volume of rade for 1,2,, being he rades in he VWAP period. An execuion rader who has o execue a large buy order during he period of one rading day can spli up he order ino smaller bis and seek o obain a final VWAP which is close o he marke VWAP. This way he will know ha he combined order was execued a reasonable prices given he volailiy and liquidiy condiions in he marke. This is a more useful benchmark han he simple average price. Define as he sraegy inra day cumulaive volume and as oal final volume. Marke cumulaive and oal volume are denoed by and, respecively. The sraegy s inra day relaive volume can hen be wrien as /, and marke inra day relaive volume as /. Here he analysis is limied o buy orders, and unlike in secion 5.2 above, he variable is now normalized beween 0 and 1, where 0 means ha nohing has been raded, and 1 means ha he enire order has been raded and he operaion is done. An opimal VWAP sraegy is a sraegy ha minimizes he expeced difference beween marke VWAP and raded VWAP. This can be expressed as where is he conrolled rading sraegy and is he marke. Konishi (2002) derives a saic opimal execuion sraegy ha minimizes he norm of min EVWAPM VWAPx (5.20) AEF Thesis Page 27

28 5. Background Par II: Trading algorihms In his framework, prices follow sandard Brownian moion wihou drif of he form, (5.21) Under his assumpion, for a single sock rade, if price volailiy is independen of marke rading volume, he opimal execuion sraegy is deermined only by he expeced marke rading volume disribuion and is independen of expecaions regarding he magniude and ime dependency of price volailiy (KONISHI, H., 2002). The deails of he analysis will no be pursued here. Insead a generalizaion of he approach ha models inra day volume as a Cox process will be presened. Inra day volume as a Cox process McCulloch (2007) shows ha if inra day volume is modeled as a Cox (doubly sochasic) poin process hen inra day relaive volume may be modeled as a doubly sochasic binomial poin process. Based on his idea, as well as he resuls of Konishi (2002), McCulloch and Kazakov (2007) derive an opimal VWAP rading sraegy ha akes price drif ino accoun. Prices are assumed o evolve as a semi maringale of he form, where is price drif, is a maringale and is he iniial price. The minimum VWAP risk rading problem is generalized ino he opimal VWAP rading problem using a mean variance framework as in secion The resuling opimal sraegy is given by x maxe (5.22) where is a Lagrange muliplier, and is inerpreed as he VWAP raders risk aversion coefficien. McCulloch and Kazakov (2007) show ha for all feasible VWAP rading sraegies, here is always residual VWAP risk. The residual risk can be wrien as and is proporional o he price variance of he sock and he variance of he relaive volume process. Empirical esing shows ha relaive volume variance is proporional o he inverse of sock final rade coun raised o he power of min T σ d The relaive volume is he raio of a random sum specified by he doubly sochasic binomial poin σ K. process as he ground process over he non random sum of all rade volumes. I is assumed ha Page 28 AEF Thesis

29 5. Background Par II: Trading algorihms final rade volume is known in he informaion filraion of, which is clearly an unrealisic assumpion. In pracice his value mus be forecased. This assumpion is no made in Konishi (2002), and is a disadvanage of he approach in McCulloch and Kazakov (2007). Implemenaion of he opimal VWAP sraegy is done by dividing each day ino bins. Bins are designed by dividing he VWAP rading period 0, ino ime periods wih he bin boundary imes for bin denoed as and. So 0. Each bin (ime inerval) mus be large enough o allow he rading sysem o reach a specific proporional amoun x of he oal size of he order a he end of each bin. One way of dividing he bins is o use equal volume bins so 1/,. Addiional VWAP risk from using discree volume bins depends on he number of bins as. Figure 5 4 depics he opimal rading rajecory using equal volume bins or opimal bins wih he coninuous soluion superimposed. Using opimal bins is a reasonable approximaion o he coninuous soluion. Figure 5 4: The coninuous soluion is compared o 10 opimal and 10 equal volume bins. The x axis is ime and he y axis is relaive inra day volume on he inerval {0,1}. (MCCULLOCH, J. and Kazakov, V., 2007). AEF Thesis Page 29

30 5. Background Par II: Trading algorihms 5.3 Speculaive algorihms Overview The aim of speculaive algorihms is o profi from changes in prices of raded asses by rading according o specific rules. These rules are ypically based on marke inpus, such as a live feed of marke prices. The algorihm processes his live inpu and creaes buy and sell signals accordingly. In order for he algorihm o be successful, he rade signals mus be imely and reliable. The frequency of rading is arbirary and depends on he naure of he algorihm. I may be a high frequency equiy sraegy ha rades several imes per minue, or a managed fuures sraegy ha ake a sraegic posiion once per monh. Speculaive algorihms may be based on many differen sraegies. Three caegories of ypical sraegies are momenum, relaive value and microsrucure sraegies. Momenum sraegies aemp o idenify and follow rends in marke prices by using saisical measures such as a moving average cross over. They perform well in a marke environmen ha is characerized by srong rends ha are persisen over ime. If marke prices are moving side ways or show wildly oscillaing behavior, momenum sraegies will no perform well. Relaive value sraegies compare he price of one or more securiies o he price of one or more oher securiies and rade hem agains each oher when he price difference (or raio or oher relaive measure) diverges from hisorical norms. The cheap or under valued securiies are bough (long posiion), and he dear or over valued securiies are sold (shor posiion). The sraegy is based on he idea ha if he relaive pricing of he securiies has diverged from he hisorical norm, hey will converge again in he fuure. The obvious risk is ha he circumsances or facors ha dicaed he pricing of he securiies in he pas will no longer do so in he fuure. Therefore, i is possible ha he relaive pricing of he securiies will never reurn o he hisorical level on which he rading sraegy is based, and may coninue o diverge. This is known as a regime change, and is he primary risk of relaive value rading. Microsrucure sraegies aemp o exploi he mechanics of elecronic markes. The archiecure of some markes allows informaion o be exraced and aced upon in a way ha is difficul o achieve wihou he help of an algorihm. A good example of his is he elecronic limi order book. The deph of he limi order book varies over ime and beween socks and exchanges, bu i is ypically Page 30 AEF Thesis

31 5. Background Par II: Trading algorihms repored as he five or en bes bids and offers a a given poin in ime, along wih he quaniies of shares o be bough or sold. This informaion gives an idea abou he supply and demand in he marke a differen prices and how he balance changes over ime. In a fas and efficien marke, such as he marke for shares of large companies, he limi order book is updaed very frequenly, and i is a challenge for mos raders o process his informaion, le alone observe i wih accuracy. A compuer algorihm may in his case be useful as i can rapidly process he informaion in he limi order book and execue rades based on i. A popular order ype in elecronic markes is he iceberg. Iceberg orders reveal only a fracion of oal volume a a ime, replenishing as rades are execued. They are used o minimize rade impac by hiding he inenions of he rader from oher marke paricipans. Anecdoal evidence 6 suggess ha iceberg orders have become increasingly common in he marke for fuures conracs on ineres raes. As a consequence, he limi order book of for example Bund 7 or Euribor 8 conracs reveals less volume a he bid and offer han wha is acually readily available from marke paricipans. In , Bund and Euribor conracs wih mauriy 1 3 monhs ino he fuure ypically had a oal of conracs on he bid and ask. By December 2008 he volume had fallen o a few hundred conracs, largely because of iceberg orders. Open ineres and raded volume in he conracs has fallen as well, bu no nearly a he same rae, see Figure 11 1 in Appendix 11.2 for an illusraion. This makes he analysis of hisorical limi order books more difficul (TRAGSTRUP, L., 2008). An example of a microsrucure rading sraegy is o make use of limi order books ha include soploss orders (TEITELBAUM, R., 2007). A sop loss order is an order ha raders submi o sell below he curren bes bid or buy above he curren bes offer o close a losing long or shor posiion, respecively. An algorihm searches for a paricularly large sop loss sell order close o a psychologically significan price level such as $15.00, when he sock is rading a, for insance, $ The algorihm hen submis a subsanial sell order a, say, $15.01 hoping ha he sop loss order a $15.00 will be hi. If he order a $15.00 is execued, i will place significan downward 6 Auhor s inerview wih Senior Execuion Trader Lars Tragsrup, Danske Markes (a division of Danske Bank) on December 30, Fuures conrac for he delivery of EUR 100,000 noional principal of German governmen bonds wih a mauriy of 7 10 years a a fuure dae. 8 Fuures conrac for a 3 monh deposi wih a noional value of EUR 1,000,000. AEF Thesis Page 31

32 5. Background Par II: Trading algorihms pressure on he sock. The algorihm hen submis a ake profi order a a pre deermined level, for example a $ If $15.01 is reached, bu $15.00 is never reached, he algorihm may submi a sop loss order of is own, a for example $ The same algorihm may be used o rade many differen socks as i only requires ha sop loss orders are made publicly available, which may be he case for all socks raded on a given exchange Pairs rading Pairs rading will be used as an example of a speculaive relaive value rading algorihm. The basic idea behind pairs rading is o rade wo socks ha move ogeher over ime in a sysemaic way. If hey drif apar o a specific pre deermined exen, he cheaper sock is purchased and he more expensive sock is shored (sold). Proceeds from shoring he second sock should finance mos of he iniial purchase. Then he rader wais for he wo prices o converge owards heir hisorical price difference. If and when ha happens he posiion is closed a a profi. The risk is ha he wo socks coninue o diverge furher, and never reurn o heir hisorical price difference. In ha case he rader loses money. The reason ha he wo socks should move ogeher is ha hey may share common facors in he sense of equilibrium asse pricing such as arbirage pricing heory (CAMPBELL, J. Y. e al., 1997), or hey are coinegraed in he sense of Engle and Granger (1987). Asse pricing can be viewed in absolue and relaive erms. The pricing of a sock in absolue erms is done by discouning fuure cash flows a a discoun rae ha reflecs he company s risk. I is nooriously difficul o find an accurae price and here is a wide margin of error due o he uncerainy involved in forecasing he cash flows and deermining an appropriae discoun facor. Relaive pricing is based on he Law of One Price, which Ingersoll (1987) defines as he proposiion ha wo invesmens wih he same payoff in every sae of naure mus have he same curren value. (GATEV, E. e al., 2006). Two invesmens wih similar fuure payoffs should herefore rade for a similar value. This may be rue for wo similar socks. The similariy of value will vary over ime, bu he relaionship will be meanrevering o he exen ha he fundamenals driving he wo socks don change. This is a criical poin in he evaluaion of he risk of pairs rading, as he greaes vulnerabiliy of he sraegy is srucural breaks. Srucural breaks are poins in ime a which a significan change in he Page 32 AEF Thesis

33 5. Background Par II: Trading algorihms fundamenal valuaion of a given securiy akes place. This is ypically due o a company specific even such as a lawsui, a new produc invenion, or a surprising earnings announcemen. If he basis for he hisorical relaionship beween wo socks changes significanly, here is no reason why heir relaive value in he fuure should resemble he pas Coinegraion A heoreical basis for pairs rading is found in coinegraion. When a linear combinaion of wo ime series of socks prices is saionary, while he wo individual ime series are non saionary (which is usually he case), he wo socks are said o be coinegraed. Suppose ha combinaions of ime series of sock prices obey he equaion:, for (5.23) Where is he price of sock a ime, is he regression coefficien of sock on sock, and is a covariance saionary error erm in he sense of Shumway and Soffer (2006). Assuming ha are covariance saionary afer differencing once, he price vecor is inegraed of order 1 wih coinegraing rank (ENGLE, R. F. and Granger, C., 1987). Thus, here exis r linearly independen vecors,, such ha are weakly dependen. In oher words, r linear combinaions of prices will no be driven by he k common non saionary componens. The nonsaionary componens are in his case k socks in he populaion of socks ha are redundan in he process of creaing coinegraing vecors. The coinegraing rank of he individual price vecors is no used explicily in he creaion of pairs rading sraegies in his paper. Ye he concep serves o show ha in a given populaion of socks, here may be several possible linear combinaions of socks wih a coinegraing relaionship. Noe ha his inerpreaion does no imply ha he marke is inefficien, raher i says ha cerain asses are weakly redundan, so ha any deviaion of heir price from a linear combinaion of he prices of oher asses is expeced o be emporary and revering (GATEV, E. e al., 2006). The idea ha a linear combinaion of wo socks may be covariance saionary, may be inerpreed as saying ha a coinegraing vecor may be pariioned in wo pars, such ha he wo corresponding porfolios are priced wihin a covariance saionary error of each oher. Given a large enough AEF Thesis Page 33

34 5. Background Par II: Trading algorihms populaion of socks, his saemen is empirically valid and provides he basis for idenifying pairs of socks suiable for pairs rading (GATEV, E. e al., 2006). A research noe on pairs rading from Kauphing Bank recommends he use of coinegraion ess for pair selecion (BOSTRÖM, D., 2007). The argumen is ha he more significan he coinegraion es is, he more likely i is ha he pairs rade will work. Two ess for coinegraion are menioned, he Engle Granger es and he Durbin Wason es 9. The Engle Granger es is based on he Dickey Fuller uni roo es for saionariy. I sars by regressing one ime series,, on anoher, : α β (5.24) The residuals from his regression are hen used o perform he regression: β ε (5.25) where denoes he difference operaor and ε is a whie noise error erm. The saisic of he β parameer in (5.25) is called he au saisic 10, he criical values of which can be found in Gujarai (2003) and in mos saisical sofware packages. If he absolue value of he saisic obained is larger han is criical au value, he residuals are inegraed of order 0, 0, ha is, hey are saionary, and he wo series and are coinegraed. Engle and Granger (1987) compare various measures of saionariy as a means of esing for coinegraion and conclude ha he Dickey Fuller es is he mos powerful in a saisical sense. They noe ha if he daa is auocorrelaed he augmened Dickey Fuller (adf) es should be used. The adf es assumes ha he observed ime series is driven by a uni roo zero drif process, i.e. an ARIMA(P,1,0) model wih P auoregressive erms. I is based on he regression ζ Δ ζ Δ ζ Δ ε (5.26) for some AR(1) coefficien 1, and a number of lags. If he observed ime series is a random walk wih drif, a consan erm can be included in he above regression. The sock price daa used in secion 8.2 is normalized, however, so he consan erm is no differen from zero in a saisical sense. Therefore he model in equaion (5.26) is correcly specified. 9 See Appendix 11.3 for a definiion of he Durbin Wason es. 10 Noe ha here is no connecion o Kendall s au disribuion. Page 34 AEF Thesis

35 5. Background Par II: Trading algorihms Bosröm (2007) uses coinegraion ess as one amongs a number of saisical ess o evaluae he qualiy of a pairs rade. I.e. he back esing procedure described in he paper requires each pair o have Engle Granger and Durbin Wason es saisics above cerain pre specified levels. A similar mehod will be used in secion 8.2, specifically he MATLAB funcion dfartes, which performs an augmened Dickey Fuller es assuming zero drif in he underlying process Pair selecion and rading signals There are several possible approaches o choosing pairs. One way is he mehod of esing for coinegraion described above. Anoher is he use of a minimum disance crierion beween he normalized prices of he socks in a given populaion. This mehod is used by boh Gaev e al 1997 and Perlin (2007). The firs sep of he mehod is o normalize he price series of each sock in he populaion of socks (he populaion can be chosen arbirarily as he members of a sock index or all socks raded on a given exchange, for example). In his way, socks wih differen price levels may be compared in a consisen way. This may be wrien as where is he price of sock a ime, and are he mean and sandard deviaion of he price series of sock, respecively, and is he normalized price. We denoe he normalized price of he pair of by, and he difference beween he wo as. To find a suiable pair for a sock in a populaion of n oher socks, we find he sock ha minimizes he sum of he squared differences ( norm) for 1,2,,: min. In a given populaion of socks, here may be several socks ha have a similarly low minimum disance value, bu for he purpose of his paper, only he sock wih he lowes value is chosen. Alhough his mehod of finding pairs is sricly defined, i is possible o guess owards he characerisics of possible pairs. Depending on he chosen populaion of socks, wo candidaes for a pairs rade are probably in he same indusry, rade on he same sock exchange, and share oher feaures such as scale of operaion, geography, and marke value. Once pairs have been idenified, he disance beween each arge sock and is pair is evaluaed on a daily basis. For some pre specified consan, when a pairs rade is opened. Depending on wheher is posiive or negaive, he arge sock is sold or bough, respecively, and he opposie posiion is aken in is pair. The uni of he difference in normalized prices can be vaguely AEF Thesis Page 35

36 5. Background Par II: Trading algorihms inerpreed as he number of sandard deviaions beween he wo (because boh prices have been normalized). Therefore a logical level for would be 2, as his would imply a wo sandard deviaion difference o he hisorical normalized spread, which can be considered a low probabiliy even (a approximaely he 5% level if he normalized spread has a normal disribuion). Anoher possibiliy is o define as for some consan, where. denoes sandard deviaion. In his way, he barrier level will be unique for each pair, and will reflec he specific volailiy of. Again, a logical value for is 2. The size of he wo posiions may be deermined in various ways. One opion is o use linear regression o deermine he weigh of he pair sock,, as if i were a hedge: α (5.27) Anoher opion is for boh o have he same iniial marke value. The size of he wo posiions will change over ime as he marke prices of he wo socks change. They may eiher be rebalanced or lef alone, he risk is ha one posiion will become much larger or smaller han he oher, so he overall marke exposure becomes eiher posiive or negaive. This is paricularly clear when a large porfolio of pairs is raded simulaneously. The regression mehod of weighing posiions suggess an alernaive approach o deermining he price disance. The residual from equaion (3.30) is he deviaion of from given he esimaed parameers α i and. I may be used as a rading signal by defining a barrier level in he same way as described above. The mehod can be used on regular prices as well, ha is and, as he consan erm α i akes ino accoun he difference in levels. Secion 8.2 will presen empirical resuls based on he mehod of normalized prices and pair posiions wih equal marke value Mulivariae pairs rading The idea of pairs rading can be exended o rading a porfolio of one more socks agains anoher porfolio of socks (PERLIN, M. S., 2007b). Using he noaion from above for normalized prices his can be expressed as where is some funcion of a marix wih informaion ha explains. This informaion could be any economic variable, bu if i isn he price of a securiy, i canno be raded. If his is he case i is sill possible o creae a sraegy ha rades ourigh Page 36 AEF Thesis

37 5. Background Par II: Trading algorihms agains he signal in, bu he sraegy will no be marke neural. Since his paper focuses on marke neural relaive value sraegies, only radable securiies will ener he marix. The normalized price of arge sock may be compared o he normalized prices of a porfolio of oher socks, yielding he expression 1 1 (5.28) where is a consan, is an error erm, and, 1,.., are he weighs given o each sock. To simplify, 1. (5.29) The socks in he porfolio, which we shall call he M pair porfolio, may be found in various ways. One approach is o compare wih each candidae for individually, by means of a minimum disance crierion or a coinegraion es. Alernaively OLS may be used on various combinaions of socks in, o find which combinaion bes explains (highes R 2 ). Bu since boh and are non saionary (he process of normalizaion doesn affec uni roo non saionariy), he problem of spurious regression arises, i.e. he migh be non saionary. One way o ge around he problem is o use discree or log reurns of and (i.e. no he normalized series and ), where is a whie noise error erm. 1. (5.30) Bu if here is a coinegraing relaionship beween and is M pair porfolio, in (5.29) may be saionary. In ha case, i is possible o use he coefficiens for saisical inference. The M pair porfolio may be creaed in several ways. One obvious approach is o use he socks wih leas ( norm) disance o individually, as was done for a single sock in secion Anoher approach is o use an ieraive procedure ha maximizes he degree of coinegraion beween and. The firs sock in will be chosen using he minimum disance crierion. Each subsequen sock will be chosen o minimize he augmened Dickey Fuller es saisic of he error AEF Thesis Page 37

38 5. Background Par II: Trading algorihms erm in (5.29). The benefi of he mehod is ha i ensures ha and are coinegraed 11. The downside is ha i is difficul o say a priori wheher his will acually improve he algorihm. I is no cerain ha he addiional socks added o conain useful informaion. In order o rade he M pair porfolio, we mus scale he in (5.29) so ha hey sum o one. This is done by dividing each weigh by he sum of he weighs o obain /. The sum o one, so scaling he posiions by hese parameers ensures ha he marke exposure of he M pair porfolio will be equivalen o he arge sock. 12 Perlin (2006b) also suggess using a correlaion weighing scheme ha calculaes he weighs as / for 1,..,. I is difficul o see wha he advanage of his mehod should be, excep ha he weighs will reflec he relaionship beween each and wihou he effec of he oher. An alernaive approach o mulivariae pairs rading is o use sae space mehods o exrac a signal from one or more socks agains which o rade. Sae space models allow he esimaion of unobserved processes based on observaions. This unobserved process, which is called a signal, can be based on one or more socks including he reference sock. Secion 6 will presen he concep of sae space models, and secion 8.3 will use such a model for mulivariae pairs rading. 11 To he exen ha he augmened Dickey Fuller es works. 12 The sraegy will only be marke neural iniially. Price flucuaions from day o day will change he magniudes of he various posiions, bringing he sraegy ou of balance. Unless he posiions are rebalanced on a daily basis, ne marke exposure will be differen from zero. Page 38 AEF Thesis

39 6. Background Par III: Sae space models 6 BACKGROUND PART III: STATE SPACE MODELS 6.1 Inroducion This secion will provide general background on sae space models, and illusrae he properies ha are used in secion 8.3 on mulivariae pairs rading. Sae space models were originally invened as a ool o rack he posiion and rajecory of space craf. Given a se of inpus from various sensors and racking devices, such as velociy and azimuh, i was necessary o esimae he unobserved quaniies of posiion and rajecory in a compuaionally efficien way. Furhermore, hese esimaes had o be coninuously updaed as new observaions came in from he sensors and racking devices. This led o he Kalman filer, a mehod of recursive updaes of a sysem of equaions which is he underlying consruc of sae space models. The benefis of sae space models are heir inheren flexibiliy and scope of applicaion, and heir compuaional efficiency is a major benefi in erms of numerical esimaion of parameers. Examples of applicaions include srucural models of rend and seasonaliy, exponenial and spline smoohing, as well as sochasic volailiy. Sae space models can be considered an alernaive o he ARIMA sysem of analysis of Box and Jenkins (see fx Box e al 1994). ARIMA models require ha he ime series used as inpu are covariance saionary, so i is ypically necessary o derend daa by aking one or more differences. This is no required in he sae space approach where daa characerisics such as rend and seasonaliy may be modeled explicily. This is a fundamenal difference beween he wo mehods. Furhermore i is ineresing o noe ha ARIMA analysis may be expressed and esimaed in sae space form. 6.2 The linear Gaussian sae space model This secion will describe he linear Gaussian sae space model. The descripion is adaped from Durbin & Koopman (2001), bu will also include elemens from Shumway & Soffer (2006) and Welch & Bishop (2006). The general linear Gaussian sae space model can be wrien in he form y Z T R 1 ~ N 0, H ~ N(0, Q ) 1,..., n (6.1) AEF Thesis Page 39

40 6. Background Par III: Sae space models where y is a p x 1 vecor of observaions and is an unobserved m x 1 vecor called he sae vecor. The idea underlying he model is ha he developmen of he sysem over ime is deermined by according o he second equaion of (6.1), bu because canno be observed direcly, an esimae is made of based on he observaions y. The firs equaion of (6.1) is called he observaion equaion, and he second is called he sae equaion. Z is a p x m marix called he observaion marix, and T is he sae evoluion marix wih dimensions m x m. In mos applicaions including he ones in his paper, R is he ideniy marix. The marices Z, T, R, H and Q are eiher assumed known or esimaed, depending on how he model is consruced. Typically, some or all of he elemens of hese marices will depend on elemens of an unknown parameer vecor, which can be esimaed wih an opimizaion algorihm. The error erms and are assumed o be serially independen, and independen of each oher over ime. The iniial sae vecor 1 is assumed o be, N a1 P 1 independenly of,..., 1 n and,..., 1 n, where a 1 and p 1 may be assumed known or esimaed. Noe ha he firs equaion of (6.1) has he srucure of a linear regression model where he coefficien vecor changes over ime. The second equaion represens a firs order vecor auoregressive model, he Markovian naure of which accoun for many of he elegan properies of he sae space model. This general specificaion is a powerful and flexible ool ha makes he analysis of a wide range of problems possible. The main poin is ha a vecor of one or more underlying signals can be esimaed using a vecor of observaions y. I is possible o include addiional known inpus in boh he observaion and sae equaion, bu his is no used in he subsequen analysis and will herefore no be described here. The mulivariae case is a sraigh forward exension where he disurbances are wrien as where and ~ N 0, ~ N 0, are p x p and m x m covariance marices. The disurbances may be independen (diagonal covariance marices) or correlaed insananeously across series. Page 40 AEF Thesis

41 6. Background Par III: Sae space models 6.3 An example of a srucural model Shumway & Soffer (2006) model he quarerly earnings of Johnson & Johnson using a simple srucural model: y T S u where T is he rend componen, S is he seasonal and u is a disurbance erm. The rend is allowed o increase exponenially, ha is T T 1 w1, where 1. The seasonal componen is modeled as S S 1S2 S3 w2, which corresponds o assuming ha he seasonal componen is expeced o sum o zero over a period of four quarers. To express his in sae space form we define T, S, S, S equaion becomes as he sae vecor so he observaion 1 2 y T S S S 1 2 u and he sae equaion T T1 w1 S S 1 w 2 S S 2 0 S S where 11 and The parameers o be esimaed are 11, he noise variance in he observaion equaion, 11 and 22 he model variances corresponding o he rend and seasonal componens, and, he ransiion parameer ha models he growh rae. An iniial guess has o be given for he parameers. The iniial mean of is specified as 0.5,.3,.2,.1 wih diagonal covariance marix 0.01 for i 1,..., 4. Iniial sae covariance is specified as and 22.1, corresponding o relaively low ii AEF Thesis Page 41

42 6. Background Par III: Sae space models uncerainy in he rend compared wih he seasonal. The measuremen error covariance is sared a Growh is abou 3% per year so is sared a Using he expecaion maximizaion algorihm (see secion 6.4.4) he ransiion parameer sabilized a 1.035, which is exponenial growh wih an annual inflaion rae of approximaely 3.5% (see Shumway & Soffer (2006) for furher deails). Noe ha he iniial guess values for he parameers in he model are chosen raher arbirarily, and ha hey may have a subsanial impac on where he esimaion algorihm converges. Because of his, an elemen of rial and error in he esimaion of sae space models is ineviable. 6.4 The Kalman filer The Kalman filer is a recursion ha allows he calculaion of fuure sae esimaes based on he curren esimae and observaion, and an iniial guess of he sae mean and covariance. Based on he given parameers of he sae space model in quesion, he Kalman filer derives opimal esimaes of he unobserved signal(s) ha are modeled as an auoregressive process in he sae equaion. This secion will show wha he Kalman filer recursions look like. The presenaion closely follows Durbin & Koopman (2001) and uses elemens of Shumway & Soffer (2006) as well as Welch & Bishop (2006). Denoe he se of observaions y,..., 1 y by Y, hen he Kalman filer allows he calculaion of and P Var Y a E Y 1 1 y given 1 Y and F Varv given a and P. Define v as he one sep forecas error of 1 1. The recursion equaions are hen given by 13 v y Za K TPZF 1 a Ta K v 1 for 1,..., n F Z PZ H L T K Z P TPTRQR 1 (6.2) Noe here ha a 1 has been obained as a linear funcion of he previous value a and v. K is known as he Kalman gain. The key advanage of he recursions is ha we do no have o inver a (p x 13 See Appendix 11.4 for a derivaion. Page 42 AEF Thesis

43 6. Background Par III: Sae space models p) marix o fi he model each ime he h observaion comes in for 1,..., n; we only have o inver he (p x p) marix F and p is usually much smaller han n. The updae of he sae esimae ha akes place in a 1 Ta Kv can be seen as wo discree seps; firs a projecion of he curren sae ino he fuure ( Ta ), and hen a correcion ha akes ino accoun he new (or incoming) observaion ( Kv). Time updae projecion Observaion updae correcion Figure 6 1: The Kalman filer recursion loop. In a similar fashion, he error covariance P is firs projeced ino he fuure, and hen correced. Using L T KZ, he error covariance recursion P 1 TPT RQR can be wrien as P TPTRQRTP KZ. This can also be seen as wo discree seps; firs a projecion of he 1 curren sae error covariance ino he fuure ( TPT RQR ), and hen a correcion ha akes ino accoun he new observaion ( TP KZ ). The process is summarized in Figure 11 2 in Appendix Dimensions of sae space model (3.12) Vecor y p x 1 Z m x 1 T p x 1 H r x 1 R Marix Q a m x 1 1 P 1 p x m m x m p x p m x r r x r m x m Dimensions of Kalman filer Vecor v p x 1 F Marix K L M a m x 1 P a m x 1 P p x p m x p m x m m x p m x m m x m Table 6 1: The marix dimensions of he sae space model and he Kalman filer. (DURBIN, J. and Koopman, S., 2001) AEF Thesis Page 43

44 6. Background Par III: Sae space models To compue he conemporaneous sae vecor esimae E Y and is associaed error variance marix, which we denoe by a and P respecively, he conemporaneous filering equaions can be used. They are a reformulaion of he equaions in (6.2). v y Z a a a M F v a 1 Ta 1 for 1,..., n. F Z PZ H M PZ P P M F M 1 P TP TRQR 1 (6.3) The Kalman smooher and Disurbance smoohing As shown above, he Kalman filer is a forward recursion ha derives an esimae for given all observaions up o ime. I is also possible o esimae given he enire series 1 y,..., y in a n backwards recursion. This is called he Kalman smooher and, as he name suggess, provides a more accurae fi o he observed daa. While he Kalman smooher is no used direcly in he analysis in his paper, a paricular aspec of i called disurbance smoohing is used for parameer esimaion. The backward recursions for sae smoohing are given by r ZF v Lr ˆ a Pr for n,...,1, N ZF Z LN L 1 1 V P PN P 1 (6.4) wih r n 0, Nn 0, and N, V m x m marices. We wrie he smoohed esimaes of he disurbance vecors and as ˆ E y and ˆ E y. I can be shown ha ˆ Hu where he smoohing error u is defined as u F v Kr. The smoohed esimae of he sae disurbance, ˆ, is defined as ˆ QRr.The 1 recursions for he smoohed disurbances and heir variance can be summed up as Page 44 AEF Thesis

45 6. Background Par III: Sae space models 1 ˆ H(F v Kr) ˆ QRr for n,...,1, 1 Var y H H F K N K H Var y Q Q RN R Q (6.5) These definiions will be used for parameer esimaion in he nex secion Maximum likelihood esimaion Following Durbin & Koopman (2001), he likelihood funcion of he Gaussian linear sae space model can be wrien as np 1 log L y log 2 log F v F v 2 2 n 1 1 I is known as he predicion error decomposiion. The quaniies v and F oupu from he Kalman filer so log L y can be calculaed simulaneously wih he Kalman filer speeding up he calculaion process. This is convenien for numerical esimaion of he unknown parameers, as many funcion evaluaions are (ypically) made before log L y is minimized. Define as a vecor of one or more elemens of he sysem marices Z, T, H, R and Q ha have o be esimaed using maximum likelihood. The dependence of he log likelihood on can be wrien as log Ly. In his paper a Quasi Newon mehod of solving equaions is used o minimize log L y using he BFGS algorihm in MATLAB by means of he funcion fminunc. Newon s mehod solves he equaion using he firs order Taylor series for some rial value, where log L y (6.6) AEF Thesis Page 45

46 6. Background Par III: Sae space models wih 2 2 ' log L y By equaing (6.6) o zero we obain a revised value from he expression This process is repeaed unil i converges or unil a swich is made o anoher opimizaion mehod. The gradien 1 deermines he direcion of he sep aken o he opimum and he Hessian modifies he size of he sep. I is possible o oversae he size he maximum in he direcion deermined by he vecor 1, 2 1 and herefore i is common pracice o include a line search along he gradien vecor wihin he opimizaion process. We obain he algorihm s, where various mehods are available o find he opimum value for s, which is usually found o be beween 0 and 1. The BFGS algorihm calculaes an approximaion of he Hessian 2 and updaes i a each new value of using he recursion * * * g g 1 1 gg 2 2 s g g g where g is he difference beween he gradien 1 and he gradien for a rial value of prior o * and 1 g 2 g. The BFGS mehod ensures ha he approximae Hessian marix remains negaive definie. The score vecor L y 1 log / specifies he direcion in he parameer space along which a search should be made. The score vecor akes he form Page 46 AEF Thesis

47 6. Background Par III: Sae space models where ˆ, ˆ 1 n log L y 1 1 log H log 1 r ˆˆ Q Var yh 2 1, Var y and Var y 1 ˆ ˆ yq 1 r Var are obained for as in secion (6.4.1). ^ denoes ha he parameer has been replaced by is maximum likelihood esimae Sandard error of maximum likelihood esimaes The disribuion of ˆ for large n is approximaely ˆ ~ N,, where 2 log L Thus he sandard error of maximum likelihood esimaes are given as he square roo of he diagonal of, diag, where is he inverse of he negaive Hessian. The Hessian marix is ypically calculaed using finie difference mehods, and may resul in negaive elemens in he marix due o approximaion and rounding errors. In ha case i may be concluded ha he model is missspecified, or he absolue value of diag may be aken before he square roo, o obain approximae sandard errors (DURBIN, J. and Koopman, S., 2001) The expecaion maximizaion algorihm An alernaive mehod of parameer esimaion is he expecaion maximizaion (EM) algorihm. The EM algorihm iniially converges faser han numerical opimizaion wih BFGS, bu slower near he maximum (or minimum) (SHUMWAY, R. H. and Soffer, D. S., 2006). Therefore i is aracive o iniially use EM and hen swich o BFGS once he speed of convergence begins o fall. The analysis in his paper, however, only uses BFGS as complicaions arise wih he EM algorihm when resricing which parameers in he sysem marices should be changed, and which parameers should remain a a pre specified value. The EM algorihm changes all elemens in he error covariance marices R, including he elemens off he diagonals, which is no always desirable. For example, H and AEF Thesis Page 47

48 6. Background Par III: Sae space models independence beween individual sae and observaion vecors canno be forced, by seing values off he diagonals o zero. Page 48 AEF Thesis

49 7. Analysis Par I: High frequency equiy daa 7 ANALYSIS PART I: HIGH FREQUENCY EQUITY DATA This firs par of he analysis invesigaes equiy ick daa from London Sock Exchange using vecor auoregression as described in secion 4.3. The aim is o exrac informaion abou he dynamics of rading from ransacion daa. 7.1 Daa descripion The daa se used in he empirical analysis is ick daa from London Sock Exchange obained from Reuers Daascope. The daa se includes all rade and quoe evens for Anglo American PLC during March of Every even is coupled wih a ime samp and a qualifier (idenificaion code). Error evens and oher abnormal evens are also included in he daa se and are also idenified by qualifiers. The daa se conains numerous elecronically generaed evens such as an exchange calculaed inra day volume weighed average price (VWAP). Appendix 11.1 provides deails of he various even ypes Descripion of a represenaive sock: Anglo American PLC Figure 7 1 is an example of ick daa observaions of Anglo American PLC from London Sock Exchange. The daa are here shown as comma separaed values, and he daa values in rows 60, 75 and 76 are explained in Table 7 1. AEF Thesis Page 49

50 7. Analysis Par I: High frequency equiy daa Figure 7 1: Represenaive ick daa for Anglo American PLC on March 3, 2008 Source: Reuers Daascope Descripion Reuers ag Row 60 value Row 75 value Row 76 value Sock quoe #RIC AAL.L AAL.L AAL.L Dae Dae[G] 03 MAR MAR MAR 2008 Time Time[G] 08:00: :00: :00: GMT offse GMT Offse Trade / quoe Type Quoe Trade Trade Traded price Price 3161 Traded volume Volume 502 VWAP VWAP Quoed bid Bid Price 3159 Quoed bid size Bid Size Quoed ask Ask Price 3170 Quoed ask size Ask Size Qualifiers Qualifiers A[LSE] OB VWAP[GEN] Table 7 1: Column labels for Anglo American PLC ick daa. A ypical quoe, rade and VWAP even is included. Source: Reuers Daascope Page 50 AEF Thesis

51 7. Analysis Par I: High frequency equiy daa As an example of a limi order book, he able below shows he 10 bes bid and ask quoes a he ime corresponding o row 74 in Figure 7 1: Anglo American PLC Limi order book Dae March 3, 2008 Time 08:00:40.42 Bid Volume Ask Volume Dealing wih ransacion ime There are a leas four possible approaches o dealing wih ransacion ime: 1. Hasbrouck s mehod (HASBROUCK, J., 1991): Give each rade or quoe even a new index, bu wih wo excepions: a. The appearance of a quoe revision wihin 5 seconds prior o a ransacion is a considered a reporing anomaly and he quoe is resequenced afer he rade b. Quoe revisions occurring wihin 15 seconds afer a rade are given he same subscrip as he rade 2. Simple mehod: Give each even in he ick daa a new index. This implies rades and quoes will never be simulaneous, and herefore neiher will rade evens and quoe revisions. 3. Alernaive: Give each even in he ick daa a new index. Bu rades and quoes wih he same ime samp (a he millisecond level) are lumped ogeher wih he same subscrip. Trades wih he same ime samp and price are also combined. 4. Mehod (2) and (3) can be furher modified by removing all quoe evens wih no revision. This may reduce he concern of sale quoes, and can be used o gauge he imporance of sale quoes by comparing resuls from a VAR analysis ha uses all quoe evens, and one ha AEF Thesis Page 51

52 7. Analysis Par I: High frequency equiy daa doesn. This modificaion will obviously creae larger clock ime gaps beween he ransacion ime indices. The daa used in Hasbrouck (1991) is from 1989, and is almos cerainly less frequen han he FTSE 100 daa from used in his paper. I is unforunaely no menioned in Hasbrouck s paper how many observaions here were in he daa se, namely he 62 rading days of he firs quarer of For he empirical analysis in his paper, modificaion (1.a) above seems dubious. If he quoe revision 5 seconds before a given rade is, for insance, righ afer he previous rade, why should i hen be sequenced afer he second rade? Modificaion (1.b) seems biased owards emphasizing he causaliy running from rades o quoe revisions. Rewriing equaions (4.6) and (4.8) using rade indicaors insead of signed volume x and include consan erms we obain he following VAR(p) sysem 0 0, (7.1) , (7.2) 1 One paricular problem ha may arise due o modificaion (a) is ha i may bias he magniude of he coefficien of in model (7.1). Hasbrouck considers he posiive coefficien of in he regression of on lagged and o be paricularly imporan, as i is he average quoe revision immediaely subsequen o a rade (wihin 15 seconds). The coefficiens of he subsequen lags (>0) are hen he effec of rades on quoe revisions beyond 15 seconds, each by an incremen of ransacion ime 1. So he acual clock ime beween he ransacion ime incremens changes considerably for lags greaer han 0. This could be a source of bias in he analysis owards emphasizing he immediae impac of rades on quoe revisions. Anoher issue is how o deal wih muliple rades wih he same price and possibly he same ime samp. The rades are given each heir own index because i canno safely be assumed ha he rades are made beween he same wo counerparies. I is possible ha one of he wo counerparies is he same for he simulaneous rades (same ime samp), bu no necessarily boh counerparies. Therefore each rade even mus remain separae, as i conains informaion abou 1 Page 52 AEF Thesis

53 7. Analysis Par I: High frequency equiy daa he behavior of agens in he marke ha will be los if he rades are aggregaed and given he same ransacion ime index. Each separae even can be seen as a unique decision by a marke paricipan o iniiae a rade or pos a quoe and herefore conains informaion. I should be noed ha he FTSE 100 daa conains many repeiive price quoes and herefore many quoe revisions will be zero. This is no a major concern, however, because he mehod of vecor auoregression ress on he assumpion of covariance saionariy which is no impaired by a ime series dominaed by zeros (or any oher numerical value for ha maer) (HASBROUCK, J., 2007). 7.3 Empirical analysis of ick daa To es he model specificaion of secion 4.3, he VAR sysem (4.6) (4.8) will be esimaed using five lags. For quoe revision r and signed rade indicaor we wrie: , (7.3) , (7.4) The inpu daa is all rade and quoe evens of Anglo American PLC in March AEF Thesis Page 53

54 7. Analysis Par I: High frequency equiy daa Daa inpu Company Anglo American PLC Marke value (bn ) 38.0 Avg rade volume Sock quoe AAL.LN Median rade volume 383 Sar dae 03 Mar 08 Low price 2, End dae 31 Mar 08 High price 3, Trading days 19 Average price 3, Observaions 1,253,636 VWAP 3, Evens removed 195,692 Min spread 1 Evens used 1,057,944 Max spread 72 Trade evens 234,832 Avg spread 2.6 Quoe evens 823,112 Median spread 2 Nonevens 571,939 Min ime incremen 0 Toal volume 149,037,283 Max ime incremen Avg daily volume 7,844,068 Mean ime incremen Min rade volume 1 Median ime incremen Max rade volume 2,053, Volume in shares, prices in GBp and ime incremens in seconds The resuls from he esimaion are he following: Lag Coeff. T sa Lag Coeff. T sa a c a c a c a c a c a c b d b d b d b d b d Table 7 2: Resuls from vecor auoregression using quoe revisions and signed rade indicaors from Anglo American PLC in March Page 54 AEF Thesis

55 7. Analysis Par I: High frequency equiy daa Coefficien sums p 5 10 a b c d Table 7 3 All he esimaed coefficiens are saisically significan a he 5% level. The value of for eq. (7.3) is low, bu ypical for analysis using high frequency daa (HASBROUCK, J., 1991). The negaive signs of he coefficiens are evidence of reversal in quoe revisions. The posiive signs of he and coefficiens show ha prices rise as a response o buying, bu rades do no immediaely reverse direcion. Signed rades acually show posiive auocorrelaion, which is inconsisen wih microsrucure models of invenory conrol. If invenory conrol consideraions were dominan, he coefficiens would be negaive, as marke makers would raise prices in response o purchases, and lower prices in response o sales in order o replenish invenories (O'HARA, M., 1997). This would have an effec of rade reversal. A possible explanaion for his inconsisency wih microsrucure heory is ha he radiional role of marke makers has been replaced by elecronic limi order books. Invenory consideraions have been replaced by shor erm rade momenum caused by algorihms ha divide rades ino several orders, or use iceberg orders o he same effec. The calculaions were repeaed using 10 lags and he resuls are shown Table 7 3. rises only slighly for each equaion, bu he sums of he coefficiens change significanly. Table 11 1 in Appendix 11.5 shows deailed resuls of he esimaed coefficiens. All and coefficiens are significan a he 5% level while he and coefficiens are significan o lag 7 and 6 respecively. To undersand he shor erm dynamics of he VAR model, we calculae α, from eq. (4 11). I is he response of o a ime 0 rade innovaion,,, of 1 uni (while, 0 ) and can be wrien as α, r. Figure 7 2 shows ha he adjusmen is rapid, afer only 5 periods he majoriy of he response has aken place. In response o a rade, he quoe midpoin has been revised higher by GBp 0.46 on average afer weny periods of ransacion ime. I is possible o inerpre he response funcion α, for eq. (7.4) as he likelihood ha a rade even will be followed by anoher rade even afer periods. If α, 0 he sign of he rade is persisen, whereas α, 0 indicaes rade reversal. In his case α, , which implies ha he AEF Thesis Page 55

56 7. Analysis Par I: High frequency equiy daa likelihood ha a rade even will be followed by anoher rade wih he same sign wihin 20 periods is 86.34% (TSAY, R. S., 2005). Noe ha he calculaion of α, involves he wo way dynamics beween and, he iniial rade innovaion causes a quoe revision which has an impac on fuure rades ha in urn impac fuure quoes and so on. Cumulaive response r_ x_ Transacion ime () Figure 7 2: The response of and hrough period 20 o a uni rade innovaion a ime. I is possible ha he response funcion α, is affeced more by larger rades han by small. To see if his is he case we esimae model (7.7) (7.8) again using he signed volume in place of. Cumulaive rading aciviy 40, , , , x_ r_ (R.A.) Cumulaive quoe revision Transacion ime () Figure 7 3: The response of and hrough period 20 o a purchase of 2 mln shares a ime. The above figure shows he impac of a ime 0 purchase of 2 mln shares. A rade of his magniude is on average followed by addiional purchases of 33,900 shares and an upwards quoe revision of GBp Page 56 AEF Thesis

57 7. Analysis Par I: High frequency equiy daa Subconclusion The resuls from he vecor auoregression analysis confirm hose of Hasbrouck (1991), even hough Hasbrouck s analysis was done wih ransacion daa from 1989, 19 years older han he daa used for his analysis. Noing he limied scope of he daa sample here used, we may conclude ha alhough markes have become more elecronically driven over he pas few decades, he basic dynamics beween prices and rades remain largely unchanged. AEF Thesis Page 57

58 8. Analysis Par II: Speculaive algorihms 8 ANALYSIS PART II: SPECULATIVE ALGORITHMS Of he hree ypes of speculaive algorihms described in secion 5.3, he firs wo will be illusraed using empirical daa, namely a momenum sraegy and a relaive value sraegy. The goal is o illusrae how speculaive algorihms can be consruced, and o deermine wheher hey can make saisically significan posiive profis ou of sample. The process of designing speculaive algorihms is arbirary; here are many ways of approaching he problem, and many soluions. The same sraegy may be implemened in several ways, of course wih sligh variaions in resuls. Differen variaions of he same sraegy will perform well under cerain marke condiions, and worse in ohers. The opimal sraegy for a given securiy or marke is herefore likely o evolve over ime along wih changing marke condiions. The founder of a prominen algorihmic rading hedge fund explains he process of combining old sraegies wih new ones in a coninuous process: Wha you need o do is pile hem up. You need o build a sysem ha is layered and layered. And wih each new idea, you have o deermine, Is his really new, or is his somehow embedded in wha we've done already? So you use saisical ess o deermine ha, yes, a new discovery is really a new discovery. Okay, now how does i fi in? Wha's he righ weighing o pu in? And finally you make an improvemen. Then you layer in anoher one. And anoher one (LUX, H., 2000). 8.1 Momenum sraegies using moving averages As described in secion 5.3, momenum sraegies are designed o exploi he rending behavior of markes. To undersand he concep of a momenum sraegy we firs define a rend. An arihmeic random walk wih a deerminisic rend, or drif, may be wrien as µx w µ w (8.1) where is he drif erm and w is whie noise. 0 1,, w ~0,1 Page 58 AEF Thesis

59 8. Analysis Par II: Speculaive algorihms x =x -1 + x =+x x Figure 8 1: A random walk (RW) and random walk wih drif. For he RW wih drif,.,,.,, The sandard model of sock prices menioned in secion is based on geomeric Brownian moion as Time (8.2) where dz denoes a Wiener process (HULL, J. C., 2006). The rend is now muliplied by he sock price, bu he effec of having a posiive or negaive rend is he same as in Figure 8 1. In (8.2) he rend is consan, bu we may consider a model in which is a funcion of ime by wriing This model may be approximaed in discree ime by (8.3) Δ Δz Δz Δ ~ 0,1 (8.4) I is impossible o predic when he drif parameer changes, bu i is possible o observe wheher i is posiive, negaive or zero over a given period of ime. Therefore, if we assume ha here is some persisence in he sign of, ha is, is auocorrelaed, we may posiion ourselves o ake advanage of fuure posiive or negaive drif in he price. A possible specificaion for is he AR(p) process AEF Thesis Page 59

60 8. Analysis Par II: Speculaive algorihms (8.5) 1 where is a whie noise error process. One way of quanifying he exisence and direcion of a rend is o use exponenially weighed moving averages (EMAs) 14. The exponenial moving average x can be wrien as he infinie sum x λ1 λ x w (8.6) where x is he underlying ime series and w denoes a whie noise error process. Depending on he smoohing parameer, less or more weigh is given o new observaions. may be defined as 2/ 1 where he consan dicaes he responsiveness of he EMA o new observaions. Lower values of give a more responsive EMA by increasing. This can be seen more clearly in he recursive represenaion 15 given by x 1 λx λx x 0, 0λ 1 (8.7) (SHUMWAY, R. H. and Soffer, D. S., 2006) 16. Using wo EMAs wih differen values of, he more responsive called he leading EMA and he less responsive called he lagging EMA, a simple rading rule buys he securiy when he leading EMA is higher han he lagging EMA and vice versa. Figure 8 2 shows wo EMAs wih 10,25 superimposed on daily observaions of he FTSE 100 index wih he rading posiion on any given day shown below. The sraegy is eiher long one uni of he index, or shor one uni, i.e. he sraegy is exposed o changes in marke prices a all imes. The values of were chosen arbirarily and he resuling profi and loss of he rading rule for a more exended period is shown in Figure 8 3. Trading coss are ignored and i is assumed ha i is possible o rade he index a he closing price of each day They are someimes abbreviaed EWMA. 15 See Appendix 11.6 for a derivaion. 16 Noe ha he EMA equaions in Shumway & Soffer (2006), (3.132) and (3.133) have here been modified o correspond o more common versions by subsiuing 1 for and updaing he EMA a ime, x, using he conemporary new observaion a ime, x. 17 This could be done using exchange raded funds (ETFs) or index fuures. In his case he acual index values are used for illusraive purposes. Page 60 AEF Thesis

61 8. Analysis Par II: Speculaive algorihms Price EMA N=10, M=25 FTSE 100 EMA 10 EMA Day Posiion Day Figure 8 2: EMA rading rule applied o he FTSE 100 Index from January 19, 2007 o March 28, P&L, N=10, M=25 Cumulaive profi Day Figure 8 3: The cumulaive profi and loss of he EMA sraegy in Figure 8 2 from May 2, 2000 o Augus 1, AEF Thesis Page 61

62 8. Analysis Par II: Speculaive algorihms The reurn from he rading rule over his period is 16.68% ( 2.19% annualized) wih a Sharpe raio 18 of To find which combinaion of leading and lagging EMAs provides he highes Sharpe raio over he evaluaion period, a back es was run for a large number of combinaions of and. Sharpe raio hea map N M Figure 8 4: FTSE 100 back es: Sharpe raio hea map and surface for EMA sraegy wih varying values of and. Source: Auhor s calculaions, (KASSAM, A., 2008). Figure 8 4 shows a hea map and surface plo of he Sharpe raios obained wih various combinaions of and. Clearly large values of and perform bes, and he opimal soluion is 61,98 wih a Sharpe raio of and a cumulaive reurn of 82.43% or 7.55% annualized. The hea map is upper riangular because mus always be smaller han. Figure 11 3 in Appendix 11.7 shows a graph of he FTSE 100 index wih he opimal EMAs and he cumulaive reurn superimposed. So far he analysis has been based on daily observaions of he FTSE 100 index. The index has rended srongly during his period, and his creaes a bias owards momenum sraegies wih large values of. Oher asses show a differen kind of rending behavior, or perhaps even mean revering behavior. This is ofen he case for ineres raes such as he price of he German 10 year governmen bond or Bund. Applying he same procedure as above o daily observaions of Bund fuures gives opimal 18 The Sharpe raio is here defined as 252/ where is he average daily reurn and is he sandard deviaion of he daily reurns. I.e. he risk free rae is no subraced from. This is also known as he informaion raio. Page 62 AEF Thesis

63 8. Analysis Par II: Speculaive algorihms values of 3,23 which is significanly less han for he FTSE 100 index. The cumulaive reurn is in his case 26.51% (7.05% annualized) wih a Sharpe raio of Figure 8 5 shows a hea map and surface plo of he Sharpe raios obained wih various combinaions of and for Bund fuures prices sampled daily. The much higher Sharpe raio shows ha he Bund daa is beer suied o EMAmomenum sraegies han he FTSE 100 index when sampled daily. Sharpe raio hea map 20 1 N M Figure 8 5: Bund back es: Sharpe raio hea map and surface for EMA sraegy wih varying values of and. Source: Auhor s calculaions, (KASSAM, A., 2008) EMA momenum sraegy wih differen sampling frequencies I is ineresing o consider he effec of differen sampling frequencies on he performance of he EMA momenum sraegy. As described in secion 4.3.1, he mos deailed level of marke informaion, namely ick daa, is usually sampled in bins o show price movemens in fixed ime inervals. The securiy markes ha generae he ick daa rend over ime, bu he rends may go in differen direcions on differen ime scales. A a given poin in ime he Bund may be rending upward on a monhly basis, downward on a daily basis, upward on a 15 minue basis, and so on. Like he concep of collecing rade daa in bins, he concep of a rend can be seen as a scaling phenomenon i can be observed across he enire range of sampling frequencies. AEF Thesis Page 63

64 8. Analysis Par II: Speculaive algorihms Therefore, a sraegy ha is designed o exploi rending behavior, such as he EMA momenum sraegy, should be calibraed o find he mos profiable sampling inerval o operae in. This can be done by esing he sraegy for differen values as well as differen sampling frequencies. In his case i is done using Bund fuures daa wih sampling frequencies ranging from 1 minue o 11 hours (a full rading day). The Sharpe raios for various combinaions of he hree parameers, and sampling frequency is depiced in Figure 8 6. The daa sample spans from Sepember 15, 2002 o March 21, Figure 8 6: Iso surface 19 of Sharpe raios for various sampling frequencies. Legend for Sharpe raios: Blue { }; yellow { }; red { }. Source: Auhor s calculaions, (KASSAM, A., 2008). The performance of he EMA momenum sraegy shows grea variey across he range of sampling frequencies, and many combinaions of he parameers yields Sharpe raios above 1.4 (shown in red in Figure 8 6). Indeed for every sampling frequency here is a combinaion of values for he lead and lag EMAs ha yields a Sharpe raio above 1.4. The highes Sharpe raio obained in he sample 19 An iso surface is a 3D surface represenaion of poins wih equal values in a 3D daa disribuion. I is he 3D equivalen of a conour line. Page 64 AEF Thesis

65 8. Analysis Par II: Speculaive algorihms was 2.05 and resuled from, 8,12 wih a sampling frequency of 5 minues. The corresponding cumulaive reurn was 29.46% or % on an annualized basis, also he highes in he sample. This is a remarkably high number for a sraegy wih annual volailiy of 4.93%. Bu due o he high sampling frequency, he sraegy rades very frequenly, namely 1,159 imes per year on average. Assuming ha ransacions coss are 0.03%, his is equivalen o a loss of 34.77% per year, and wipes ou any posiive gains from rading. This shows ha while here may be aracive algorihmic rading opporuniies a high sampling frequencies, ransacion coss quickly become a big issue. Figure 5 6 shows anoher iso surface of Sharpe raios now including ransacion coss of 0.03% per rade. Figure 8 7: Iso surface of Sharpe raios for various sampling frequencies including ransacion coss. Legend for Sharpe raios: Blue { }; yellow { }; red { }. Source: Auhor s calculaions, (KASSAM, A., 2008). Like in Figure 8 6 only posiive Sharpe raios are included, and now here are significanly less daa poins. The larges concenraion of high Sharpe raios is found a low sampling frequencies coupled wih relaively high values. Oher high Sharpe raios are seen a very high sampling frequencies for AEF Thesis Page 65

66 8. Analysis Par II: Speculaive algorihms low values of and. The highes Sharpe raio is and resuled from, 10,72 wih a sampling frequency of 120 minues. This sraegy also has he highes cumulaive reurn which is 15.66% or 4.31% annualized wih volailiy of 4.88% per annum. I has very aracive properies, paricularly he rade off beween risk and reurn, and is abiliy o make money in rising and falling markes. And since i is based on fuures prices, i is easy o leverage (a ypical margin requiremen for Bund fuures is 2 5% of he noional amoun of one conrac). The invesigaion here has been done in sample, in he sense ha no raining period was used o deermine which combinaion of he parameers, and o use ou of sample. Ye, he sraegy is remarkably sable over ime in differen marke environmens for a large number of combinaions of he parameers. A more horough back esing of he EMA momenum sraegy would require ou ofsample esing, however, and include a more realisic reamen of rade execuion and ransacion coss. Trade execuion can be made more realisic by using hisorical bid/ask prices (raher han he close ask as in his analysis), as hose are he bes esimaes of prices ha can acually be raded a a given poin in ime. Marke deph and varying liquidiy condiions over ime should also be aken ino accoun. The imporance of his will depend on he desired amoun of marke exposure aken wih he sraegy. As an example, he deph of he marke in Bund fuures places limis on how many conracs can be sold or purchased a any given ime wihou moving he bid/ask spread. Ye relaive o oher markes, such as he markes for individual socks, he Bund fuures marke is highly liquid, and will herefore be suiable for he EMA momenum sraegy. Bu even in liquid markes, he EMAmomenum algorihm should be combined wih an opimal execuion algorihm ha seps in once a rading signal has been creaed o minimize he implemenaion shorfall. 8.2 Univariae pairs rading This secion will presen empirical resuls from a pairs rading back es rouine. In order o resemble an insiuional seing, he rading algorihm will search for pairs in a large populaion of socks and rade he pairs simulaneously as a porfolio. The resuls will show wheher i has been possible o earn posiive excess reurns from pairs rading over an exended period of ime. Page 66 AEF Thesis

67 8. Analysis Par II: Speculaive algorihms The empirical analysis uses 75 socks in he FTSE 100 index raded on London Sock Exchange as he populaion in which o search for pairs (he remaining 25 were no acively raded during he enire daa period). The daa se is 2085 daily observaions of closing prices from May 2, 2000 o Augus 1, As explained in secion 5.3.4, rading signals are creaed when he difference beween he normalized price of a sock in he daa se and is pair exceeds a cerain pre specified barrier level b. The sraegy uses a rolling window in which o rain he algorihm; ha is o find a pair for each sock based on he minimum disance crieria. This rolling window may be for insance 252 rading days which is equivalen o a calendar year. For each rading period saring on he final day in he iniial raining window, he difference in normalized prices is calculaed for each pair. If he difference exceeds he barrier level b, hen a rading signal is creaed for he firs rading period. If d is posiive, he sock being considered is rading more expensive han is pair, so a shor posiion is aken in his paricular sock and a long posiion is aken in is pair. If d is negaive, he opposie happens, so he sock being considered is bough (long posiion) and is pair is sold (shor posiion). The pairs are reevaluaed a fixed inervals during he daa period. The inerval may be anyhing from a few weeks o a year; a shor inerval will keep he pairs and posiions up o dae, bu will incur high ransacion coss, while a long inerval risks having oudaed posiions in a changing marke environmen. Several differen lenghs of raining windows (TW) and evaluaion periods (EP) will be aemped. The iniiaed posiions are kep unil one of wo evens: 1. Convergence beween he sock and is pair is achieved, i.e. he difference in normalized prices goes o zero or below. 2. The pair has no converged afer wo periods subsequen o he evaluaion period in which he pairs rade was opened. So he sraegy has a memory of maximum hree evaluaion periods in he case where he pairs rade is opened on he firs day in an evaluaion period. Virually no posiions ake longer han wo periods o converge, so i is no an unreasonable limiaion. An example of a pairs rade is shown in Figure 8 8. The disance is shown on he y axis, he red lines denoe he barrier levels , he arge sock (g) is Naional Grid PLC and is pair is AEF Thesis Page 67

68 8. Analysis Par II: Speculaive algorihms Libery Inernaional PLC. The ime period is from Ocober 13, 2004 o April 12, Five rades are made over he period for a profi of 120,600 wih an exposure of 2mln, which is approximaely 12% annualized. The coinegraion saisic of he pair is , which is well below he criical value of The squared disance in normalized price space, for his pair, is close o he minimum of (Briish Land Co PLC also wih Libery Inernaional PLC) for he period and can be compared o a mean of for all pairs in he period. 1 Disance Close Shor g Close Shor g Close -0.8 Long g Time Figure 8 8: Disance beween Naional Grid PLC (g) and Libery Inernaional PLC. Evaluaion period number 8/15 using TW = 250 days and EP = 125 days. I is worh noing ha his is he only period of 15 in which hese wo socks form a pair, and ha Naional Grid PLC is he pair of 15 oher socks during he daa period. The back es rouine The univariae pairs rading back es rouine is depiced in Figure 11 4 in Appendix 11.8, and a descripion of each sep in he rouine follows: 1. The firs sep is o preallocae memory o he various vecors and arrays ha keep rack of prices, posiions and oher variables over ime. Long g Page 68 AEF Thesis

69 8. Analysis Par II: Speculaive algorihms 2. The nex sep is o make iniial calculaions of daily reurns for he socks during he raining window. 3. Then he main loop is iniiaed, which repeas for each day beginning on he las day of he iniial raining window. The firs day is also he beginning of he firs evaluaion period, so he pair of each sock is idenified using he minimum disance crieria. Duplicae pairs are found and removed. Individual socks may be paired wih more han one oher sock, however. 4. The qualiy of each pair is evaluaed by eiher 1) calculaing he correlaion beween hisorical price reurns of he arge sock and is pair or 2) calculaing he augmened Dickey Fuller coefficien o deermine he degree of coinegraion beween he wo. If he resuls are saisfacory, he given pair is considered acive. 5. For each pair 1,, he barrier level is calculaed and compared o he price difference. If he condiions in sep 4 are fulfilled and, hen he appropriae posiion is opened in pair. is calculaed as for 0 (calculaed using he rolling window). 6. Sep 5 is repeaed on each day unil he beginning of he nex evaluaion period on which we sar over wih sep 3. Pairs, barrier levels and posiions from he previous periods are remembered in order o preserve open posiions ha haven converged. The memory of he algorihm is wo evaluaion periods in addiion o he one in which he posiion was opened. 7. A he beginning of each new day, all posiions are updaed according o marke movemens. 8. When he end of he daa se is reached he posiions are aggregaed and resuls and risk key figures are calculaed. 9. Finally, he resuls are repored. Feaures of he implemenaion Gross marke exposure is measured as: Gross marke exposure = long posiions + shor posiions 0 (8.8) Ne marke exposure is measured as: Ne marke exposure = long posiions + shor posiions (8.9) AEF Thesis Page 69

70 8. Analysis Par II: Speculaive algorihms When a pair is opened, a posiion of 1 is aken in boh he arge sock (+1) and is pair ( 1). So if a pair is opened on day 1, ne marke exposure will be 0. On each day he posiions are adjused in size according o marke movemens. For example, if he arge sock rises by 10% on he following day, he posiion is now 1.1. If he pair sock is unchanged, ne marke exposure a he end of he day will be 0.1. The resuls are repored in he equivalen unis. So for example, a cumulaive reurn of 30 for he sraegy implies ha he profi is equal o 15 imes wha was iniially invesed in each individual pair. Transacion coss are fixed a 0.1% of he nominal amoun invesed when a posiion is opened, and again when i is closed. If he posiion has increased in magniude, he cos of closing he posiion will be higher in absolue erms. The chosen value is arbirary and includes bid ask spread and any fixed rading fees. I may be biased o he downside, especially if he amoun invesed in each pair is large enough o move he bid ask spread. Managing capial The performance of he pairs rading sraegy may be repored in several differen ways. In his analysis hree mehods will be used. Raw excess reurn Reurn o commied capial Reurn from a fully invesed sraegy The raw excess reurn is he profi generaed by he sraegy using he marke exposure dicaed by he sraegy. I shows how successful he sraegy has been in absolue erms ( serling) wihou aking ino accoun he amoun of capial invesed in he sraegy. The reurn o commied capial compares he profi generaed by he sraegy o a prespecified iniial invesmen ha is considered large enough o susain he marke exposure aken by he sraegy. The commied capial mus be large enough o cover any margin requiremens by a broker, and gives a fairly realisic idea of he reurns ha may be generaed in an insiuional seing. An invesmen bank would mos likely measure commied capial in erms of regulaory capial, bu i is hard o speculae in nominal erms wha his may be, so using commied capial in absolue erms Page 70 AEF Thesis

71 8. Analysis Par II: Speculaive algorihms provides an inuiive alernaive. Commied capial does no earn he risk free rae of ineres in imes of lile or no marke exposure. The fully invesed reurn measures performance under he assumpion ha capial allocaed o he sraegy is adjused daily o mach he gross marke exposure of he sraegy. I gives a less realisic view of performance han he commied capial sraegy, as i unrealisic ha a hedge fund or rading desk is allocaed capial on a one day basis. Bu i allows us o compare he performance of pairs rading o he performance of holding he marke porfolio wih he same gross exposure. Resuls The pairs rading sraegy was iniially back esed on he enire sample of price observaions, wih no upper limi on he number of possible pairs (in he populaion of 75 socks). The following parameers were used in he iniial back es: Barrier parameer 2 Minimum required coinegraion es saisic (adf) 3 Rolling windows of beween 25 and 500 days (required o be 2 EP) 20 Evaluaion periods beween 4 and 125 days The bes performing sraegies had rolling windows beween 50 and 150 days, and evaluaion periods beween 5 and 25 days. The opimal sraegy had TW = 150 days and EP = 7 days. I earned over he daa period wih an annualized informaion raio of The maximum drawdown using commied capial of 25 was 7.85% on a single day. See Appendix 11.8 for addiional graphs of he resuls. Figure 8 9 shows he annual performance of he opimal sraegy compared o a long posiion in he FTSE 100 index wih equivalen risk defined as daily reurn volailiy. 20 The resricion is caused by programming issues. Sraegies ha didn fulfill his crierion weren used. They are found in he lower lef corner of he hea maps in Appendix The annualized mean reurn was 5.27 wih a sandard deviaion of AEF Thesis Page 71

72 8. Analysis Par II: Speculaive algorihms Pairs rading Pairs rading (R.A.) FTSE 100 (R.A.) % % % % Excess reurn % 40% Pc reurn % 0.0 0% % % Figure 8 9: Annual reurn from sraegy wih TW = 150, EP = 7 and commied capial of 25. R.A. = righ axis. Looking a he wo bes sraegies, we now ry o vary he barrier parameer and find he following: E.P. = 6 E.P. = 7 E.P. = 6 (R.A.) E.P. = 7 (R.A.) Excess reurn Required rades Barrier parameer a_i 0 Figure 8 10: The effec of changing he barrier parameer for wo differen evaluaion periods (E.P.). R.A. = righ axis. Disabling he coinegraion screening Figure in Appendix 11.8 shows he effec of disabling coinegraion screening for he sraegy wih TW = 150, EP = 7 and 2. The marke exposure of he sraegy rises as more pairs are raded, and so does excess reurn. Bu he annualized informaion raio drops from 1.75 o 1.51 and maximum drawdown rises from 3.32 o 5.75, i.e. he radeoff beween risk and reurn has Page 72 AEF Thesis

73 8. Analysis Par II: Speculaive algorihms deerioraed. Similar resuls were found for oher combinaions of window and evaluaion period, which is evidence ha explici esing for coinegraion improves univariae pairs rading. Long / shor posiions and marke neuraliy The oal excess reurn of he opimal sraegy, TW = 150 and EP = 7, is no evenly divided beween long and shor posiions. Long posiions earned (82.7%) and shor posiions 7.00 (17.3%) of he oal excess reurn. The resul is common o all pairs rading sraegies ha were back esed, ha is, long posiions consisenly ouperform shor posiions for differen combinaions of TW, EP and. This shows ha he specific pairs rading sraegy esed here is a good sock picker, while a he same ime hedging is bes profiably. To es he degree of marke neuraliy alpha and bea wih respec o he FTSE 100 index are calculaed using he regression R P, αβr M, r, 1,,. (8.10) where R P, is he daily reurn of he pairs rading sraegy, R M, is he daily reurn of he FTSE 100 index, and r is he daily over nigh risk free rae of ineres. The resuls are shown below Regression resuls Parameer Esimae T sa Es. annual. α 1.52E β 5.42E ρr P,,R M, Sraegy Marke Sharpe raio Boh parameers are saisically significan a he 1% level, so using some degree of bea exposure, he sraegy produces annual alpha of 3.91%. Linear correlaion wih marke reurns is fairly low a 13.94%, while he risk adjused (volailiy adjused) reurn of he sraegy is much higher han he marke. The reurns have skewness of 0.42 and kurosis of The average skewness of he reurns of he FTSE 100 socks in he sample is 0.15 and he kurosis is The equivalen figures for he pairs 22 The Sharpe raio is here defined as, /, where 252 is he number of rading days per year. AEF Thesis Page 73

74 8. Analysis Par II: Speculaive algorihms rading sraegies using many differen combinaions of TW and EP are 0.19 and 5.68, which is no far from he individual socks. The reurns from he pairs rading sraegy hus have a heavier ail han a normal disribuion, and hey consis of many slighly negaive daily reurns and few large posiive daily reurns. Ne marke exposure The ne marke exposure of each pair rade is zero on he day he rade is opened, bu becomes differen from zero as he value of he arge sock and is pair changes. Mos of he ime his marke exposure is rivial in magniude for he aggregae porfolio, bu i may occasionally spike due o large price changes in single socks. Ineresingly, his ne exposure is almos always negaive (ne shor) see Figure The mos likely explanaion is ha here is more posiive han negaive momenum in he socks in he sample. Say ha a hypoheical sock A (wih pair B) rises significanly and is price diverges from B by a magniude large enough o be shored by he algorihm. If sock A keeps rising faser han sock B during he following days (or weeks) he shor posiion in sock A will grow larger han he corresponding long posiion in sock B, and he pair will become ne shor he marke. The endency of he porfolio as a whole o be ne shor he marke is an indicaion ha here is a posiive momenum effec in he socks in he sample (which is sronger han a possible negaive momenum effec). This hypohesis could be esed by applying he EMA momenum sraegy o he socks in he sample and see if (any) profis are generaed primarily by long or shor posiions. I is emping o sugges ha he negaive ne marke exposure is a consequence of all socks rising in general. Bu Figure 8 11 shows ha here doesn seem o be any clear connecion beween general marke movemens and he ne marke exposure of he sraegy. Page 74 AEF Thesis

75 8. Analysis Par II: Speculaive algorihms 100 Gross marke exposure serling Day 1 Ne marke exposure serling Day 7000 FTSE 100 index Index value Day Figure 8 11: Gross and ne marke exposure of pairs rading sraegy wih parameers: TW = 150 days, EP = 5 days, cumulaive reurn = Improving he univariae pairs rading algorihm There are several ways o improve he pairs rading algorihm. The algorihm used for his analysis is quie simple, and in pracice many enhancemens would need o be made. The mos imporan of which are: The daa se of daily observaions used in his analysis does no include dividends. The implicaion is ha every price change in he daase caused by a dividend paymen will resul in an incorrec reurn calculaion. For example, if a sock pays a dividend on January 5, he AEF Thesis Page 75

76 8. Analysis Par II: Speculaive algorihms share price will fall by an amoun equal o he dividend per share, compared o he sock price on he previous rading day. Thus a negaive reurn is recorded even hough he acual reurn may have been zero or posiive. The exen of his problem is deermined by he number of socks paying dividends and he size of hose dividends. The bias in he resuls will be limied by he fac ha he algorihm will someimes be long and someimes shor dividend paying socks. 23 The use of bid ask quoes and possibly hisorical limi order books raher han las raded quoes. Las raded quoes are an approximaion of he acual price ha may be raded a he end of each day. However, i is no unrealisic ha he algorihm will be able o rade during he closing aucion and obain a price close o he las raded, wih a bias owards worse prices due o he bid ask bounce. The inclusion of shor selling coss o he analysis. The assumpion of equal ransacion coss for long and shor posiions should be modified o ake ino accoun he higher funding coss of shor selling, and possibly he exisence of shor selling resricions on some socks. The use of sop loss orders o avoid holding pairs ha show no signs of convergence. Insead of holding each pair unil convergence or he passing of hree evaluaion periods, a sop loss order may be placed a a predeermined price spread. If he given pair keeps diverging insead of converging in price, i is closed down. Alhough sop loss orders are ypically used by praciioners, here is no guaranee ha hey will improve he algorihm (NATH, P., 2003). How o place hem opimally is also a challenge. Nah (2003) akes he approach of opening pair rades a he 25 h and 75 h perceniles of daily disances ( ) and sops a he 5 h and 95 h perceniles, respecively. Trade convergence is defined by he median of. Sub conclusion Univariae pairs rading is capable of generaing posiive excess reurns ha are uncorrelaed wih he long only reurn of he populaion of socks used. The sraegy is highly sensiive o he parameers used. Changing he lengh of he raining window, he ime beween reevaluaion of pairs, or he barrier parameer makes a large impac on resuls. Neverheless, rying many differen combinaions 23 There is a risk, of course, ha he algorihm iself is biased owards shoring dividend paying socks, and going long nondividend paying socks. Page 76 AEF Thesis

77 8. Analysis Par II: Speculaive algorihms of parameers an opimal sraegy was found. In addiion, i was shown ha lower values of he barrier parameer yields higher profis. This seems counerinuiive, as one would guess ha opening rades a larger deviaions from equilibrium values would yield beer resuls. The conclusion is ha despie he higher ransacion coss, i is profiable o ener many quick rades being on immediae reversals in price movemen. The minimum disance crierion is successful a finding profiable pairs. Screening each pair for a high coinegraion value furher ensures he qualiy of he pairs, and increases he informaion raio of he sraegy. I is also found ha a porfolio of many open pair rades ends o have a negaive ne marke exposure (if posiions aren rebalanced on a daily basis). This suggess ha he socks in he sample have sronger posiive reurn momenum han negaive. The performance of pairs rading in his paricular daa sample shows ha afer four good years from 2001 o 2004, pairs rading hi a rough pach from 2005 o Performance seems o have picked up again in 2008, maybe hanks o he higher price volailiy creaed by he financial crisis of Mulivariae pairs rading In his secion we will es he mulivariae pairs rading mehods presened in secion by means of wo main approaches. One ha makes use of sae space mehods and one ha doesn. They boh exrac a signal,, from he M pair porfolio of sock a ime. The size of he M pair porfolio is arbirary, and is a parameer of he algorihm. The disance in price space beween arge sock and is signal is defined as (8.11) Based on his signal he algorihm rades in he same way as he univariae algorihm, so he main difference lies in he calculaion of. Figure in Appendix 11.9 depics he srucure of he mulivariae back es rouine. can be calculaed by means of OLS as described in secion 5.3.5: AEF Thesis Page 77

78 8. Analysis Par II: Speculaive algorihms where denoes esimaed parameers. 1 (8.12) Alernaively a sae space mehod may be esimaed. The aim is o exrac an underlying signal from he M pair porfolio. We remember he observaion and sae equaions (6.1) from secion 6.2: y Z T R 1 ~ N 0, H ~ N(0, Q ) 1,..., n (8.13) Wih socks in our M pair porfolio we wrie he observaion equaion as and he sae equaion as where 1,,, , 1 2 2, (8.14) 0 1 1,1 2, , 2, (8.15) 0, 0 and ,,,, are he normalized prices of he socks in he M pair porfolio and, is he normalized price of he arge sock (T), i.e. he sock we are rading he M pair porfolio agains. The variable of ineres is, which is he underlying signal driving he observaions in. We will use, as he for arge sock o calculae. In order o esimae he model based on he observed daa 1,,,, we mus make an iniial guess of he various parameers and he iniial sae mean and sae covariance marix. The guess will be,, 0.5 ;, 1 ;,,, 2 ; 0 and where denoes he ideniy marix. The parameers are esimaed using maximum likelihood as described in secion 6.2.,, Page 78 AEF Thesis

79 8. Analysis Par II: Speculaive algorihms Resuls I was no possible o include posiion memory in he mulivariae pairs rading rouine. Therefore all posiions are closed down a he end of each evaluaion period. This has a negaive impac on performance because each open rade has less ime o converge owards equilibrium, i.e. 0. Iniially an M pair porfolio size of 3 socks is used. The socks are found using he minimum disance crierion described in secion Using he OLS based mehod of calculaing in equaion (8.12) various combinaions of TW and EP were esed using barrier parameer 1.5. The resuls are shown in Figure in Appendix The opimal sraegy uses TW=200 and EP=40. I produces an excess reurn of 10.0 wih an informaion raio of Transacion coss a have become a concern. The EP is significanly longer han he opimal univariae sraegy (EP=7), as a higher EP usually means less rading aciviy and lower ransacion coss. Figure 8 12 shows he effec of varying he barrier parameer. A higher resuls in less rading aciviy, lower ransacion coss and a higher informaion raio. Revenue Trans. coss Excess reurn Info. raio serling Barrier parameer a_i 0.40 Figure 8 12: Resuls for various barrier parameer values. TW=200 and EP=40. AEF Thesis Page 79

80 8. Analysis Par II: Speculaive algorihms In an aemp o enhance he sraegy we ry o change he way he socks in he porfolio are chosen. Insead of using he minimum disance crierion (MD), we use he coinegraion based (CI) mehod described in secion The resuls are shown in Figure G.E. CI (R.A.) G.E. MD (R.A.) E.R. CI E.R. MD serling serling Figure 8 13: Resuls using wo differen approaches o creae he porfolio. G.E. = gross exposure, E.R. = excess reurn, CI = coinegraion based mehod, MD = minimum disance, R.A. = righ axis. Mehod CI MD Excess reurn ( ) Annual volailiy ( ) Informaion raio Avg. marke exposure ( ) The CI mehod akes more risk, bu has a worse riskreurn radeoff as measured by he informaion raio. The jagged edges of he marke exposure in he above figure reflec ha all posiions are closed down a he end of each evaluaion period. We may also aemp o enhance he sraegy by changing he number of socks in he M pair porfolio. Table 8 1 repors resuls. In he space of mulivariae sraegies, 2 dominaes he oher possibiliies. Bu he univariae pairs rading sraegy performs significanly beer han he bes mulivariae sraegy The memory feaure of he univariae sraegy has been disabled o be on an even fooing wih he mulivariae sraegy. I.e. posiions are no carried ino new evaluaion periods. Page 80 AEF Thesis

81 8. Analysis Par II: Speculaive algorihms M Revenue ( ) Trans. coss ( ) Excess reurn ( ) Required rades 6,460 9,128 12,706 16,349 19,371 22,776 Informaion raio Max. drawdown ( ) Table 8 1: The effec of varying he number of socks in he M pair porfolio. TW = 200, EP = 40 and.. Sae space resuls Revenue ( ) 2.28 Transacion coss ( ) 7.08 Excess reurn ( ) 4.81 Annual volailiy ( ) 0.96 Informaion raio 0.67 Avg. marke exposure ( ) Required rades 3,542 SS models esimaed 1,026 and he marke exposure is unevenly disribued over ime. We now es he sae space (SS) approach o see if i is an improvemen of he above. Using he same parameers as above, namely TW = 200 and EP = 40, he resuls are no encouraging. The barrier parameer is lowered o 1, bu his doesn help much. Alhough he sraegy has posiive revenue over he daa sample, ransacion coss are oo high. The SS sraegy rades significanly less han he OLS sraegy, G.E. (R.A.) Revenue Trans. coss E.R. Performance serling Gross exposure serling Figure 8 14: Resuls for sae space sraegy. TW = 200, EP = 40 and. G.E. = gross exposure, E.R. = excess reurn, R.A. = righ axis AEF Thesis Page 81

82 8. Analysis Par II: Speculaive algorihms As an example, in he firs evaluaion period, sock 16 mached up wih socks 42,64, The esimaed sae space sysem wih esimaed parameers looks like,,,, , , 0 1,, ,, 1 0, The resuls are obained afer 2 ieraions and 121 funcion evaluaions, and he final likelihood value is Comparing he esimaed parameers of he differen models, here was wide dispersion in he esimaed. Bu he error variance and were mos of he ime close o 1 and 1.66 respecively. A possible reason for he bad earnings performance of he sraegy is ha he esimaed signal, resembles he arge sock oo closely. This migh be because he maximum likelihood funcion is very jagged and i is difficul for he BFGS algorihm o find he global minimum. In some cases esimaion is no possible because individual marices in he sae space recursions come oo close o singulariy. 26 The disance is very sable over ime compared o univariae pairs rading, so here are relaively few rade signals. The average linear correlaion beween and is 94.23% which is a very high value, and considerably higher han he corresponding values for he oher approaches o pairs rading. Sub conclusion The mulivariae pairs rading sraegies esed in his secion underperformed he univariae approach significanly. Each addiional sock in he porfolio doesn conribue enough new informaion o he pairs rade o make up for he higher ransacion coss. Forming pairs using a coinegraion based approach does no improve performance, and neiher does he use of sae space mehods. I is possible ha he M pair formaion approach can be improved by calculaing he disance or degree of coinegraion using all possible combinaions of socks in he 25 Sock 16 is Briish Peroleum Plc which operaes in he oil & gas indusry. Arranged in order of increasing norm in normalized price space, socks 42,64,53 are Lonmin Plc (basic maerials), Smih & Nephew Plc (healhcare producs) and Reed Elsevier Plc (media). 26 The back es rouine skips M pairs wih his problem by using a marix reciprocal condiion number esimae in he Kalman filer. This is done wih he MATLAB funcion rcond. Page 82 AEF Thesis

83 8. Analysis Par II: Speculaive algorihms M pair. The combinaion wih he lowes disance, he highes degree of coinegraion, or a weighed average of he wo is hen chosen. Noe ha for values of 2 i is a very ime consuming procedure. I is possible ha he sae space model in (8.13) (8.14) can be improved by using a differen specificaion. One possibiliy is o separae he socks in he M pair porfolio ino differen groups depending on heir characerisics (eiher saisical or fundamenal, such as indusry). The observaion equaion of such a specificaion wih 4 could ake he form wih sae equaion where 1, 2, 3, 4,, , ,, (8.16) 1, , 2, , (8.17), , 0 0 0, The signal would in his case be,. This would leave open he quesion of how o find he, socks 1,2,3,4. AEF Thesis Page 83

84 9. Conclusion 9 CONCLUSION The subjecs covered in his paper have been chosen o give he reader a broad inroducion o marke microsrucure and rading algorihms. Boh subjecs are exensive and challenging, and ools from several fields wihin finance, economics and saisics have been applied. The analysis has ouched on boh heoreical and empirical aspecs of marke microsrucure and rading algorihms. The academic lieraure in marke microsrucure is rich, and he possibiliies for empirical research are many. Secion 4.3 showed how a simple microsrucure model can be esed empirically using vecor auoregression echniques. Several challenges arose due o he high frequency naure of he empirical daa, as he number of observaions is very high. Neverheless, he analysis was able o confirm he findings of Hasbrouck (1991). Quoes are revised in response o rading, and rading is done in response o changes in quoes, giving rise o a wo way dynamic relaionship beween he wo evens. While quoe revisions were self correcing, rade evens had posiive auocorrelaion. An imporan feaure of he model is ha privae (asymmeric) informaion is revealed as he rade innovaion. The following secion on he opimal execuion of porfolio ransacions showed how a relaively simple model of inra day rading can give much inuiion on he challenges of rading large amouns of shares in a shor period of ime. Almgren and Chriss (2001) show ha he problem of minimizing implemenaion shorfall can be solved as a quadraic opimizaion problem using a quadraic uiliy funcion. An imporan conclusion of he analysis is ha he opimal execuion sraegy is saic over ime. I.e., due o marke efficiency i is no possible o improve he execuion process by forecasing prices. Even if i were possible o predic he drif of he price process, or if prices exhibi serial correlaion, he benefis of including such informaion in an execuion sraegy are oo few. The volume weighed average price can be used as an execuion benchmark. Furhermore, using he analysis of McCulloch and Kazakov (2007) i was shown ha if inra day volume is modeled as a Page 84 AEF Thesis

85 9. Conclusion doubly sochasic binomial poin process, mean variance analysis can be used o find an opimal VWAP rading sraegy. I has been shown ha exponenial moving averages can be used o capure drif in he price process of radable insrumens, known as price momenum. A comprehensive back es procedure was carried ou by varying he sensiiviy of he EMAs and he price sampling frequency. The sraegy was capable of generaing posiive excess reurns wih an annualized informaion raio in excess of 1. Using a minimum disance crierion o pair socks from a large populaion, as suggesed by Gaev e al (2006), univariae pairs rading also generaed posiive excess reurns. The majoriy of he reurns were earned in he beginning of he sample period, however, showing ha he sraegy is unsable over ime. The use of explici esing for coinegraion beween individual socks was shown o improve he informaion raio significanly. Mulivariae pairs rading was less successful. The higher ransacion coss involved in rading one sock agains a porfolio of socks ouweighed he benefis. Aemping o improve he procedure by choosing socks in he M pair porfolio using a coinegraion measure was unsuccessful. The use of sae space mehods was no successful eiher, as he rade signal creaed by he sae space model resembled he arge sock oo much. Finally, an alernaive specificaion of he sae space model was suggesed. Suggesions for furher research Much ineresing analysis can be carried ou using hisorical ick daa, for example, o see how he resuls from he VAR framework differ beween socks in differen markes and over ime. Anoher possibiliy is o back es an opimal execuion algorihm on hisorical inra day daa. The back esing of speculaive algorihms was carried ou using daily las raded price quoes. The realism of he analysis can be grealy improved by using inra day bid and ask quoes. In heory, pairs rading should be possible using high frequency daa in ransacion ime. Sae space mehods would be paricularly suiable o such a sraegy, as i is possible o rea nonrading evens as missing observaions. AEF Thesis Page 85

86 10. Sources 10 SOURCES ALEXANDER, C. and A. DIMITRIU The Coinegraion Alpha: Enhanced Index Tracking and Long Shor Equiy Marke Neural Sraegies. ISMA Cenre, Universiy of Reading, UK. ALMGREN, R Execuion Coss. Encyclopedia of Quaniaive Finance. ALMGREN, R. and N. CHRISS Opimal Execuion of Porfolio Transacions. Journal of Risk. 3, pp ALMGREN, R., C. THUM, E. HAUPTMANN, and H. LI Equiy Marke Impac. RISK. 18, pp BOSTRÖM, D Marke Neural Sraegies. Kauphing Bank. Sockholm. BOX, G., G. JENKINS, and G. REINSEL Time Series Analysis, Forecasing and Conrol 3rd ediion. San Francisco: Holden Day. CAMPBELL, J. Y., A. W. LO, and A. C. MACKINLAY The Economerics of Financial Markes. Princeon, New Jersey: Princeon Universiy Press. DB Deusche Bank auobahn. [online]. Available from World Wide Web: <hp:// DURBIN, J. and S. KOOPMAN Time Series Analysis by Sae Space Mehods. Oxford: Oxford Universiy Press. ENGLE, R. F. and C. GRANGER Co Inegraion and Error Correcion: Represenaion, Esimaion, and Tesing. Economerica. 55(2), pp ENGLE, R. and J. RUSSELL The Auoregressive Condiional Duraion Model. Economerica. 66, pp FAMA, E. F Efficien capial markes: A review of heory and empirical work. Journal of Finance. 25, pp FAMA, E. F Efficien Capial Markes: II. The Journal of Finance. 46(5), pp GATEV, E., W. N. GOETZMANN, and K. G. ROUWENHORST Pairs Trading: Performance of a Relaive Value Arbirage Rule. Yale ICF Working Paper No GUJARATI, D. N Basic Economerics (Fourh Ediion). New York: McGraw Hill/Irvin. Page 86 AEF Thesis

87 10. Sources HASBROUCK, J Empirical Marke Microsrucure. New York: Oxford Universiy Press. HASBROUCK, J Measuring he Informaion Conen of Sock Trades. Journal of Finance. XLVI(1), pp HULL, J. C Opions, Fuures and Oher Derivaives 6h Ediion. Upper Saddle River, New Jersey: Prenice Hall. INGERSOLL, J Theory of Financial Decision Making. New Jersey: Rowman and Lilefield. KASSAM, A The MahWorks Recorded Webinar: Algorihmic Trading wih MATLAB for Financial Applicaions. [online]. [Accessed 25 Sepember 2008]. Available from World Wide Web: <hp:// pe=file> KIM, K Elecronic and Algorihmic Trading Technology. Burlingon, MA: Elsevier. KONISHI, H Opimal slice of a VWAP rade. Journal of Financial Markes. 5, pp LUX, H Inerview wih James Simons, founder of Renaissance Technologies Corp. [online]. [Accessed 28 June 2008]. Available from World Wide Web: <hp://chineseschool.nefirms.com/abacus hedge funds Jim Simons.hml> MCCULLOCH, J Relaive Volume as a Doubly Sochasic Binomial Poin Process. Quaniaive Finance. 7, pp MCCULLOCH, J. and V. KAZAKOV Opimal VWAP Trading Sraegy and Relaive Volume. Quaniaive Finance Research Cenre, Research Paper Series 201. Universiy of Technology, Sydney. NATH, P High Frequency Pairs Trading wih U.S. Treasury Securiies. London Business School. O'HARA, M Marke Microsrucure Theory. Malden, MA: Blackwell Publishing. PERLIN, M. S. 2007a. Evaluaion of a Pairs Trading Sraegy a he Brazilian Financial Markes. ICMA/Reading Universiy Working Paper. PERLIN, M. S. 2007b. M of a kind: A Mulivariae Approach a Pairs Trading. ICMA/Reading Universiy Working Paper. AEF Thesis Page 87

88 10. Sources SHUMWAY, R. H. and D. S. STOFFER Time Series Analysis and Is Applicaions (Second Ediion). New York: Springer. TEITELBAUM, R Simons a Renaissance Cracks Code, Doubling Asses (Updae1). [online]. [Accessed 13 June 2008]. Available from World Wide Web: <hp:// TRAGSTRUP, L Inerview wih he auhor on 30 December Copenhagen. TSAY, R. S Analysis of Financial Time Series. Hoboken, New Jersey: John Wiley & Sons. TURQUOISE Turquoise Homepage. [online]. Available from World Wide Web: <hp:// WELCH, G. and G. BISHOP An Inroducion o he Kalman Filer. Deparmen of Compuer Science Universiy of Norh Carolina a Chapel Hill. WIKIPEDIA. 2008c. Erlang (programming language). [online]. Available from World Wide Web: <hp://en.wikipedia.org/wiki/erlang_(programming_language)> WIKIPEDIA. 2008a. Financial Informaion exchance. [online]. Available from World Wide Web: <hp://en.wikipedia.org/wiki/fix_proocol> WIKIPEDIA. 2008b. Markes In Financial Insrumens Direcive. [online]. Available from World Wide Web: <hp://en.wikipedia.org/wiki/mifid> Page 88 AEF Thesis

89 11. Appendix 11 APPENDIX 11.1 Even ypes in he FTSE 100 ick daa Supplemen o secion 7.1: The daa se is obained from Reuers Daascope and he following informaion is from he Reuers Daascope documenaion. Each line in he daase will be called an even. There may be more han one even a a given poin in ime. Some evens in he daa se have one or more qualifiers ha classify he even. There are five ypes of qualifier: Generic, price, secondary price, irregular rade and exchange specific. The qualifiers are coupled wih a TAG ha gives specific informaion abou he qualifier. For example, he mos ypical qualifier is A[LSE], where A[] is he qualifier and LSE is is ag. The wo main even ypes of ineres are rades and quoes, as hey are he only evens used for calculaions. To prepare he daa se for calculaions i mus be cleaned from unwaned evens. The qualifiers were used o idenify hose evens, and make he process easy (possible). The main evens of ineres are rade evens. Their qualifiers are: A[]: Auomaic rade generaed by he sysem hrough auomaic execuion. O[]: Non proeced porfolio. As opposed o proeced porfolio evens ha are no repored. E[]: Error correcion. L[]: Lae repored rade. N[]: Overnigh execuion. n[]: SEATS non risk rade Ohers The vas majoriy of rades are auomaic (A[]). Non proeced porfolio rades (O[]) are ofen raded a price levels far from he mos recen quoe midpoin a he ime hey are repored. Therefore hey were excluded from he calculaions if he raded price was more han 1% from he previous quoe. Oher imporan evens are: OB[]: Inra day Opening order book price derived from order driven rade. CLS[]: Marke closed AEF Thesis Page 89

90 11. Appendix AUC[]: Aucion call periods. They announce he beginning of opening aucions in he morning and closing aucions in he afernoon. Auomaically generaed generic evens are (agged wih GEN ): Oher OB[] order book evens such as High and Low inra day price. OB VWAP[]: The order book volume weighed average price. This even is repored every ime a rade is execued from he consolidaed order book Volume and open ineres of Bund and Euribor fuures Supplemen o secion 5.3.1: Bund Euribor Bund (R.A.) Euribor (R.A.) Traded volume (mln) Open ineres (mln) Figure 11 1: Monhly raded volume and open ineres of Bund and Euribor fuures on Eurex. R.A. = righ axis. Source: Bloomberg. Page 90 AEF Thesis

91 11. Appendix 11.3 Addiional co inegraion ess Supplemen o secion 5.3.3: The Durbin Wason es for coinegraion uses he d saisic obained from he regression α β (11.1) The d saisic is defined as. The null hypohesis is ha 0 which implies ha has a uni roo and ha and are no coinegraed. This is because 21 and is defined as 1 in he regression δy u. So if is abou 1 hen is abou 0. The criical values for he d saisic are 0.511, and a he 1, 5 and 10% level respecively. If he observed d saisic is below i may hen be concluded a he 1% level ha 0, has a uni roo, and ha and are no coinegraed (GUJARATI, D. N., 2003). AEF Thesis Page 91

92 11. Appendix 11.4 Derivaion of Kalman filer recursions and illusraion Supplemen o secion 6.4: The following is adaped from Durbin & Koopman (2001): The derivaion requires only some elemenary properies of mulivariae regression heory: Suppose ha, and are random vecors of arbirary orders ha are joinly normally disribued wih means and covariance marices Σ for,, and wih 0 and Σ 0. The symbols,,, and are employed in his lemma for convenience and have no relaion o oher pars of he paper. Lemma 11.1 Proof: See Durbin & Koopman (2001). The linear Gaussian sae space model is wrien as,, Σ Σ Σ ~0, ~0, ~, 1,,. (11.2) Le denoe he se of pas observaions,,. The objecive is o obain he condiional disribuion of given for 1,,. Because all disribuions are normal, saring a 1 and building up he disribuions of and recursively, i is easy o show ha,,, and,,,. Here we derive he Kalman filer in (11.2) for he case where he iniial sae is, where and are known. Our objecive is o obain he condiional disribuion of given for 1,, where,,. Since, we have (11.3) Page 92 AEF Thesis

93 11. Appendix (11.4) for 1,,. Le. Then is he one sep forecas error of given. When and are fixed hen is fixed and vice versa. Thus,. Using Lemma (11.1) we have,, (11.5) where,, and by definiion of. Here,,, and. We assume ha is nonsingular. Subsiuing in (11.3) and (11.4) we ge, 1,,, (11.6) wih. We observe ha has been obained as a linear funcion of he previous value and, he forecas error of given. Using Lemma (11.1) again we have,,,. Subsiuing in (11.4) gives, 1,...,, (11.7) wih. The recursions (11.6) and (11.7) consiue he Kalman filer for model (11.2) (DURBIN, J. and Koopman, S., 2001). AEF Thesis Page 93

94 11. Appendix Illusraion of he Kalman filer recursion process Based on heir iniial esimaes he sae esimae and error covariance a and P are firs projeced ino he fuure, and hen correced using he Kalman gain. A superscrip denoes an esimae ha has been projeced, bu no ye correced. Iniial esimaes of a and P Time updae ( predicion ) (1) Projec he sae ahead a 1 Ta (2) Projec he error covariance ahead P TPT RQR 1 Observaion updae ( correcion ) (1) Compue he Kalman gain K TPZ /( ZPZH ) TPZF 1 (2) Updae esimae wih observaion y a 1 a 1 Kv (3) Updae he error covariance P P TP K Z 1 1 Figure 11 2: The Kalman filer recursion. (WELCH, G. and Bishop, G., 2006) Page 94 AEF Thesis

95 11. Appendix 11.5 Empirical analysis of ick daa Supplemen o secion 7.3 Lag Coeff. T sa Lag Coeff. T sa a c a c a c a c a c a c a c a c a c a c a c b d b d b d b d b d b d b d b d b d b d R^ Table 11 1: Oupu from VAR analysis of Anglo American PLC daa of March AEF Thesis Page 95

96 11. Appendix 11.6 Derivaion of he exponenial moving average Supplemen o secion 8.1: To show ha x λ1 λ x w has he recursive represenaion x 1 λx λx for x 0 and 0 1 we wrie x x λ1 λ x λ1λ x λx 1λ λ1 λ x λ1λ x λx 1λ λ1 λ x λ1 λ x So x 1 λx λx. Page 96 AEF Thesis

97 11. Appendix 11.7 EMA momenum sraegies Supplemen o secion 8.1: P&L FTSE 100 Index EMA(61) EMA(98) 8, , , , , , , % 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 10.00% 20.00% Figure 11 3: The FTSE 100 Index wih an EMA(61), EMA(98), and he cumulaive reurn of he sraegy superimposed. AEF Thesis Page 97

98 11. Appendix 11.8 Univariae pairs rading Supplemen o secion 8.2: Preallocae memory Iniial calculaions Updae pairs Updae posiions Main loop Screen for opporuniies Open / close posiions Calculae risk figures Repor resuls Figure 11 4: Univariae pairs rading back es rouine. Page 98 AEF Thesis

99 11. Appendix E.R. a_i=0.1 E.R. a_i=0.1 (N.T.) E.R. a_i=2 E.R. a_i=2 (N.T.) serling Figure 11 5: Excess reurn and excess reurn wihou ransacion coss (N.T.) of pairs rading sraegies wih TW = 150, EP = 7, and. or. G.E. a_i=0.1 G.E. a_i=2 L.R. a_i=0.1 (R.A.) L.R. a_i=2 (R.A.) serling Leverage raio Figure 11 6: Gross marke exposure (G.E.) and leverage raio (L.R.) using commied capial of he same wo sraegies,. or. AEF Thesis Page 99

100 11. Appendix 60 Number of coinegraed pairs per evaluaion period Percenage of possible pairs coinegraed L i f i d i l Figure 11 7: The sock populaion is large enough o creae a sufficien number of coinegraed pairs in each evaluaion period (TW = 150, EP = 7). Evaluaion period Excess reurn Window Figure 11 8: Hea map of excess reurns for various combinaions of window and evaluaion period lenghs Page 100 AEF Thesis

101 11. Appendix Evaluaion period Evaluaion period Informaion raio Window Figure 11 9: Hea map of annualized informaion raios. Commied capial max drawdown pc Window Figure 11 10: Hea map of maximum drawdown of commied capial. The daa is shown in absolue values AEF Thesis Page 101

102 11. Appendix G.E. w/o coin. G.E. E.R. w/o coin. E.R serkubg serling Figure 11 11: Gross exposure (G.E.) and excess reurn (E.R.) wih or wihou coinegraion screening. Page 102 AEF Thesis

103 11. Appendix 11.9 Mulivariae pairs rading Supplemen o secion 8.3: Preallocae memory Iniial calculaions Updae pair porfolios Updae posiions Main loop Calculae mulivariae signals Open / close posiions Screen for opporuniies Calculae risk figures Repor resuls Figure 11 12: Mulivariae pairs rading back es rouine. AEF Thesis Page 103

104 11. Appendix Evaluaion period Excess reurn Window Evaluaion period Informaion raio Window Evaluaion period Commied capial max drawdown pc Window Figure 11 13: Back es using OLS mehod o find. Page 104 AEF Thesis