Convergence of Binomial Large Investor Models and General Correlated Random Walks

Transcription

1 Covergece of Biomial Large Ivesor Models ad Geeral Correlaed Radom Walks vorgeleg vo Maser of Sciece i Mahemaics, Diplom-Wirschafsmahemaiker Urs M. Gruber gebore i Georgsmariehüe. Vo der Fakulä II Mahemaik ud Naurwisseschafe der Techische Uiversiä Berli zur Erlagug des akademische Grades Dokor der Naurwisseschafe Dr. rer. a. geehmige Disseraio. Guacher: Prof. Dr. Mari Schweizer Eidgeössische Techische Hochschule Zürich Prof. Dr. Alexader Schied Techische Uiversiä Berli Prof. Dr. Rüdiger Frey Uiversiä Leipzig Tag der wisseschafliche Aussprache: 17. Sepember 2004 Berli 2004 D 83

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3 Absrac This hesis sudies he problem of opio pricig via replicaio by a large ivesor whose radig affecs he sock price. We formulae ad solve his quesio firs i a biomial seig. The we cosider a suiably scaled sequece of such biomial large ivesor models ad prove heir covergece owards a coiuous-ime diffusio. This requires ha we aalyze carefully boh he covergece of he large ivesor s sraegy fucios ad he sochasic process of he uderlyig fudameals. The covergece of he laer is derived from a ew covergece resul for geeral correlaed radom walks. I each sigle ime sep, we model he sock price as a fucio of ime, some fudameals ad he large ivesor s sock holdigs, ad we assume ha he fudameals describe a radom walk. We aalyze i deail he price mechaism which models how he large ivesor s rades affec prices ad elaborae o he imporace of a fair price sysem as a heoreical bechmark. This ca be used o defie implici rasacio losses ad he real value of a large ivesor s porfolio. We derive codiios which preve paper-value ad real-value arbirage opporuiies for he large ivesor ad show he exisece ad uiqueess of a replicaio sraegy for a give coige claim. As a cosequece of is feedback o he sock price, his sraegy is i geeral oly give implicily by a fixed poi heorem. To sudy he covergece of a sequece of biomial large ivesor models, we rescale he fudameals as i Dosker s heorem. I a firs sep, we he show ha he covergece of he large ivesor s sraegy fucios is implied by heir covergece a mauriy. The limi fucio is ideified as he soluio of a o-liear fial value problem. By a suiable sraegy rasform, his ca be simplified o a perurbaio of a liear problem i a fair marke. We he prove he covergece i disribuio of he biomial large ivesor models uder wo differe regimes of marigale measures. Because he rasiio probabiliies for he fudameals uder hese measures ypically deped o he large ivesor s sock holdigs before ad afer his rade, we have o exed classical covergece resuls o a seig wih geeral correlaed radom walks. For geeral correlaed radom walks, he direcio of he ex move depeds o ime, he curre posiio ad he direcio of he previous move. Usig Dosker s scalig, we prove he covergece i disribuio of a sequece of such walks owards a diffusio limi, ad we explicily ideify he diffusio coefficies. I urs ou ha i compariso o he classical case, boh volailiy ad drif are reiforced due o he correlaio bewee he icremes of he discree walks. I paricular, we obai a covergece resul for exisig large ivesor models from he lieraure. Moreover, our sudy highlighs he imporace ad ifluece of he choice of price mechaism.

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5 Zusammefassug Die vorliegede Arbei berache das Problem der Opiosbewerug durch Replikaio für eie Großivesor, der de Akiepreis durch sei Hadel beeifluß. Diese Frage wird zuers i eiem biomiale Rahme formulier ud gelös. Im Aschluß uersuche wir eie geeige skaliere Folge vo solche biomiale Großivesormodelle ud beweise ihre Kovergez gege ei zeiseiges Diffusiosmodell. Dazu müsse wir sowohl die Kovergez der Sraegiefukioe des Großivesors als auch de sochasische Prozeß, der die zugrudeliegede Fudamealdae modellier, sorgfälig beschreibe. Die Kovergez der Modelle erhale wir aus eiem eue Kovergezresula für allgemeie korreliere Irrfahre. Für jede eizele Zeipuk modelliere wir de Akiepreis als eie Fukio vo Zei, gewisse Fudamealdae ud dem Akiebesiz des Großivesors, ud wir beschreibe die Fudamealdae durch eie biäre Irrfahr. Wir uersuche deaillier de Preismechaismus, der de Eifluß des Großivesors auf de Akiekurs modellier, ud arbeie die Bedeuug eies faire Preissysems als heoreischer Bechmark heraus. Dieser ka da beuz werde, um implizie Trasakiosverluse ud de Realwer eies Großivesorporefeuilles zu defiiere. Wir ewickel Bediguge, die Papierwer- ud Realwer- Arbirage ausschließe, ud beweise die Exisez ud Eideuigkei vo Replikaiossraegie für ei gegebees Edporefeuille. Wege ihrer rückkoppelde Wirkug auf de Akiepreis is diese Sraegie im allgemeie ur implizi durch eie Fixpuksaz gegebe. Um die Kovergez eier Folge vo biomiale Großivesormodelle zu berache, reskaliere wir de Prozeß der Fudamealdae wie im Saz vo Dosker. Zuächs zeige wir da, daß die Kovergez der Sraegiefukioe des Großivesors aus ihrer Kovergez am Fälligkeisermi folg. Die Grezfukio ergib sich als Lösug eies ich-lieare Edwerproblems, welches durch eie geeigee Sraegierasformaio auf eie Sörug eies lieare Problems i eiem faire Mark reduzier werde ka. Im Aschluß beweise wir die Vereilugskovergez der biomiale Großivesoremodelle uer zwei verschiedee Regime vo Marigalmaße. Weil die Übergagswahrscheilichkeie für de Fudamealdaeprozeß uer diese Maße i der Regel vom Akiebesad des Großivesors vor ud ach seier Trasakio abhäge, müsse wir dazu klassische Kovergezresulae auf allgemeie korreliere Irrfahre erweier. Für allgemeie korreliere Irrfahre häg die Richug des ächse Schries vo Zei, momeaer Posiio ud der Richug des leze Schries ab. We eie Folge solcher Irrfahre wie bei Dosker skalier wird, zeige wir, daß sie i Vereilug gege eie Diffusiosprozeß kovergier, desse Diffusioskoeffiziee wir explizi beschreibe. Dabei sell sich heraus, daß im Vergleich zum klassische Fall sowohl Volailiä als auch Drif durch die Korrelaio zwische de Zuwächse der Irrfahre versärk werde. Isbesodere erhale wir ei Kovergezresula für besehede Großivesormodelle aus der Lieraur. Darüber hiaus uersreich usere Arbei die Bedeuug ud de Eifluß, de die Wahl des Preismechaismus ha.

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7 Coes Iroducio 3 1 The Large Ivesor i Discree Time The Marke Mechaism i a Sigle Time Poi Roud-Trips ad a Fair Price The Class of Geeral Price Fucios Exisece ad Uiqueess of a Fair Price The Bechmark Price A Traslaio Ivariace for Expoeial Price Fucios Trasacio Losses The Trasacio Loss Fucio Two Desirable Properies for Trasacio Loss Fucios The Local Trasacio Loss Rae The Biomial Muli-Period Large Ivesor Marke Model The Geeral Dyamic Large Ivesor Price Sysem A Biomial Model for he Fudameals The Large Ivesor s Porfolio Sraegy The Evoluio of he Sock Price The Value of a Porfolio Sraegy Replicaio Defiiios Replicaio of Sar-Covex Coige Claims Paper Value Replicaio Sar-Cocave Porfolios Examples of Large Ivesor Price Fucios Recursive Equaios for Value ad Sraegy No Arbirage ad Marigale Measures No Arbirage for he Large Ivesor Examples of Admissible Tradig Sraegies Three Kids of Marigale Measures Recursive Schemes for he Value Fucios The Value Processes as Super-Marigales Recursive Schemes for he Sraegy Fucio Coecios o Models wih Trasacio Coss Markes wih a Muliplicaive Equilibrium Price Fucio The Sraegy Trasform The Recursive Schemes Revisied Tradig a he Bechmark Price

8 2 CONTENTS 3 Covergece of he Sraegy Fucios Hölder Spaces ad Discree Derivaives The Case wihou Trasacio Losses The Limiig PDEs for he Sraegy Fucios ad heir Trasforms Covergece of he Trasformed Sraegy Fucios Covergece of he Sraegy Fucios Covergece of a Subsequece of Sraegy Fucios The Geeral Case Exisece of a Soluio o he Limiig PDE Covergece of he Trasformed Sraegy Fucios Covergece of he Sraegy Fucios The Limi of he Real Value Fucios Covergece of he Real Value Fucios The Fial Value Problems for Sraegy ad Real Value Revisied Compariso wih Sadard Models Covergece of he Biomial Model Covergece for Geeral Correlaed Radom Walks Covergece uder he p-marigale Measures Geeral Assumpios ad Defiiios Exisece of he p-marigale Measures Covergece of he Fudameals Covergece of he Large Ivesor Price ad he Paper Value Processes The Coiuous-Time Paper Value Fucio The Coiuous-Time Sochasic Model Covergece uder he s-marigale Measures Diffusio Limis for Geeral Correlaed Radom Walks Resuls o Homogeeous Correlaed Radom Walks Our Resuls for Geeral Correlaed Radom Walks Proof of he Mai Covergece Theorem Codiioal Momes of he Correlaed Radom Walk Approximaios for he Auxiliary Fucios Tighess Covergece of he Codiioal Local Momes Employig he Coiuiy i he Time Variable Fial Preparaory Seps Proof of he Mai Covergece Theorem Collecio of Saed Assumpios 272 Bibliography 277

9 Iroducio Sochasic models for opio pricig ca be raced back o he hesis of Bachelier 1900 from he begiig of he las ceury. Bachelier modelled he sock price by a Browia moio wih drif ad he calculaed opio prices as expeced values uder he real-world probabiliy measure. To preve egaive sock prices, Samuelso 1965 proposed o model reurs by a Browia moio so ha he sock price iself becomes a geomeric Browia moio. I ha seup, Black ad Scholes 1973 argued ha he price for a opio o he sock mus coicide wih he price of a replicaig porfolio i sock ad bod. By showig ha replicaig sraegies for Europea calls ad pus exis, hey derived heir celebraed pricig formula. Mero 1973, 1977 exeded he resuls of Black ad Scholes i may direcios. Cox ad Ross 1976 ad more geerally Harriso ad Kreps 1979 showed ha i coras o Bachelier s approach, he Black-Scholes opio price ca be compued as he expecaio of he fial payoff uder he risk-eural measure uder which he sock ears he riskless rae of reur. Aroud he same ime Cox, Ross ad Rubisei 1979 developed a discree approximaio of he Black-Scholes model. They showed ha he Black-Scholes price for a Europea call ca be obaied as he limi of he uique arbirage-free call opio prices i a sequece of suiably scaled biomial models if he ime sep goes o zero. I each biomial model, he opio price ca be foud by elemeary mahemaics because a ay ode of he biomial ree, he required sock ad bod holdigs are deermied by a self-fiacig codiio from he wo possible opio values a he ex odes. This gives a recursio o calculae simulaeously he opio values ad he hedgig sraegy from he fial payoff values a mauriy. The classical lieraure assumes ha all ivesors have he same iformaio, ha he marke is complee i he sese ha every coige claim is aaiable by some replicaig radig sraegy, ha he marke is fricioless, ad ha all ivesors ac as price akers. Of course, hese assumpios oly give a very idealized picure of realiy. To amed his, he Black-Scholes aalysis has bee exeded i umerous ways o accomodae for example icompleeess, rasacio coss, shor-sale cosrais, or asymmeric iformaio. Also he price-akig assumpio has bee relaxed, firs i a discree biomial model by Jarrow 1994, ad he i a coiuous-ime model by Frey 1998 ad ohers. I his hesis, we firs exed he class of price mechaisms cosidered by hese auhors ad he show i his more geeral seig ha a sequece of discree biomial models similar o Jarrow 1994 coverges o a coiuous model which geeralizes Frey If he large ivesor acs as a price aker, his reduces o he Cox-Ross-Rubisei 1979 resul. Jarrow 1992, 1994 sars wih a geeral model for he sock price process i a discree biomial large ivesor model. To exclude marke maipulaio geeraed by radig i sock ad bod, he assumes ha he sock price depeds o ime, some fudameals, ad o he curre bu o o he previous sock holdigs of he large ivesor. If he large ivesor is allowed o rade i a derivaive of he sock, he markes for he sock ad is derivaive mus be i sychroy o preve marke maipulaio sraegies for he large ivesor. Jarrow 1994 he shows by example ha i such a sychroous marke, he price 3

10 4 INTRODUCTION ad he large ivesor s hedgig sraegy for a Europea call ca sill be derived as i he Cox-Ross-Rubisei model by a backward recursio. I coras o he sadard biomial model, however, he sock prices i he recursive formulæ deped o he large ivesor s hedgig sraegy, which iduces addiioal volailiy. Frey 1998 sars wih a geeral reacio fucio which describes i a emporary equilibrium he sock price as a fucio of ime, some fudameal value modelled as a geomeric Browia moio, ad he large ivesor s sock holdigs. He coceraes o he replicaio price a which a large ivesor ca perfecly replicae a opio wih a sufficiely smooh payoff. The replicaig sraegy is give via a marigale represeaio for he fial opio value, where he radig sraegy appears o oly as iegrad, bu also i he iegraor of he sochasic iegral which describes he large ivesor s cumulaive gais from rade. Frey 1998 rasforms his sochasic represeaio io a quasi-liear fial value problem for he large ivesor s sraegy fucio, paramerized by ime ad fudameals. He proves exisece ad uiqueess of soluios o ha fial value problem ad discusses he qualiaive differece of he large ivesor s replicaig sraegy compared o he correspodig hedgig sraegy i he classical Black-Scholes model. While Jarrow ad Frey work wih a exeral fudameal sae variable, Schöbucher ad Wilmo 2000 ad Sircar ad Papaicolaou 1998 use he feedback perurbed price process as he observable process. Schöbucher ad Wilmo 2000 sudy he price dyamics i illiquid markes ad is reacio o he radig sraegy of a large ivesor. Their aalysis leads o a parial differeial equaio for he replicaio paper value of a opio, which is equivale o Frey s 1998 descripio via he associaed sraegy. Sircar ad Papaicolaou 1998 derive he same parial differeial equaio ad perform a exesive asympoic ad umerical sudy by cosiderig he olieariy as a perurbaio o he classical Black- Scholes parial differeial equaio. Weak covergece quesios for discree large ivesor models have already bee examied by Frey ad Sremme 1997 ad Bierbaum These auhors assume i heir resuls he covergece boh of he sraegy fucios used by he large ivesor ad of he discree fudameal price processes. I coras, we derive here he covergece of hese wo sequeces direcly from he opio replicaio resul i he discree biomial models. I relaio o he exisig lieraure o large ivesor models, his hesis makes wo mai coribuios. We firs iroduce ad aalyze i deail a exesive class of discree biomial models for opio replicaio ad opio valuaio wih a large ivesor. The we show ha hese discree models coverge i disribuio o cerai diffusio models. Similarly as i Jarrow 1994 or Frey 1998, he equilibrium price o iervals where he large ivesor does o rade is modelled as a fucio of ime, some fudameal value ad he large ivesor s sock holdigs. I discree ime, we carefully explore he price mechaism which deermies he sock price a which he large ivesor acually rades. This icludes a ivesigaio of rades a he iiial ad fial radig daes, ad covers boh permae ad emporary price impacs which may resul from he large ivesor s aciviy. The correspodig coiuous-ime limis provide ew isighs io he assumpios abou he price mechaism i exisig large ivesor models. I he discree model, he biomial ree for he releva sock prices is sill recombiig if he large ivesor uses pah-idepede radig sraegies. We derive exisece ad uiqueess of such a sraegy which replicaes a give coige claim. If he sock price does o compleely adjuss o a order of he large ivesor before i is execued, he large ivesor s replicaig sraegy is o give explicily as i Cox, Ross ad Rubisei 1979, bu oly as a soluio o a fixed poi problem. Is o-lieariy makes he subseque covergece aalysis more difficul o hadle ad forces us o derive quie precise asympoic error esimaes.

11 INTRODUCTION 5 Oe impora isigh ha emerges from our aalysis is ha i large ivesor models, he mai focus should be o he radig sraegy ad o o he value process. I paricular, he self-fiacig codiio is a codiio o he sraegy, ad i uiquely deermies he laer from a give fial posiio. Bu as already observed by Jarrow 1992 ad Schöbucher ad Wilmo 2000, here are a leas wo differe mehods o assess a large ivesor s sraegy: is paper value ad is real value. We give a ew ierpreaio of he real value as he porfolio liquidaio value uder a fair price sysem i he large ivesor marke. We also derive codiios o he radig sraegy which exclude paper-value arbirage, ad we show ha a large class of price sysems does o allow real-value arbirage opporuiies. The covergece i disribuio of our biomial large ivesor models o a coiuous-ime diffusio is proved i wo seps. We firs show ha he large ivesor s sraegy fucios coverge owards some limi fucio ad he use his o show ha he sequece of fudameal processes coverges i disribuio. The above limi fucio is he soluio of a geerally highly o-liear fial value problem; his ca be subsaially simplified by a suiable iegral rasform. The rasformed problem ca he be viewed as a perurbaio of a liear problem i a large ivesor marke wih a fair pricig sysem. To obai he covergece of he fudameal processes i a sequece of large ivesor models, we have o sudy geeral correlaed radom walks, for which he direcio of he ex move depeds o ime, he curre posiio ad he direcio of he previous move. We prove ha a sequece of such correlaed radom walks which are scaled as i Dosker s heorem coverges o a diffusio limi, ad we ideify he diffusio coefficies. The volailiy ad drif of he limi process are reiforced due o he correlaio bewee he icremes of he discree radom walks. These resuls are of idepede mahemaical ieres ad cosiue aoher mai coribuio of he hesis. The basic covergece heorem for geeral correlaed radom walks is firs applied o prove he covergece i disribuio of a sequece of biomial models uder he associaed p- marigale measures uder which boh he large ivesor sock price ad he paper value of he large ivesor s porfolio are marigales. The rasiio probabiliies for he fudameals ca here deped o he wo previous values, sice he large ivesor sock price is a fucio of he large ivesor s sock holdigs boh before ad afer his rade. Hece we eed he full sregh of he heorem. For he paricular class of models where he sock price compleely adjuss o a order of he large ivesor before ha is execued, he rasiio probabiliies oly deped o he las value of he fudameals. Here we esablish ha a suiably scaled sequece of Jarrow s 1994 biomial models coverges o Frey s 1998 model uder he p-marigale measures. We ow give a more deailed overview of he various chapers i his hesis. I Chaper 1, we prese he discree-ime biomial model of a large ivesor marke. Like Frey 1998, we model he equilibrium sock price i he marke as a fucio of ime, of some fudameal value, ad of he curre sock holdigs of he large ivesor. However, his price is oly valid if he large ivesor is iacive a his poi i ime, ad oe mus model very precisely wha happes whe he large ivesor rades a o-ifiiesimal umber of shares. The large ivesor price a which he large ivesor ca acually sele his rades is defied as a weighed average of equilibrium prices so ha he price sysem i he large ivesor marke is described by a pair ψ, µ of a equilibrium price fucio ψ ad a price deermiig measure µ. The choice of µ represes he way ha he marke reacs o he large ivesor s order. Our seig covers he mechaisms used i he papers of Frey ad Sremme 1997, Bierbaum 1997, Frey 1998, Sircar ad Papaicolaou 1998, Schöbucher ad Wilmo 2000, Josso, Keppo ad Meg 2004 ad Baum 2001, who all assume ha he marke immediaely ad fully adjuss o a order of he large ivesor. I also icludes he

12 6 INTRODUCTION oher exreme where he large ivesor ca rade a he old equilibrium price wih he price oly adjusig direcly afer he large ivesor s rade; his mechaism is implied i Plae ad Schweizer 1998 whe swichig from he ih o he i + 1h model. I addiio o he above permae price impacs we ca also model emporary price impacs i he spiri of Bersimas ad Lo 1998, Bersimas, Hummel ad Lo 2000, Almgre ad Chriss 1999, 2000, Huberma ad Sazl 2004, 2003, Baksei 2001, or Çei, Jarrow ad Proer For a praciioer s view o he marke mechaism i he presece of a large ivesor see also Taleb A key role is played by he heoreical bechmark price defied as he arihmeic average of all equilibrium sock prices which correspod o a fixed sock posiio of he large ivesor lyig bewee he sock holdigs before ad afer his rade. I saisfies he properies of a fair price ad ca be used o specify implied rasacio losses relaive o he bechmark price. This provides a lik o small ivesor models wih rasacio coss as sudied i discree ime by Boyle ad Vors 1992 ad Opiz Sice he sock price is affeced by he large ivesor s sock holdigs, i is o a priori clear how o value a porfolio of he large ivesor. Chaper 1 iroduces wo differe coceps for his. Oe is a mark-o-marke approach which simply uses he las price see o he marke by he large ivesor o assess his complee sock holdigs. The porfolio value obaied from his valuaio is called paper value as i Jarrow 1992 ad Schöbucher ad Wilmo Frey 1998 also implicily values he large ivesor s porfolio by meas of he paper value. The secod approach cosiders he real value of he large ivesor s porfolio, defied as he heoreical liquidaio price he large ivesor could achieve if he sold his porfolio wihou ay rasacio losses. This coicides wih he real value of Schöbucher ad Wilmo 2000 who defie his as he limi of successive small block rades. Hece we obai a ew ierpreaio of why he real value is a good proxy for he acual value of he large ivesor s porfolio. Afer fixig he oe-period price mechaism, we exed he large ivesor model o a dyamic muli-period seig where he fudameals are give by a biomial radom walk. This exeds he biomial models of Jarrow 1994 ad Baksei For pah-idepede radig sraegies, he biomial ree described by he vecor of possible large ivesor ad equilibrium sock prices is sill recombiig. For a suiable class of coige claims, his allows oe o deermie a self-fiacig opio replicaio sraegy alog he lies of Cox, Ross ad Rubisei Bu i coras o he explici Cox-Ross-Rubisei case, if he sock price does o compleely adjus o a order of he large ivesor before is execuio, he large ivesor s sraegy is oly give as he soluio o a fixed poi problem for which we show exisece ad uiqueess. Moreover, oe mus carefully ivesigae he behavior of he large ivesor sock price a he iiial ad fial radig daes. I he Cox-Ross-Rubisei model, he discoued value process of ay self-fiacig radig sraegy is a marigale uder he uique measure which makes he discoued price process a marigale. This gives a recursive formula for he value fucio ad also shows ha his small ivesor marke is free of arbirage. Chaper 2 coais similar resuls for our large ivesor marke, bu we ow eed o disiguish bewee paper value ad real value arbirage. For a aural class of self-fiacig radig sraegies, he paper value process is a marigale uder he p-marigale measure, he uique measure which urs he large ivesor price process io a marigale, ad so his class is free of paper-value arbirage. However, he p-marigale measure is highly depede o he large ivesor s radig sraegy. If he equilibrium price fucio has a muliplicaive srucure, he real value process is always a supermarigale uder he s-marigale measure, which is he marigale measure for he associaed small ivesor price process. Hece such a marke srucure permis o

13 INTRODUCTION 7 real-value arbirage opporuiies. Sufficie codiios o exclude arbirage for he large ivesor have bee give i differe models by Jarrow 1992, 1994, Bak 1999, Baum 2001, Bak ad Baum 2004, Huberma ad Sazl 2004, Çei, Jarrow ad Proer The marigale propery of he paper value has already bee used by Frey 1998 o deermie he replicaig sraegy for a coige claim i his coiuous-ime model. I Chaper 2, we also derive a implici differece equaio of secod order for he large ivesor s sraegy fucio ξ i he h biomial model. This forms he basis of he subseque covergece aalysis i Chaper 3. We he explai he similariies wih he proporioal rasacio cos models of Boyle ad Vors 1992 ad Musiela ad Rukowski These sugges o focus o muliplicaive price sysems, where he impac from he large ivesor s sock holdigs eers he equilibrium price i a muliplicaive way. For such price sysems he recursios for he sraegy fucio ad he real value process simplify cosiderably. If he large ivesor does o face ay rasacio losses, he real value process eve becomes a marigale uder he s-marigale measure, every coige claim is aaiable, ad we ca explicily calculae is replicaig sraegy. Chaper 3 is devoed o he covergece of a sequece {ξ } IN of sraegy fucios from biomial large ivesor models as he ime sep goes o zero. Uder suiable assumpios, he limi fucio ϕ mus saisfy a parial differeial equaio which is he coiuous aalogue of he differece equaio for ξ. Togeher wih he covergece a he fial dae, his gives a fial value problem for ϕ, ad we prove exisece ad uiqueess of a soluio. The we show ha he covergece of he discree sraegy fucios {ξ } IN o ϕ follows from he covergece of heir values immediaely before ad a mauriy o he correspodig values of ϕ. I geeral, he fial value problem for ϕ is highly o-liear, bu i ca be simplified o a quasi-liear problem by workig wih rasformed fucios g = g ξ ad γ = g ϕ. If he price sysem excludes ay isaaeous rasacio gais or losses, he rasformed problem is eve liear, each g ca be calculaed by a explici recursive scheme from is values a ad immediaely before mauriy, ad he limi γ saisfies a liear fial value problem. Thus exisece ad uiqueess of soluios o he fial value problem as well as he covergece of he rasformed sraegy fucios follow from classical resuls. If he price sysem does o preve rasacio losses, however, he recursive scheme for g remais implici ad he fial value problem for γ is oly quasi-liear. We adap a proof by Frey 1998 o show ha eve i his seig he fial value problem for γ sill has a soluio if he boudary values a mauriy do o become oo large. We he geeralize he covergece resul for {g } IN o his geeral case ad rasform he resuls back io correspodig resuls for he sraegy fucios {ξ } IN ad heir limi ϕ. The deailed ivesigaio of he sraegy as a fucio of ime ad fudameals is o eeded i he sadard Cox-Ross-Rubisei model; sice he sock price is o affeced by he sraegy, he covergece of he value process ca be show wihou usig he covergece of he sraegy. Bu i he large ivesor model, covergece of he sraegy fucios is esseial o deduce he covergece i disribuio of he biomial large ivesor models. Covergece of he sraegy fucios is also a key assumpio for he covergece resuls of Frey ad Sremme 1997 ad Bierbaum Oce he covergece of he sraegy fucios is show, we use his o derive a similar covergece resul for he real value fucios. The parial differeial equaio saisfied by heir limi v resembles he srucure kow from coiuous-ime models wih proporioal rasacio coss as he coiuous-ime limis of he models of Lelad 1985 ad Boyle ad Vors 1992 or he coiuous-ime model of Barles ad Soer This srucural resemblace has also bee observed by Frey 2000.

14 8 INTRODUCTION Ad like i he Black-Scholes model, he limiig sraegy fucio ϕ is a rasform of he spaial derivaive of he real value fucio v. If he price sysem does o iduce rasacio losses, he parial differeial equaio for v agai becomes liear ad basically reduces o he Black-Scholes equaio. Such a behavior was also discovered by Josso, Keppo ad Meg I Chaper 4, we ivesigae he covergece i disribuio of a sequece of biomial large ivesor models. Apar from he covergece of he sraegy fucios, he oher key assumpio for he covergece resul i Frey ad Sremme 1997 is he covergece of he discree-ime fudameal processes o a coiuous-ime diffusio. We give codiios o he parameers of he biomial models which acually imply he covergece i disribuio of he fudameal processes, ad we explicily deermie he coefficies of he limiig diffusio i erms of he price sysem ψ, µ ad he limiig sraegy fucio ϕ. I is he sraighforward o prove by he coiuous mappig heorem he covergece of all oher model-releva processes like price, sraegy ad value. Of course, covergece i disribuio always depeds o he uderlyig probabiliy measures, ad his becomes a issue for he large ivesor model. We show covergece uder wo differe regimes: he p-marigale measures ad he s-marigale measures. Uder esseially he same assumpios which guaraee he covergece of he sraegy fucios i Chaper 3, we ca apply a covergece heorem for geeral correlaed radom walks from Chaper 5 o deduce he desired covergece. I he paricular case where he equilibrium price compleely adjuss o a order of he large ivesor before he order is acually execued, he covergece uder he p-marigale measures implies he covergece of a suiably scaled versio of Jarrow s 1994 model o he model of Frey By wriig he paper value i he limi model as a fucio of ime ad sock price, we also obai a o-liear parial differeial equaio for he coiuous paper value fucio which geeralizes he correspodig parial differeial equaios of Schöbucher ad Wilmo 1996, 2000, Sircar ad Papaicolaou 1998, Frey 2000, ad Frey ad Paie The siuaio uder he s-marigale measures is cosiderably simpler, ad he limi of he fudameals is jus a Browia moio wih drif. I he absece of a large ivesor, he resuls uder he p- ad s-marigale measures coicide ad we recover he covergece of he Cox-Ross-Rubisei models o he Black-Scholes model as a special case. The key igredie for he resuls i Chaper 4 is a covergece heorem for geeral correlaed radom walks. This is a mahemaical coribuio of idepede ieres which is preseed i Chaper 5. For a correlaed radom walk, he direcio of he ex move depeds o is il, i.e., o he direcio of he move i he previous sep. Bu for our applicaio i Chaper 4, we eed geeral correlaed radom walks where he direcio of he ex move ca also deped o ime ad he curre posiio i space. If a sequece of such walks is scaled as i Dosker s heorem ad if for each possible direcio of he radom walk s previous move, he rasiio probabiliies coverge o a possibly differe limi fucio, our mai covergece heorem for geeral correlaed radom walks saes ha he rescaled sequece coverges i disribuio o he soluio of a sochasic differeial equaio. The geeraor of he laer is explicily give, ad a posiive correlaio bewee he direcio of wo successive moves icreases he volailiy of he limi process. As a corollary, we show ha geeral correlaed radom walks ca be used o approximae geeral diffusio processes via a recombiig biomial ree. While he proof of he mai covergece heorem is based o sadard ideas, he deails become raher ricky ad ivolved because he correlaio bewee wo successive icremes of he radom walk eed o vaish asympoically. I is esseial o carefully selec proper leses o look a our radom walk, sice is behavior a a microscopic level o ime iervals

15 INTRODUCTION 9 of order Oδ 2 is very differe from he large picure o ime iervals of legh Oδ which prevails i he limi. Correlaed radom walks, which were iroduced by Gillis 1955 ad Moha 1955, are impora objecs o heir ow ad have a variey of applicaios ouside mahemaical fiace. A overview of some of he lieraure is give i Secio 5.1. Bu up o ow, research has almos exclusively focused o ime- ad space-homogeeous correlaed radom walks which are much easier o hadle ha he geeral ihomogeeous case. Thus our covergece resuls oiceably exed resuls of Reshaw ad Hederso 1981, Szász ad Tóh 1984, Tóh 1986, Opiz 1999 ad Mauldi, Moicio ad Weizsäcker 1996 o he covergece of homogeeous correlaed radom walks. Ackowledgeme I is a grea pleasure o hak my advisor M. Schweizer for ecouragig me o sar his work ad for his may valuable commes ad suggesios. He geerously shared wih me his wide kowledge of probabiliy heory ad mahemaical fiace ad coiuously suppored me whe he compleio of he hesis ook loger ha expeced. I would also like o express my deep graiude o R. Frey ad A. Schied for readily accepig he ask of beig co-examiers. Special haks go o E. Waymire who firs iroduced me o he field of fiacial mahemaics ad o V. Schmid for his ecourageme. My colleagues ad frieds P. Bak, D. Baum, B. D lugaszewska, D. Becherer, R. Dahms, S. Dereich, G. Dimiroff, F. Esche, U. Hors, B. Niederhauser, T. Rheiläder, S. Weber ad may ohers meri a special oe of haks for umerous discussios ad all heir compee ad moral suppor. Fiacial suppor by he Deusche Forschugsgemeischaf via Graduierekolleg Sochasische Prozesse ud Probabilisische Aalysis ad via SFB 373 Quaifikaio ud Simulaio Ökoomischer Prozesse as well as idealisic suppor via Bischöfliche Sudieförderug Cusauswerk is graefully ackowledged.

16 10

17 Chaper 1 The Large Ivesor i Discree Time I his chaper we prese he discree, biomial model of a large ivesor marke, which forms he basis of our covergece aalysis i Chaper 3 ad 4. A he begiig, we have o describe he marke mechaism i some more deail. The marke is supposed o be i a Walrasia equilibrium as log as he large ivesor does o rade. I is he esseial o model very precisely he sock price movemes whe he large ivesor rades a oifiiesimal umber of shares, ad because of is imporace, we sar wih such a model i a saic world. Our discussio will reveal he sigificace of a cerai bechmark price which ca he be used i order o specify implied rasacio losses i he large ivesor marke model. Afer havig described he price mechaism i a sigle ime poi, we ur o a geeral dyamic muli-period large ivesor marke model, which also depeds o ime ad o he evoluio of some exeral fudameals give by a biomial radom walk, ad defie self-fiacig radig sraegies ad porfolio values i a way similar o small ivesor marke models. However, we have o differeiae bewee he paper value ad he real value. Uder cerai assumpios o he price sysem of our large ivesor model saed i erms of he associaed rasacio losses we he prese he class of sar-covex coige claims, which are defied i erms of he large ivesor s fial sock holdigs a mauriy, ad we show ha all hose coige claims are aaiable by a replicaig radig sraegy. A similar resul holds for he replicaio of a cerai paper value. I coras o he sadard Cox-Ross-Rubisei model, he correspodig replicaig sraegies will i geeral oly be give as soluios o a fixed poi equaio, ad he derivaio of a exisece ad uiqueess resul for his fixed poi equaio is a ceral resul of his chaper. Las bu o leas we give examples of large ivesor price sysems which saisfy he assumpios eeded for hese resuls, ad we show ha our large ivesor model coais he Cox-Ross-Rubisei model as a special case. 1.1 The Marke Mechaism i a Sigle Time Poi I a large ivesor fiacial marke here exiss oe large ivesor who ca affec he sock prices by his radig. Because of he large ivesor s ifluece o he sock prices, he sock price will vary depedig o he rades of he large ivesor, eve if he ime ad he fudameal iformaio a his ime is kep cosa. Especially, allowig he large ivesor o perform several subseque rades a oe poi i ime, he large ivesor migh eve realize immediae arbirage opporuiies due o price maipulaio echiques, which a small ivesor cao apply. 11

18 12 CHAPTER 1. THE LARGE INVESTOR IN DISCRETE TIME Before we develop a fully dyamic large ivesor model where he large ivesor s sock price also depeds o ime ad some fudameal iformaio a his ime, we will firs focus o he price mechaism i a sigle ime poi, i.e. before some ew fudameal iformaio arrives. I such a siuaio we carve ou codiios o geeral large ivesor s price fucios which esure ha ay roud-rip of he large ivesor excludes ay rasacio gais or losses. These codiios are saisfied by he bechmark price, which is cosruced as he mea of equilibrium prices. For he impora class of expoeial equilibrium price fucios, he bechmark price eve is he uique price fucio which excludes boh immediae rasacio gais ad immediae rasacio losses. Thus, le us assume ha a some fixed ime or i some ime ierval i which o ew iformaio arrives he marke here exis some fucio S : IR 2 IR such ha for each ξ 1, ξ 2 IR he large ivesor is faced wih a per-share price of Sξ 1, ξ 2 whe shifig his sock posiio from ξ 1 o ξ 2. Supposig ha he rades of he large ivesor are woud off much faser ha ew iformaio appears i he marke, we ca ake for graed ha he large ivesor ca coduc several rasacios accordig o his price buildig mechaism. I a idealized world, all rasacios do o ake ime a all, such ha he large ivesor ca perform ifiiely may rasacios Roud-Trips ad a Fair Price Depedig o he paricular form of he price mechaism described by S : IR 2 IR he large ivesor migh profi or suffer from buyig ad sellig socks. I his secio we look for codiios which a price fucio S : IR 2 IR has o saisfy o be a fair price i ha he large ivesor does o gai or lose ay moey from roud-rips. I order o sar, suppose ha he large ivesor iiially holds ξ shares of sock. The he could buy α 0 shares a a oal price of αsξ, ξ+α ad he sell hese α shares immediaely a a oal price of αsξ + α, ξ. Overall, his ivesme coss him C + ξ, α := α Sξ, ξ + α Sξ + α, ξ. If he coss C + ξ, α are egaive, his meas ha he large ivesor receives he amou C + ξ, α as a resul of his wo rasacios. I his case he large ivesor could repea he game over ad over ad basically ear ay posiive amou oe ca imagie; his would be a immediae arbirage opporuiy for he large ivesor. Similarly, he large ivesor could also sell α socks ad he re-buy hem leadig o oal coss of C ξ, α := α Sξ α, ξ Sξ, ξ α. The wo sraegies described above are simple forms of roud-rips. A roud-rip is a sraegy o buy ad sell socks i such a way ha he iiial sock posiio is re-aaied a he ed. I mahemaical erms we use he followig defiiio: Defiiio 1.1. A deermiisic k-sep roud-rip is a vecor α IR k which saisfies k i=1 α i = 0. For all k IN we deoe he space of all k-sep roud-rips by R k. The coss associaed wih a roud-rip α R k sarig a level ξ IR are give by C k ξ, α := k i=1 α i S ξ + i 1 α j, ξ + j=1 i α j. Remark. By he defiiio of R k i is obvious ha R k is he k 1-dimesioal space orhogoal o he vecor 1, 1,..., 1 r IR k. j=1

19 1.1. THE MARKET MECHANISM IN A SINGLE TIME POINT 13 A large ivesor wih a iiial sock holdig ξ who applies he roud-rip α R k chages his sock holdigs accordig o he scheme k 1 ξ ξ + α 1 ξ + α 1 + α 2 ξ + α j ξ. Like i he exemplary buy-ad-sell case, i is clear ha if he large ivesor sars wih a sock posiio ξ IR ad here exis some k IN ad some roud-rip α R k such ha he associaed coss C k ξ, α are sricly egaive, he he large ivesor has a arbirage opporuiy. A fair price mechaism i a large ivesor marke would be a price mechaism which excludes ay isaaeous rasacio gais ad rasacio losses from roud-rips. Thus, a fair price fucio S : IR 2 IR should saisfy k Ck ξ, α := i=1 i 1 α i S ξ + α j, ξ + j=1 j=1 i α j = 0 for all ξ IR, α R k, ad k IN 1.1 j=1 Isead of verifyig 1.1 for all k IN i suffices o check i for k = 3, as he followig proposiio shows: Proposiio 1.2. Codiio 1.1 holds for all k IN if ad oly if for all ρ [0, 1] ad ξ, d IR. ρs ξ, ξ + ρd + 1 ρs ξ + ρd, ξ + d = S ξ + d, ξ 1.2 Proof. I is clear ha 1.2 is ecessary for 1.1, sice he former follows from he laer by akig α = ρd, 1 ρd, d r R 3. I remais o show ha 1.2 for all ρ [0, 1] ad ξ, d IR is also sufficie for 1.1. For his reaso le us fix k IN ad suppose ha 1.2 holds for all ρ [0, 1] ad ξ, d IR. Sice he oly roud-rip α R 1 is α = 0, i is clear ha 1.1 holds for k = 1. For k = 2 a roud-rip α R 2 mus have he form α = d, d for some d IR. The 1.1 is implied by 1.2 wih ρ = 1 which gives he symmery S ξ, ξ + d = S ξ + d, ξ for all ξ, d IR. 1.3 Le us ow come o he case k = 3 ad ake some α R 3. Wihou loss of geeraliy we ca assume α i 0 for all i {1, 2, 3}, oherwise we are back i he case k = 2. By he defiiio of R 3 we have α 1 + α 2 + α 3 = 0. The oe of he α i s has he opposie sig of he wo ohers, i.e. here exis a i {1, 2, 3} wih sg α i = sg αj for j i. We will firs assume i = 3 ad se d = α 3. Sice 3 i=1 α i = 0, α i 0 for i {1, 2, 3}, ad i = 3 we have hece we see ha α 1 = ρd ad α 2 = 1 ρd for some ρ 0, 1, C 3ξ, α = ρds ξ, ξ + ρd + 1 ρds ξ + ρd, ξ + d ds ξ + d, ξ. Due o 1.2 his erm vaishes, hus we have proved 1.1 for k = 3 if i = 3. The cases i = 1 ad i = 2 follow similarly. For example if i = 1 we ca se ˆα 1 = α 2, ˆα 2 = α 3, ad ˆα 3 = α 1 as well as ˆξ = ξ + α 1 o coclude from 3 i=1 ˆα i = 0 ha C 3ξ, α = ˆα 3 S ˆξ + ˆα1 + ˆα 2, ˆξ + ˆα 1 S ˆξ, ˆξ + ˆα1 + ˆα2 S ˆξ + ˆα1, ˆξ + ˆα 1 + ˆα 2 = 0

20 14 CHAPTER 1. THE LARGE INVESTOR IN DISCRETE TIME which of course simplifies o C 3ξ, α = 3 i 1 ˆα i S ˆξ + ˆα j, ˆξ i + ˆα j = C3ˆξ, ˆα. 1.4 i=1 j=1 j=1 I sill remais o prove 1.1 for k > 3. Here we are goig o use a iducive argume. Le us assume ha for some k > 3 we have show k 1 i 1 α i S ξ + α j, ξ + i=1 j=1 i α j = 0 for all ξ IR ad α R k j=1 The we have o show ha 1.1 eve holds for all ξ IR ad α R k. Thus, le us fix ξ IR ad some roud-rip α k R k. We he fragme his k-sep roud-rip io oe k 1-sep roud-rip α k 1 ad a 3-sep roud-rip β 3 by defiig α k 1 R k 1 by ad he vecor β 3 R 3 by The we have for 1 i k 2: for i = k 1: ad for i = k: k 2 αi k 1 = αi k for 1 i k 2 ad α k 1 k 1 = αj k, k 2 β1 3 := αj k, β2 3 = αk 1 k, ad β3 3 = αk k. j=1 i 1 αi k S ξ + αj k, ξ + j=1 i j=1 k 2 k 1 αk 1 ξ k S + αj k, ξ + j=1 k 1 αk ξ k S + αj k, ξ + j=1 j=1 k j=1 α k j α k j α k j Fially, by he defiiios of α k 1 k 1 ad β3 1 α k 1 k 1 S k 2 ξ + j=1 k 1 j, ξ + α k 1 j=1 α k 1 j Thus 1.1 wih α = α k is equivale o k 1 i 1 i αi k 1 S ξ + αj k 1, ξ + i=1 j=1 j=1 = αi k 1 S = β 32S = β 33S i 1 ξ + ξ + ξ + j=1 j=1 α k 1 j, ξ + 1 βj 3, ξ + j=1 2 βj 3, ξ + j=1 2 j=1 3 j=1 β 3 j β 3 j i j=1,. α k 1 j ad he wo-dimesioal case 1.3, we ge α k 1 j = β 3 1S ξ + β 3 1, ξ = β 3 1S ξ, ξ + β i 1 βi 3 S ξ + βj 3, ξ + i=1 j=1 i j=1 β 3 j, = 0 ad his holds rue because of he iducio hypohesis 1.5 ad he case for k = 3. q.e.d.

21 1.1. THE MARKET MECHANISM IN A SINGLE TIME POINT 15 Remark. Jarrow 1992 derives sufficie codiios for he o-exisece of marke maipulaio sraegies i discree muli-period large ivesor markes. Oe of hese codiios preves ay radig sraegies where he large ivesor esablishes a red ad he rades agais i before he marke collapses. I our model which was resriced o a sigle ime poi, or a leas a ime ierval i which o ew iformaio occurs ad solely he large ivesor has a sigifica impac o he sock price, he codiio 1.1 excludes such marke maipulaig radig sraegies. Of course, he cosa price fucio S : IR 2 IR wih S c for some c IR saisfies he codiios 1.1 ad 1.2. Before we prese o-rivial price fucios which saisfy hese codiios as well, we should ake a closer look a how we wa o model he price mechaism i he large ivesor marke so as o search for fucios wihi he proper class The Class of Geeral Price Fucios I his secio we will prese a class of price fucios S : IR 2 IR which is used o describe he sock price mechaism i a sigle ime poi. The class preseed is moivaed by a aalysis of he marke reacio o rades of he large ivesor, ad allows for various iformaio srucures bewee he small ad he large ivesor. Thus, le us assume ha S : IR 2 IR is some price fucio, such ha he large ivesor is faced wih a price of Sξ 1, ξ 2 whe he shifs his sock holdigs from ξ 1 o ξ 2. If he large ivesor holds ξ IR shares ad does o rade, he sock price i he marke will be fξ := Sξ, ξ. This price fξ ca be viewed as he Walrasia equilibrium price i a marke where ξ shares are held by he large ivesor ad he large ivesor has o addiioal demad or supply. The res of he shares is assumed o be held by he small ivesors, ad hus he price fξ could have bee derived from he cumulaive excess demad fucio of he small ivesors for ay give cosa sock posiio ξ IR of he large ivesor. Now suppose ha for some ξ 1, ξ 2 IR he large ivesor chages his sock posiio from ξ 1 o ξ 2 shares. I his case he old Walrasia equilibrium a he price fξ 1 is disurbed ad he marke will move owards he ew equilibrium a he price fξ 2. I remais o model i more deail how he rasiio akes place from he old o he ew equilibrium, ad i paricular from he old equilibrium price fξ 1 o he ew equilibrium price fξ 2. Especially, we are ieresed a which per-share price he large ivesor ca rade he ξ 2 ξ 1 socks eeded o shif his sock holdigs from ξ 1 o ξ 2 shares. This quesio basically leads back o he quesio how he iformaio abou he large ivesor s rade is oiced by he marke paricipas. Depedig o he iformaio srucure he small ivesors ca adap heir sock holdigs more or less quickly o he sock holdigs which hey prefer i he ew equilibrium. We will illusrae his wih wo simple examples. Example 1.1. Suppose ha our marke cosiss of he large ivesor ad ifiiely may ifiiesimally small ivesors. As log as he large ivesor holds ξ 1 IR socks, we are i he old equilibrium a he sock price fξ 1 ; hus each small ivesor is willig o buy ad sell a ifiiesimal share of he sock a a price fξ 1 per sock. Wihou loss of geeraliy le us assume ha ξ 1 < ξ 2, i.e. he large ivesor was o buy ξ 2 ξ 1 socks. He could achieve his goal by eerig separae coracs wih all he small ivesors o buy a ifiiesimal small amou of socks from each of he ifiiely may small ivesors, such ha overall he has bough ξ 2 ξ 1 socks. I his case he small ivesors oice he disurbace of he old equilibrium wih a cerai delay, ad he large ivesor ca realize a per-share price of fξ 1. Afer he large ivesor s rade he small ivesors have o adjus heir idividual sock holdigs accordig o heir idividual excess demad fucios, such ha he ew equilibrium price fξ 2 will quickly

22 16 CHAPTER 1. THE LARGE INVESTOR IN DISCRETE TIME emerge. The large ivesor ca realize he price fξ 1 sice he socks are exchaged before he small ivesors are aggregaed. I he ex example he iformaio srucure is reversed ad he demad of he small ivesors is aggregaed before he large ivesor s rade is execued. Example 1.2. We oce agai cosider a marke wih oe large ivesor ad ifiiely may ifiiesimally small ivesors, bu ow suppose ha he large ivesor does o or cao eer io coracs wih each small ivesor separaely, bu buys he ξ 2 ξ 1 > 0 socks eeded o shif his sock holdigs from ξ 1 o ξ 2 a he sock exchage. Sice he large ivesor was o shif his porfolio regardless of he sock price he ca obai, he has o place a ulimied order. Due o he addiioal demad for socks a he sock exchage here will be much more small ivesors whose sell orders ca be acceped o maximize he volume of sales, ad he price fixed by he broker will be fξ 2. Sice a similar reasoig works if he large ivesor sells socks, we ca coclude ha if he small ivesors are aggregaed before he socks are exchaged he large ivesor ca oly realize a price fξ 2. Depedig o he realisic problem, i is easy o hik of cases where we have a siuaio i bewee he wo exremes of he precedig examples: Perhaps o all of he small ivesors are of he same size, here migh be a few larger oes wih whom he large ivesor is willig o eer io separae coracs o a par of he socks he is goig o buy or sell, ad he migh buy/sell he res a possibly differe sock exchages. If we kow for each ξ IR he equilibrium sock price fξ which would appear i he marke wheever he large ivesor held he fixed amou of ξ shares, he we ca model he price mechaism i he large ivesor marke by iroducig a price-deermiig probabiliy measure µ o IR, which reflecs he iformaio srucure bewee he small ivesors ad he large ivesor. Namely, we he model he price fucio S : IR 2 IR which describes he per-share price of he large ivesor s rasacio of ξ 2 ξ 1 shares o chage his sock holdigs from ξ 1 o ξ 2 by Sξ 1, ξ 2 = f 1 θξ 1 + θξ 2 µdθ for all ξ1, ξ 2 IR. 1.6 Wihi his seig we recover he price-buildig mechaism of Example 1.1, where he price-deermiig measure is he Dirac measure δ 0 coceraed i 0, ad he mechaism of Example 1.2 where µ = δ 1, i.e. he Dirac measure coceraed i 1. Mos aural are price-deermiig measures which lie bewee hese wo Dirac measures, i.e. which are probabiliy measures o [0, 1]. However i akig for example µ = δ x for some x > 1, we ca also model price dyamics, where he marke a firs overreacs because of he sudde addiioal large ivesor s supply or demad, sice here is o eough isaaeous liquidiy i he marke. Remark. The represeaio 1.6 ca be rewrie as Sξ, ξ + α = We will use boh represeaios i he sequel. fξ + θα µdθ for all ξ, α IR. 1.7 I order o guaraee he exisece of he price fucios of he form 1.6 we suppose ha he fucio f : IR IR describig he equilibrium sock prices is Lebesgue-measurable ad locally bouded, ad ha he price-deermiig measure µ is such ha he iegral i 1.6 exiss for all ξ 1, ξ 2 IR. This leads o he followig defiiios:

23 1.1. THE MARKET MECHANISM IN A SINGLE TIME POINT 17 Defiiio 1.3. Le MIR deoe he se which coais all probabiliy measures o IR. For every Lebesgue-measurable ad locally bouded fucio f : IR IR he family Sf of geeral price fucios based o f is give by he se of all fucios S : IR 2 IR saisfyig 1.6 for some price-deermiig measure µ Mf := { µ MIR : The class of geeral price fucios is he give by f 1 θξ 1 + θξ 2 µdθ < for all ξ 1, ξ 2 IR}. 1.8 S := { S S Sf for some Lebesgue-measurable ad locally bouded f : IR IR }. Moreover, we defie he class S e of expoeial price fucios as he se S e := { S S Sf for some f : IR IR wih fξ = a + be cξ for all ξ IR }. The class of expoeial price fucio urs ou o be a well-suied subclass of geeral price fucios. Especially, all expoeial price fucios are eiher bouded from above if b 0 or from below if b 0, ad i paricular if boh b 0 ad a 0, he each large ivesor price S Sf which is geeraed from he equilibrium price fucio fξ = a + be cξ for all ξ IR is oegaive. Provided ha eve a = 0 ad b > 0, he sock price is eiher cosa if c = 0 or for every x > 0 here exiss some posiio ξ of shares held by he large ivesor, such ha he equilibrium price i he marke becomes fξ = x. Of course, he equilibrium fucio f associaed o a expoeial price fucio S S e is always moooe, ad i is sricly moooe if b, c IR\{0}. Remark. I Secio we will geeralize he equilibrium sock price f so ha i also depeds o ime ad some sochasic process which describes marke fudameals. The defiig equaio 1.6 for he large ivesor s sock price is geeralized accordigly. This will give us a sochasic ad dyamic model for he sock price which is comparable o he usual models of he sock price i large ivesor models as for example i Jarrow 1992, 1994 or Frey Compared o hese models we have modelled more precisely he price mechaism a a radig dae for he large ivesor, ad we have subsaially exeded he class of price mechaism cosidered. For his reaso, we will already pause here o discuss how he sock price i a large ivesor marke is modelled i he lieraure. Jarrow 1992, 1994 models he sock price as a reacio fucio o he large ivesor s rades ad hus assumes ha he marke compleely adjus o a order of he large ivesor before his order is execued. While Jarrow 1992, 1994 sars his discussio wih a very geeral discree sock price process which ca deped o he eire hisory of he large ivesor s sraegy, he ca oly prove absece of arbirage for he large ivesor if he sock price process is idepede of he large ivesor s pas holdigs. I a marke i coiuous ime, Frey ad Sremme 1997 use a marke clearig codiio of zero oal excess demad as iroduced by Föllmer ad Schweizer 1993 o obai a Walrasia equilibrium sock price. By his meas, hey implicly suppose ha he marke adjuss as well o he large ivesor s order before i is acually execued. The model of Frey ad Sremme 1997 has bee applied ad exeded by Frey 1998, Sircar ad Papaicolaou 1998, Plae ad Schweizer 1998, Bierbaum 1997, Baum 2001 ad Bak ad Baum Schöbucher ad Wilmo 1996, 2000 describe i deail he price mechaism; hey assume ha firs he small ivesors ad he large ivesor simulaeously submi a order ad ha he he equilibrium price is deermied. Closely relaed are he models of Kyle 1985 ad Back 1992 for a fiacial marke wih a isider. I all hese models he implied price-deermiig measure is give by µ = δ 1. I Cviaić ad Ma 1996 ad he successioal papers of Buckdah ad Hu 1998 ad Cuoco ad Cviaić 1998 he large ivesor s sock holdigs do o affec he sock price