Stochastic Control in Limit Order Markets

Transcription

1 Sochasic Conrol in Limi Order Markes Curve Following, Porfolio Liquidaion and Derivaive Valuaion D I S S E R T A T I O N zur Erlangung des akademischen Grades Dr. rerum nauralium im Fach Mahemaik eingereich an der Mahemaisch-Naurwissenschaflichen Fakulä II Humbold-Universiä zu Berlin von Dipl.-Mah. Felix Naujoka geboren am in Dresden Präsiden der Humbold-Universiä zu Berlin: Prof. Dr. Jan-Hendrik Olberz Dekan der Mahemaisch-Naurwissenschaflichen Fakulä II: Prof. Dr. Elmar Kulke Guacher: 1. Prof. Dr. Ulrich Hors 2. Prof. Dr. Peer Bank 3. Prof. Dr. Abel Cadenillas eingereich am: Tag der mündlichen Prüfung:

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3 Ich widme diese Arbei meiner Familie.

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5 Absrac Tradiional lieraure on financial markes assumes perfecly liquid markes, so ha an arbirary number of shares can be raded a any ime, and rading has no impac on marke prices. If only limied liquidiy is available for rading, hese assumpions are no always saisfied. In his hesis we sudy a new class of sochasic conrol problems and analyse opimal rading sraegies in coninuous ime in illiquid markes, wih a focus on limi order markes. The firs chaper addresses he problem of curve following in a limi order marke. We consider an invesor who wans o keep his sock holdings close o a given sochasic arge funcion. Applicaions include porfolio liquidaion, hedging and algorihmic rading. We consruc he opimal sraegy which balances he penaly for deviaing and he cos of rading. Trading sraegies comprise boh (absoluely coninuous) marke and passive orders. We firs esablish a priori esimaes on he rading sraegy which allow o prove exisence and uniqueness of an opimal conrol. The opimal rading sraegy is hen characerised in erms of he soluion o a coupled forward backward sochasic differenial equaion (FBSDE) involving jumps via a sochasic maximum principle. Analysing he FBSDE, we give a second characerisaion in erms of buy and sell regions. In he case of quadraic penaly funcions he FBSDE admis an explici soluion. The imporan applicaion of porfolio liquidaion is sudied in deail. Finally, we discuss some counerexamples where marke and passive orders have differen signs. In he second chaper, we allow for a larger class of admissible conrols including he economically more realisic case of discree marke orders. Using echniques of singular sochasic conrol, he resuls of he firs chaper are exended o a wo-sided limi order marke wih emporary marke impac and resilience, where he bid ask spread is also conrolled. We now face an opimisaion problem wih consrains, since passive buy and sell orders are modelled separaely and boh are nonnegaive processes. We firs show exisence and uniqueness of an opimal conrol. In a second sep, a suiable version of he sochasic maximum principle is derived which yields a characerisaion of he opimal rading sraegy in erms of a nonsandard coupled FBSDE. We show ha he opimal conrol can be characerised via buy, sell and no-rade regions. Unlike in he firs par, we now ge a nondegenerae no-rade region, which implies ha marke orders are only used when he spread is small. Specifically, we consruc a hreshold for he spread in erms of he adjoin process. This allows o describe precisely when i is opimal o cross he bid ask spread, a fundamenal problem of algorihmic rading. We also show ha he conrolled sysem can be described in erms of a refleced BSDE. As an applicaion, we solve he porfolio liquidaion problem wih passive orders. When markes are illiquid, opion holders may have an incenive o increase heir porfolio value by using heir impac on he dynamics of he underlying. This problem is addressed in he hird chaper in he framework of sraegically ineracing marke paricipans. We provide a mahemaical framework o consruc opimal rading sraegies under marke impac in a muli-player exension of he model of Chaper 1. Specifically, we consider a financial marke model wih several players ha hold European coningen claims and whose rading has an impac on he price of he underlying. We esablish exisence and uniqueness of equilibrium resuls for risk-neural and CARA invesors and show ha he equilibrium dynamics can be v

6 characerised in erms of a coupled sysem of non-linear PDEs. For he linear cos funcion, we obain a (semi) closed form soluion. Analysing his soluion, we show how marke manipulaion can be reduced. Keywords: Sochasic conrol, Maximum principle, BSDEs, Illiquid markes. vi

7 Zusammenfassung Eine implizie Annahme vieler klassischer Modelle der Finanzmahemaik is, dass jederzei beliebige Mengen eines Werpapiers ohne Preiseinfluß gehandel werden können. Is die Menge der zum Handeln verfügbaren Liquidiä beschränk, so is diese Annahme nich immer erfüll. In dieser Disseraion lösen wir eine neue Klasse sochasischer Konrollprobleme und konsruieren opimale zeiseige Handelssraegien in illiquiden Märken, insbesondere Limi-Order-Märken. Wir benuzen Mehoden der sochasischen und singulären Konrollheorie. Im ersen Kapiel berachen wir einen Invesor in einem Limi-Order-Mark, der sein Porfolio möglichs nahe an einer gegebenen sochasischen Zielfunkion halen möche. Jede Transakion is mi Liquidiäskosen verbunden, gesuch is also diejenige Handelssraegie, die gleichzeiig die Abweichung vom Zielporfolio und die Handelskosen minimier. Typische Anwendung sind Porfolioliquidierung, Hedging und algorihmisches Handeln. Die Klasse der zulässigen Sraegien umfass akive und passive ( marke und limi ) Orders. Wir zeigen zunächs eine a-priori Abschäzung an die Konrolle und anschließend Exisenz und Eindeuigkei einer opimalen Sraegie. Wir beweisen eine Version des sochasischen Maximumprinzips und leien dami eine nowendinge und hinreichende Bedingung für Opimaliä miels einer gekoppelen sochasischen Vorwärs-Rückwärsgleichung her. Anschließend beweisen wir eine zweie Charakerisierung der opimalen Sraegie miels Kauf- und Verkaufregionen. Die Form dieser Regionen in Abhängigkei von der Zielfunkion wird im Deail analysier. Den Spezialfall quadraischer Sraffunkionen lösen wir explizi, dies liefer insbesondere eine Lösung des Porfolioliquidierungsproblems. Abschließend zeigen wir miels dreier Gegenbeispiele, dass passive und akive Orders verschiedene Vorzeichen haben können. Im zweien Kapiel verallgemeinern wir die Klasse der zulässigen Sraegien und erlauben insbesondere diskree Markorders. Miels Mehoden und Techniken der singulären Konrollheorie erweiern wir die Resulae des ersen Kapiels auf zweiseiige Limi-Order-Märke, in denen der Preiseinfluß einer Order nur langsam abnimm. Insbesondere modellieren wir den Spread und seine Abhängigkei von der Handelssraegie explizi. Dies führ zu einem Konrollproblem mi Nebenbedingungen, da passive Kauf- und Verkauforders separa als nichnegaive Prozesse modellier werden. Wie zuvor zeigen wir Exisenz und Eindeuigkei einer opimalen Sraegie. Im zweien Schri beweisen wir eine Version des Maximumprinzips im singulären Fall, die eine nowendige und hinreichende Opimaliäsbedingung liefer. Daraus leien wir eine weiere Charakerisierung miels Kauf-, Verkaufs- und Nichhandelsregionen ab. Wir zeigen, dass Markorders nur benuz werden, wenn der Spread klein genug is. Dami können wir präzise beschreiben, wann ein Überqueren des Spreads sinnvoll is und beanworen dami eine fundamenale Frage des algorihmischen Handels. Wir schließen dieses Kapiel mi einer Fallsudie über Porfolioliquidierung ab. Das drie Kapiel hemaisier Markmanipulaion in illiquiden Märken. Wenn Transakionen einen Einfluß auf den Akienpreis haben, dann können Opionsbesizer dami den Wer ihres Porfolios beeinflussen. Wir analysieren opimale Sraegien im Mehrspielerfall, indem wir sraegische Inerakion in das Modell aus dem ersen Kapiel einführen. Wir berachen mehrere Agenen, die europäische Derivae halen und den Preis des zugrundeliegenden Werpapiers beeinflussen. Wir beschränken vii

8 viii uns auf risikoneurale und CARA-Invesoren und zeigen die Exisenz eines eindeuigen Gleichgewichs, das wir miels eines gekoppelen Sysems nichlinearer PDEs charakerisieren. Für lineare Kosenfunkionen leien wir die Lösungen explizi her. Abschliessend geben wir Bedingungen an, wie diese Ar von Markmanipulaion verhinder werden kann. Schlagwörer: Sochasische Konrollheorie, Maximumprinzip, BSDEs, Illiquide Märke.

9 Conens Inroducion and Main Resuls Acknowledgemens Curve Following in Illiquid Markes Inroducion The Conrol Problem Main Resuls Exisence of a Soluion The Sochasic Maximum Principle The Cos-Adjused Targe Funcion Examples Curve Following wih Signal Porfolio Liquidaion Bid-Ask Spread and he Independence of he Jump Processes When o Cross he Spread: Curve Following wih Singular Conrol Inroducion The Conrol Problem Exisence of a Soluion The Sochasic Maximum Principle Buy, Sell and No-Trade Regions Link o Refleced BSDEs Examples Porfolio Liquidaion wih Singular Marke Orders Porfolio Liquidaion wih Singular and Passive Orders An Example Where I Is Opimal Never To Trade On Marke Manipulaion in Illiquid Markes Inroducion The Model Price dynamics and he liquidiy premium The Opimisaion Problem A Priori Esimaes Soluion for Risk Neural Invesors Soluion for CARA Invesors How o Reduce Manipulaion ix

10 Conens A. Appendix 129 A.1. Auxiliary Resuls for Chaper A.2. Auxiliary Resuls for Chaper A.3. Auxiliary Resuls for Chaper A.3.1. An Exisence Resul A.3.2. Proof of Proposiions and A.3.3. Burgers Equaion Concluding Remarks Symbols and Noaion x

11 Inroducion and Main Resuls Sandard financial marke models assume ha asse prices follow an exogenous sochasic process and ha all ransacions can be seled a he prevailing price wihou any impac on marke dynamics. In recen years here has been an increasing ineres in illiquid markes, where hese assumpions are relaxed. In such markes, rading has an impac on prices and every rading sraegy incurs liquidiy coss. In his hesis we consider sochasic conrol problems in coninuous ime arising in he conex of illiquid markes. In paricular we solve he problems of curve following, porfolio liquidaion and marke manipulaion under marke impac. Illiquid Markes Kyle 1985 singles ou hree measures of liquidiy: Tighness, he cos of urning around a posiion over a shor period of ime, deph, he size of an order flow innovaion required o change prices a given amoun, and resiliency, he speed wih which prices recover from a random, uninformaive shock. Liquidiy risk can also be defined as he addiional risk due o he iming and size of a rade as in Çein e al. 2004, bu a clear-cu definiion of liquidiy is no available, o he bes of our knowledge. A lack of liquidiy migh be due o asymmeric informaion (as in Kyle 1985 and Back 1992), he presence of large invesors (Frey and Sremme 1997 and Bank and Baum 2004) or an imbalance in supply and demand (as in Çein e al. 2004). Liquidiy risk affecs he replicaion of derivaives (Jarrow 1994 or Çein e al. 2009), plays a role in algorihmic rading (see for insance Bersimas and Lo 1998 and Almgren and Chriss 2001) and may lead o marke manipulaion (Jarrow 1994 and Huberman and Sanzl 2005). Limi Order Markes Almos all modern exchanges are organised as elecronic limi order markes. These are designed as coninuous double aucions, see O Hara 1995 and Parlour and Seppi 2008 for a deailed discussion. In such markes, wo ypes of orders are available. Limi orders are submied for fuure execuion and are sored in he limi order book. Each limi order indicaes he inenion o buy (or sell) a cerain quaniy of he asse for a cerain price. Marke orders are submied for immediae execuion, hey are mached wih oher raders ousanding limi orders and hence change he level of supply and demand. Invesors can hus provide liquidiy using limi orders or consume liquidiy by means of a marke order. As a resul, he cash proceeds from a large order depend crucially on he order placemen sraegy. In general, limi orders yield a beer price han marke orders, bu heir execuion is uncerain. Moreover, a sequence of small 1

12 Conens marke orders migh be realised a beer prices han a single large marke order. Mos exising models however only allow for one ype of orders, ypically marke orders. We refer he reader o Gökay e al for a recen survey. In conras o sandard financial models, in limi order markes here are wo prices. Limi sell orders are available a prices higher han or equal o he bes ask price, and limi buy orders are available a prices lower han or equal o he bes bid price. The difference of hese wo prices is called bid ask spread. Prices are discree and hence he bes bid and bes ask are muliples of a fixed ick size. A ypical ick size is 0.01 cen (also denoed one basispoin). On he microscopic level, he order book can be described as a high-dimensional prioriy queueing 1 sysem. Such models have been proposed among ohers by Kruk 2003 and Oserrieder Models of his kind are mainly designed o sudy he long run disribuion of available liquidiy and prices, bu hey are ofen oo involved for he analysis of opimal rading sraegies. Marke Impac In limi order markes here is ypically a limied amoun of liquidiy available on each price ick. Thus, a large marke order moves he curren bes bid (or bes ask) and widens he spread. As a resul, rading has an impac on prices. There is a large body of empirical lieraure on price impac 2, le us only menion Kraus and Soll 1972, Holhausen e al. 1987, Holhausen e al. 1990, Biais e al and Almgren e al There is ypically a disincion beween insananeous (recovers insanly), emporary (recovers gradually) and permanen (does no recover) price impac. Insead of describing he inerplay of supply and demand on a microscopic level, many mahemaical papers on opimal order execuion ake a macroscopic view and model he marke impac direcly. Obizhaeva and Wang 2005, e.g., assume a coninuous disribuion of available liquidiy wih a consan order book heigh, whereas Alfonsi e al allow for more general shape funcions. In a second sep, he resilience is modelled, i.e. how fas new limi orders arrive inside he bid ask spread afer liquidiy was consumed. Almgren and Chriss 2001 for insance assume insan recovery, whereas Obizhaeva and Wang 2005 consider a model wih finie resilience. A furher simplificaion concerns he class of admissible rading sraegies ( conrols ). Mos papers only allow for one ype of orders, ypically marke orders. Some auhors, such as Almgren and Chriss 2001 and Schied and Schöneborn 2008, only consider absoluely coninuous rading sraegies, ohers allow for coninuous and discree rades, le us menion Obizhaeva and Wang 2005 and Predoiu e al A noable excepion is Kraz 2011, who considers porfolio liquidaion in a primary exchange and a dark pool in he muli-asse case. While his resuls for he single asse case are qualiaively similar o wha will be derived in Chaper 1, he opimisaion problem we consider here is more general and he soluion echnique is differen. Unlike Kraz 2011 we disinguish beween passive buy and sell orders (boh of hem being nonnegaive) and as a resul we face a consrained 1 There are ypically prioriy rules for limi order execuion wih respec o price and ime. 2 Here we mean he impac of marke orders. The impac of limi orders is less well undersood, see however Cebiroglu and Hors

13 Conens opimisaion problem in Chaper 2. Many exising models for price impac are designed o sudy porfolio liquidaion, which describes he ask of selling a large number of shares of a given asse in a shor period of ime such ha he expeced liquidiy coss are minimised. This is a fundamenal problem of algorihmic rading and may serve as a building block for more involved algorihms. Moreover, his problem is closely relaed o he valuaion of a large porfolio of asses under liquidiy risk. In limi order markes, differen valuaion mehods make sense: Marked-o-marke (i.e. valuaion under he bes bid price), insan liquidaion (which involves no volailiy risk, bu high liquidiy coss) or valuaion under he opimal liquidaion sraegy (which reflecs a balance beween revenue and risk). One of he firs papers in he mahemaical finance lieraure concerned wih porfolio liquidaion is Bersimas and Lo They solve he porfolio liquidaion problem for a baske of socks and a risk-neural invesor in a discree ime model wih permanen and insananeous price impac. Almgren and Chriss 2001 exend his model o riskaverse invesors by considering a mean variance opimisaion crierion and sill find closed form soluions. They allow for nonlinear insananeous price impac and exend he model o coninuous ime in Almgren 2003; an empirical calibraion can be found in Almgren e al The permanen price impac is aken o be linear, i is shown in Huberman and Sanzl 2004 ha his is necessary o preven quasi-arbirage. Schied and Schöneborn 2007b give he soluion firs for exponenial uiliy and hen for general uiliy funcions in Schied and Schöneborn The common feaure of all hese models is ha hey allow for permanen and insananeous, bu no emporary price impac. I is assumed ha he order book recovers insanly afer a rade. Gaheral 2010 and Gaheral e al also allow for emporary price impac, so ha he ransacion price a ime does no only depend on he order a, bu possibly on he rading sraegy in 0,. They consider differen decay funcions and derive condiions o exclude liquidiy-induced arbirage. In heir model, here is one price which is influenced by he invesor s buy and sell rades. Thus, if he price reacs only slowly o pas rades, a fas sequence of buy and sell rades migh lead o quasi-arbirage. This is of course no desirable, and we avoid his by considering wo price processes in our model. In Chaper 2 we suppose ha marke buy orders increase he bes ask price and marke sell orders decrease he bes bid price, hen each rade incurs nonnegaive liquidiy coss and here is no liquidiy-induced arbirage. The aricles menioned above resric he analysis o absoluely coninuous rading sraegies. However, real world rading is discree, and a large discree rade may have a subsanial impac on he bes quoes. Obizhaeva and Wang 2005 ake his ino accoun and solve he porfolio liquidaion problem in a block-shaped order book for singular conrol processes, so ha boh coninuous and discree marke orders are allowed. I urns ou ha he opimal sraegy for a risk-neural invesor is composed of iniial and erminal discree rades and a consan rae of coninuous rading in beween. Generalisaions o arbirary shape funcions for he order book are given in Alfonsi e al as well as Predoiu e al Fruh 2011 reas he case of sochasic order book heigh. These papers focus on porfolio liquidaion and herefore only consider a one-sided order book. In addiion, hey only allow for marke orders. In he presen 3

14 Conens work, we solve a more general opimisaion problem in a wo-sided order book model and we also allow for passive orders. Seup and Economic Conribuion In he firs wo chapers we consider he problem of curve following. We ake he perspecive of an invesor rading in a limi order marke who wans o keep his sock holdings close o a sochasic arge funcion. He faces a radeoff beween he penaly for deviaing and he coss of rading. This is a quie general framework which covers an array of ineresing applicaions. We already discussed porfolio liquidaion above, here he arge funcion migh be chosen o be idenically equal o zero. A second applicaion is hedging, in his case he arge funcion represens a prespecified hedging porfolio. Le us also menion invenory managemen, where a rader (e.g. a marke maker, a broker or an invesmen bank) receives random orders from cusomers. Incoming orders lead o jumps in sock holdings ( invenory ) and he rader needs o rebalance his porfolio by rading in he open marke. More generally, he curve follower may serve as a par of an algorihmic rading plaform, where he arge funcion is he oupu of some higher-level program. A ypical example is rading a volume-weighed average prices (VWAP). The opimisaion problem oulined above is relaed o he well sudied problem of racking a sochasic process, also known as he monoone follower problem. Among ohers, Beneš e al. 1980, Karazas and Shreve 1984 and Karazas e al solve he problem of racking Brownian Moion wih finie fuel, using mehods of singular sochasic conrol. This is exended o more general sochasic processes and a dynamic fuel consrain in Bank In he finance lieraure, Leland 2000 considers a siuaion where an invesor aims o mainain fixed proporions of his wealh in a given selecion of socks, in a marke where here are proporional ransacion coss. Pliska and Suzuki 2004 reformulae he problem in a marke wih fixed and proporional ransacion coss and compue explici sraegies using mehods of impulse conrol. An exension o he mulidimensional case can be found in Palczewski and Zabzyck Mos of he papers menioned above use he dynamic programming approach and solve he associaed Hamilon-Jacobi-Bellman equaion. For he verificaion argumen, one ypically needs smoohness of he value funcion, which is no easy o prove in he presen framework since he forward diffusion is no uniformly parabolic. Alernaively, he weaker concep of viscosiy soluions can be used, bu his gives less informaion on he conrol. In he firs wo chapers of his hesis, we shall insead prove suiable versions of he sochasic maximum principle, since i does no require regulariy of he value funcion and provides a direc represenaion of he opimal conrol in erms of a forward backward sochasic differenial equaion (FBSDE). Analysing he FBSDE, we hen derive necessary and sufficien condiions of opimaliy in erms of buy, sell and no-rade regions. This allows o describe he srucure of he opimal rading sraegy quie explicily and, in special cases, even in closed form. Our mahemaical framework is flexible enough o allow for passive orders. Passive 4

15 Conens orders are undersood as orders wihou price impac and wih uncerain execuion, such as limi orders or orders placed in a dark venue or crossing nework. Dark pools are rading venues associaed o a classical exchange. Liquidiy available in he dark pool is no openly displayed and rades are only execued if maching liquidiy is available. Dark pools can be used o reduce he liquidiy coss due o marke impac, see Hendersho and Mendelson 2000, Kraz and Schöneborn 2009, Kraz 2011 and he references herein for furher deails. In conras o he model of Kraz 2011, in Chaper 2 we model passive buy and sell orders separaely. This covers siuaions where he probabiliy of execuion is differen for passive buy and sell orders. In our model, he arge funcion which may be influenced by sochasic signals. This offers a large degree of flexibiliy in inpus and allows for a complicaed arge driven by differen marke phenomena such as he order book heigh or he bid ask spread. I urns ou ha for general arge funcions i is necessary o assume ha he sochasic signal is independen from passive order execuion, since marke and passive orders in his case have he same sign. We consruc explici examples which show ha, if his assumpion is relaxed, rading simulaneously on differen sides of he marke migh be opimal. Specifically, he opimal sraegy may be composed of marke buy and passive sell orders, which is no a desirable feaure. To he bes of our knowledge, his problem has no been addressed in he lieraure on illiquid markes, since mos papers eiher only allow for one ype of orders or only consider porfolio liquidaion where i is clear a priori ha only sell orders and no buy orders are used. Unforunaely, if he signal represens bid ask spread, i is no independen from limi order execuion. I follows ha he signal canno be inerpreed as bid ask spread, in general. We will show however ha in he imporan case of porfolio liquidaion he undesirable feaure described above does no occur, so our framework covers porfolio liquidaion wih sochasic spread. The marke model of Chaper 2 allows o answer he quesion of when o cancel a passive order and submi a marke order insead ( cross he spread ). For small spread sizes, rading is relaively cheap and submiing a marke order migh be beneficial. For large spreads, marke orders are expensive and i migh be opimal o place only passive bu no marke orders and wai unil he spread recovers. This decision is relevan for rading algorihms, and o our bes knowledge no soluion is ye available in he mahemaical lieraure on limi order markes. This is mainly due o he fac ha no exising paper explicily models he bid ask spread and allows for boh marke and passive orders. For insance, Obizhaeva and Wang 2005 and Predoiu e al do no allow for passive orders. They do model he spread, bu in a way ha submiing marke orders is always beneficial and i is never opimal o sop rading. Fruh 2011 allows for sochasic order book heigh, hen i is opimal o sop rading when he marke is oo hin. Again, he analysis is resriced o marke orders. In he second chaper we consruc a hreshold for he bid ask spread and show ha submiing a marke order is opimal if and only if he spread is smaller han his hreshold. The opimisaion problems and models discussed hus far are designed for a single player. In illiquid markes every rader poenially moves prices and ineresing problems of sraegic ineracion arise. Typically invesors wan o reduce marke impac, e.g. 5

16 Conens when liquidaing a large porfolio. However under cerain circumsances i migh even be beneficial o move prices. Specifically, invesors holding opions wih cash delivery and illiquid underlying can drive up he sock price a mauriy and hus increase heir payoff. Kumar and Seppi 1992 call such rading sraegies punching he close. We model his siuaion in a muli-player framework in Chaper 3 by inroducing sraegic ineracion ino he model of he firs chaper. We hen characerise opimal rading sraegies in form of a Nash equilibrium. Deliberaely manipulaing prices is no legal, however i is ineresing from a mahemaical perspecive. Moreover, a beer undersanding of marke manipulaion may help o deec and preven i. Differen noions of marke manipulaion have been discussed in he lieraure, le us menion shor squeezes, he use of privae informaion or false rumours. We refer he reader o Kyle 1985, Back 1992, Jarrow 1994, Allen and Gale 1992, Pirrong 2001, Du and Harris 2005, Kyle and Viswanahan Closes o our seup is he paper by Gallmeyer and Seppi They consider a binomial model wih hree periods and finiely many risk-neural agens holding call opions on an illiquid underlying. Assuming a linear permanen price impac and linear ransacion coss, and assuming ha all agens are iniially endowed wih he same derivaive hey prove he exisence of a Nash equilibrium rading sraegy. We shall exend heir resuls o a coninuous ime diffusion model, allowing for a more general liquidiy cos erm and differen endowmens. We consruc he soluion for risk-neural and risk-averse invesors and characerise i in erms of a coupled sysem of parial differenial equaions. A deailed analysis of he soluion allows o show how marke manipulaion can be reduced. In urns ou ha for zero sum games, i.e. for offseing payoffs, he agens aggregae manipulaion sraegy is zero. We furher show ha manipulaion can be reduced by increasing he number of informed compeiors. Similarly, spliing a produc and selling i o several cusomers may be beer han selling i o a single agen. We close he hird chaper by showing ha derivaives wih physical delivery do no induce marke manipulaion in he sense of punching he close. Mahemaical Resuls of Chaper 1 In Chaper 1 we sar wih he curve following problem in coninuous ime. We allow for passive and marke orders, where he laer are resriced o absoluely coninuous rading sraegies as in Almgren and Chriss This simplificaion allows us o concenrae on he radeoff beween accuracy and liquidiy coss, he generalisaion o singular marke orders is given in Chaper 2. The main difficulies in he firs chaper are due o he presence of jumps in he sae variables and he fac ha passive orders incur no liquidiy coss, so he sandard characerisaion as poinwise maximisers of he Hamilonian does no apply. As indicaed above, our aim is o prove a suiable version of he sochasic maximum principle which provides a characerisaion of he opimal conrol. As a firs sep, we derive a priori esimaes on he conrol by comparing he problem o a simpler linear quadraic regulaor problem whose soluion is consruced explicily via Riccai equaions. Using our a priori esimaes, we hen apply a Komlós 6

17 Conens argumen which provides he exisence and uniqueness of an opimal rading sraegy. In a second sep, we define he adjoin equaion, a backward SDE which involves he opimal rading sraegy. We hen show ha he opimisaion problem is convex and explicily compue he Gâeaux derivaive of he performance funcional. Nex, we apply argumens based on Cadenillas and Karazas 1995 and Cadenillas 2002 o derive a necessary and sufficien condiion for opimaliy. We noe ha he resuls herein canno be applied direcly o our seing, since hey only allow for linear sae dynamics (whereas he sochasic signal driving he arge funcion in our model has more general dynamics). Our version of he sochasic maximum principle involves a coupled forward backward SDE, composed of a forward equaion for he sae process and a backward equaion for he adjoin process, as well as a poinwise opimaliy condiion on he conrol. Consrucing he opimal soluion is hus equivalen o solving a fully coupled FBSDE. The opimal marke order is hen given as he poinwise minimiser of he Hamilonian, a funcion involving he sae variable as well as he adjoin process. The opimal passive order however is characerised only implicily by a condiion on he soluion of he backward equaion. To obain more insigh ino he srucure of he opimal rading sraegy, in a hird sep we idenify a (sochasic) hreshold in erms of sock holdings. This hreshold is defined in erms of he value funcion and we call i he cos-adjused arge funcion. We show ha if sock holdings are above (below) his funcion, i is opimal o sell (resp. buy). I follows ha rading is always direced owards his funcion, and no owards he original arge funcion as migh be expeced. Inuiively, he cos-adjused arge funcion represens he expeced fuure evoluion of he arge curve, weighed agains expeced rading coss. This funcion is key, i separaes he buy from he sell region, and we discuss is dependence on he inpu parameers in deail. In he example secion, we show ha he FBSDE wih a general signal and a general arge funcion can be solved in closed form if he penaly funcions are quadraic. As one applicaion, we explicily solve he porfolio liquidaion problem allowing for he simulaneous use of marke and passive orders. While he discree ime case is solved in Kraz and Schöneborn 2009, o he bes of our knowledge our soluion is he firs 3 in coninuous ime. We close he firs chaper wih a echnical secion, where we show why i is necessary o assume ha he signal process is independen of passive order execuion. Specifically i urns ou ha if his assumpion is weakened, i migh be opimal o use marke buy and passive sell orders a he same ime and hus rade on boh sides of he marke simulaneously, a feaure which is raher undesirable from he praciioner s poin of view. Mahemaical Resuls of Chaper 2 The resuls of he firs chaper are derived under he hypohesis of absoluely coninuous marke orders. Trading in real markes is discree hough. We herefore exend he model of Chaper 1 o singular marke orders in he second chaper, so ha coninuous 3 Simulaneously o our work, Kraz 2011 exended he resuls from Kraz and Schöneborn 2009 o coninuous ime using mehods differen o ours. 7

18 Conens and discree rading is allowed. The conrol now comprises absoluely coninuous passive orders as well as singular marke orders. The challenge is ha here are now wo sources of jumps, represening passive order execuion and singular marke orders. The singular naure of he marke order complicaes he analysis considerably, i canno be characerised as he poinwise maximiser of he Hamilonian as in he absoluely coninuous case. In addiion, here are now consrains on he conrol, since boh passive buy and sell orders are nonnegaive processes. We sar wih he marke model inroduced in Obizhaeva and Wang 2005, and exend i by allowing for rading on boh sides of he marke and also include passive orders. As before, marke orders have an impac on prices, hey consume liquidiy and increase he bid ask spread emporarily. The spread hen narrows gradually and prices slowly recover o normal levels. In conras o he firs chaper, he spread is now conrolled, and rading coss a ime depend on he whole rading sraegy in 0,. Our firs mahemaical resul is an a priori esimae on he conrol. For he proof, we reduce he curve following problem o a simpler opimisaion problem wih quadraic cos erms and zero arge funcion and hen apply a scaling argumen. The a priori esimae hen provides he exisence and uniqueness of an opimal sraegy via a Komlós argumen. We go on o prove a suiable version of he sochasic maximum principle, which yields a characerisaion of he opimal conrol in erms of a coupled forward backward SDE which now involves singular erms. The proof combines argumens from Cadenillas and Haussmann 1994 wih ideas developped in he firs chaper. We noe ha he singular maximum principle derived in Cadenillas and Haussmann 1994 does no cover he presen siuaion as i does no allow for jumps, sae-dependen singular cos erms as well as general dynamics for he sochasic signal. The maximum principle given in Øksendal and Sulem 2010 includes jumps, bu can also no be applied direcly as i only allows for singular bu no absoluely coninuous conrols (which are needed here for he passive order). Our maximum principle provides a characerisaion of opimaliy which is quie implici, and for his reason we prove a second characerisaion in erms of buy, sell and no-rade regions. In conras o Chaper 1, i now urns ou ha here is a nondegenerae no-rade region where he coss of rading are larger ha he penaly for deviaing. This region is defined in erms of a hreshold for he bid ask spread. We show ha spread crossing is opimal if he spread is smaller han or equal o he hreshold. If i is larger, hen no marke orders should be used and rading sops. This resul allows o characerise precisely when spread crossing is opimal for a large class of opimisaion problems, a novel resul in he mahemaical lieraure on limi order markes. We also show ha marke orders are submied in order o keep he conrolled sysem inside (he closure of) he no-rade region, so ha is rajecory is refleced a he boundary of he no-rade region. To make his precise, we show ha he conrolled sysem can be inerpreed as he soluion o a refleced BSDE. Due o he presence of jumps in he sae process and due o he singular naure of he conrol, i is in general difficul o solve he coupled FBSDE explicily. For quadraic penaly funcions and zero arge however we are able o consruc he soluion in closed form. As one applicaion, we solve he porfolio liquidaion problem wih passive orders, which exends he resul from Obizhaeva and Wang 2005 o a siuaion wih boh marke and passive orders. The 8

19 Conens new feaure is ha he opimal sraegy is hen no longer deerminisic, bu is adaped o passive order execuion. As a second illusraion we provide a furher example where i is opimal never o use marke orders. Mahemaical Resuls of Chaper 3 In he hird chaper we inroduce sraegic ineracion ino he model of Chaper 1. We consider a finie se of agens and assume ha each of hem is endowed wih a fixed European opion wih cash selemen, for insance a Call opion. We assume for simpliciy ha he endowmen is fixed and he opion does no rade. The underlying asse is illiquid, is price depends on he rading sraegy of all he agens. As in Almgren 2003 we assume ha he permanen price impac is linear. In addiion, here is an insananeous price impac which is modelled by a general liquidiy cos funcion. Passive orders have no price impac, so we remove hem and only allow for marke orders. Each agen has o balance he gain from driving he sock price ino a favourable direcion agains he liquidiy coss of rading, aking ino accoun his compeiors sraegies. We se his up as a sochasic differenial game and look for soluions in he form of a Nash equilibrium. We consider he cases of risk-neural invesors and riskaverse agens wih exponenial uiliy funcions separaely. In boh cases he preference funcionals are ranslaion invarian. I is hen no necessary o keep rack of each agen s rading coss, which simplifies he analysis. The firs sep is o esablish a priori esimaes on he conrols, he proof is based on a linear growh condiion on he payoffs. The mehods used in he firs wo chapers are no applicable here, since he opimisaion problem for each agen is no necessarily convex. Insead, we use he dynamic programming approach. The agens value funcions can be described by a coupled sysem of Hamilon-Jacobi-Bellman PDEs. In conras o he preceding chapers, he forward diffusion is no degenerae and he HJB PDEs are uniformly parabolic. However, sandard resuls of exisence and uniqueness of a smooh soluion o his coupled PDE do no apply since we work on an unbounded sae space. Insead, we give a direc proof based on argumens from Taylor 1997 which explois our a priori esimaes on he conrols. For he special case of linear cos funcions, we show ha he coupled sysem of PDEs can be solved explicily. We shall analyse hese closed form soluions in deail in order o derive condiions on how marke manipulaion can be avoided. I urns ou ha he aggregae rading speed converges o zero if he number of informed compeiors (wihou endowmen) increases. We also show ha in he case of physically seled Call opions, he opimal rading sraegy for each agen is zero, so manipulaion is no beneficial. 9

20

21 Conens Acknowledgemens Firs and foremos I would like o hank my supervisor Prof. Dr. Ulrich Hors. His guidance and suppor hroughou he years made his hesis possible. The hird chaper is based on join work wih Ulrich and is acceped for publicaion in Quaniaive Finance. I also would like o hank Prof. Dr. Peer Bank and Prof. Dr. Abel Cadenillas for agreeing o co-examine his hesis. Special hanks go o Dr. Nicholas Wesray. The firs chaper is based on join work wih Nick and is acceped for publicaion in Mahemaics and Financial Economics. Nick always ook he ime for discussions and consrucive commens. Financial suppor from Deusche Bank is graefully acknowledged; he Quaniaive Producs Laboraory was an inspiring working environmen. I hank Dr. Marcus Overhaus, Prof. Dr. Peer Bank, Prof. Dr. Ulrich Hors and Almu-Mirjam Birsner who worked so hard o keep he QPL running. This hesis grealy benefied from he ineracion wih praciioners from Deusche Bank, in paricular Dr. Andy Ferraris, Dr. Boris Drovesky and Dr. Chrisopher Jordinson. Moreover, I would like o hank my friends and colleagues a QPL, in paricular Anje Fruh, Karin Eichmann, Gökhan Cebiroglu, Dr. Mikhail Urusov and Dr. Torsen Schöneborn, for many valuable discussions and he pleasan working amosphere. Finally I wish o express my graiude o my family and my fiancee for heir love and suppor. 11

22

23 1. Curve Following in Illiquid Markes 1.1. Inroducion In modern financial insiuions, due o exernal regulaion as well as clien preferences, here are ofen imposed rading arges which should be followed. These can ake he form of a curve giving desired sock holdings over he course of some ime horizon, one could hink of a day. In an idealised seing one would simply rade so as o say exacly on he arge. Prevening his is he associaed coss, hus one has o balance he wo conflicing objecives of ensuring minimal deviaion from he prespecified arge and concurrenly minimising rading coss. In he presen chaper we address and solve he curve following problem using echniques of sochasic conrol. In paricular, we prove exisence and uniqueness of an opimal conrol and hen give a characerisaion via he sochasic maximum principle. Conrols include boh marke and passive orders. Our opimal sysem is described by a fully coupled forward backward sochasic differenial equaion (FBSDE) and in special cases we provide closed form soluions. The main difficulies are due o he presence of jumps and he fac ha passive orders incur no liquidiy coss, so he usual opimaliy crierion via he maximisaion of he Hamilonian does no apply. Insead, we derive a characerisaion via buy and sell regions. A ypical problem in he mahemaical lieraure on price impac is ha of how o opimally liquidae a given sock holding and we menion firs he paper of Almgren 2003 in which he formulaes a coninuous ime model for emporary and permanen marke impac. He allows for absoluely coninuous marke orders and derives explici soluions o he liquidaion problem. This work has become very popular wih praciioners as well as forming he basis for subsequen research aricles including Almgren e al. 2005, Schied and Schöneborn 2008 and Almgren Our model in he presen chaper is build on Almgren s model and exends i o general cos funcions as well as passive orders. In his limi order book model we consider he problem of curve following and consruc he rading sraegy which balances he penaly for deviaing agains he liquidiy coss of rading. In he special case of racking a Brownian Moion his is known as he monoone follower problem and has been discussed in Bayrakar and Egami 2008, Beneš e al and Karazas e al. 2000, among ohers. In he finance lieraure, Leland 2000 considers a siuaion where an invesor aims o mainain fixed proporions of his wealh in a given selecion of socks, in a marke where here are proporional ransacion coss. His soluion has a local ime componen as in Davis and Norman Pliska and Suzuki 2004 reformulae he problem in a marke wih fixed and proporional ransacion coss. Using echniques of impulse conrol, hey compue explici sraegies and his ime, due o he presence of fixed coss, here is no local ime phenomenon. In addiion, hey calculae some sensiiviies. Le us also menion 13

24 1. Curve Following in Illiquid Markes Palczewski and Zabzyck 2005 who exend he model of Pliska and Suzuki 2004 o he mulidimensional case when he underlying prices are Markovian. From an economic perspecive, our firs major conribuion over he aricles menioned above is ha we allow for he use of passive orders, which are undersood as orders wihou price impac and wih a random execuion. We hink of hem as a reducedform model for limi orders or dark pools. The second conribuion is he inroducion of a arge which may depend on an array of sochasic signals. This offers a large degree of flexibiliy in inpus and allows for a arge driven by differen marke phenomena. Relevan applicaions include racking he oupu of an algorihmic rading program, porfolio liquidaion, invenory managemen and hedging. Le us now describe he mahemaical resuls in more deail. The firs sep owards a soluion is an a priori esimae on he conrol. For he proof, we reduce he curve following problem o a simpler problem wih quadraic cos erms and wihou arge funcion, for which a soluion via Riccai equaions can be conruced explicily. Our a priori esimae hen allows o prove exisence and uniqueness of an opimal rading sraegy via a Komlós argumen. We go on o derive a suiable version of he sochasic maximum principle wih jumps. The proof is based on ideas from Cadenillas and Karazas 1995 and Cadenillas However, heir resuls canno be direcly applied o he presen framework since hey only allow for linear dynamics of he sae variables, while in our case he SDE for he signal may be nonlinear. Our maximum principle provides a necessary and sufficien condiion of opimaliy in erms of a FBSDE, which is composed of a forward equaion for he sae process, a backward equaion for he adjoin process and a poinwise opimaliy condiion on he conrol. Consrucing he opimal rading sraegy is hen equivalen o solving a coupled FBSDE. The moivaion for using such echniques is due o he fac ha our model has a degenerae forward diffusion componen and is herefore no uniformly parabolic. This means ha sandard argumens which may imply a smooh soluion o he Hamilon-Jacobi-Bellman (HJB) equaion do no apply. Secondly, our ineres is no in he value funcion per se, bu primarily in he opimal conrol, abou which one ges more informaion wih he presen mehods. Our work also conribues o he sochasic conrol lieraure by showing ha i is possible o describe very clearly he srucure of he problem by analysing probabilisically he corresponding FBSDE raher han he HJB equaion via viscosiy soluion echniques. Specifically, we provide a deailed analysis of he forward backward equaion which yields a second characerisaion of opimaliy in erms of buy and sell regions. We show ha here is a hreshold in erms of sock holdings above which i is opimal o sell and below which we buy. We call his funcion he cos-adjused arge funcion, i represens he expeced fuure evoluion of he arge, weighed agains expeced rading coss. I urns ou ha sock holdings should be kep close o his funcion, and no o he original arge funcion. For quadraic penaly and liquidiy cos funcions, we are able o solve he conrolled sysem in (semi-)closed form. We also provide an explici soluion o he porfolio liquidaion problem, which is he firs 1 in coninuous ime al- 1 Simulaneously o our work, a similar soluion was derived in Kraz 2011 using differen echniques. 14

25 1.2. The Conrol Problem lowing for boh marke and passive orders. In he final secion some counerexamples are provided in closed form o illusrae ha marke and passive orders migh have differen signs if he signal is no independen from passive order execuion. In economic erms, his corresponds o rading on differen sides of he marke simulaneously, which is no a desirable feaure. To our bes knowledge, his problem has no ye been addressed in he mahemaical lieraure on limi order markes since mos exising papers only allow for one ype of orders. The ouline of his chaper is as follows, Secion 1.2 derives he model as well as inroducing he arge funcions, sochasic signal and conrol problem. Secion 1.3 conains our main resuls, Secions 1.4, 1.5 and 1.6 discuss he proofs. We consider he quadraic case and some applicaions in Secion 1.7 and close in Secion 1.8 wih some counerexamples. Pars of his chaper are published in Naujoka and Wesray The Conrol Problem We consider a erminal ime T ogeher wih a filered probabiliy space (Ω, F, {F(s) : s 0, T }, P ) saisfying he usual condiions of righ coninuiy and compleeness. Assumpion The filraion is generaed by he following hree muually independen processes, 1. A d-dimensional Brownian Moion W. 2. A one-dimensional Poisson process N wih inensiy λ. 3. A compound Poisson process M wih compensaor m(dθ)d, where m(r k ) <. We consider an invesor whose sock holdings are governed by he following SDE, dx u (s) = u 1 (s)n(ds) + u 2 (s)ds, (1.1) for s, T and wih X u () = x. The conrol process u is an R 2 -valued process and chosen in he following se, U { u L 2 (, T Ω) : u 1 predicably measurable and u 2 progressively measurable }. The inerpreaion of u is as follows. The invesor places a passive order of size u 1, when a jump of N occurs he order is execued and he porfolio adjuss accordingly. For ease of exposiion we consider only full liquidaion. The componen u 2 represens he marke order, inerpreed here as a rae as in Almgren 2003; more general marke orders will be considered in Chaper 2. The invesor can hus ake and provide liquidiy. We use he noaion u L 2 o denoe he L 2 (, T Ω)-norm of a conrol, where will be undersood from he conex. To keep a disincion we use R n for he Euclidean norm of an n-dimensional vecor, while is reserved for real numbers. Inequaliies wih respec o random variables are assumed o hold a.s. 15

26 1. Curve Following in Illiquid Markes In addiion o he conrolled process X u, here is an unconrolled n-dimensional vecor Z wih dynamics given by dz(s) = µ(s, Z(s))ds + σ(s, Z(s))dW (s) + γ (s, Z(s ), θ) M(ds, dθ), (1.2) R k for s, T and wih Z() = z. Observe ha we wrie M(0, s A) M(0, s A) m(a)s for he compensaed Poisson maringale; similarly Ñ N λs. The funcions µ and σ ake values from, T R n and are valued in R n and R n d respecively, while γ akes values from, T R n R k and is valued in R n. The vecor Z denoes a collecion of n facors which may influence he coss of rading as well as he arge curve o be followed, however i is no affeced by he rading sraegy of he invesor. Le us now inroduce he performance funcional, T J(, x, z, u) E g(u 2 (s), Z(s)) + h ( X u (s) α(s, Z(s)) ) ds (1.3) + f ( X u (T ) α(t, Z(T )) ) X u () = x, Z() = z. The funcion α :, T R n R is he arge funcion and h and f penalise deviaion from he arge. The cos funcion g capures he liquidiy coss of marke orders and we now give a heurisic derivaion. Trading akes place in a limi order marke, which is characerised by a benchmark price D and a collecion of oher raders ousanding limi orders. We assume ha he process (D(s)) s T is a nonnegaive maringale. A a given insan s, here are limi sell orders available a prices higher han D(s) and limi buy orders a prices lower han D(s). The invesor s marke buy order is mached wih prevailing limi orders and execued a prices higher han D(s). The more volume he rader demands, he higher he price paid per share, ha is o say here is an increasing supply curve, as in Çein e al Similarly, marke sell orders are execued a prices lower han D(s) and he price per share is decreasing in he volume sold. The invesor may also use passive orders, hese are placed and fully execued a D(s). A passive order always achieves a beer price, however is ime of execuion is uncerain. Given a marke order u 2, recall here inerpreed as a rae, ogeher wih he sochasic signal Z, he above consideraions lead us o define he asse price as S(s, Z(s), u 2 (s)) = D(s) + g ( u 2 (s), Z(s) ), (1.4) where g capures he insananeous price impac of he marke order per uni. We assume ha u 2 g(u 2, z) is increasing and such ha g(0, z) = 0. The cash flow over he inerval, T is given by T CF(u) = T T u 2 (s)s(s, Z(s), u 2 (s))ds + u 2 (s)d(s) + u 2 (s) g ( u 2 (s), Z(s) ) ds + u 1 (s)d(s )N(ds) T u 1 (s)d(s )N(ds), 16

27 1.2. The Conrol Problem where we assume all he necessary condiions for he above sochasic inegrals o exis. The premium paid due o no being able o rade a he benchmark price, he cos of rading over he inerval, T, is hen given by T T T CF(u) u 2 (s)d(s)ds u 1 (s)d(s )N(ds) = u 2 (s) g ( u 2 (s), Z(s) ) ds. Defining he liquidiy cos funcion g as g(u 2, z) u 2 g(u 2, z) gives precisely he erm in (1.3). Remark There are wo naural inerpreaions of he passive order. The firs would be as an order placed in a dark venue, where he underlying level of liquidiy is unobservable, see Hendersho and Mendelson 2000 and he references herein for furher deails. Le us also menion Kraz 2011 who discuss porfolio liquidaion in he muli-asse case in he presence of a dark venue. For he special case of a single asse, hey have porfolio dynamics similar o ours. A second inerpreaion of he passive order is a sylised version of a limi order where placemen is only a he benchmark price and here is no ime prioriy and only full execuion. Remark Le us compare he presen seing wih he lieraure. Wihou passive orders, our approach is close o Rogers and Singh In heir model, absolue liquidiy coss are capured by a convex, nonnegaive loss funcion. If we se g(u 2, z) = κu 2 2 for some κ > 0, we recover he model of Almgren However herein here is an addiional permanen price impac, which is undesirable in he presen case. In Chaper 3 we consider opions wih illiquid underlying where rading does have a permanen impac. In his case, marke manipulaion may be beneficial. In he presen model we assume ha rading only has an insananeous price impac, i.e. he order book recovers insanly afer a rade. A marke wih emporary price impac (i.e. finie resilience) will be discussed in Chaper 2. In ha model, he bid ask spread depends on he rading sraegy and recovers only gradually afer a rade. We now proceed o he main problem of ineres. The value funcion associaed o our opimisaion problem is defined as v(, x, z) inf u U J(, x, z, u). In he sequel we slighly abuse noaion and wrie J(u) J(, x, z, u) if (, x, z) 0, T R R n is fixed. The curve following problem is hen defined o be Problem Find û U such ha J(û) = min u U J(u). To ensure exisence of an opimal conrol we need some assumpions on he inpu funcions. We remark ha here and hroughou he consans may be differen a each occurrence. 17

28 1. Curve Following in Illiquid Markes Assumpion Each funcion ψ = f( ), g(, z), h( ) saisfies: 1. The funcion ψ is sricly convex, nonnegaive, C 1 and normalised in he sense ha ψ(0) = In addiion, ψ has a leas quadraic growh, i.e. here exiss ε > 0 such ha ψ(x) ε x 2 for all x R. In he case of g his is supposed o be uniform in z. 3. The funcions µ, σ and γ are Lipschiz coninuous, i.e. here exiss a consan c such ha for all z, z R n and s, T, µ(s, z) µ(s, z ) 2 R n + σ(s, z) σ(s, z ) 2 R n d + γ(s, z, θ) γ(s, z, θ) 2 R k R nm(dθ) c z z 2 R n. In addiion, hey saisfy sup µ(s, 0) 2 R n + σ(s, 0) 2 R + γ(s, 0, θ) 2 n d s T R k R nm(dθ) <. 4. The arge funcion α has a mos polynomial growh in he variable z uniformly in s, i.e. here exis consans c α, η > 0 such ha for all z R n, sup α(s, z) c α (1 + z η R ). n s T 5. The funcions f and h have a mos polynomial growh. Remark Le us briefly commen on hese assumpions. The nonnegaiviy assumpion is moivaed by he fac ha rading is always cosly ogeher wih i never being desirable o deviae from he arge. Taking f and h normalised is no loss of generaliy, his may always be achieved by a linear shif of f, h and α. The convexiy and quadraic growh condiion lead naurally o a convex coercive problem which hen admis a unique soluion. A ypical candidae for he penaly funcion is f(x) = h(x) = x 2, which corresponds o minimising he squared error. We also noe ha our framework is flexible enough o cover nonsymmeric penaly funcions, e.g. if falling behind he arge curve is penalised sronger han going ahead. Once exisence and uniqueness of he opimal conrol has been esablished we shall need furher assumpions for a characerisaion of opimaliy via an FBSDE. Assumpion We require he exisence of a consan c such ha 1. The derivaives f and h have a mos linear growh, i.e. for all x R f (x) + h (x) c(1 + x ). 18

29 1.3. Main Resuls 2. The cos funcion g has polynomial syle growh, i.e. for all u 2 R u 2 g u2 (u 2, ) c(1 + g(u 2, )). 3. The cos funcion g saisfies a subaddiiviy condiion, i.e. for all u 2, w 2 R g(u 2 + w 2, ) c (1 + g(u 2, ) + g(w 2, )). 4. Consan deerminisic conrols have finie cos, in paricular for all u 2 R, T E g(u 2, Z(s))ds <. Remark We need he linear growh on he derivaives of f and h o ensure ha we can solve he adjoin BSDE. In paricular his essenially limis us o quadraic penaly funcions h and f. For he cos funcion g, one example saisfying he above assumpions would be o se g(u 2, Z) = cu 2 arcan(u 2 ) + u 2 2(Z + ε), for some ε > 0, where Z is a nonnegaive mean-revering jump process. We hink of Z as modelling he inverse order book heigh. The funcion u 2 arcan(u 2 ) represens a smooh approximaion o u 2 and he consan c > 0 represens bid ask spread. This represens a model wih fixed spread and sochasic order book heigh. In he presen seing, we are mos ineresed in processes on, T Ω and wrie ha a given propery (P) holds ds dp a.e. on B" for a measurable subse B, T Ω when (P) holds for he resricion of he measure ds dp o B Main Resuls Having formulaed he problem and inroduced he necessary assumpions, we can now give our main resuls of he presen chaper. Theorem The funcional u J(u) is sricly convex for u U. If Assumpion holds hen for any iniial riple (, x, z) 0, T R R n here is an opimal conrol, unique ds dp a.e. on, T Ω. We pospone he proof o Secion 1.4. To characerise he opimal conrol û and he corresponding sae process ˆX Xû we define he following BSDE on, T, he adjoin equaion, dp (s) = h ( ) ˆX(s) α(s, Z(s)) ds + Q(s)dW (s) + R1 (s)ñ(ds) (1.5) + R 2 (s, θ) M(ds, dθ), R k 19

30 1. Curve Following in Illiquid Markes P (T ) = f ( ˆX(T ) α(t, Z(T ) ). Theorem Le Assumpions and hold. Then 1. The above BSDE has a unique soluion for all saring riples (, x, z) 0, T R R n. 2. A conrol û is opimal if and only if ds dp a.e. on, T Ω, a) û 2 (s, ω) is he poinwise minimiser of u 2 g(u 2, Z(s, ω)) P (s, ω)u 2. b) P (s, ω) + R 1 (s, ω) = 0. Remark The second par of Theorem is essenially a version of he sochasic maximum principle and we now describe how his relaes o hose in he lieraure. Our resuls are mos similar o Cadenillas 2002, however in his seing one requires ha (in our noaion) he join process (X, Z) have dynamics which are joinly affine as funcions of (X, Z) and conrol u. This is no necessarily he case for only Lipschiz µ, σ and γ, so ha we are ouside he scope of he resuls herein. The aricle Tang and Li 1994 considers he case where he dynamics of (X, Z) need no be affine, as in he presen aricle, however hey require ha he conrol saisfies he following inegrabiliy condiion sup E u(s) 8 R 2 <, s T which excludes he L 2 -framework considered here. Finally we menion Ji and Zhou 2006, where he auhors allow for square inegrable conrols and non-affine dynamics bu have no jumps, so ha again heir resuls do no cover he presen siuaion. The proof of he second iem relies on he sochasic maximum principle and is deal wih in Secion 1.5. The characerisaion given in Theorem is sill raher implici, we can describe û 1 more precisely and for his require he following definiion. Definiion The cos-adjused arge funcion α is defined o be he poinwise minimiser (wih respec o x) of he value funcion, α(, z) arg min v(, x, z). x R The fac ha α is well defined is a consequence of he convexiy of v as well as Lemma where i is shown ha v has a leas quadraic growh in x. The nex heorem shows ha rading is direced owards he cos-adjused arge funcion, moivaing is definiion. Theorem Le Assumpions and hold, hen 1. The opimal passive order is given ds dp a.e. on, T Ω by û 1 (s, ω) = α(s, Z(s, ω)) ˆX(s, ω). 20

31 1.4. Exisence of a Soluion 2. Define he buy region via R buy { (s, x, z), T R R n : x < α(s, z) }, as well as he se where he opimal sae process is valued in R buy, A buy = {(s, ω), T Ω : (s, ˆX(s, } ω), Z(s, ω)) R buy. Then we have ha û 1, û 2 > 0, ds dp a.e. on A buy. The symmeric resul holds for he sell region, R sell, defined similarly wih > replacing <. 3. For he corresponding boundary ses, R no rade { (s, x, z), T R R n : x = α(s, z) }, A no rade {(s, ω), T Ω : (s, ˆX(s, } ω), Z(s, ω)) R no rade, we have û 1 = û 2 = 0, ds dp a.e. on A no rade. The proof of his resul and a discussion of furher properies of he cos-adjused arge funcion are given in Secion Exisence of a Soluion The aim of his secion is o esablish exisence of an opimal conrol. This is done in several seps, firs some a priori esimaes on he growh of he value funcion are esablished. These are hen used o show ha i is sufficien o consider a subse of conrols wih a uniform L 2 -norm bound. This hen permis he use of a Komlós argumen o consruc he opimal conrol. We begin wih some esimaes from he heory of SDEs. Lemma Le X u and Z have dynamics (1.1) and (1.2) respecively. 1. For every p 2 here exiss a consan c p such ha for every 0, T we have E sup Z(s) p R Z() = z n s T c p (1 + z p R n ). 2. There exiss a consan c x such ha for any u U we have E sup X u (s) 2 X u () = x s T c x ( 1 + u 2 L 2 ). In paricular, X u has square inegrable supremum for all u U. Proof. Iem (1) is a well known esimae on he soluion of an SDE wih Lipschiz coefficiens, see for example Barles e al Proposiion 1.1. Le us now prove iem 21

32 1. Curve Following in Illiquid Markes (2). For s, T we have using (1.1) and Jensen s inequaliy X u (s) 2 s s = x + 2 u 1 (r)n(dr) + u 2 (r)dr ( s s ) 3 x 2 + u 1 (r) 2 N(dr) + u 2 (r) 2 dr T T ) c x (1 + u 1 (r) 2 N(dr) + u 2 (r) 2 dr. We now use Lemma A.1.3 in he appendix and relabel he consan o ge for each s, T E sup 0 s T X u (s) 2 ( T T ) X u () = x c x 1 + u 1 (r) 2 dr + u 2 (r) 2 dr. Since he cos and penaly funcions have quadraic growh in x and he SDE for X u is linear in he conrol u, i is sensible, a leas inuiively, ha conrols wih large L 2 -norm canno be opimal. Specifically, an a priori esimae on he conrol will be derived in Lemma This esimae is necessary for he proof of our firs main resul, he exisence of an opimal rading sraegy. As a prerequisie for our esimae, we now esablish a quadraic growh esimae on he value funcion. The proof relies on he quadraic growh assumpions on he penaly and liquidiy cos funcions, which allow o reduce he curve following problem o a simpler opimisaion problem whose soluion is known. Lemma There exis consans c 0, c 1, η > 0 such ha for all (, x, z) 0, T R R n. v(, x, z) c 0 x 2 c 1 (1 + z η R n ), Proof. To ease noaion we wrie he expecaion in (1.3) as E,x,z. Using he quadraic growh of f, g, h yields T v(, x, z) ε inf E,x,z u 2 (s) 2 + ( X u (s) α(s, Z(s)) ) 2 ds u U + ( X u (T ) α(t, Z(T )) ) 2. Nex an applicaion of he inequaliy (a b) a2 b 2 provides v(, x, z) ε 2 inf u U E,x,z T u 2 (s) 2 + X u (s) 2 ds + X u (T ) 2 22

33 1.4. Exisence of a Soluion T εe,x,z α(s, Z(s)) 2 ds + α(t, Z(T )) 2. The polynomial growh of α coupled wih Lemma allows us o wrie v(, x, z) ε T 2 inf E,x,z u 2 (s) 2 + X u (s) 2 ds + X u (T ) 2 c 1 (1 + z η u U R ). n I now remains only o esimae he infimum. This erm may be inerpreed as a sochasic conrol problem wih quadraic penaly and cos funcions and zero arge. I is known ha such a conrol problem admis an analyic soluion via Riccai equaions. In paricular we have ha v(, x, z) ε 2 a()x2 c 1 (1 + z η R n ), for a funcion a given by he soluion of he differenial equaion a (s) = a 2 (s) 1 + λa(s), s, T, a(t ) = 1. Solving explicily for a one finds ha i is monoone and ha a() > 0. If we se his complees he proof. c 0 ε min{a(), a(t )} > 0, 2 Using he preceding esimae on he value funcion, we now show ha i is enough o consider rading sraegies which saisfy a uniform L 2 -norm bound. We noe here ha from Assumpion ogeher wih Lemma we deduce ha J(0) <, which is needed in he proof of he following resul. Lemma There is a consan c max such ha u 2 L 2 c max implies ha u canno be opimal. Proof. For a conrol u U we wan o show ha we may bound J(u) from below in erms of u 2 L 2. For he marke order u 2 we have T T J(u) E g(u 2 (s), Z(s))ds εe u 2 (s) 2 ds, (1.6) where we have used Assumpion (2). The esimae in erms of he passive order u 1 is slighly more involved. Le τ 1 denoe he firs jump ime of he Poisson process N afer, an exponenially disribued random variable wih parameer λ, and se τ τ 1 T. The funcions f, g and h are nonnegaive, combining his wih he definiion of v as an infimum we derive τ J(u) =E,x,z g(u 2 (s), Z(s)) + h (X u (s) α(s, Z(s))) ds 23

34 1. Curve Following in Illiquid Markes + E,x,z J(τ, X u (τ), Z(τ), u) E,x,z v(τ, X u (τ), Z(τ)), where J in he above is evaluaed a conrols on he sochasic inerval τ, T. Noing he nonnegaiviy of v his implies he lower bound J(u) E,x,z 1 {τ1 <T }v(τ 1, X u (τ 1 ), Z(τ 1 )). Applying firs he growh esimaes from Lemma 1.4.2, hen combining he inequaliy 1 {τ1 <T } Z(τ 1 ) η R sup Z(s) η n R, n s T wih Lemma provides he exisence of a consan c 1,z such ha J(u) c 1,z + c 0 E,x,z 1 {τ1 <T } X u (τ 1 ) 2, where c 0 > 0 is as in Lemma We may wrie X u (τ 1 ) = X u (τ 1 ) + u 1 (τ 1 ) and observe ha on he se {τ 1 < T } we have he relaion τ1 X u (τ 1 ) = x + u 2 (s)ds. Using he inequaliy (a + b) a2 b 2 wice, ogeher wih he Jensen inequaliy, we ge J(u) c 1,x,z + c 0 E 1 {τ1 <T } u 1 (τ 1 ) 2 τ1 c 2 E u 2 (s) 2 ds, for some consan c 2 > 0, where we drop he subscrip {, x, z}. In ligh of inequaliy (1.6) we derive ( 1 + c 2 ε ) J(u) c 1,x,z + c 0 E 1 {τ1 <T } u 1 (τ 1 ) 2. An applicaion of he law of oal expecaion and relabelling he consans provides he esimae J(u) c 1,x,z + c 0 T λe u 1 (s) 2 e λ(s ) ds. We apply he uniform bound e λ(s ) e λ(t ) for s, T in he above, hen combine wih (1.6) o see ha J(u) c 1,x,z + c 0 u 2 L 2. 24

35 1.4. Exisence of a Soluion In paricular if u 2 L 2 c max J(0) c 1,x,z c 0 + 1, hen we see ha J(u) > J(0) and he conrol u is clearly no opimal. We remark ha a generalisaion of he growh esimae on he value funcion from Lemma o he case of singular conrols will be derived in Lemma Similarly, an a priori esimae which also covers he singular conrol case is given in Lemma Before compleing he proof of Theorem 1.3.1, we recall a definiion and refer he reader o Proer 2004 for furher deails. Definiion A sequence of processes (Y n ) n N defined on, T Ω and valued in R converges o a process Y :, T Ω R uniformly on compacs in probabiliy (UCP) if, for all ε > 0, ( ) lim P n sup Ys n Y s > ε s T We may now complee he proof of our firs main resul, he exisence and uniqueness of an opimal rading sraegy. The proof combines our a priori esimae on he conrol wih a Komlós argumen. Proof of Theorem The sric convexiy of J is a direc consequence of he sric convexiy of f, g and h. From he previous lemma i follows ha (using he noaion herein) if we se hen U cmax = 0. { } u U : u 2 L 2 c max, inf J(u) = inf J(u). u U u U cmax We ake a sequence of minimising processes (u n ) n N U cmax. Due o he uniform bound on he L 2 -norms we may proceed as in Beneš e al Theorem 2.1 o find a subsequence (also indexed by n) ogeher wih a process û :, T Ω R 2 such ha 1 lim n n n u j = û j=1 ds dp a.e. on, T Ω. To be precise, we noe here ha he superscrips index he sequence and he subscrips he componens of he process. Due o Karazas and Shreve 1991 Proposiion 1.2 we may assume ha û 2 is progressively measurable, whereas he predicabiliy of û 1 follows as in Applebaum 2009 Lemma In paricular we deduce firs he appropriae measurabiliy of û U and hen from Faou s lemma ha 25

36 1. Curve Following in Illiquid Markes û U cmax. Before proving he opimaliy of û we firs show some convergence resuls. For n N we se We have he following esimae, ū n 1 n u j and n X n 1 n X uj. n j=1 j=1 E sup X n (s) X m (s) s T T T E ū n 2 (s) ū m 2 (s) ds + E ū n 1 (s) ū m 1 (s) N(ds). Via he de-la-vallée-poussin Theorem, a consequence of he uniform bound on he L 2 - norms is ha (ū n ) n N also converges in L 1 (, T Ω) o û. I now follows ha ( X n ) n N is Cauchy in D, he space of càdlàg processes equipped wih he UCP opology. Hence here exiss a process ˆX such ha X n converges o ˆX. Here by convergence we mean ha lim E sup X n (s) ˆX(s) = 0. n s T In paricular he above argumen implies ha ˆX = Xû up o indisinguishabiliy and is well defined. For he opimaliy, applying Faou s lemma ogeher wih he convexiy of f, g and h gives J(û) lim inf n 1 n n j=1 J(u j ) = inf u U J(u). Turning o uniqueness, suppose û and ū are opimal conrols. The sric convexiy of u 2 g(u 2, z) implies ha û 2 = ū 2 ds dp a.e. on, T Ω. We also have Xû = Xū ( ) ds dp a.e. since oherwise J û+ū 2 < J(û) = J(ū) due o he sric convexiy of x h(x). An applicaion of Lemma A.1.1 now provides ū = û ds dp a.e. on, T Ω The Sochasic Maximum Principle In his secion we are concerned wih he proof of Theorem In paricular we show he exisence of a soluion o he adjoin equaion as well as providing a characerisaion of he opimal conrol. To avoid difficulies relaed o conrols for which he performance funcional is no finie we define a subclass of conrols given by U adm {u U : J(u) < }. 26

37 1.5. The Sochasic Maximum Principle I is clear ha û U adm, ha minimising J over U adm is equivalen o minimising over U and ha for any u U adm we have T E g(u 2 (s), Z(s))ds <. We shall use hese properies hroughou. We begin by recalling he adjoin BSDE on, T given in equaion (1.5), ( ) dp (s) =h ˆX(s) α(s, Z(s)) ds + Q(s)dW (s) + R 1 (s)ñ(ds) + R 2 (s, θ) M(ds, dθ), R ( k ) P (T ) = f ˆX(T ) α(t, Z(T ). Remark As before Q and R are R d and R 2 -valued respecively. Sricly speaking he adjoin equaion should be for a vecor (P 1,..., P n+1 ), where P 1 saisfies he BSDE above. However when one wries down he full sysem one sees ha, due o he fac ha he conrol does no ener he signal Z, P 1 and (P 2,..., P n+1 ) are decoupled. Moreover, we shall see ha he opimaliy crierion only involves P 1, hus i is sufficien o omi (P 2,..., P n+1 ) and o consider only he above BSDE. Due o he simple srucure of he adjoin equaion, he proof of exisence and uniqueness of a soluion is sraighforward, we only give i here for compleeness. Lemma There is a soluion (P, Q, R) o he adjoin BSDE such ha E sup s T T + T P (s) 2 + R 1 (s) 2 ds + Q(s) 2 R d ds T R 2 (s, θ) 2 m(dθ)ds <. R k I is unique amongs riples (P, Q, R) saisfying he above inegrabiliy crierion. Proof. The funcions f, h and α have, respecively, linear and polynomial growh. Applying Lemma we see ha he process T Φ(s) E h ( ) ˆX(r) α(r, Z(r)) dr + f ( ) ˆX(T ) α(t, Z(T )) F s is a square inegrable maringale on, T. Since he filraion is generaed by he Brownian moion and he (compound) Poisson processes by Tang and Li 1994 Lemma 2.3 we deduce he exisence of processes Q, R 1 and R 2 such ha T T T E Q(r) 2 R dr + R d 1 (r) 2 dr + R 2 (r, θ) 2 m(dθ)dr <, R k 27

38 1. Curve Following in Illiquid Markes such ha for s, T we have s s Φ(s) = Φ() + Q(r)dW (r) + A calculaion now shows ha s s R 1 (r)ñ(dr) + R 2 (r, θ) M(dr, dθ). P (s) Φ(s) + h ( ) ˆX(r) α(r, Z(r)) dr T = E h ( ) ˆX(r) α(r, Z(r)) dr + f ( ) ˆX(T ) α(t, Z(T )) F s s is he required soluion. Uniqueness follows from he uniqueness in he maringale represenaion heorem. I remains o show ha P has inegrable supremum. For his, we firs apply Doob s inequaliy o he maringale Φ, hen use he linear growh of f, h and Lemma o ge E sup s T Φ(s) 2 c 1 E Φ(T ) 2 T c 1 E h ( ) ˆX(r) α(r, Z(r)) 2 dr + f ( ) ˆX(T ) α(t, Z(T )) 2 ) c 1 (1 + E ˆX(s) 2 + z η R <. n sup s T Now from he definiion of P i follows ha T E P (s) 2 c 2 (E Φ(s) 2 + E sup s T This complees he proof. sup s T h ( ˆX(r) α(r, Z(r)) ) 2 dr ) <. The sochasic maximum principle explois he convexiy of he performance funcional J ogeher wih he fac ha he minimum of J can be characerised using he subgradien inequaliy, which allows us o give an explici condiion for he opimaliy of a conrol û. Given conrols u, w U adm, he Gâeaux derivaive of J is defined as J J(w + ρu) J(w) (w), u = lim. ρ 0 ρ The following lemma provides he explici formula for he Gâeaux derivaive of he performance funcional. We give a proof based upon Cadenillas and Karazas 1995 Lemma 1.1, however we avoid he measurable selecion argumen herein. In his chaper he saring poin (, x, z) is fixed and we herefore wrie E for E,x,z. 28

39 1.5. The Sochasic Maximum Principle Lemma For u, w U adm he Gâeaux derivaive of J is given by T J ( (w), u =E X u (s) x ) h ( X w (s) α(s, Z(s)) ) + u 2 (s)g u2 (w 2 (s), Z(s)) ds + E ( X u (T ) x ) f (X w (T ) α(t, Z(T ))). Proof. We firs noe ha for s, T, ρ 0, 1 and u, w U adm we have X w+ρu (s) =x + s ( w2 (r) + ρu 2 (r) ) s ( dr + w1 (r) + ρu 1 (r) ) N(dr) =X w (s) + ρ(x u (s) x). Using he fac ha he signal is unaffeced by he conrol ogeher wih he mean value heorem we may compue, for ρ 0, 1, J J(w + ρu) J(w) (w), u = lim ρ 0 ρ { T 1 ( = lim E X u (s) x ) h ( X w (s) + ζρ ( X u (s) x ) α(s, Z(s)) ) dζds ρ 0 0 T 1 +E u 2 (s)g u2 (w 2 (s) + ζρu 2 (s), Z(s)) dζds +E ( X u (T ) x ) f ( X w (T ) + ζρ ( X u (T ) x ) α(t, Z(T )) ) dζ Due o he convexiy of he funcions f, g(, z) and h, exacly as in Cadenillas and Karazas 1995 Lemma 1.1, he inegrands are all decreasing as ρ decreases so ha all limis are well defined as ρ 0. The saemen of he lemma will follow from he monoone convergence heorem once we show ha T ( E X u (s) x ) h ( X w (s) + ( X u (s) x ) α(s, Z(s)) ) ds (1.7) +E T u 2 (s)g u2 (w 2 (s) + u 2 (s), Z(s)) ds +E ( X u (T ) x ) f ( X w (T ) + ( X u (T ) x ) α(t, Z(T )) ) <. Using he linear growh of h ogeher wih he Young inequaliy one can find a consan c such ha for s, T ( X u (s) x ) h ( X w (s) + X u (s) α(s, Z(s)) ) c ( 1 + X u (s) 2 + X w (s) 2 + α(s, Z(s)) 2). The righ hand side is inegrable over, T Ω hanks o he growh esimaes from Lemma and he polynomial growh of α. An idenical argumen may be applied o }. 29

40 1. Curve Following in Illiquid Markes he erm involving f. Thus o complee he proof we need an esimae for he second erm in (1.7). The subgradien inequaliy ogeher wih he nonnegaiviy of g implies ha for s, T he following holds, u 2 (s)g u2 (w 2 (s) + u 2 (s), Z(s)) g (u 2 (s), Z(s)) + ( w 2 (s) + u 2 (s) ) g u2 (w 2 (s) + u 2 (s), Z(s)). Applying now Assumpion (2) and (3) and observing ha u, w U adm complees he proof. Having consruced a formula for he Gâeaux derivaive we may now urn o characerising he opimal conrol. As a prerequisie for some algebraic manipulaions of he Gâeaux derivaive, le us compue P X. From he inegraion by pars formula we derive, for a conrol u U adm and s, T, s P (s)x u (s) P ()X u () s = X u (r )Q(r)dW (r) + s + s ψ(r, u(r))dr R k X u (r )R 2 (r, θ) M(dr, dθ), (P (r ) + R 1 (r)) u 1 (r) + X u (r )R 1 (r) Ñ(dr) where we have used ha N, M = 0, a consequence of he independence of M and N ogeher wih Applebaum 2009 Proposiion , as well as seing ψ(r, u(r)) P (r)u 2 (r) + λu 1 (r)p (r ) + R 1 (r) + X u (r)h ( ˆX(r) α(r, Z(r)) ). We rewrie he above as Y u (s) = P ()X u () + L u (s), (1.8) where he local maringale par L u is defined for s, T by s L u (s) s + X u (r )Q(r)dW (r) + s R k X u (r )R 2 (r, θ) M(dr, dθ) (P (r ) + R 1 (r)) u 1 (r) + X u (r )R 1 (r)ñ(dr). and he non-maringale par Y u is defined by s Y u (s) P (s)x u (s) ψ(r, u(r))dr. We now verify ha L is a rue maringale. Lemma For all u U adm he process L u is a maringale saring in 0. Proof. For he coninuous local maringale par, he resul follows from he Burkholder- 30

41 1.5. The Sochasic Maximum Principle Davis-Gundy and Hölder inequaliies noing ha X u has square inegrable supremum and Q L 2 (, T Ω). For each of he remaining erms we may apply Lemma A.1.3, where he appropriae inegrabiliy follows from he fac ha X u and P have square inegrable suprema and Q, R 1, R 2 as well as u are square inegrable. We can now urn o he proof of our second main resul, he sochasic maximum principle. I is based on ideas given in Cadenillas 2002 and exended o he case where he dynamics are no affine in (X, Z), as discussed in Remark We noe however ha our heorem is no a sraighforward exension hereof. We mus verify firsly ha Cadenillas 2002 Assumpions 3.1 and 4.1 hold in his new seing and secondly ha one can sill derive an apriori L 2 -esimae on he conrol for our choice of cos funcions and dynamics. This moivaes he presen deailed derivaion of he maximum principle. Proof of Theorem We are minimising a convex funcional over U adm, so by Ekeland and Témam 1999 Proposiion a necessary and sufficien condiion for opimaliy of û is ha J (û), u û 0 for all u U adm. (1.9) Due o Lemma we know ha L u is a maringale saring in 0 for all u U adm. In paricular from equaion (1.8) we have ha EY u (T ) Y û(t ) = 0. The definiion of Y u ogeher wih he erminal condiion in (1.5) allows us o wrie his as 0 =E P (T ) X u (T ) ˆX(T T ) ψ(s, u(s)) ψ(s, û(s)) ds =E f ( ) ˆX(T ) α(t, Z(T )) X u (T ) ˆX(T ) T ψ(s, u(s)) ψ(s, û(s)) ds. Using he known form of he Gâeaux derivaive we derive J (û), u û = J (û), u û + E Y u (T ) Y û(t ) T = E u 2 (s) û 2 (s) g u2 (û 2 (s), Z(s)) P (s) ds λe T u 1 (s) û 1 (s) (P (s ) + R 1 (s))ds, (1.10) for every conrol u U adm. A consequence of (1.9) and (1.10) is ha û is opimal if and only if we have ds dp a.e. on, T Ω, 1. u 2 (s) û 2 (s) g u2 (û 2 (s), Z(s)) P (s) 0 u = (u 1, u 2 ) U adm, 2. P (s ) + R 1 (s) = 0. 31

42 1. Curve Following in Illiquid Markes I remains o show ha he firs iem is equivalen o û 2 being he poinwise minimiser of he funcion u 2 g(u 2, Z(s, ω)) P (s, ω)u 2. (1.11) By Assumpion consan deerminisic marke orders û 2 (s, ω) u 2 R incur finie liquidiy coss, in paricular hey are admissible and iem (1) is equivalen o u 2 û 2 (s) g u2 (û 2 (s), Z(s)) P (s) 0 u 2 R. This is he subgradien condiion for he sricly convex funcion defined in (1.11), so û 2 is indeed he poinwise minimiser and he proof is complee. Corollary The opimal marke order saisfies ds dp a.e. 1. û 2 (s) = g u2 (, Z(s)) 1 (P (s)). 2. If P (s) > 0 hen û 2 (s) > 0. Similarly, if P (s) < 0 hen û 2 (s) < 0 and if P (s) = 0 hen û 2 (s) = 0. Proof. By assumpion we have ha g(, z) is sricly convex and hus g u2 (, z) is sricly increasing for each fixed z R n. This funcion is normalised since g(, z) admis is minimum a zero. Moreover, he range of g u2 (, z) is R which is due o Assumpion 1.2.5(2). I is now clear ha his funcion is a bijecion from R ino R, is inverse exiss and is also a bijecion. Iem (1) is now a direc consequence of Theorem 1.3.2(2). Iem (2) is rue since g u2 (, z) is sricly increasing and normalised The Cos-Adjused Targe Funcion In Secion 1.5 we have esablished a necessary and sufficien condiion for he opimaliy of a given conrol. Unforunaely in he case of he passive order i is far from explici. In he presen secion we prove Theorem 1.3.5, which provides a more explici characerisaion of opimaliy in erms of buy and sell regions. These regions are defined in erms of he cos-adjused arge funcion as given in Definiion As a firs sep, we show ha he passive order is of he form û 1 = α ˆX, so i is a sell (buy) order if sock holdings are above (resp. below) he cos-adjused arge funcion. For he proof of his resul, we need he following esimae. Proposiion There exis consans c α and η such ha for each z R n, sup α(s, z) c α (1 + z η R ). n s T Proof. Choosing he zero conrol u 0 and using he polynomial growh of he funcions f, h and α as well as Lemma we see ha here exis c 1 and η 1 such ha v(, 0, z) J(, 0, z, 0) c 1 (1 + z η 1 R n ). 32

43 1.6. The Cos-Adjused Targe Funcion If we now apply Lemma we find furher consans c 2 > 0, c 3 and η 2 such ha v(, α(, z), z) c 2 α(, z) 2 c 3 (1 + z η 2 R n ). Since α is he poinwise minimiser of v wih respec o x, combining he above inequaliies and relabelling consans provides he resul. The following proposiion shows ha he opimal passive order is direced owards he cos-adjused arge funcion. Proposiion The opimal passive order û 1 is given ds dp a.e. on, T Ω by û 1 (s, ω) = α(s, Z(s, ω)) ˆX(s, ω). Proof. We consider he process X which is defined in erms of he opimal marke order û 2 by he following SDE on, T, d X(s) ( = û 2 (s)ds + α(s, Z(s )) X(s ) ) N(ds), X() = x, and wan o show ha he conrol ũ defined for s, T by ) ( α(s, Z(s )) X(s ) ũ(s), û 2 (s) is admissible. The predicabiliy and progressive measurabiliy are sraighforward and hus we need only check he L 2 -naure of he conrol, which is a consequence of he following esimae, sup X(s) 2 c s T ( 1 + T ) û 2 (s) 2 ds + sup α(s, Z(s)) 2, s T ogeher wih Proposiion and Lemma Now le us prove ha such a sraegy is in fac opimal. We le τ 1 be he firs jump ime of N afer and τ τ 1 T. By he dynamic programming principle we have τ E,x,z τ E,x,z g(û 2 (s), Z(s)) + h( X(s) ( α(s, Z(s)))ds + v τ, X(τ), ) Z(τ) g(û 2 (s), Z(s)) + h( ˆX(s) ( α(s, Z(s)))ds + v τ, ˆX(τ), Z(τ)). On he sochasic ime inerval, τ) we have ha he opimal rajecories ˆX and X coincide, and on he se {τ 1 > T } we have equaliy in he above. If we define he se A {τ 1 T } hen he above inequaliy leads o ( E,x,z v τ 1, X(τ ) ( 1 ), Z(τ 1 ) 1 A E,x,z v τ 1, ˆX(τ ) 1 ), Z(τ 1 ) 1 A ( = E,x,z v τ 1, ˆX(τ ) 1 ) + û 1 (τ 1 ), Z(τ 1 ) 1 A. 33

44 1. Curve Following in Illiquid Markes Independence of N and M ogeher wih Applebaum 2009 Proposiion implies Z(τ 1 ) = Z(τ 1 ) so ha by he consrucion of he process X we ge ( E,x,z v τ 1, X(τ ) ( 1 ), Z(τ 1 ) 1 A = E,x,z v τ 1, α ( τ 1, Z(τ 1 ) ), Z(τ 1 ) )1 A. We combine his wih he fac ha τ 1 is exponenially disribued o derive ( 0 E,x,z (v τ 1, α ( τ 1, Z(τ 1 ) ) ) (, Z(τ 1 ) v τ 1, ˆX(τ ) ) 1 ) + û 1 (τ 1 ), Z(τ 1 ) 1 A =E,x,z T λe (v λ(r ) ( r, α(r, Z(r)), Z(r) ) v ( r, ˆX(r ) + û 1 (r), Z(r) )) 1 A dr. Since α is he poinwise minimiser of he value funcion wih respec o x, he inegrand on he righ hand side is nonposiive, and even sricly negaive on he se { (r, ω) û 1 (r, ω) ũ 1 (r, ω) }. This proves ha û 1 = ũ 1 ds dp a.e. on, T Ω and complees he proof. The preceding proposiion shows ha passive sell (buy) orders are used if and only if sock holdings are above (resp. below) he cos-adjused arge funcion. The aim is now o esablish he corresponding resul for he opimal marke order. The proof relies on a careful analysis of he FBSDE and some echnical resuls which are saed in Lemma as well as Proposiions and We denoe by ( ˆX,x,z, Z,z, P,x,z ) he soluion o he coupled FBSDE given by equaions (1.1), (1.2) and (1.5), sared a (, x, z) 0, T R R n. We now show ha ˆX,x,z is monoone in x. Lemma If x < y hen ˆX,x,z (s) ˆX,y,z (s) for each s, T. Proof. As above we denoe by τ 1 he firs jump ime of N afer ime. If here is a jump in, T hen by Proposiion ˆX,x,z (τ 1 ) = ˆX,y,z (τ 1 ) = α(τ 1, Z(τ 1 )). Due o he uniqueness of he soluion o he FBSDE for any iniial daa we derive he flow propery, exacly as in Pardoux and Tang 1999 Theorem 5.1, ˆX s, ˆX,x,z (s),z,z (s) (r) = ˆX,x,z (r), s r T. (1.12) This implies ha ˆX,x,z and ˆX,y,z coincide on τ 1, T. Before a jump of N, ˆX,x,z and ˆX,y,z evolve coninuously and we define he sopping ime τ 2 inf { s : ˆX,x,z (s) = ˆX,y,z (s) } τ 1 T. 34

45 1.6. The Cos-Adjused Targe Funcion By coninuiy ˆX,x,z < ˆX,y,z in s, τ 2 ) and ˆX,x,z = ˆX,y,z in τ 2, τ 1 T again hanks o he flow propery. The following resul is classical in he sudy of fully coupled FBSDEs, see for insance Bender and Zhang 2008 Corollary 6.2. Since we have jumps we provide a proof. Proposiion There exiss a deerminisic measurable funcion ϕ :, T R R n R such ha for s, T we have P,x,z (s) = ϕ(s, ˆX,x,z (s), Z,z (s)). Proof. When = 0 since P 0,x,z is adaped and he filraion is generaed by he (compound) Poisson processes and he Brownian moion we have ha P 0,x,z (0) is consan so ha he map (x, z) P 0,x,z (0) is well defined. Using a ime shif argumen exacly as in El Karoui e al. 1997b Proposiion 4.2 one can show ha P,x,z () is deerminisic so ha he map ϕ(, x, z) = P,x,z () is well defined. Using he flow propery (1.12) we see ha for s, T as required. P,x,z (s) = P s, ˆX,x,z (s),z,z (s) (s) = ϕ(s, ˆX,x,z (s), Z,z (s)), We now show ha ϕ (or equivalenly P ), viewed as a funcion of x, is sricly decreasing and normalised a α. Combining his wih he represenaion û 2 = gu 1 2 (P ) from Corollary will hen lead o buy and sell regions which are separaed by α. Proposiion For all s, T and z R n he map x ϕ(s, x, z) is sricly decreasing. Moreover we have ϕ (s, α(s, z), z) = 0. Proof. Using Proposiion and he definiion of he adjoin equaion (1.5) we have he represenaion T ( ) P,x,z () = ϕ(, x, z) = E,x,z h ˆX,x,z (s) α(s, Z,z (s)) ds (1.13) E,x,z f ( ) ˆX,x,z (T ) α(t, Z,z (T )). Suppose x < y, hen from Lemma ogeher wih he càdlàg propery of he pahs of ˆX,x,z and he fac ha h, f are normalised and sricly increasing, i follows ha ϕ(s, x, z) ϕ(s, y, z). We observe ha ϕ(, x, z) = ϕ(, y, z) would imply ˆX,x,z (s) = ˆX,y,z (s) ds dp a.e. on, T Ω so ha ˆX,x,z and ˆX,y,z would be indisinguishable by Lemma A.1.1, which conradics ˆX,x,z () = x < y = ˆX,y,z (). To prove he second claim, le τ 1 denoe he firs jump ime of N afer and define τ τ 1 T. Due o he independence of N and M, we see from Applebaum

46 1. Curve Following in Illiquid Markes Proposiion ha hey do no jump a he same ime. In paricular, using ha τ 1 is exponenially disribued wih parameer λ, we may wrie E P (τ) 2 (P = E (τ ) + R1 (τ) ) 2 T = E λe λ(s )( P (s ) + R 1 (s) ) 2 ds = 0. The final equaliy follows since P (s )+R 1 (s) = 0, ds dp a.e. on, T Ω by Theorem and we have dropped he superscrips as we now consider a fixed saring poin (, x, z) 0, T R R n. Using Proposiion we may wrie his as 0 =E T =E P (τ) 2 = E A consequence of Proposiion is ha ( ϕ τ, ˆX(τ), ) 2 Z(τ) ( λe λ(s ) ϕ s, ˆX(s), ) 2 Z(s) ds. ˆX(τ) = ˆX(τ ) + û 1 (τ) = α(τ, Z(τ )) = α(τ, Z(τ)), so we have T 0 = E λe λ(s ) ϕ (s, α(s, Z(s)), Z(s)) 2 ds, and hus ϕ (s, α(s, Z(s)), Z(s)) = 0 ds dp a.e. on, T Ω. Since he process P (and hence ϕ) is càdlàg, an argumen as in Lemma A.1.1 now shows ϕ (s, α(s, Z(s)), Z(s)) = 0 for all s, T. We are now in a posiion o prove our hird main resul, Theorem 1.3.5, which provides a necessary and sufficien condiion of opimaliy in erms of buy and sell regions. The proof is now basically a consequence of Proposiions and Proof of Theorem The firs asserion is he conen of Proposiion To prove he second par, firs recall he buy region R buy { (s, x, z), T R R n : x < α(s, z) }. Using Proposiion we conclude ha for (s, ω) such ha (s, ˆX(s, ω), Z(s, ω)) is in he buy region we have P (s, ω) > 0, recall ha P (s, ω) and P (s, ω) are equal ds dp a.e. on, T Ω. An applicaion of Corollary 1.5.5(2) shows ha in his case û 2 (s) > 0 ds dp a.e. The fac ha û 1 > 0 ds dp a.e. for such (s, ω) is a consequence of he definiion of he buy region ogeher wih Proposiion The proof for he sell and no-rade regions is symmeric. One paricular consequence of Theorem is he following: If sock holdings are 36

47 1.6. The Cos-Adjused Targe Funcion above he cos-adjused arge funcion, hen i is opimal o use marke sell orders; if hey are below, i is opimal o use buy orders. Only if sock holdings and he cosadjused arge funcion agree, no marke orders are used. In his sense, he no-rade region consiss of only one poin and is degenerae. Equivalenly, we have Technically, his is due o he represenaion û 2 (s) = 0 if and only if P (s) = 0. (1.14) û 2 (s) = g u2 (, Z(s)) 1 (P (s)) from Corollary coupled wih he sric convexiy and smoohness of he cos funcion u 2 g(u 2, ). These assumpions also imply ha he marginal coss of rading are zero, i.e. for each z R n we have lim g u 2 (u 2, z) = g u2 (0, z) = 0. (1.15) u 2 0 The following counerexample shows ha if he marginal coss of rading are nonzero, he consideraions above do no hold. In his example, here is a fixed posiive bid ask spread which makes marke orders less aracive. We remark ha in Chaper 2 we will consider a model wih emporary price impac and resilience. In ha case, we also have a posiive spread and i will urn ou ha he no-rade regions is hen no degenerae (as in he presen case). Example Le us illusrae ha Theorem and in paricular relaion (1.14) are no longer rue if he assumpion of g being C 1 is dropped. We consider he liquidiy cos funcion defined by g(u 2, z) = c u 2 + εu 2 2 for a consan c > 0 represening bid ask spread as in Remark u 2 g(u 2, ) is no C 1 and he marginal coss of rading are given by lim g u2 (u 2, z) = c > 0. u 2 0, u 2 0 The funcion In his case one can sill prove a maximum principle and show ha he opimal marke order is he poinwise minimiser of u 2 g(u 2, Z(s)) P (s)u 2, which is now given by û 2 (s) = 1 2ε sign(p (s))( P (s) c ) +. I follows ha if P (s) c hen û 2 (s) = 0 and (1.14) does no hold. We have now esablished a necessary and sufficien condiion of opimaliy in erms of buy and sell regions, which are defined via he cos-adjused arge funcion. In order o gain furher insigh ino he srucure of hese regions, he remainder of his secion is devoed o some qualiaive properies of he funcion α. Specifically, we show 37

48 1. Curve Following in Illiquid Markes ha he map α α is ranslaion invarian and preserves orderings and boundedness. Moreover, in he case when α is a deerminisic funcion α coincides wih α if and only if α is consan. We firs demonsrae ha he map α α is monoone. This propery is naural, a larger arge funcion canno correspond o a smaller cos-adjused arge funcion. Proposiion If α(s, z) β(s, z) for all (s, z), T R n hen we have α(s, z) β(s, z) for all (s, z), T R n. Proof. We prove he claim by conradicion and assume here exiss ( 0, z 0 ), T R n wih α( 0, z 0 ) < β( 0, z 0 ) so ha one may choose x 0 wih α( 0, z 0 ) < x 0 < β( 0, z 0 ). We denoe by ( ˆX α, P α ) and ( ˆX β, P β ) he opimal pairs for he problem sared a ( 0, x 0, z 0 ) wih arges α and β, respecively. Observe firs ha by Proposiion we have P α ( 0 ) < 0 < P β ( 0 ). sopping ime τ α,β inf { } s 0, T : P α (s) P β (s) Define he as he firs ime ha P α is larger han or equal o P β, wih he convenion inf =. To deduce a conradicion we mus firs esablish several properies of he sopping ime τ α,β. If τ 1 denoes, as in Lemma 1.4.3, he firs jump ime afer 0 of he Poisson process N hen from Proposiions and we have P α (τ 1 ) = P β (τ 1 ) = 0. Thus we conclude τ α,β τ 1. (1.16) The waiing ime unil he firs jump of N or M is exponenially disribued and since P α, P β evolve coninuously in he absence of jumps we have τ α,β > 0. (1.17) We now wan o compare he processes ˆX α and ˆX β up o he sopping ime τ α,β. The funcion g(, z) is assumed o be smooh and sricly convex for fixed z R n, in addiion i has uniform quadraic growh in u 2. This implies ha g u2 (, z) is inverible wih a well defined sricly increasing inverse, for all z R n. Thus we deduce ha ds dp a.e. on ( 0, τ α,β T ) Ω û α 2 (s) = g u2 (, Z(s)) 1 (P α (s)) g u2 (, Z(s)) 1 (P β (s)) = û β 2 (s). 38

49 1.6. The Cos-Adjused Targe Funcion Using (1.16) we have ha for s 0, τ α,β ) which shows ha s s ˆX α (s) = x 0 + û α 2 (r)dr x 0 + û β 2 (r)dr = ˆX β (s), (1.18) 0 0 ˆX α (s) α(s, Z(s)) ˆX β (s) β(s, Z(s)). (1.19) Finally, consider he se {τ α,β > T }. On his se we have P α (T ) < P β (T ), hus using he erminal condiion of he BSDE (1.5) we see f ( ˆXα (T ) α(t, Z(T )) ) > f ( ˆXβ (T ) β(t, Z(T )) ). However comparing his wih (1.19) and noing τ α,β > T we ge a conradicion as f is sricly increasing. Thus we also have Le us now derive a conradicion. We wrie using (1.19) τ α,β T. (1.20) τα,β E 0,x 0,z 0 P α (τ α,β ) P α ( 0 ) = E 0,x 0,z 0 h ( ˆXα (s) α(s, Z(s)) ) ds 0 τα,β E 0,x 0,z 0 h ( ˆXβ (s) β(s, Z(s)) ) ds 0 = E 0,x 0,z 0 P β (τ α,β ) P β ( 0 ). This implies E 0,x 0,z 0 P α (τ α,β ) P β (τ α,β ) P α ( 0 ) P β ( 0 ) < 0, bu P α (τ α,β ) P β (τ α,β ) by definiion of τ α,β, which is he desired conradicion. Nex, we show ha he map α α is ranslaion invarian. Proposiion For any consan c, if β = α + c hen β = α + c. Proof. Le us denoe by v α and v β he value funcions corresponding o he arges α and β, respecively. We use ha u X u is affine o deduce T v α (, x, z) = inf E,x,z g(u 2, Z(s)) + h ( X u (s) + c α(s, Z(s)) c ) ds u U + f ( X u (T ) + c α(t, Z(T )) c ) = v β (, x + c, z). 39

50 1. Curve Following in Illiquid Markes Where he final line follows from using ranslaion properies of he expecaion ogeher wih he definiion of v β. Since he cos-adjused arge is defined o be he resul follows. α(, z) = arg min v α (, x, z), x R In he case ha α is independen of z, we can say more abou he srucure of he cos-adjused arge. Specifically, if he arge funcion is consan (e.g. in he case of porfolio liquidaion) hen i agrees wih he cos-adjused arge funcion. In he more ineresing case of nonconsan arge, hese wo funcions are no he same. Proposiion Le α be independen of z and coninuously differeniable in. Then α α if and only if α is consan. Proof. Suppose α c, a consan. If x = c hen he conrol u 0 yields J(, x, z, u) = 0. Since J is nonnegaive we see ha v(, c, z) = 0 and ha v(, x, z) > 0 for x c. This implies ha α(, z) = arg min x v(, x, z) = c. This proves he if -par. We prove he opposie implicaion by conradicion. Suppose ha α α and α is no consan, by coninuiy here is a global minimum and maximum on 0, T. A leas one of hem is aained a some 0 (0, T, and we only consider he case ha α aains a maximum a 0 (he case of a minimum is symmeric). Now here is δ > 0 such ha α is sricly increasing on 0 δ, 0. For he remainder of he proof we assume s 0 δ, 0 ). We denoe by ˆX he process ˆX 0 δ,α( 0 δ),z sared a he poin ( 0 δ, α( 0 δ), z ) and P he corresponding soluion o he backward equaion. The crucial observaion is ha sock holdings are never above he arge funcion, i.e. ˆX(s) α(s), (1.21) for each s 0 δ, 0 ). Indeed, if τ denoes a jump ime of he Poisson process N, hen by Proposiion and he assumpion α α we have ˆX(τ) = α(τ, Z(τ )) = α(τ). In paricular ˆX does no jump above α on he ime inerval 0 δ, 0 ). Furhermore, if here is no jump and if we have ˆX(s) = α(s) = α(s) hen by Theorem (3) we have ds dp a.e. on 0 δ, 0 Ω û 2 (s, ω) = 0 < α (s), as α is smooh and sricly increasing on 0 δ, 0 ) by assumpion. In oher words, if sock holdings are on he cos-adjused arge funcion, no marke orders are used and rading sops. The implicaion is ha ˆX does no cross α from below and (1.21) holds. As ˆX is never above α (and hus never above α) he monooniciy propery of P given 40

51 1.6. The Cos-Adjused Targe Funcion in Proposiion implies for s 0 δ, 0 ). However from he definiion of P we have P (s) 0, (1.22) E 0 δ,α( 0 δ),z P (s) P ( 0 δ) s h ( ) ˆX(r) α(r) dr 0. =E 0 δ,α( 0 δ),z 0 δ The las inequaliy follows from noing ha ˆX(r) α(r) 0 and ha h is increasing and normalised. Rearranging he above inequaliy we see ha E 0 δ,α( 0 δ),z P (s) P ( 0 δ). (1.23) In addiion we have P ( 0 δ) = 0, since ˆX sars on he cos-adjused arge funcion. Combining (1.22) and (1.23) we now see ha P (s) = 0 on he whole ime inerval 0 δ, 0 ). An applicaion of Corollary now shows ha we have û 2 (s) = 0 ds dp a.e. on 0 δ, 0 ), i.e. no marke orders are used. Moreover, from P (s) = 0 and Proposiion i follows ha ˆX(s) = α(s) = α(s), (1.24) a.e. on 0 δ, 0 ) so ha û 1 (s) = α(s) ˆX(s) = 0 a.e. and passive orders are also no used. This implies ha ˆX has pahs which are almos surely consan on he inerval 0 δ, 0 ). However by assumpion α is sricly increasing on his inerval, so (1.24) provides he necessary conradicion. As a corollary we show ha he cos-adjused arge funcion can be bounded above (below) by he maximum (resp. minimum) of he arge funcion. This is naural, he cos-adjused arge should no exceed he maximum of he arge funcion. Corollary Le (s, z), T R n. We have he following esimae, inf r T inf α(r, y) α(s, z) sup y Rn r T Proof. We only prove he firs inequaliy and define c inf r T inf α(r, y). y Rn sup α(r, y). y R n If c =, here is nohing o prove. Since he funcion α has polynomial growh, we may assume c R. From Proposiion we have α(s, z) c for all (s, z), T R n and c = c from Proposiion

52 1. Curve Following in Illiquid Markes We now move on o consider some special cases Examples In his secion, we shall show ha Theorem may be used o derive a closed form soluion for he opimal conrol when he penaly and cos funcions are quadraic. In his case our problem becomes one of quadraic linear regulaor ype, which have been well sudied in he lieraure, see Yong and Zhou 1999 Chaper 6 for an overview. The novely in he presen applicaions is he inerpreaion of he jumps in erms of passive order execuion Curve Following wih Signal The following proposiion gives an example under which we can find he opimal conrol (semi)explicily. Here we have quadraic penaly and liquidiy cos funcions and a general signal which feeds ino he arge funcion. Proposiion Le = 0 and g(u 2, z) = κu 2 2 for a consan κ > 0, h(x) = f(x) = x2 and α be any coninuous funcion saisfying Assumpion Suppose ha he signal is defined by dz(s) = µ(s, Z(s))ds + σ(s, Z(s))dW (s), Z(0) = z. where µ and σ are bounded coninuous real-valued funcions wih σ δ > 0 for some consan δ. Suppose in addiion ha µ, σ, α are Hölder coninuous for some exponen less han 1, uniformly in. Then he opimal conrol is given ds dp a.e by û 1 (s, ˆX(s ), Z(s)) û 2 (s, ˆX(s), Z(s)) b(s, Z(s)) = a(s) ˆX(s ), = a(s) ( ) b(s, Z(s)) 2κ a(s) ˆX(s), (1.25) where he funcions a and b saisfy linear PDEs and can be given in (semi)explici form. Proof. We firs noe ha one can check by direc compuaion ha he funcions g(u 2, ) = κu 2 2 and f(x) = h(x) = x 2 saisfy Assumpions and We know from Theorem ha ds dp a.e. on 0, T Ω, he opimal marke order is given by he poinwise minimiser of u 2 κu 2 2 P (s)u 2. 42

53 1.7. Examples Thus we compue ha u 2 (s) = (2κ) 1 P (s). This leads o he following coupled FBSDE, d ˆX(s) = (2κ) 1 P (s)ds + û 1 (s)n(ds), dp (s) = 2 ( ( )) ˆX(s) α s, Z(s) ds + Q(s)dW (s) + R1 (s)ñ(ds), ˆX(0) = x, P (T ) = 2 ( (1.26) ( )) ˆX(T ) α T, Z(T ). We have omied he process Z in he above as i may be solved independenly of (P, ˆX) and hen regarded as an inpu. I follows from Theorem ha any soluion of he above for which P (s )+R 1 (s) = 0, ds dp a.e. on 0, T Ω provides he opimal conrol. Moreover, since he coupled FBSDE is in one o one correspondence wih he opimal conrol (again by Theorem 1.3.2) here is a mos one soluion. We make he ansaz P (s) = a ( s, Z(s) ) ˆX(s) + b ( s, Z(s) ), (1.27) for deerminisic funcions a and b o be deermined. A consequence of he above ansaz is ha he jumps of P are equal o a imes he jumps of ˆX. In paricular we know ha û1 should ensure P (s ) + R 1 (s) = a ( s, Z(s) ) ˆX(s ) + û 1 (s) + b ( s, Z(s) ) = 0. (1.28) This gives he following form for he passive order, û 1 ( s, ˆX(s ), Z(s) ) = b(s, Z(s)) a ( s, Z(s) ) ˆX(s ). Insering his ino (1.26) leads o he FBSDE which we hope o solve by he above ansaz. We proceed similarly o he four sep scheme of Ma e al. 1994, applying Iô s formula o he expression for P from equaion (1.27) and using he definiion of he opimal conrols we derive ha dp = (a ˆX s + µa z + 1 ) 2 σ2 a zz ( 1 + 2κ a a ˆX + b λb λa ˆX + (b s + µb z + 1 ) 2 σ2 b zz ds, ( ) ds + σ ˆXaz + b z ) ds ( ) b dw (s) a + ˆX Ñ(ds) where we suppress he argumens (s, z) of a, b, µ, σ and ˆX for breviy. These dynamics mus coincide wih hose of equaion (1.26) so ha maching he coefficiens we derive ha he funcions a and b should solve he following parial differenial equaions on 43

54 1. Curve Following in Illiquid Markes 0, T R, a s + µa z σ2 a zz λa + 1 2κ a2 2 = 0, a(t, z) = 2, (1.29) b s + µb z σ2 b zz λb + 1 ab + 2α = 0, b(t, z) = 2α(T, z), (1.30) 2κ where we suppress he (s, z) for noaional simpliciy. The equaion (1.29) for a can be solved independenly of z as a sandard Riccai equaion, ( ) 2ζ a(s) = κ λ + ζ 1 c a e ζs, (1.31) where we define ζ λ 2 + 4/κ and choose c a such ha he boundary condiion a T is saisfied. A calculaion shows ha a(s) < 0 for all s 0, T, so he erms in (1.25) are well defined. The PDE (1.30) for b hen becomes a hea equaion wih a source erm and bounded Hölder coninuous coefficiens for which Friedman 1964 Theorem gives he exisence of a soluion as well as a semi-explici formula in erms of Green s funcions. Using he Feynman-Kac formula we can give a more explici probabilisic soluion which beer displays is relaion wih he original problem. Namely we have T ( r ) ( ) T b(s, z) = 2E s,z exp ρ(w)dw α(r, Z(r))dr + exp ρ(w)dw α(t, Z(T )), s s s where we use he noaion E s,z as in Lemma and he funcion ρ(s) a(s) 2κ λ for s 0, T. Remark In he case of quadraic cos funcions here is a clear economic inerpreaion of he conrols. The funcion b encodes he expeced fuure moion, appropriaely discouned by λ and a 2κ, of he arge funcion wih respec o he disribuion of he signal Z. This hen feeds ino he cos-adjused arge funcion via he raio b a. We see ha his is forward looking, evolves in ime and reacs o changes in Z. A passive order is placed and coninuously adjused so ha when a jump occurs he sock holdings move o he cos-adjused arge funcion. Simulaneously marke orders are used wih a rae a(s) 2κ proporional o he amoun held in he passive order. Such a conrol srucure is inuiive, for fixed (s, z) rading slows down as one approaches he cos-adjused arge α and speeds up as one moves away. Pu anoher way, when he agen has sock holdings near he funcion α he reduces he rading in marke orders, preferring o wai for passive order execuion. The parameer κ describes how expensive rading is, when i is large he agen uses less marke orders and relies more on passive orders. This again coincides wih rading sraegies seen in markes wih low liquidiy. We poin ou a final poin on he naure of he soluion. We have chosen he funcions f(x) = h(x) = ηx 2, wih he value η = 1. For general η one can simply scale he value funcion by η 1 o reduce o he curren seing, wih parameer κ = κ η. Now we can 44

55 1.7. Examples Figure 1.1.: Targe funcion (blue), cos-adjused arge (red) and a ypical rajecory of sock holdings (black). inerpre η as urgency parameer, a key feaure presen in all modern rading algorihms, his conrols how close one should adhere o he arge and hus decides he allocaion beween marke and passive orders. Figure 1.1 illusraes he relaion of arge, cos-adjused arge and a ypical rajecory of sock holdings Porfolio Liquidaion We consider an invesor who wans o sell x sock shares over he inerval 0, T, however rading incurs liquidiy coss. The quesion hen becomes wha he opimal sraegy should be. The consrucion of such a rading program has received much aenion recenly, see for insance Almgren and Chriss 2001, Obizhaeva and Wang 2005 as well as Schied and Schöneborn Wih a lile exra work one can embed his in he presen seup. When we assume, as in Almgren 2003, ha marke orders incur quadraic rading coss and here is a quadraic penaly on sock holdings we are led o he following formulaion, v(0, x) inf J(0, x, u), u U 0,X u (T )=0 T J(0, x, u) E 0,x κ u 2 (s) 2 + X u (s) 2 ds, 0 where κ > 0 and we have omied a signal for ease of exposiion. Observe he new feaure in he presen opimisaion problem is ha we now have a binding consrain, we are ineresed in only hose conrols for which X u (T ) = 0. 45

56 1. Curve Following in Illiquid Markes Proposiion The opimal conrol is given ds dp a.e. by û 1 (s, ˆX(s )) = ˆX(s ), û 2 (s, ˆX(s)) = a(s) 2κ ˆX(s), where a is given by wih ζ λ 2 + 4/κ. ( ) 2ζ a(s) = κ λ + ζ 1 e ζ(s T ), (1.32) Proof. Le n N and define he sequence of approximae opimisaion problems wihou binding consrain, bu wih an addiional penaly erm a ime T, by v n (0, x) inf u U 0 J n (0, x, u), T J n (0, x, u) E 0,x κ u 2 (s) 2 + X u (s) 2 ds + n X u (T ) 2. 0 Applying he mehods of he previous subsecion we see ha he opimal conrol corresponding o v n is given ds dp a.e. by û n 1 (s, ˆX n (s )) = ˆX n (s ), û n 2 (s, ˆX n (s)) = an (s) 2κ ˆX n (s), where he a n are defined by ( ) a n 2ζ (s) κ λ + ζ 1 c n e ζ(s T ), wih he consan c n chosen so ha a n (T ) = 2n. A calculaion shows ha he funcions a n converge owards a and hence he opimal conrols û n converge o û, for a and û given in he saemen of he proposiion. We may rewrie he performance funcional associaed o he porfolio liquidaion problem as T J(0, x, u) = E 0,x κ u 2 (s) 2 + X u (s) 2 ( ds + δ {R\{0}} X u (T ) ), 0 where δ {R\{0}} is he indicaor funcion in he sense of convex analysis. This leads o he following inequaliy, J n (0, x, u) J(0, x, u) for all x R, u U 0. (1.33) Nex we show ha he sock holdings associaed o û = lim n u n saisfy he liquidaion 46

57 1.7. Examples conrain ˆX(T ) = 0. Is dynamics are given by d ˆX(s) = a(s) 2κ ˆX(s)ds ˆX(s )N(ds), ˆX(0) = x. If N jumps in 0, T we clearly end up wih zero sock holdings, hus we are reduced o showing ha ( ) 1 T ˆX(T ) = x exp a(s)ds 1 T τ1 = 0, 2κ 0 where τ 1 is again he firs jump ime of N. Using he explici formula for a from (1.32), his equaliy can be verified. Before we prove he opimaliy of û, we need o show ha he inegrand of he performance funcional J n converges. Indeed, we have ds dp a.e. { lim κ û n n 2 (s) 2 + Xûn (s) 2} = κ û 2 (s) 2 + ˆX(s) 2. A compuaion shows ha he sock holdings associaed o û n are given by ( s Xûn (s) =x exp 0 =x exp ) ( s û 2 (r)dr 1 {s<τ1 } = x exp 0 ( ) (λ ξ)s cn exp(ξ(s T )) 1 1 c n exp( ξt ) {s<τ1 }, and a furher compuaion shows ha he erminal penaly saisfies { lim n X û n (T ) 2} = 0. n ) 1 2κ an (r)dr 1 {s<τ1 } Now an applicaion of Faou s Lemma ogeher wih (1.33) yields for any u U 0 T J(0, x, û) =E 0,x κ û 2 (s) 2 + ˆX(s) 2 ds 0 lim inf n E 0,x T κ û n 2 (s) 2 + Xûn (s) 2 ds + n Xûn (T ) 2 0 = lim inf n J n (0, x, û n ) lim inf n J n (0, x, u) J(0, x, u). This shows ha û is opimal and complees he proof. Remark The opimal sraegy is o simulaneously place all ousanding shares as a passive order and sell using marke orders wih he rae 1 2κ a(s), proporional o curren sock holdings. A similar resul is obained in Kraz Remark Our main resuls, Theorems and 1.3.5, can be used o obain a soluion o he porfolio liquidaion problem under quie weak condiions, e.g. for 47

58 1. Curve Following in Illiquid Markes general cos and penaly funcions and a general sochasic signal Z. The process Z could represen e.g. he order book heigh, hereby exending Almgren 2003 o a seing wih sochasic liquidiy parameers as well as passive orders. The example discussed in Proposiion provides an explici soluion and herefore needs srong assumpions (consan parameers, no spread, quadraic coss, quadraic penaly) Bid-Ask Spread and he Independence of he Jump Processes One choice for he sochasic signal Z would be bid ask spread. However, in our model a jump of N represens a liquidiy even which execues he invesor s passive order. In real markes a liquidiy even which execues passive orders migh also emporarily widen he bid ask spread on one side of he book. This is in conras wih our requiremen ha Z and N be independen. In he presen secion we shall show ha when one relaxes his assumpion an ineresing feaure occurs, i migh be opimal o place passive sell and marke buy orders a he same ime. From a praciioner s poin of view, rading on differen sides of he marke simulaneously is no desirable and we discuss his furher in he presen secion. To he bes of our knowledge, his problem has no been addressed in he lieraure on illiquid markes, since mos papers eiher only allow for one ype of orders or only consider porfolio liquidaion where i is clear a priori ha only sell orders and no buy orders are used. We now drop he assumpion of Z and N being independen and suppose ha he dynamics of Z inroduced in (1.2) are replaced by dz(s) = µ(s, Z(s))ds + σ(s, Z(s))dW (s) + γ (s, Z(s ), θ) M(ds, dθ) + δ (s, Z(s )) N(ds), R k for some funcion δ :, T R n R n Lipschiz in z, uniformly in s. Using an idenical proof o ha of Theorem one can show exisence and uniqueness of a soluion as well as he characerisaion of Theorem 1.3.2(2). In his secion we presen examples o show ha Theorem 1.3.5(2) is no longer valid, in paricular i can be opimal o place boh marke buy (sell) and passive sell (buy) orders simulaneously. When one inerpres he process û 1 as an order placed in a dark pool, he above behaviour amouns o selling in he dark venue and buying in he visible venue (or vice versa). This phenomenon arises because when he passive order is execued, he signal jumps as well and he passive order foresees his, whereas he marke order does no so ha hey may have differen signs. However when û 1 is inerpreed as a limi order his behaviour is equivalen o he invesor placing liquidiy on he buy (sell) side and hen consuming i hemselves. This is raher counerinuiive and is no economically raional or realisic, hus we conclude from our examples ha in he general seing i is necessary o reain he assumpion of independence beween N and Z. Hence for arbirary arge funcions Z may no be 48

59 1.8. Bid-Ask Spread and he Independence of he Jump Processes Figure 1.2.: Targe funcion (blue) and sock holdings (black) for Example Before a jump, marke buy orders are used o reduce deviaion. A he same ime, a passive sell order is placed such ha sock holdings and signal boh jump o zero in he case of execuion. inerpreable as spread. However by imposing specific condiions on α we show ha, even when independence does no hold, he opimal conrol does no exhibi he undesirable behaviour described above. Thus in cerain circumsances an inerpreaion of Z as spread is compaible wih he noion of û 1 as a limi order. Our firs example shows ha if he signal eners he arge bu no he cos funcion hen one may simulaneously buy and sell wih posiive probabiliy. Example Suppose he arge funcion is given by α(, z) = z and he sochasic signal saisfies dz(s) = Z(s )N(ds), Z(0 ) = z > 0. The signal is piecewise consan, before a jump of he process N i akes he value z, aferwards i akes he value 0 (and says here). We assume as above ha N is a Poisson process wih inensiy λ > 0. Le he cos and penaly funcions be given by g(u 2, z) = κu 2 2 for some κ > 0 and h(y) = y2. The invesor wans o minimise he deviaion of sock holdings from he arge funcion, where his sock holdings saisfy as above dx u (s) = u 2 (s)ds + u 1 (s)n(ds), X u (0 ) = x (0, z). The performance funcional is now defined as T J(, x, z, u) E,x,z κu 2 (s) 2 + (X u (s) Z(s)) 2 ds. 49

60 1. Curve Following in Illiquid Markes Applying he same ideas as in he proof of Proposiion 1.7.1, we firs noe ha equaion (1.28) urns ino P (s ) + R 1 (s) = a(s) ˆX(s ) + û 1 (s) + b ( s, 0 ) = 0, where he coefficien b is evaluaed a 0 since he signal jumps o 0 in case of a jump. In paricular he new feaure for he passive order is ha he signal Z is no evaluaed before, bu afer he jump of N. Inuiively, a jump of N execues he passive order and les he signal jump o zero. The opimal conrol can hen be compued as above as û 1 (s, ˆX(s ), b(s, 0) Z(s )) = a(s) ˆX(s ), û 2 (s, ˆX(s ), Z(s )) = a(s) ( ) b(s, Z(s )) 2κ a(s) ˆX(s ). (1.34) The economic inerpreaion of he opimal conrols is idenical wih ha of Remark save for where he cos-adjused arge funcion is evaluaed afer he jump ime of N. As above, he coefficiens a and b are given by he following differenial equaions The explici soluions are given by a s λa + 1 2κ a2 2 = 0, a(t ) = 0, b s λb + 1 ab + 2z = 0, b(t, z) = 0. 2κ ( a(s) =κ λ + ζ b(s, z) = a(s)z = κz 2ζ 1 c a e ζs ), ( λ + ζ ) 2ζ 1 c a e ζs, where he inegraion consan c a is chosen such ha he erminal condiion a(t ) = 0 is me and we se ζ λ 2 + 4/κ. I can be verified ha a is sricly negaive on 0, T ). I now follows ha he cos-adjused arge funcion is given by, for s 0, T ), b(s, z) α(s, z) = = z. (1.35) a(s) In paricular, he cos-adjused arge funcion agrees wih he arge funcion. Combining (1.34) and (1.35) we see ha he opimal conrol is û 1 (s, ˆX(s ), Z(s )) û 2 (s, ˆX(s ), Z(s )) = ˆX(s ), = a(s) ( Z(s ) 2κ ˆX(s ) ). Recall ha ˆX(0 ) = x < z = Z(0 ) and ha a < 0. Before a jump of N, he dynamics 50

61 1.8. Bid-Ask Spread and he Independence of he Jump Processes of X u are now given by d ˆX(s) = a(s) ( z 2κ ˆX(s ) ) ds, which is a mean revering process wih mean z. I grows monoonically owards z wihou reaching i. The fac ha ˆX < z a.e. before he firs jump of N implies ha û2 > 0, i.e. marke buy orders are used. If a jump of N occurs, he cos-adjused arge funcion jumps o zero. A he same ime, he passive order is execued and all socks are sold. To sum up, we have ds dp a.e. before a jump û 1 < 0 and û 2 > 0. Figure 1.2 illusraes he dynamics of he opimal rajecories. The preceding example illusraes ha i migh be opimal o use marke buy and passive sell orders a he same ime. In his example, he signal only influences he arge funcion, bu no he liquidiy coss. One migh conjecure ha such counerinuiive behaviour can be excluded in a model where he signal affecs he liquidiy coss, bu no he arge funcion. Unforunaely, his is wrong, as illusraed by he following example. In addiion i provides an explici soluion o a curve following problem wih a regime shif in liquidiy. Example This is a sylised model of liquidiy breakdowns. There are imes wih high liquidiy (low liquidiy coss) and wih sparse liquidiy (high liquidiy coss). This regime shif migh be riggered by a news even or a very large rade. Consider an invesor who wans o keep his sock holdings close o a deerminisic funcion α. The opimisaion problem is T inf E 0,x,z Z(s)u 2 (s) 2 + (α(s) X u (s)) 2 ds. u U 0 0 Here Z is a liquidiy parameer, we migh hink of he inverse order book heigh. The higher Z, he more expensive marke orders are. We assume ha Z can only ake wo values. In he firs sage (before a jump of he Poisson process N), Z equals κ 1 > 0. In he second sage (afer he firs jump of N), Z akes he value κ 2 > 0 and remains here unil mauriy. This assumpion is made for simpliciy. I allows o solve he opimisaion problem in wo sages, corresponding o he wo possible values of Z. The dynamics of Z are hen given by dz(s) = (κ 2 Z(s )) N(ds), Z(0 ) = κ 1. As above, we model he invesor s sock holdings by dx u (s) = u 2 (s)ds + u 1 (s)n(ds), X u (0) = x. We will solve he above opimisaion problem in wo sages, before and afer a jump 51

62 1. Curve Following in Illiquid Markes of he Poisson process N. In he second sage, he signal is consan, Z κ 2. The performance funcional and he value funcion are defined by T J(, x, u) E,x κ 2 u 2 (s) 2 + (α(s) X u (s)) 2 ds, v(, x) inf u U J(, x, u). The HJB equaion associaed o his opimisaion problem is { } 0 = inf u R 2 v s + u 2 v x + κ 2 u (α x) 2 + λ v(, x + u 1 ) v(, x), wih erminal condiion v(t, x) = 0. We ry he following quadraic ansaz for v: v(, x) = 1 2ā()x2 + b()x + c(), for coefficiens ā, b, c : 0, T R. The opimal conrol as well as he cos-adjused arge funcion in he second sage hen saisfy û 1 (s, ˆX(s )) û 2 (s, ˆX(s )) α 2 (s) = α 2 (s) ˆX(s ), = ā(s) ( α 2 (s) 2κ ˆX(s ) ), 2 b(s) = ā(s), where he coefficiens ā, b, c are given as he soluion o he following sysem of Riccai equaions ā s 1 ā λā = 0, ā(t ) = 0, 2κ 2 bs 1 ā b 2α λ b = 0, b(t ) = 0, 2κ 2 c s 1 4κ 2 b2 + α 2 λ b 2 2ā = 0, c(t ) = 0. The closed form soluion o his sysem of ODEs is given by 2ζ ā(s) = κ 2 ( λ ζ + 1 cāe ζs T ( b(s) = 2α(r) exp 1 2κ 2 c(s) = s T s ), r ( 1 4κ 2 b2 (r) + α 2 (r) λ b 2 (r) 2ā(r) s ) ā(w)dw λ(r s) dr, ) dr, 52

63 1.8. Bid-Ask Spread and he Independence of he Jump Processes where he inegraion consan cā is chosen such ha he erminal condiions ā(t ) = 0 is saisfied and we defined ζ λ 2 + 4/κ 2. Now ha we have solved he opimisaion problem in he second sage, le us consider he firs sage (before he firs jump of N). We hen have Z κ 1. Le τ 1 denoe he firs jump ime of N and τ τ 1 T. The opimisaion problem under consideraion is now τ inf E 0,x κ 1 u 2 (s) 2 + (α(s) X u (s)) 2 ds + v (τ, X u (τ ) + û 1 (τ)). u U 0 0 By definiion, τ 1 is exponenially disribued wih densiy φ(s) λe λs. We proceed as in Pham 2009 Secion and rewrie he opimisaion problem as T inf E 0,x u U T λe λs ( s 0 λe λs ( T 0 κ 1 u 2 (r) 2 + (α(r) X u (r)) 2 ) dr + v (s, X u (s ) + û 1 (s)) ds ) κ 1 u 2 (r) 2 + (α(r) X u (r)) 2 dr + v (T, X u (T )) ds. Changing he order of inegraion and combining wih v(t, x) = 0 and leads o r λe λs ds = e λr β(r) T inf E 0,x u U 0 0 (e λr κ 1 u 2 (r) 2 + (α(r) X u (r)) 2 + λe λr v (r, X u (r ) + û 1 (r)) ) dr T ( = inf E 0,x β(r) κ 1 u 2 (r) 2 + (α(r) X u (r)) 2 ) + φ(r) v (r, X u (r ) + û 1 (r)) dr. u U 0 0 The HJB equaion associaed o his opimisaion problem is { } 0 = inf u R 2 v s + u 2 v x + βκ 1 u β(α x) 2 + φ(s) v(s, x + u 1 ), wih erminal condiion v(t, x) = 0. The poinwise minimisers are û 1 = arg min v(s, x) x = b(s) x R ā(s) x, 1 û 2 = v x (s, x), 2β(s)κ 1 and we remark ha for s 0, T ) ā(s) is sricly posiive, so ha he above expressions 53

64 1. Curve Following in Illiquid Markes are well defined. The HJB equaion in he firs sage urns ino ( 1 0 = v s vx 2 + β(s)(α x) 2 + φ(s) b(s) 2 ) 4β(s)κ 1 2ā(s) + c(s). As above, we ry a quadraic ansaz for v: v(, x) = 1 2 a()x2 + b()x + c(), for coefficiens a, b, c : 0, T R. The opimal conrol as well as he cos-adjused arge funcion in he firs sage hen saisfy û 1 (s, ˆX(s )) = α 2 (s) ˆX(s ) û 2 (s, ˆX(s )) = a(s) ( α 1 (s) 2β(s)κ ˆX(s ) ), 1 (1.36) α 1 (s) = b(s) a(s), where he coefficiens a, b, c are given by 1 a s a 2 + 2β(s) = 0, a(t ) = 0, 2β(s)κ 1 1 b s ab 2β(s)α(s) = 0, b(t ) = 0, 2β(s)κ 1 1 ( c s b 2 + β(s)α(s) 2 + φ(s) b ) 2 (s) 4β(s)κ 2ā(s) + c(s) = 0, c(t ) = 0. 1 This sysem admis he following soluion 2ζ a(s) = β(s)κ 1 ( λ ζ + 1 c a e ζs T b(s) = 2α(r)2β(r) exp c(s) = s ( T s ( 1 2κ 1 ), r s ) a(w) β(w) dw dr, ( 1 4β(r)κ 1 b(r) 2 + β(r)α(r) 2 + λβ(r) b(r) 2 2ā(r) + c(r) where he inegraion consan c a is chosen such ha he erminal condiion a(t ) = 0 is me and we se ζ λ 2 + 4/κ 1. To conclude, we see from (1.36) ha in he firs sage he opimal marke order is direced owards he cos-adjused arge funcion of he firs sage, bu he opimal passive order is direced owards he cos-adjused arge funcion of he second sage. The reason for his asymmeric behaviour is ha he passive order akes ino accoun he jump of N, i.e. i foresees he regime shif from he firs o he second sage. A consequence is ha (in general) here are regions where marke buy and passive sell orders are used, )) dr, 54

65 1.8. Bid-Ask Spread and he Independence of he Jump Processes Figure 1.3.: Cos-adjused arge funcions for Example before (red) and afer (dashed red) a jump of he signal. The blue curve is he arge funcion. Before a jump, marke orders are used o keep he sock holdings close o he red curve. Simulaneously, passive orders are direced owards he dashed red curve. In he shaded region, marke sell and passive buy orders are used, or vice versa. and vice versa. To give a concree example we simulaed hese funcions for he arge funcion α(s) = (2s 1) 2, inensiy λ = 10, mauriy T = 1 and liquidiy parameers κ 1 = 0.01 and κ 2 = 1, see Figure 1.3. Examples and show ha counerinuiive rading migh occur if he signal eners he cos, bu no he penaly funcion, and vice versa. The reader migh conjecure ha a sufficien condiion o exclude his is ha he arge funcion is deerminisic and sricly decreasing and sock holdings sar above he arge funcion. Heurisically, he rajecory of sock holdings should hen also be decreasing, so ha only sell orders are used. The following counerexample shows ha his conjecure is wrong. Example We remain in he framework of Example and choose he arge funcion α(s) = anh(100(0.5 s)) + 1. Then α is deerminisic and sricly decreasing. Furhermore, we choose he mauriy T = 1, inensiy λ = 10 and liquidiy cos parameers κ 1 = 10 2 (in he firs sage) and κ 2 = 10 4 (in he second sage). The sock holdings are assumed o be above he arge funcion iniially, i.e. ˆX(0) = α(0) Figure 1.4 shows a plo of he arge funcion (blue) as well as he cos-adjused arge funcions in he firs (red) and second (dashed red) sage. The opimal rajecory of sock holdings in he absence of jumps is given in black. We see ha he sock holdings eners he shaded region where α 1 (s) < ˆX(s) < α 2 (s). In his region, marke sell and passive buy orders are used. The reason is he following: In he firs sage, rapid rading is (relaively) expensive and so i is opimal o reduce sock holdings slowly and sar selling early. In he second sage, rading is (relaively) cheap, so i is opimal o say very close o he arge func- 55

66 1. Curve Following in Illiquid Markes Figure 1.4.: Cos-adjused arge funcions for Example before (red) and afer (dashed red) a jump of he signal. The blue curve is he arge funcion. The black curve represens he opimal sock holdings if no jump occurs. We see ha his funcion eners he shaded region, where marke sell and passive buy orders are opimal. ion. If sock holdings are in he shaded area and a jump happens, i is opimal o use a passive buy order which ake sock holdings up o he dashed red curve again and hen closely follow he blue curve. In conclusion, having a sricly decreasing arge funcion and sock holdings which sar above he arge is no sufficien o exclude counerinuiive opimal conrols. The preceding counerexamples show ha, in general, he signal canno be inerpreed as bid ask spread. We conclude his secion wih a posiive resul and show ha in he case of a consan arge funcion, passive and marke orders always have he same sign, i.e. he counerinuiive resuls described above do no occur. This exends Proposiion o he case where he signal is no independen from passive order execuion. The implicaion is ha for consan arge funcions, e.g. for he imporan case of porfolio liquidaion, we may indeed inerpre Z as bid ask spread. Specifically, he cos funcion discussed in Remark may be generalised o g(u 2, Z) = Z 1 u 2 arcan(u 2 ) + u 2 2(Z 2 + ε), where Z 1 and Z 2 are mean-revering posiive jump processes represening spread and he inverse order book heigh, respecively. Such a cos funcion exends he model from Almgren 2003 o sochasic liquidiy parameers. Lemma Le α(s, z) = c for all (s, z), T R n and some c R. Then u 1 (s) and u 2 (s) have he same sign ds dp a.e. Proof. We only consider he case c = 0 and we firs assume X u (0) > 0. We will use hroughou ha each of he funcions ψ g(, z), h and f is sricly convex and aains is minimum a zero. In paricular ψ (x) 0 if x 0. 56

67 1.8. Bid-Ask Spread and he Independence of he Jump Processes A firs consequence is ha J(, 0, z, 0) = 0 for each (, z) 0, T R n and J(, 0, z, u) > 0 for each u U which is no idenically zero. This implies v(, 0, z) = 0 and v(, x, z) > 0 for each x 0 since v is sricly convex and nonnegaive. I now follows ha α(, z) = arg min v(, x, z) = 0. x R An argumen as in he proof of Proposiion shows ha he opimal passive order saisfies ds dp a.e. û 1 (s) = ˆX(s). A consequence of 0 = J(, 0, z, 0) < J(, 0, z, u) is ha if he opimal rajecory of sock holdings ˆX his zero, i says here unil mauriy. Le τ 1 denoe he firs jump ime of he Poisson process N. A he jump ime, we have ˆX(τ) = ˆX(τ ) + û 1 (τ) = 0. ˆX sars above zero and evolves coninuously before τ 1. I is now clear ha ˆX 0 ds dp a.e. Combining his wih he explici represenaion of he adjoin process (1.13) we ge P 0 ds dp a.e. An argumen as in he proof of Theorem yields ha a.e. û 2 (s) = g u2 (, Z(s)) 1 (P (s)) 0. In conclusion, boh he opimal passive and marke order are nonnegaive. A similar argumen shows ha hey are nonposiive if X u (0) < 0 and zero if X u (0) = 0. 57

68

69 2. When o Cross he Spread: Curve Following wih Singular Conrol 2.1. Inroducion In he previous chaper we solved he problem of curve following in an order book model wih insananeous price impac and absoluely coninuous marke orders. In paricular i was assumed ha rades have no lasing impac on fuure prices. However in limi order markes he bes bid and bes ask prices ypically recover only slowly afer large discree rades. In he presen chaper we herefore consider a wo-sided limi order marke model wih emporary marke impac and resilience, where he price impac of rading decays only gradually. Trading sraegies now include coninuous and discree rades, so ha we are in he framework of singular sochasic conrol. The addiional difficuly is ha here are now wo sources of jumps, represening passive order execuion and singular marke orders. The singular naure of he marke order complicaes he analysis considerably, he opimal marke order canno be characerised as he poinwise maximiser of he Hamilonian as in he absoluely coninuous case. Moreover, we now face an opimisaion problem wih consrains, since passive buy and sell orders are modelled separaely and boh are nonnegaive. Mehods of singular conrol have been applied in differen fields including he monoone follower problem as in Beneš e al. 1980, he consumpion-invesmen problem wih proporional ransacion coss in Davis and Norman 1990 and finie fuel problems as in Karazas e al Mos of hem rely on he dynamic programming approach or a direc maringale opimaliy principle. Here we shall prove a version of he sochasic maximum principle. In conras o dynamic programming, his does no require regulariy of he value funcion and provides informaion on he opimal conrol direcly. Maximum principles for singular sochasic conrol problems can be found in Cadenillas and Haussmann 1994, Øksendal and Sulem 2001 and Bahlali and Mezerdi 2005, among ohers. Unforunaely hese resuls canno direcly be applied o he opimisaion problem under consideraion, since i involves jumps and sae dependen singular cos erms. The recen paper Øksendal and Sulem 2010 provides necessary and sufficien maximum principles for jump diffusions wih parial informaion. Despie being fairly general, heir seup does no cover he paricular model we consider; insead we give a direc proof based on Cadenillas and Haussmann 1994 and he ideas developped in Chaper 1. As in he previous chaper we consider an invesor who wans o minimise he deviaion of his sock holdings from a prespecified arge funcion, where he laer is driven by a vecor of unconrolled sochasic signals. The applicaions we have in mind are index 59

70 2. When o Cross he Spread: Curve Following wih Singular Conrol racking, porfolio liquidaion, hedging and invenory managemen. Our problem can be seen as an exension of he monoone follower problem o an order book framework wih marke and passive orders. In our model, he invesor faces a radeoff beween he penaly for deviaion and he liquidiy coss of rading in he order book, i.e. he coss of crossing he spread and buying ino he order book. The invesor s marke orders widen he spread emporarily; he gap hen aracs new limi orders from oher marke paricipans and he spread recovers. The key decision he rader has o ake is he following: If he spread is small, rading is cheap and a marke order migh be beneficial. For large spreads however i migh be beer o sop rading and wai unil he spread recovers. When o cross he spread is a fundamenal quesion of algorihmic rading in limi order markes. An equivalen quesion would be when o conver a limi ino a marke order. To he bes of our knowledge, he problem of when o cross he bid ask spread has no been addressed in he mahemaical finance lieraure on limi order markes. Obizhaeva and Wang 2005 and Predoiu e al. 2010, for insance, consider porfolio liquidaion for a one-sided order book wih iniial spread zero; in his case i is opimal never o sop rading. Our order book model is inspired by Obizhaeva and Wang 2005, a model which has recenly been generalised o arbirary shape funcions by Alfonsi e al as well as Predoiu e al and sochasic order book heigh in Fruh While he menioned aricles focus on porfolio liquidaion, we consider here he more general problem of curve following and herefore need a wo-sided order book model. In addiion, we allow for passive orders. These are orders wih random execuion which do no induce liquidiy coss, such as limi orders or orders placed in a dark venue. Our firs mahemaical resul is an a priori esimae on he conrol. For he proof, we reduce he curve following problem o an opimisaion problem wih quadraic penaly and wihou arge funcion and hen use a scaling argumen. This resul provides he exisence and uniqueness of an opimal conrol via a Komlós argumen. Nex we prove a suiable version of he sochasic maximum principle and characerise he opimal rading sraegy in erms of a coupled forward backward sochasic differenial equaion (FBSDE). The proof builds on resuls from Cadenillas and Haussmann 1994 and exends hem o he presen case where we have jumps, sae-dependen singular cos erms and general dynamics for he sochasic signal. Nex we give a second characerisaion of opimaliy in erms of buy, sell and no-rade regions. I urns ou ha here is always a region where he coss of rading are larger han he penaly for deviaing, so ha i is opimal o sop rading when he conrolled sysem is inside his region. This is in conras o he previous chaper, where only absoluely coninuous rading sraegies are allowed and a smoohness condiion on he cos funcion is imposed. I was shown in Chaper 1 ha under hese condiions he no-rade region is degenerae, so ha he invesor always rades. In he presen model he no-rade region is defined in erms of a hreshold for he bid ask spread. We show ha spread crossing is opimal if he spread is smaller han or equal o he hreshold. If i is larger, hen no marke orders should be used. The hreshold is given explicily in erms of he FBSDE and as a resul, we can precisely characerise when spread crossing is opimal for a large class of opimisaion problems. We will see ha marke orders are applied such ha he conrolled sysem 60

71 2.2. The Conrol Problem remains inside (he closure of) he no-rade region a all imes, and ha is rajecory is refleced a he boundary. To make his precise, we show ha he adjoin process ogeher wih he opimal conrol provides he soluion o a refleced BSDE. In general i is difficul o solve he coupled forward backward SDE (or he corresponding Hamilon-Jacobi-Bellman quasi variaional inequaliy, henceforh HJB QVI) explicily. This is due o he Poisson jumps (leading o nonlocal erms) and he singular naure of he conrol. For quadraic penaly funcion and zero arge funcion hough he soluion can be given in closed form. This corresponds o he porfolio liquidaion problem in limi order markes and exends he resul of Obizhaeva and Wang 2005 o rading sraegies wih passive orders. The new feaure is ha he opimal sraegy is no deerminisic, bu adaped o passive order execuion, and he rading rae is no consan bu increasing in ime. The remainder of his chaper is organised as follows: We describe he marke environmen and he conrol problem in Secion 2.2 and show in Secion 2.3 ha a unique opimal conrol exiss. We hen provide wo characerisaions of opimaliy, firs via he sochasic maximum principle in Secion 2.4 and hen via buy, sell and no-rade regions in Secion 2.5. The link o refleced BSDEs is presened in Secion 2.6 and we conclude wih some examples in Secion The Conrol Problem Le (Ω, F, {F(s) : s 0, T }, P ) be a filered probabiliy space saisfying he usual condiions of righ coninuiy and compleeness and T > 0 be he erminal ime. Assumpion The filraion is generaed by he following muually independen processes, 1. A d-dimensional Brownian Moion W, d Two one-dimensional Poisson processes N i wih inensiies λ i for i = 1, A compound Poisson process M on 0, T R k wih compensaor m(dθ)d where m(r k ) <. Trading akes place in a wo sided limi order marke. Limi order markes are a special ype of illiquid markes; we give a deailed discussion of heir properies in he inroducory chaper of his hesis. We posulae he exisence of hree price processes: he benchmark price (a nonnegaive maringale), he bes ask price (which is above he benchmark) and he bes bid price (which is below he benchmark). On he buy side of he order book liquidiy is available for prices higher han he bes ask price, and we assume a block shaped disribuion of available liquidiy wih consan heigh 1 κ 1 > 0. This assumpion is also made in Obizhaeva and Wang 2005; i is key for he curren approach as i leads o linear dynamics for he bid ask spread. Similarly, liquidiy is available on he sell side for prices lower han he bes bid. We assume a block shaped disribuion of liquidiy available on he sell side wih consan heigh 1 κ 2 > 0. The 61

72 2. When o Cross he Spread: Curve Following wih Singular Conrol invesor s rades have a emporary impac 1 on he bes bid and ask prices, his will be made more precise below. The benchmark price is hypoheical and canno be observed direcly in he marke. As in Chaper 1, i represens he fair price of he underlying or a reference price in he absence of liquidiy coss. We assume ha he benchmark price is unconrolled. A permanen price impac as in Almgren and Chriss 2001 can easily be incorporaed here, bu we do no include i and insead focus on he radeoff beween accuracy and liquidiy coss. A sylised snapsho of he order book and a ypical rajecory of he price processes are ploed in Figure 2.1. The invesor can apply marke buy (sell) orders o consume liquidiy on he buy (sell) side of he order book. His cumulaed marke buy (sell) orders are denoed by η 1 (η 2, respecively). These are nondecreasing càdlàg processes, and hence we allow for coninuous as well as discree rades and denoe by η i (s) η i (s) η i (s ) 0 for s 0, T and i = 1, 2 he jumps of η i. Such conrol processes are more general han absoluely coninuous rading sraegies and hey seem beer suied o describe real world rading sraegies. Real world rading is purely discree and coninuous sraegies are merely an analyical device, hey represen he limi of a sequence of small discree rades. The addiional difficuly is hen o cope wih he jumps in he conrol. The Hamilon-Jacobi-Bellman equaion associaed o a singular sochasic conrol problem is a quasi variaional inequaliy and no only a parial differenial equaion, cf. Øksendal and Sulem 2001 Chaper 5. In addiion, jus as in he previous chaper, he invesor can use passive buy (sell) orders u 1 (u 2, respecively). We assume ha hey are placed and fully execued a he benchmark price. Thus a passive order always achieves a beer price han he corresponding marke order, however is execuion is uncerain. We hink of hem as orders placed in a dark venue or as a sylised form of limi orders. Marke and passive orders represen aking and providing liquidiy. The class of admissible conrols is now defined for 0, T as { U (η, u) :, T Ω R 2 + R 2 + η i( ) = 0, E η i (T ) 2 + T η i is nondecreasing, càdlàg and progressively measurable and } u i is predicably measurable, for i = 1, 2. u i (r) 2 dr <, Each conrol consiss of he four componens η 1, η 2, u 1, u 2, each of hem being nonnegaive. In paricular, we face an opimisaion problem wih consrains. We noe ha η 1 (s) (resp. η 2 (s)) denoes he marke buy (resp. sell) orders accumulaed in, s. In conras, u 1 (s) (resp. u 2 (s)) represens he volume placed as a passive buy (resp. sell) 1 A fundamenal propery of illiquid markes is ha rades move prices. There is a large body of empirical lieraure on he price impac of rading, we refer he reader o Kraus and Soll 1972, Holhausen e al. 1987, Holhausen e al. 1990, Biais e al and Almgren e al

73 2.2. The Conrol Problem (a) (b) Figure 2.1.: (a) This sylised snapsho of he order book shows he bes bid, benchmark and bes ask price as well as liquidiy ha is available (dark) and consumed (ligh). (b) Here we see a ypical evoluion of he price processes over ime. The bes ask (red) is above he benchmark price (dashed black), which is above he bes bid price (blue). Marke buy (resp. sell) orders lead o jumps in he bes ask (resp. bid) price. In he absence of rading, he bes ask and bes bid converge o he benchmark. order a ime s, T. We remark ha from η i (T ) < a.s. for i = 1, 2 i follows ha r,t η i(r) < a.s. and we have he following decomposiion of he singular erm ino a coninuous and a pure jump par for s, T : η i (s) = ( η i (s) r,s ) η i (r) + r,s η i (r). Having defined he admissible conrols, le us now specify he price dynamics. As poined ou in he inroducory chaper of his hesis, a model for a limi order marke should saisfy he following properies: The bes ask price is larger han or equal o he bes bid price. The invesor s marke buy orders have a emporary impac on he bes ask price, bu no on he bes bid. The invesor s marke sell orders have a emporary impac on he bes bid price, bu no on he bes ask. The impac of a rade on he price decays over ime (resilience). In he absence of rading, boh processes are (or converge o) maringales. The invesor s passive orders do no move prices. More precisely, he price impac of passive orders as compared o he impac of marke orders is negligible. 63

74 2. When o Cross he Spread: Curve Following wih Singular Conrol Specifying price dynamics wih hese properies is no sraighforward. We find i more convenien o work wih he buy and sell spreads insead. Specifically, we denoe by X 1 he disance of he bes ask price o he benchmark price and call his process he buy spread. As in Obizhaeva and Wang 2005 and Alfonsi e al we assume exponenial recovery of he buy spread wih resilience parameer ρ 1 > 0. The dynamics of he buy spread are hen given for s, T by s X 1 (s) X 1 ( ) = ρ 1 X 1 (r)dr + κ 1 dη 1 (r), X 1 ( ) = x 1 0.,s As a convenion, we wrie,s for inegrals wih respec o he singular processes η i for i = 1, 2 o indicae ha possible jumps a imes s and are included. Similarly, he sell spread X 2 is defined as he disance of he bes bid price o he benchmark price and i saisfies s X 2 (s) X 2 ( ) = ρ 2 X 2 (r)dr + κ 2 dη 2 (r), X 2 ( ) = x 2 0.,s An immediae consequence is ha he spreads X 1 and X 2 are nonnegaive, mean revering and he price sysem hus defined saisfies he properies given above. Remark In he lieraure he bid ask spread is ypically defined as he disance of he bes ask from he bes bid price; in our noaion his process is given by X 1 + X 2. In he seminal paper Kyle 1985 hree measures of liquidiy are defined, all of which are capured in he model we propose. Deph, he size of an order flow innovaion required o change prices a given amoun, is given by he parameers κ 1 and κ 2 which denoe he inverse order book heigh. Resiliency, he speed wih which prices recover from a random, uninformaive shock, is capured by he resilience parameers ρ 1 and ρ 2. Finally, ighness, he cos of urning around a posiion over a shor period of ime, can be measured in erms of he bid ask spread X 1 + X 2. We remark ha he spread dynamics specified above are linear in he conrol and he sae variable. This is necessary o compue he Gâeaux derivaive of he cos funcional, which hen allows o derive a necessary and sufficien condiion for opimaliy. General shape funcions for he order book lead o nonlinear dynamics, which are no compaible wih he curren approach. If he passive orders are inerpreed as limi orders placed on he benchmark price, hey do change he bes quoes. Sricly speaking, in his case X 1 and X 2 represen he buy and sell spreads modulo he invesor s own limi order. As in he previous chaper, he dynamics of he benchmark price is no imporan, aside from he fac ha his process is a nonnegaive maringale. Then i does no conribue o he expeced rading coss and he liquidiy coss only depend on he buy and sell spread. 64

75 2.2. The Conrol Problem We assume ha, in he absence of rading, he spreads converge back o zero. More generally, hey migh converge o any number z R +. This is equivalen o inroducing an addiional fixed spread of size z as in Obizhaeva and Wang However, our assumpion is no loss of generaliy, since an addiional fixed spread can be incorporaed in he presen model by replacing X i by X i + z in he performance funcional defined in (2.1) below, for i = 1, 2. We shall need a hird sae process X 3 represening he invesor s sock holdings. They are he sum of he marke and passive buy orders less he marke and passive sell orders, and hus for s, T given by X 3 (s) X 3 ( ) =,s X 3 ( ) = x 3 R. s s dη 1 (r) dη 2 (r) + u 1 (r)n 1 (dr) u 2 (r)n 2 (dr),,s A jump of he Poisson process N i represens a liquidiy even which execues he passive order u i, for i = 1, 2. For simpliciy we consider full execuion only, his assumpion is also made in Kraz 2011 and in he firs chaper of his hesis. We define he vecor X (X i ) i=1,2,3 and wrie X = X η,u if we wan o emphasise he dependence of he sae process on he conrol. Noe ha here are wo sources of jumps, he Poisson processes and he discree marke orders. More precisely, he jump of he sae process a ime s, T is given by X 1 (s) X 1 (s ) X(s) = X 2 (s) X 2 (s ) = N X(s) + η X(s) X 3 (s) X 3 (s ) 0 κ 1 η 1 (s) 0 + κ 2 η 2 (s). u 1 (s) N 1 (s) u 2 (s) N 2 (s) η 1 (s) η 2 (s) The formulaion of he curve following problem is close o he previous chaper. We briefly describe he relevan changes here. The rader wans o minimise he deviaion of his sock holdings o a prespecified arge funcion α :, T R n R. This funcion depends on a vecor of unconrolled sochasic signals Z wih dynamics given for s, T by s Z(s) Z( ) = s µ(r, Z(r))dr + s + σ(r, Z(r ))dw (r) R k γ(r, Z(r ), θ) M(dr, dθ), Z( ) = z R n. We hink of Z as a sochasic facor which drives he arge funcion, i migh represen a sock price index, he price of some underlying or some oher kind of risk facor. As above we denoe he compensaed Poisson maringale by M(0, s A) M(0, s A) m(a)s; similarly Ñi N i λ i s for i = 1, 2. 65

76 2. When o Cross he Spread: Curve Following wih Singular Conrol Having defined he sae processes and heir respecive dynamics, le us now specify he opimisaion crierion. The performance funcional is defined for (, x, z) 0, T R 3 R n and a conrol (η, u) U as J(, x, z, η, u) E,x,z X 1 (r ) + κ 1 2 η 1(r) +,T T dη 1 (r) + X 2 (r ) + κ 2,T h (X 3 (r) α(r, Z(r))) dr + f (X 3 (T ) α(t, Z(T ))) 2 η 2(r) dη 2 (r). (2.1) There are four cos erms represening he conflicing ineress of liquidiy coss and accuracy and we now explain hem briefly. The firs wo erms on he righ hand side of (2.1) capure rading coss of marke buy (sell) orders, i.e. he coss of crossing he spread and buying (selling) ino he order book. Specifically, an infiniesimal marke buy order dη 1 (r) is execued a he bes ask price, so ha he coss of crossing he spread are given by X 1 (r )dη 1 (r). A discree buy order η 1 (r) eas ino he block shaped order book and shifs he spread from X 1 (r ) o X 1 (r ) + κ 1 η 1 (r). Is liquidiy coss are given by ( X 1 (r ) + κ ) 1 2 η 1(r) η 1 (r). In paricular, he jump par in (2.1) is undersood as, for i = 1, 2 E η i (r)dη i (r) = E,T r,t ( E r,t ( η i (r)) 2 ) 2 η i (r) E η i (T ) 2 <. (2.2) The las wo erms on he righ hand side of (2.1) penalise deviaion from he arge funcion, he erm involving h is referred o as running coss, while he erm involving f represens erminal coss. As a shorhand, we someimes wrie J(η, u) J(, x, z, η, u) if (, x, z) 0, T R 3 R n is fixed. The opimisaion problem under consideraion is Problem Minimise J(η, u) over (η, u) U. For (, x, z) 0, T R 3 R n he value funcion is defined as v(, x, z) = inf J(, x, z, η, u). (η,u) U Remark Problem is a singular sochasic conrol problem. Maximum principles for singular conrol are derived for insance in Cadenillas and Haussmann 1994, Øksendal and Sulem 2001 and Bahlali and Mezerdi However, he above problem is no covered by heir resuls for several reasons. Firsly, i involves jumps. Secondly, he singular cos erms,t Xi (r ) + κ i 2 η i(r) dη i (r) for i = 1, 2 depend on he sae 66

77 2.2. The Conrol Problem variable and on he jumps of he conrol, which is no he case in he usual formulaion. The sandard seup only allows for cos erms of he form T k(s, ω)dη(s). In financial applicaions, e.g. in Davis and Norman 1990, he inegrand k is ofen inerpreed as proporional ransacion coss. In he illiquid marke model we propose he buy and sell spreads play he role of he ransacion cos parameers, and hey are hemselves conrolled. A hird difficuly in he presen model is ha he absoluely coninuous conrol u (he passive order) does no incur rading coss, so he sandard characerisaion as he poinwise maximiser of he Hamilonian does no apply. The recen aricle Øksendal and Sulem 2010 provides necessary and sufficien maximum principles for he singular conrol of jump diffusions, where he singular cos erm may depend on he sae variable. However, hey do no allow for erms like,t η i(r)dη i (r), hey do no incorporae absoluely coninuous conrols (which are needed in he presen framework for he passive order u) and heir sufficien condiion is based on a convexiy condiion on he Hamilonian which is no saisfied in our specific case. Insead we give a direc proof based on Cadenillas and Haussmann 1994 and ideas used in he previous chaper. To ensure exisence and uniqueness of an opimal conrol, we impose he following assumpions. Here and hroughou, we wrie c for a generic consan, which migh be differen a each occurrence. Assumpion The penaly funcions f, h : R R are sricly convex, coninuously differeniable, normalised and nonnegaive. 2. In addiion, f and h have a leas quadraic growh, i.e. here exiss ε > 0 such ha f(x), h(x) ε x 2 for all x R. 3. The funcions µ, σ and γ are Lipschiz coninuous, i.e. here exiss a consan c such ha for all z, z R n and s, T, µ(s, z) µ(s, z ) 2 R n + σ(s, z) σ(s, z ) 2 R n d + γ(s, z, θ) γ(s, z, θ) 2 R k R nm(dθ) c z z 2 R n. In addiion, hey saisfy sup µ(s, 0) 2 R n + σ(s, 0) 2 R + γ(s, 0, θ) 2 n d s T R k R nm(dθ) <. 4. The arge funcion α has a mos polynomial growh in he variable z uniformly in s, i.e. here exis consans c α, q > 0 such ha for all z R n, sup α(s, z) c α (1 + z q R ). n s T 5. The penaly funcions f and h have a mos polynomial growh. 67

78 2. When o Cross he Spread: Curve Following wih Singular Conrol Remark Le us briefly commen on hese assumpions. Taking f and h nonnegaive is reasonable for penaly funcions. Normalisaion is no loss of generaliy, his may always be achieved by a linear shif of f, h and α. Quadraic growh of f and h is only needed in Lemma for an a priori L 2 -norm bound on he conrol, which is hen used for a Komlós argumen. The convexiy condiion leads naurally o a convex coercive problem which hen admis a unique soluion. A ypical candidae for he penaly funcion is f(x) = h(x) = x 2, which corresponds o minimising he squared error. We require ha he signal SDE admis a unique srong soluion which has momens of all orders, he Lipschiz assumpions on µ, σ and γ are sufficien o guaranee his, however we remark ha any condiion guaraneeing exisence, uniqueness and all momens would be appropriae here. Once he exisence of an opimal conrol is esablished, we need one furher assumpion. I guaranees he exisence and uniqueness of he adjoin process. Assumpion The derivaives f and h have a mos linear growh, i.e. for all x R we have f (x) + h (x) c(1 + x ) Exisence of a Soluion The aim of he presen secion is o show ha he performance funcional is sricly convex and ha i is enough o consider conrols wih a uniform L 2 -norm bound. Combining hese resuls wih a Komlós argumen, we hen prove ha here is a unique opimal conrol. We begin wih some growh esimaes for he sae processes. Henceforh we impose Assumpion Lemma For every p 2 here exiss a consan c p such ha for every (, x, z) 0, T R 3 R n we have E,x,z sup Z(s) p R c n p (1 + z p R ). n s T 2. There exiss a consan c x such ha for any (η, u) U we have ( ) T E,x,z sup X η,u (s) 2 R 3 c x 1 + E,x,z η(t ) 2 R 2 + E u(r) 2 R s T 2 dr. In paricular, X η,u has square inegrable supremum for all (η, u) U. Proof. Iem (1) is a well known esimae on he soluion of an SDE wih Lipschiz coefficiens, see for example Barles e al Proposiion 1.1. We now prove iem (2). For s, T we have X 1 (s) 2 = x 1 s ρ 1 X 1 (r)dr + κ 1,s dη 1 (r) 2 68

79 3 x ρ x ρ 2 1 s s X 1 (r)dr sup z,r 2 + κ 2 1 η 1 (s) 2 X 1 (z) 2 dr + κ 2 1 η 1 (T ) Exisence of a Soluion where we have used Jensen s inequaliy and he fac ha η 1 is nondecreasing in he las line. An applicaion of Gronwall s Lemma now provides he exisence of a consan c > 0 such ha E,x,z sup X 1 (s) 2 c (1 + x E,x,z η 1 (T ) 2 ). s T Similar esimaes hold for X 2 and X 3. A firs consequence of he above lemma is ha he zero conrol incurs finie coss. Corollary The zero conrol incurs finie coss, i.e. for each (, x, z) 0, T R 3 R n we have J(, x, z, 0, 0) <. Proof. The polynomial growh of f, h and α implies he exisence of consans c 1, c 2 > 0 such ha T J(, x, z, 0, 0) =E,x,z h ( x 3 α(r, Z(r)) ) dr + f ( x 3 α(t, Z(T )) ) ( T c E,x,z Z(r) c 2 R dr + Z(T ) c n 2 R ). n The erms in he las line are finie due o Lemma 2.3.1(1). We now show ha he performance funcional is sricly convex in he conrol, so ha mehods of convex analysis can be applied. Proposiion The performance funcional (η, u) J(, x, z, η, u) is sricly convex, for every (, x, z) 0, T R 3 R n. Proof. From he definiion of X i for i = 1, 2 we have dη i (s) = dx i(s)+ρ i X i (s)ds κ i. We use his o rewrie he performance funcional as J(, x, z, η, u) T + 2κ 2 ρ T 1 X 1 (r) 2 dr + κ 1 X1 (T ) 2 x 2 1 =E,x,z + X 2(T ) 2 x 2 2 2κ 1 T + h (X 3 (r) α(r, Z(r))) dr + f (X 3 (T ) α(t, Z(T ))) ρ 2, X 2 (r) 2 dr κ 2. (2.3) 69

80 2. When o Cross he Spread: Curve Following wih Singular Conrol The righ hand side is sricly convex in X. Due o he fac ha (η, u) X η,u is affine, i follows ha (η, u) J(, x, z, η, u) is sricly convex. The aim in his secion is o prove exisence and uniqueness of an opimal conrol. For he proof of his resul, we need wo auxiliary lemmaa. We firs show a quadraic growh esimae on he value funcion in Lemma This exends Lemma o he singular conrol case. As above, he idea is o reduce he curve following problem o a simpler linear-quadraic regulaor problem. In conras o Lemma 1.4.2, a soluion o his LQ problem via Riccai equaions is no available and we use a scaling argumen insead. Lemma For each (, x, z) 0, T R 3 R n here are consans c 1,, c 2, c 3 > 0 such ha v(, x, z) c 1, x 2 3 c 2 (1 + z c 3 R n). Proof. The idea is o use he growh condiions on he penaly funcions o reduce he opimisaion problem o simpler linear-quadraic problem, which can hen be esimaed in erms of x 2 3. Using he quadraic growh of f and h yields v(, x, z) inf (η,u) U E,x,z,T T + X 1 (r ) + κ 1 2 η 1(r) dη 1 (r) + X 2 (r ) + κ 2,T 2 η 2(r) dη 2 (r) ε (X 3 (r) α(r, Z(r))) 2 dr + ε (X 3 (T ) α(t, Z(T ))) 2. Nex an applicaion of he inequaliy (a b) a2 b 2 leads o v(, x, z) inf E,x,z (η,u) U T +,T X 1 (r ) + κ 1 2 η 1(r) dη 1 (r) + T εe,x,z ε 2 X 3(r) 2 dr + ε 2 X 3(T ) 2 X 2 (r ) + κ 2,T 2 η 2(r) dη 2 (r) α(r, Z(r)) 2 dr + α(t, Z(T )) 2 The polynomial growh of α coupled wih Lemma provides he exisence of consans c 2, c 3 > 0 such ha v(, x, z) inf (η,u) U E,x,z,T T + X 1 (r ) + κ 1 2 η 1(r) ε 2 X 3(r) 2 dr + ε 2 X 3(T ) 2 dη 1 (r) +,T (2.4) X 2 (r ) + κ 2 2 η 2(r) dη 2 (r) c 2 (1 + z c 3 R n ). This provides an esimae of he original value funcion in erms of an easier opimisaion. 70

81 2.3. Exisence of a Soluion problem wih a quadraic penaly funcion and zero arge funcion. Economically, his may be inerpreed as a porfolio liquidaion problem. To coninue he esimae, we define he following value funcion, v 1 (, x) inf (η,u) U E,x,T T + X 1 (r ) + κ 1 2 η 1(r) dη 1 (r) + ε 2 X 3(r) 2 dr + ε 2 X 3(T ) 2.,T X 2 (r ) + κ 2 2 η 2(r) dη 2 (r) We remark ha he value funcion v 1 is monoone in X i for i = 1, 2 and X i is monoone in he saring value x i, so ha v 1 does no increase if we replace he iniial spread X i ( ) = x i 0 by zero for i = 1, 2 and only consider saring values x = (0, 0, x 3 ), i.e. v 1 (, x) v 1 (, (0, 0, x3 ) ) v 2 (, x 3 ). Le us denoe by J 2 he performance funcional associaed o he value funcion v 2. Due o x i = 0 for i = 1, 2 he mappings (η, u) X η,u i are linear and he mapping (x 3, η, u) X η,u 3 x 3 is also linear. We use his o show ha J 2 scales quadraically. We wrie for (, x 3 ) 0, T R and a scaling facor β > 0 J 2 (, βx 3, βη, βu) =E,(0,0,βx3 ) +,T X βη,βu 2 (r ) + κ 2 X βη,βu 1 (r ) + κ 1 2 β η 1(r) βdη 1 (r),t 2 β η 2(r) βdη 2 (r) T ε + X βη,βu 3 (r) 2 dr + ε X βη,βu 3 (T ) =E,(0,0,βx3 ) X βη,βu 1 (r ) + κ 1,T 2 β η 1(r) βdη 1 (r) + X βη,βu 2 (r ) + κ 2,T 2 β η 2(r) βdη 2 (r) T ε r + 2 (x β 3 + dη 1 (z) dη 2 (z) +,r,r + ε ( T 2 βx 3 + β dη 1 (z) dη 2 (z) +,T,T =β 2 E,(0,0,x3 ) X η,u 1 (r ) + κ 1,T 2 η 1(r) dη 1 (r) + X η,u 2 (r ) + κ T 2,T 2 η 2(r) dη 2 (r) + =β 2 J 2 (, x 3, η, u). r u 1 (z)n 1 (dz) ) 2 u 2 (z)n 2 (dz) dr T u 1 (z)n 1 (dz) u 2 (z)n 2 (dz) ) 2 ε 2 Xη,u 3 (r) 2 dr + ε 2 Xη,u 3 (T ) 2 71

82 2. When o Cross he Spread: Curve Following wih Singular Conrol Nex we claim ha also v 2 scales quadraically. Indeed, le (u n, η n ) U be a minimising sequence for v 2 (, x 3 ) and le β > 0 be a scaling facor. We use ha J 2 scales quadraically o wrie v 2 (, βx 3 ) = lim n J 2(, βx 3, η n, u n ) = β 2 lim n J 2(, x 3, ηn β, un β ) β2 v 2 (, x 3 ). (2.5) We now use (2.5) wih he scaling facor 1 β o ge he reverse inequaliy: β 2 v 2 (, x 3 ) = β 2 ( 1 v 2, β (βx 3) ) 1 2 β 2 v 2 (, βx 3 ) = v 2 (, βx 3 ). (2.6) β Combining (2.5) and (2.6) we see ha v 2 (, βx 3 ) = β 2 v 2 (, x 3 ). One can check ha if x 3 = 0 hen v 2 (, 0) = 0. Choosing now β = x 3 for x 3 0 we ge x 2 3 v 2(, 1), x 3 > 0 v 2 (, x 3 ) = 0, x 3 = 0 x 2 3 v 2(, 1), x 3 < 0, and defining c 1, min{v 2 (, 1), v 2 (, 1)} leads o v 2 (, x 3 ) c 1, x 2 3. Plugging his resul ino (2.4) provides he following esimae v(, x, z) c 1, x 2 3 c 2 (1 + z c 3 R n ). To prove he asserion of he lemma, i remains o show ha he consan c 1, is sricly posiive and finie for each 0, T. The proof of his resul is relegaed o Lemma A.2.1 in he appendix. We are now ready o prove an a priori esimae on he conrol, which will be needed in he Komlós argumen below. This resul exends Lemma o he singular conrol case. Lemma There is a consan K such ha any conrol wih T E,x,z η(t ) 2 R 2 + u(r) 2 R 2 dr > K canno be opimal. Proof. We firs consider he marke order η. The dynamics of X i for i = 1, 2 imply ha 72

83 2.3. Exisence of a Soluion for s, T we have X i (s) = e ρ i(s ) x i + κ i,s e ρ i(r ) dη i (r), (2.7) and hus X i (T ) κ i e ρ it η i (T ). Combining his wih (2.3) yields Xi (T ) 2 J(η, u) E,x,z x2 1 x2 2 K 1 E,x,z η i (T ) 2 K 2,x. (2.8) 2κ i 2κ 1 2κ 2 for consans K 1, K 2,x > 0. I follows ha if E,x,z η i (T ) 2 > J(0,0)+K 2,x+1 K 1 be opimal. We have J(0, 0) < due o Corollary hen η canno The esimae in erms of he passive order u is slighly more involved. Le τ i denoe he firs jump ime of he Poisson process N i afer for i = 1, 2, an exponenially disribued random variable wih parameer λ i, and se τ τ 1 τ 2 T. A he jump ime τ he sae process jumps from X(τ ) o 0 X(τ ) + N X(τ) X(τ ) + 0. u 1 (τ)1 {τ1 <τ 2 T } u 2 (τ)1 {τ2 <τ 1 T } We use he definiion of he cos funcional and he fac ha he cos erms are nonnegaive o ge J(η, u) =E,x,z +,τ) τ X 1 (r ) + κ 1 2 η 1(r) dη 1 (r) + X 2 (r ) + κ 2,τ) 2 η 2(r) dη 2 (r) h (X 3 (r) α(r, Z(r))) dr + J (τ, X(τ ) + N X(τ), Z(τ), η, u) E,x,z J(τ, X(τ ) + N X(τ), Z(τ), η, u) E,x,z v(τ, X(τ ) + N X(τ), Z(τ)), (2.9) where J in he above is evaluaed a conrols on he sochasic inerval 2 τ, T. Combining his wih Lemma we ge J(η, u) E,x,z c 1, X 3 (τ ) + N X 3 (τ) 2 c 2 (1 + Z(τ) c 3 R n). 2 More precisely, we spli he inerval, T ino he subinervals, τ and (τ, T. By definiion of he cos funcional, he singular order on he second subinerval (τ, T includes a possible jump a he lef endpoin τ, so his jump mus be excluded from he firs subinerval, τ. For his reason, he sae process direcly afer he Poisson jump in (2.9) is given by X(τ ) + N X(τ) and no by X(τ ) + N X(τ) + ηx(τ) = X(τ). 73

84 2. When o Cross he Spread: Curve Following wih Singular Conrol In view of Lemma we have E,x,z Z(τ) c 3 R n E,x,z and hus here is a consan c 2,z 0 such ha sup Z(s) c 3 R n s,t c (1 + z c 3 R n), J(η, u) c 2,z + c 1, E X 3 (τ ) + N X 3 (τ) 2. (2.10) By definiion, he sock holdings direcly afer a jump of he Poisson process are given by X 3 (τ ) + N X 3 (τ) = x 3 + η 1 (τ ) η 2 (τ ) + u 1 (τ)1 {τ1 <τ 2 T } u 2 (τ)1 {τ2 <τ 1 T }, and an applicaion of he inequaliy (a + b) a2 b 2 leads o X 3 (τ ) + N X 3 (τ) 2 1 ( ) 2 u 1 (τ 1 )1 2 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } (x3 + η 1 (τ ) η 2 (τ )) 2 1 ( ) 2 u 1 (τ 1 )1 2 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } 3 ( x η 1 (τ ) 2 + η 2 (τ ) 2) 1 ( ) 2 ( u 1 (τ 1 )1 2 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } 3 x η 1 (T ) 2 + η 2 (T ) 2). (2.11) Combining (2.10) and (2.11) we ge 1 ( 2 2 c 1,E,x,z u 1 (τ 1 )1 {τ1<τ2 T } u 2 (τ 2 )1 {τ2<τ1 T }) ( J(η, u) + c 2,z + 3c 1, x E,x,z η 1 (T ) 2 + η 2 (T ) 2). Due o equaion (2.8) we have for i = 1, 2 E,x,z η i (T ) 2 K 2,x K K 1 J(η, u), so combining he las wo displays and relabelling consans provides ( ) 2 E,x,z u 1 (τ 1 )1 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } c 1,,x,z + c 2, J(η, u). (2.12) We shall now compue he erm on he lef hand side of inequaliy (2.12). The jump imes τ i are independen and exponenially disribued wih parameer λ i, for i = 1, 2. We hus have ( ) 2 E,x,z u 1 (τ 1 )1 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } 74

85 = T T 2.3. Exisence of a Soluion λ 1 e λ 1(r 1 ) λ 2 e λ 2(r 2 ) ( u 1 (r 1 )1 {r1 <r 2 T } u 2 (r 2 )1 {r2 <r 1 T }) 2 dr1 dr 2 λ 1 e λ 1(r 1 ) λ 2 e λ 2(r 2 ) u 1 (r 1 ) 2 dr 1 dr 2, where we have used he nonnegaiviy of he inegrand in he las line and resriced inegraion o (r 1, r 2 ), T T, ). We now compue E,x,z ( u 1 (τ 1 )1 {τ1 <τ 2 T } u 2 (τ 2 )1 {τ2 <τ 1 T } T ) 2 T λ 2 e λ 2(r 2 ) dr 2 λ 1 e λ 1(r 1 ) E,x,z u 1 (r 1 ) 2 dr 1 T =e λ 2(T ) λ 1 e λ 1(r 1 ) E,x,z u 1 (r 1 ) 2 dr 1 T e λ 2(T ) λ 1 e λ 1(T ) E,x,z u 1 (r 1 ) 2 dr 1. Combining his wih equaion (2.12) and relabelling consans we ge T E,x,z u 1 (r) 2 dr c 1,,x,z + c 2, J(η, u). In paricular if T E,x,z u 1 (r) 2 dr c 1,,x,z + c 2, J(0, 0) + 1, hen we see ha J(η, u) > J(0, 0) and he conrol (η, u) is clearly no opimal. A similar esimae holds for he passive sell order u 2. Theorem There is a unique opimal conrol (ˆη, û) U for Problem Proof. Le (η n, u n ) n N U be a minimising sequence, i.e. lim n J(ηn, u n ) = inf J(η, u). (η,u) U Recall ha he singular conrol η n is a nondecreasing càdlàg process, whereas u n is absoluely coninuous. Idenifying u n wih he nondecreasing càdlàg process un (r)dr, we can also inerpre u n as a singular conrol. Due o he uniform L 2 -norm bound from Lemma we can hen apply he Komlós heorem for singular sochasic conrol given in Kabanov 1999 Lemma 3.5. I provides he exisence of a subsequence (also indexed by n) and adaped processes ˆη :, T Ω R 2 + such ha η n 1 n η i n i=1 75

86 2. When o Cross he Spread: Curve Following wih Singular Conrol converges weakly o ˆη in he sense ha for almos all ω Ω he measures η n (ω) on, T converge weakly o ˆη(ω). Similarly here is an adaped process ξ :, T Ω R 2 + such ha ū n 1 n u i n i=1 converges weakly o ξ. However i is no ye clear ha he limi ξ is absoluely coninuous wih respec o Lebesgue measure, so i is no an admissible passive order. Therefore we now fix ˆη and consider he mapping u J(ˆη, u). The sequence of conrols (u n ) n N is sill a minimising sequence, i.e. lim J(ˆη, n un ) = inf J(ˆη, u). u A Komlós argumen as in he proof of Theorem now provides he exisence of a furher subsequence (also indexed by n) and a predicable process û = û which akes values from, T Ω and is valued in R 2 + such ha ū n 1 n u i n i=1 converges o û ds dp a.e. on, T Ω. We now show ha (ˆη, û) is an opimal conrol. The weak convergence coupled wih equaion (2.7) implies ha for s, T such ha ˆη(s) = 0 we have a.s. lim X ηn,ū n n 1 (s) = lim e ρ1(s ) x 1 + κ 1 e ρ1(r ) d η n n 1 (r),s =e ρ1(s ) x 1 + κ 1 e ρ1(r ) dˆη 1 (r) = X ˆη,û 1 (s), and similarly for X 2 and X 3. Combining Faou s lemma wih he convexiy of J gives J(ˆη, û) lim inf n J( ηn, ū n 1 ) lim inf n n,s n J(η i, u i ) = i=1 inf J(η, u), (η,u) U which shows ha (ˆη, û) minimises J over U. Uniqueness is due o he sric convexiy of (η, u) J(η, u) and can be shown as in he proof of Theorem Throughou, we denoe by (ˆη, û) he opimal conrol and by ˆX = X ˆη,û he opimal sae rajecory The Sochasic Maximum Principle In he preceding secion we showed ha Problem admis a unique soluion under Assumpion We shall now prove a version of he sochasic maximum principle 76

87 2.4. The Sochasic Maximum Principle which yields a characerisaion of he opimal conrol in erms of he adjoin equaion. In he sequel, we impose Assumpion and we wrie E insead of E,x,z. The adjoin equaion is defined as he following BSDE on, T, P 1 (s) P 1 ( ) s ρ 1 P 1 (r) s Q 1 (r) P 2 (s) P 2 ( ) = ρ 2 P 2 (r) P 3 (s) P 3 ( ) h ( ˆX dr + Q 2 (r) dw (r) 3 (r) α(r, Z(r))) Q 3 (r) s R 1,1 (r) s R 2,1 (r) + R 1,2 (r) Ñ1(dr) + R 2,2 (r) Ñ2(dr) R 1,3 (r) R 2,3 (r) s R 3,1 (r, θ) + R 3,2 (r, θ) M(dr, dθ) R k R 3,3 (r, θ) dˆη 1 (r) + 1 dˆη 2 (r), (2.13),s,s 0 0 P 1 (T ) 0 P 2 (T ) = 0 P 3 (T ) f ( ˆX. 3 (T ) α(t, Z(T ))) Remark Noe ha he opimal conrol ˆη now eners he adjoin equaion, which is no he case in he usual formulaion of singular conrol problems, see e.g. Cadenillas and Haussmann We will show in Secion 2.6 ha he soluion o he BSDE defined above provides he soluion o a refleced BSDE, where he bid ask spread plays he role of he reflecing barrier. The adjoin process is hen a riple of processes (P, Q, R) defined for j = 1, 2, 3 on, T by P 1 (s) Q 1 (s) R 1,1 (s) R 1,2 (s) R 1,3 (s) P (s) P 2 (s), Q(s) Q 2 (s) and R(s) R 2,1 (s) R 2,2 (s) R 2,3 (s), P 3 (s) Q 3 (s) R 3,1 (s) R 3,2 (s) R 3,3 (s) which saisfy for i = 1, 2, 3 P i :, T Ω R, Q i :, T Ω R d, R 1,i :, T Ω R, R 2,i :, T Ω R, R 3,i :, T R k Ω R and which also saisfy he dynamics (2.13) where P is adaped and Q, R are predicable. Proposiion The BSDE (2.13) admis a unique soluion which saisfies for i = 77

88 2. When o Cross he Spread: Curve Following wih Singular Conrol 1, 2, 3 T T E sup P i (s) 2 + E Q i (r) 2 R dr + E R s T d 1,i (r) 2 dr T T +E R 2,i (r) 2 dr + E R 3,i (r, θ) 2 m(dθ)dr <. R k I is unique among riples (P, Q, R) saisfying he above inegrabiliy crierion. Proof. We claim ha he explici soluion o (2.13) is given by P 1 (s) = E (s,t e ρ1(r s) dˆη 1 (r) F s, P 2 (s) = E (s,t e ρ2(r s) dˆη 2 (r) F s, P 3 (s) = E ( ) T s h ˆX3 (r) α(r, Z(r)) dr f ( ˆX3 (T ) α(t, Z(T )) ) F s. (2.14) To show ha his is rue, we firs noe ha due o he linear growh assumpions on h and f and he growh esimaes from Lemma 2.3.1, he funcions f and h saisfy T E h ( ˆX3 (r) α(r, Z(r)) ) 2 f dr + E ( ˆX3 (T ) α(t, Z(T )) ) 2 <. Moreover, we have by assumpion for i = 1, 2 ( ) 2 E dˆη i (r) = E ˆη i (T ) 2 <.,T We only consruc he soluion for P 1, he represenaions for P 2 and P 3 follow by similar argumens. The proof proceeds in hree seps. To sar wih, we consider he following sandard BSDE wihou singular erms on, T, P 1 (s) P s 1 ( ) = + P 1 (T ) = s Q 1 (r)dw (r) + s,t R k R3,1 (r, θ) M(dr, dθ), e ρ 1(T r) dˆη 1 (r). s R 1,1 (r)ñ1(dr) + We claim ha he soluion o his BSDE is given by P 1 (s) = E e ρ 1(T r) dˆη 1 (r) F(s) Φ(s).,T R 2,1 (r)ñ2(dr) Indeed, Φ is a square inegrable maringale and by he maringale represenaion heorem 78

89 2.4. The Sochasic Maximum Principle Tang and Li 1994 Lemma 2.3 here exis unique predicable processes Q 1 :, T Ω R d, R1,1 :, T Ω R, R 2,1 :, T Ω R, R3,1 :, T R k Ω R saisfying T E Q T 1 (r) 2 R dr + E R d 1,1 (r) 2 dr T +E R T 2,1 (r) 2 dr + E R 3,1 (r, θ) 2 m(dθ)dr <, R k such ha for s, T we have s Φ(s) Φ( ) = + Φ(T ) = s Q 1 (r)dw (r) + s,t R k R3,1 (r, θ) M(dr, dθ), e ρ 1(T r) dˆη 1 (r). s R 1,1 (r)ñ1(dr) + R 2,1 (r)ñ2(dr) In a second sep, we define he process P 1 (s) P 1 (s) + e ρ 1(T r) dˆη 1 (r) = E e ρ 1(T r) dˆη 1 (r) F(s).,s (s,t This process saisfies P 1 (s) P s 1 ( ) = s Q 1 (r)dw (r) + s + P 1 (T ) =0. In a hird sep, we se R k R3,1 (r, θ) M(dr, dθ) + s R 1,1 (r)ñ1(dr) + R 2,1 (r)ñ2(dr) e ρ 1(T r) dˆη 1 (r),,s P 1 (s) e ρ 1(T s) P1 (s) = E e ρ1(r s) dˆη 1 (r) F s, (s,t Q 1 (s) e ρ 1(T s) Q1 (s), R 2,1 (s) e ρ 1(T s) R2,1 (s), R 1,1 (s) e ρ 1(T s) R1,1 (s), R 3,1 (s) e ρ 1(T s) R3,1 (s) and apply he produc rule o see ha hese processes saisfy s s s P 1 (s) P 1 ( ) = ρ 1 P 1 (r)dr + Q 1 (r)dw (r) + R 1,1 (r)ñ1(dr) 79

90 2. When o Cross he Spread: Curve Following wih Singular Conrol P 1 (T ) =0. s + R 2,1 (r)ñ2(dr) + + dˆη 1 (r),,s s R k R 3,1 (r, θ) M(dr, dθ) This proves ha he process P 1 defined above is he soluion of (he firs componen of) he adjoin equaion (2.13). I remains o show ha P 1 has inegrable supremum. We firs apply Doob s inequaliy o he maringale Φ, E sup s T Φ(s) 2 ce Φ(T ) 2 ( ) 2 = ce e ρ 1(T r) dˆη 1 (r),t ce 2ρ 1T E ˆη 1 (T ) 2 <. From he definiion of P 1 we have for s, T ( P 1 (s) = e ρ 1(T s) P1 (s) = e ρ 1(T s) Φ(s) + Combining he above wo displays leads o E sup s T ( P 1 (s) 2 c E sup s T,s e ρ 1(T r) dˆη 1 (r) ) Φ(s) 2 + e ρ 1T Ê η 1 (T ) 2) <. Φ(s) + e ρ 1T ˆη 1 (T ). Similar argumens lead o he corresponding represenaions of P 2 and P 3. Uniqueness follows from he corresponding uniqueness in he maringale represenaion heorem. The characerisaion of he opimal conrol we shall derive explois an opimaliy condiion in erms of he Gâeaux derivaive of J. Given conrols (η, u), ( η, ū) U, i is defined as J 1 ( η, ū), (η, u) = lim J ( η + εη, ū + εu) J( η, ū). ε 0 ε In our paricular case, he Gâeaux derivaive can be compued explicily. This is he conen of he following lemma. Lemma The performance funcional J : (η, u) J(η, u) is Gâeaux differeniable. Is derivaive is given by, for conrols (η, u), ( η, ū) U, J ( η, ū), (η, u) =E +,T,T X η,u 1 (r ) e ρ 1(r ) x 1 + κ 1 2 η 1(r) X η,u 2 (r ) e ρ 2(r ) x 2 + κ 2 2 η 2(r) d η 1 (r) d η 2 (r) 80

91 + +,T T 2.4. The Sochasic Maximum Principle X η,ū 1 (r ) + κ 1 2 η 1(r) dη 1 (r) + X η,ū 2 (r ) + κ 2,T ) X η,u 3 (r)h ( X η,ū 3 (r) α(r, Z(r)) 2 η 2(r) dη 2 (r) dr + X η,u 3 (T )f ( X η,ū 3 (T ) α(t, Z(T ))). Proof. The erms involving h and f can be reaed exacly as in Lemma 1.5.3, so i is enough o compue he Gâeaux derivaive of J 1 (η, u) E X 1 (r ) + κ 1 2 η 1(r) dη 1 (r).,t From equaion (2.7) i follows ha he map (η, u) X η,u 1 is affine, so for s, T, ε 0, 1 and (η, u), ( η, ū) U we have We can now compue J 1( η, ū), (η, u) 1 = lim ε 0 1 = lim ε 0 X η+εη,ū+εu 1 (s) =X η,ū 1 (s) + εκ 1 ε J 1 ( η + εη, ū + εu) J 1 ( η, ū) ε,t E,T 1 = lim ε 0 ε,t E,s e ρ 1(r ) dη 1 (r) =X η,ū 1 (s) + ε ( X η,u 1 (s) e ρ 1(s ) x 1 ). X η+εη,ū+εu 1 (r ) + κ 1 2 η 1(r) + ε κ 1 X η,ū 1 (r ) + κ 1 2 η 1(r) 2 η 1(r) d η 1 (r) X η,ū 1 (r ) + ε ( X η,u 1 (r ) e ρ1(r ) ) x 1 + κ 1 2 η 1(r) + ε κ 1 2 η 1(r) d η 1 (r) + ε =E +,T,T,T,T d ( η 1 (r) + εη 1 (r)) X η,ū 1 (r ) + κ 1 2 η 1(r) d η 1 (r) X η,ū 1 (r ) + ε ( X η,u 1 (r ) e ρ 1(r ) x 1 ) + κ 1 2 η 1(r) + ε κ 1 X η,u 1 (r ) e ρ 1(r ) x 1 + κ 1 X η,ū 1 (r ) + κ 1 2 η 1(r) This complees he proof. dη 1 (r). 2 η 1(r) d η 1 (r) 2 η 1(r) dη 1 (r) Our version of he maximum principle is based on an opimaliy condiion on he Gâeaux derivaive. As a prerequisie for some algebraic manipulaions of he Gâeaux derivaive, le us now compue d(p X) for a fixed conrol (η, u) U. Using inegraion 81

92 2. When o Cross he Spread: Curve Following wih Singular Conrol by pars, we have for s, T = P (s)x(s) P ( )X( ) s X 3 (r )h ( ˆX3 (r) α(r, Z(r)) ) dr s λ1 u 1 (r) ( P 3 (r) + R 1,3 (r) ) λ 2 u 2 (r) ( P 3 (r) + R 2,3 (r) ) dr s s s s X1 (r )Q 1 (r) + X 2 (r )Q 2 (r) + X 3 (r )Q 3 (r) dw (r) X 1 (r )R 1,1 (r) + X 2 (r )R 1,2 (r) + X 3 (r )R 1,3 (r) + u 1 (r) ( P 3 (r ) + R 1,3 (r) ) Ñ 1 (dr) X 1 (r )R 2,1 (r) + X 2 (r )R 2,2 (r) + X 3 (r )R 2,3 (r) u 2 (r) ( P 3 (r ) + R 2,3 (r) ) Ñ 2 (dr),s,s This can be wrien as X1 (r )R 3,1 (r, θ) + X 2 (r )R 3,2 (r, θ) + X 3 (r )R 3,3 (r, θ) M(dr, dθ) R k κ1 P 1 (r) + P 3 (r) dη 1 (r) + κ2 P 2 (r) P 3 (r) dη 2 (r),s X 1 (r )dˆη 1 (r) + X 2 (r )dˆη 2 (r).,s Y η,u (s) = P ( )X( ) + L η,u (s), (2.15) where we define he local maringale par L η,u for s, T by L η,u (s) s s X1 (r )Q 1 (r) + X 2 (r )Q 2 (r) + X 3 (r )Q 3 (r) dw (r) s s X 1 (r )R 1,1 (r) + X 2 (r )R 1,2 (r) + X 3 (r )R 1,3 (r) + u 1 (r) ( P 3 (r ) + R 1,3 (r) ) Ñ 1 (dr) X 1 (r )R 2,1 (r) + X 2 (r )R 2,2 (r) + X 3 (r )R 2,3 (r) u 2 (r) ( P 3 (r ) + R 2,3 (r) ) Ñ 2 (dr) R k X1 (r )R 3,1 (r, θ) + X 2 (r )R 3,2 (r, θ) + X 3 (r )R 3,3 (r, θ) M(dr, dθ), 82

93 2.4. The Sochasic Maximum Principle and he non-maringale par Y η,u for s, T by s Y η,u (s) P (s)x(s) ( ) X 3 (r)h ˆX3 (r) α(r, Z(r)) dr s λ1 u 1 (r) ( P 3 (r) + R 1,3 (r) ) λ 2 u 2 (r) ( P 3 (r) + R 2,3 (r) ) dr,s,s κ1 P 1 (r) + P 3 (r) dη 1 (r),s X 1 (r )dˆη 1 (r) X 2 (r )dˆη 2 (r).,s Le us now check ha L is indeed a maringale. κ2 P 2 (r) P 3 (r) dη 2 (r) Lemma For each (η, u) U, he process L η,u is a maringale saring in 0. Proof. We firs consider he process X 1(r )Q 1 (r)dw (r). To prove ha i is a rue maringale i is enough o check ha s E sup X 1 (r )Q 1 (r)dw (r) <. s,t An applicaion of he Burkholder-Davis-Gundy and Hölder inequaliies yields s ( T ) 1 E sup X 1 (r )Q 1 (r)dw (r) ce X 1 (r )Q 1 (r) 2 2 R dr d s,t 1 ce sup X 1 (r) 2 2 T E Q 1 (r) 2 R dr r,t d The las expression is finie due o Lemma and Proposiion Now consider he process R k X 1 (r )R 1 (r, θ) M(dr, dθ). A Hölder argumen as above shows ha T E X 1 (r)r 1 (r, θ) m(dθ)dr <. R k The maringale propery now follows from Lemma A.1.3. The remaining erms of L η,u can be reaed similarly We are now in a posiion o formulae our second main resul, he sochasic maximum principle in inegral form. Theorem A conrol (ˆη, û) U is opimal if and only if for each (η, u) U we 83

94 2. When o Cross he Spread: Curve Following wih Singular Conrol have E,T ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) 0, E,T ˆX2 (r) κ 2 P 2 (r) + P 3 (r) d (η 2 (r) ˆη 2 (r)) 0, E T u 1 (r) û 1 (r) R 1,3 (r) + P 3 (r) dr 0, E T u 2 (r) û 2 (r) R 2,3 (r) + P 3 (r) dr 0. (2.16) Proof. We proceed as in Cadenillas and Haussmann 1994 Theorem 4.1. We are minimising he convex funcional J over U, so by Ekeland and Témam 1999 Proposiion a necessary and sufficien condiion for opimaliy of (ˆη, û) is ha J (ˆη, û), (η ˆη, u û) 0 for each (η, u) U. Due o Lemma we know ha L η,u is a maringale saring in zero for each (η, u) U. In paricular from equaion (2.15) we have ha EY η,u (T ) Y ˆη,û (T ) = 0. The definiion of Y η,u ogeher wih he erminal condiion (2.13) for he adjoin equaion allows us o wrie his as ( ) 0 = E f ˆX3 (T ) α(t, Z(T )) X 3 (T ) ˆX 3 (T ) T ( ) + h ˆX3 (r) α(r, Z(r)) X 3 (r) ˆX 3 (r) dr + P 3 (r) + κ 1 P 1 (r) d(η 1 (r) ˆη 1 (r)) + +,T,T,T P 3 (r) + κ 2 P 2 (r) d(η 2 (r) ˆη 2 (r)) X 1 (r ) ˆX 1 (r ) dˆη 1 (r) + X 2 (r ) ˆX 2 (r ) dˆη 2 (r),t T + λ 1 u 1 (r) û 1 (r) P 3 (r) + R 1,3 (r) dr T λ 2 u 2 (r) û 2 (r) P 3 (r) + R 2,3 (r) dr. Combining his wih he explici formula for he Gâeaux derivaive given in Proposiion yields J (ˆη, û), (η ˆη, u û) κ 1 =E,T 2 η κ 2 1(r) ˆη 1 (r) dˆη 1 (r) +,T 2 η 2(r) ˆη 2 (r) dˆη 2 (r) + ˆX 1 (r ) + κ 1 2 ˆη 1(r) P 3 (r) κ 1 P 1 (r) d (η 1 (r) ˆη 1 (r)),t 84

95 2.4. The Sochasic Maximum Principle + ˆX 2 (r ) + κ 2,T 2 ˆη 2(r) + P 3 (r) κ 2 P 2 (r) d (η 2 (r) ˆη 2 (r)) T λ 1 u 1 (r) û 1 (r) P 3 (r) + R 1,3 (r) dr T + λ 2 u 2 (r) û 2 (r) P 3 (r) + R 2,3 (r) dr. Noe ha for i = 1, 2 we have, using he noaion from equaion (2.2), E,T =E,T + κ i 2 =E =E,T,T κ i 2 η i(r) ˆη i (r) dˆη i (r) + ˆX i (r ) + κ i,t 2 ˆη i(r) d(η i (r) ˆη i (r)) ˆX i (r )d(η i (r) ˆη i (r)) r,t η i (r) ˆη i (r) ˆη i (r) + ˆη i (r) η i (r) ˆη i (r) ˆXi (r ) + κ i ˆη i (r) d(η i (r) ˆη i (r)) ˆX i (r)d(η i (r) ˆη i (r)). Combining he above wo displays leads o J (ˆη, û), (η ˆη, u û) = E +,T,T ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) ˆX2 (r) κ 2 P 2 (r) + P 3 (r) d (η 2 (r) ˆη 2 (r)) T λ 1 u 1 (r) û 1 (r) P 3 (r) + R 1,3 (r) dr T +λ 2 u 2 (r) û 2 (r) P 3 (r) + R 2,3 (r) dr. We conclude ha (ˆη, û) is opimal if and only if for all (η, u) U we have E E E E,T,T T T ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) 0, ˆX2 (r) κ 2 P 2 (r) + P 3 (r) d (η 2 (r) ˆη 2 (r)) 0, u 1 (r) û 1 (r) R 1,3 (r) + P 3 (r) dr 0, u 2 (r) û 2 (r) R 2,3 (r) + P 3 (r) dr 0. 85

96 2. When o Cross he Spread: Curve Following wih Singular Conrol 2.5. Buy, Sell and No-Trade Regions In he preceding secion we derived a characerisaion of opimaliy in erms of all admissible conrols. This condiion is no always easy o verify. Therefore, we derive a furher characerisaion in he presen secion, his ime in erms of buy, sell and norade regions. As a byproduc, his resul shows ha spread crossing is opimal if and only if he spread is smaller han some hreshold. We sar wih he main resul of his secion, which provides a necessary and sufficien condiion of opimaliy in erms of he rajecory of he conrolled sysem ( s, ˆX(s), P (s) ) s,t. The proof builds on argumens from Cadenillas and Haussmann 1994 Theorem 4.2 and exends hem o he presen framework where we have jumps and sae-dependen singular cos erms. Theorem A conrol (ˆη, û) U is opimal if and only if i saisfies ( P ( P ) ˆX 1 (s) κ 1 P 1 (s) P 3 (s) 0 s, T ) ˆX 2 (s) κ 2 P 2 (s) + P 3 (s) 0 s, T = 1, = 1, (2.17) as well as ) P(,T 1 { ˆX 1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆη 1(r) = 0 ) P(,T 1 { ˆX 2 (r) κ 2 P 2 (r)+p 3 (r)>0} dˆη 2(r) = 0 = 1, = 1, (2.18) and ds dp a.e. on, T Ω { R1,3 + P 3 0 and (R 1,3 + P 3 ) û 1 = 0, R 2,3 + P 3 0 and (R 2,3 + P 3 ) û 2 = 0. (2.19) Proof. Firs, le (ˆη, û) be opimal and define he sopping ime ν(ω) inf { } s, T : ˆX1 (s) κ 1 P 1 (s) P 3 (s) < 0, wih he convenion inf. Consider he conrol defined by u = û, η 2 = ˆη 2 and η 1 (s, ω) ˆη 1 (s, ω) + 1 ν(ω),t (s). Then η 1 is equal o ˆη 1 excep for an addiional jump of size one a ime ν. I also is 86

97 2.5. Buy, Sell and No-Trade Regions càdlàg and increasing on, T. An applicaion of Theorem yields 0 E ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)),t ( ) =E ˆX1 (ν) κ 1 P 1 (ν) P 3 (ν) 1 {ν T } 0, which implies ha P (ν = ) = 1. This proves he firs line of (2.17), he second line follows by similar argumens. Now consider he conrol defined by u = û, η 2 = ˆη 2 and η 1 ( ) = 0, dη 1 (s, ω) 1 { ˆX1 (s,ω) κ 1 P 1 (s,ω) P 3 (s,ω) 0} dˆη 1(s, ω). Then η 1 is càdlàg and increasing on, T, since ˆη 1 is. Due o Theorem we have 0 E =E,T,T ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) ˆX1 (r) κ 1 P 1 (r) P 3 (r) 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} d ( ˆη 1(r)) 0, and in paricular 0 = E 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆη 1(r),,T which proves he firs par of (2.18), he second par follows by similar argumens. I remains o prove (2.19). Again by Theorem we have for every conrol (η, u) U T 0 E (u 1 (r) û 1 (r))(r 1,3 (r) + P 3 (r))dr. Choosing he conrol (ˆη, u) wih u 2 = û 2 and u 1 (r, ω) = û 1 (r, ω) + 1 {R1,3 (r,ω)+p 3 (r,ω)>0} we firs noe ha u 1 is predicable and we ge T 0 E 1 {R1,3 (r)+p 3 (r)>0}(r 1,3 (r) + P 3 (r))dr 0, which shows ha R 1,3 +P 3 0 ds dp a.e. Recall ha we also have û 1 0 by definiion. We now wan o show a leas one of he processes R 1,3 + P 3 and û 1 is zero. To his end, consider he conrol (ˆη, u) whose passive orders are defined by u 1 = 1 2û1 and u 2 = û 2. 87

98 2. When o Cross he Spread: Curve Following wih Singular Conrol We hen ge T 0 E (u 1 (r) û 1 (r))(r 1,3 (r) + P 3 (r))dr T =E 1 2û1(r)(R 1,3 (r) + P 3 (r))dr 0, and i follows ha ds dp a.e. we have u 1 (R 1,3 + P 3 ) = 0. The argumen for he second line in (2.18) is similar. This proves he only if par of he asserion. In order o prove he if par, le condiions (2.17), (2.18) and (2.19) be saisfied. We hen have for each (η, u) U =E E,T + E + E,T,T ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) ˆX1 (r) κ 1 P 1 (r) P 3 (r) dη 1 (r),t ˆX1 (r) κ 1 P 1 (r) P 3 (r) 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} d ( ˆη 1(r)) (2.20) (2.21) ˆX1 (r) κ 1 P 1 (r) P 3 (r) 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r) 0} d ( ˆη 1(r)). (2.22) The inegrand of (2.20) is nonnegaive due o condiion (2.17), so (2.20) is nonnegaive. The erm (2.21) is zero due o condiion (2.18). The erm (2.22) has a nonposiive inegrand and a decreasing inegraor and is herefore also nonnegaive. In conclusion, we have E ˆX1 (r) κ 1 P 1 (r) P 3 (r) d (η 1 (r) ˆη 1 (r)) 0,,T and by a similar argumen E ˆX2 (r) κ 2 P 2 (r) + P 3 (r) d (η 2 (r) ˆη 2 (r)) 0.,T Sill for arbirary (η, u) U we have using (2.19) and u 1 0 T T E u 1 (r) û 1 (r) R 1,3 (r) + P 3 (r) dr = E u 1 (r) R 1,3 (r) + P 3 (r) dr 0. By a similar argumen T E u 2 (r) û 2 (r) R 2,3 (r) P 3 (r) dr 0. An applicaion of Theorem now shows ha (ˆη, û) is indeed opimal. 88

99 2.5. Buy, Sell and No-Trade Regions The preceding heorem gives an opimaliy condiion in erms of he conrolled sysem (P, ˆX). We now show how Theorem can be used o describe he opimal marke order quie explicily in erms of buy, sell and no-rade regions. Definiion We define he buy, sell and no-rade regions (wih respec o marke orders) by { } R buy (s, x, p), T R 3 R 3 x 1 κ 1 p 1 p 3 < 0, { } R sell (s, x, p), T R 3 R 3 x 2 κ 2 p 2 + p 3 < 0, { } R n (s, x, p), T R 3 R 3 x 1 κ 1 p 1 p 3 > 0 and x 2 κ 2 p 2 + p 3 > 0. Moreover, we define he boundaries of he buy and sell regions by { } R buy (s, x, p), T R 3 R 3 x 1 κ 1 p 1 p 3 = 0, { } R sell (s, x, p), T R 3 R 3 x 2 κ 2 p 2 + p 3 = 0. Le us emphasise ha each of he hree regions defined above is open. We remark ha he ime variable s is included ino he definiion of he buy, sell and no-rade regions such ha saemens like he rajecory of he process ( s, ˆX(s), P (s) ) under he opimal conrol is inside he no-rade region make sense. Specifically, we now show ha he opimal conrol remains inside he closure of he no-rade region a all imes, i.e. i is eiher inside he no-rade region or on he boundary of he buy or sell region. Moreover, as long as he conrolled sysem is inside he no-rade region, marke orders are no used, i.e. ˆη i does no increase for i = 1, 2. Proposiion If ( s, ˆX(s), P (s) ) is in he no-rade region, i is opimal no o use marke orders, i.e. for i = 1, 2 E 1 {(r, ˆX(r),P dˆη (r)) Rn} i(r) = 0.,T 2. The opimal rajecory remains a.s. inside he closure of he no-rade region, ( (s, ) ) P ˆX(s), P (s) Rn s, T = 1. In paricular, i spends no ime inside he buy and sell regions. Proof. Iem (1) is a direc consequence of (2.18), while (2) follows from (2.17). Example The paricular case of porfolio liquidaion is solved in Subsecion In his case, he opimal sraegy is composed of discree sell orders a imes = 0, T and a consan rae of sell orders in (0, T ). Specifically, hese are chosen such ha he process ( s, ˆX(s), P (s) ) remains on he boundary beween he sell region and he no-rade region for all s 0, T. This provides an example where he conrolled sysem is on he 89

100 2. When o Cross he Spread: Curve Following wih Singular Conrol boundary of he sell region a all imes. Anoher exreme case is sudied in Subsecion There, we discuss an opimisaion problem where he conrolled sysem is always inside he no-rade region and never his he boundary, so ha i is opimal no o use marke orders a all. The above proposiion shows ha he conrolled sysem remains inside he closure of he no-rade region and marke orders are no used inside he no-rade region. This suggess ha markes orders are only used on he boundary, and we shall now make his more precise. To his end, we firs noe ha for i = 1, 2 he nondecrasing process ˆη i induces a measure on, T Ω by he following map Proposiion , s A E 1 A dˆη i (r).,s 1. We have ( ) P ˆη 1 (s) = 1 {(r, ˆX(r),P (r)) Rbuy } dˆη 1(r) s, T = 1.,s In paricular, he suppor of he measure induced by ˆη 1 is a subse of ( r, ˆX(r), ) P (r) R buy, i.e. marke buy orders are only used if he conrolled sysem is on he boundary of he buy region. 2. Similarly, we have ( ) P ˆη 2 (s) = 1 {(r, ˆX(r),P (r)) Rsell } dˆη 2(r) s, T = 1.,s In paricular, he suppor of he measure induced by ˆη 2 is a subse of ( r, ˆX(r), ) P (r) R sell, i.e. marke sell orders are only used if he conrolled sysem is on he boundary of he sell region. Proof. We only show he firs asserion. For s, T we have using ˆη 1 ( ) = 0 ˆη 1 (s) = =,s,s dˆη 1 (r) 1{ ˆX1 (r) κ 1 P 1 (r) P 3 (r)<0} + 1 { ˆX 1 (r) κ 1 P 1 (r) P 3 (r)>0} + 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)=0} dˆη1 (r). 90

101 2.5. Buy, Sell and No-Trade Regions We shall show ha erms in he second line vanish a.s. By Proposiion (2) we have ( (r, ) ) P ˆX(r), P (r) / Rbuy r, T = 1 i.e. he opimal rajecory spends no ime in he buy region, so ha a.s. for each s, T,s 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)<0} dˆη 1(r) =,s Due o equaion (2.18) we have a.s. for each s, T 0 =,T 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆη 1(r) so ha a.s. for each s, T,s,s 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆη 1(r) = 0. This shows ha a.s. for each s, T we have ˆη 1 (s) =,s 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)=0} dˆη 1(r) = 1 {(r, ˆX(r),P (r)) Rbuy } dˆη 1(r) = 0. 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆη 1(r) 0,,s 1 {(r, ˆX(r),P (r)) Rbuy } dˆη 1(r). In view of he preceding proposiions we have now achieved our main goal, namely o show when spread crossing is opimal. Specifically, here is a hreshold κ 1 P 1 + P 3 for he buy spread. If he buy spread is larger han his hreshold, i.e. he conrolled sysem is inside he no-rade region, hen he coss of marke buy orders are large as compared o he penaly for deviaing from he arge, and no marke orders are used. Noe ha he hreshold can be negaive, in his case buying is no opimal a all, irrespecive of he spread size. Marke orders are only used o preven a downward crossing of he hreshold and as a resul he buy spread is never smaller han he hreshold. In his sense, he rajecory of he conrolled sysem is refleced a he boundary of he no-rade region. This will be made more precise in Subsecion 2.6 where he link o refleced BSDEs is discussed. A similar inerpreaion holds for he sell spread, where he hreshold is given by κ 2 P 2 P 3. Proposiion shows ha he suppor of he measure induced by ˆη 1 (respecively ˆη 2 ) is concenraed on he boundary of he buy (respecively sell) region. We now show ha he suppors of hese measures are disjoin. The economic inerpreaion is ha marke buy and marke sell orders are never used simulaneously. This makes sense, since rading on boh sides of he marke a he same ime is counerinuiive. Proposiion The suppors of he measures induced by ˆη 1 and ˆη 2, respecively, are disjoin. In paricular, marke buy and marke sell orders are no used simulaneously. 91

102 2. When o Cross he Spread: Curve Following wih Singular Conrol Proof. Le us denoe by supp(ˆη i ) he suppor of he measure induced by ˆη i, for i = 1, 2. We use Proposiion o wrie 0 E 1 supp(ˆη1 )dˆη 2 (r) =E E,T,T,T 1 supp(ˆη1 )1 supp(ˆη2 )dˆη 2 (r) 1 {(r, ˆX(r),P (r)) Rbuy } 1 {(r, ˆX(r),P (r)) R sell } dˆη 2(r). (2.23) We define he sopping ime ν as he firs ime he conrolled sysem is on he boundary of he buy and on he boundary of he sell region, ν inf {r, T ( r, ˆX(r), P (r) ) } R buy R sell, wih he convenion inf =. By definiion of he boundaries, on he se ν < we hen have ˆX 1 (ν) κ 1 P 1 (ν) = P 3 (ν) = κ 2 P 2 (ν) ˆX 2 (ν). Combining his wih ˆX i 0 a.s. and P i 0 a.s. for i = 1, 2 we see ha he lef hand side is nonnegaive and he righ hand side is nonposiive, so boh erms are zero and in paricular The represenaion (2.14) now implies P 1 (ν) = P 2 (ν) = 0. 0 = E dˆη 2 (r). ν,t We combine his wih inequaliy (2.23) o ge 0 E 1 supp(ˆη1 )dˆη 2 (r) E 1 supp(ˆη1 )dˆη 2 (r) E dˆη 2 (r) = 0.,T ν,t ν,t This proves ha supp(ˆη 1 ) supp(ˆη 2 ) = Link o Refleced BSDEs In his secion we use he resuls from he preceding secion o show ha he adjoin process ogeher wih he opimal conrol is he soluion o a refleced BSDE, where he obsacle is he spread. The following definiion is aken from Øksendal and Sulem

103 2.6. Link o Refleced BSDEs Definiion Le F :, T R Ω R be a measurable funcion, L :, T Ω R be an adaped càdlàg process and G L 2. We say ha ( P, Q, R, K) is a soluion o he refleced BSDE wih driver F, reflecing barrier L and erminal condiion G on he ime inerval, T if he following holds: R 1 1. P is adaped, Q and R R 2 are predicable and hey saisfy R 3 P :, T Ω R, Q :, T Ω R d, R 1 :, T Ω R, R2 :, T Ω R, R3 :, T R k Ω R. 2. K is nondecreasing and càdlàg wih K( ) = For all s, T we have P (s) P s ( ) = s + P (T ) =G. F (r, P s (r))dr + R(r, θ)ñ(dθ, dr) Q(r)dW (r),s dk(r), 4. We have a.s. for all s, T ha P (s) L(s). 5. We have a.s. ha,t ( P (r) L(r))dK(r) = 0. The inerpreaion is as follows: By iem (4), he process P is never below he barrier L. (5) means ha he process K increases only if P is a he barrier and is fla oherwise. Le us now define he following linear combinaions of he adjoin processes: ( ) ( ) P1 κ1 P 1 P 3, P 2 κ 2 P 2 + P 3 ( ) ( ) Q1 κ1 Q 1 Q 3, Q 2 κ 2 Q 2 + Q 3 R 1,1 R1,2 R κ 1 R 1,1 R 1,3 κ 2 R 1,2 + R 1,3 2,1 R2,2 κ 1 R 2,1 R 2,3 κ 2 R 2,2 + R 2,3. R 3,1 R3,2 κ 1 R 3,1 R 3,3 κ 2 R 3,2 + R 3,3 Proposiion The process R 1,1 P 1, Q 1, R 2,1, κ 1ˆη 1 R 3,1 93

104 2. When o Cross he Spread: Curve Following wih Singular Conrol is a soluion o he refleced BSDE wih driver κ 1 ρ 1 P 1 (r) h ( ˆX 3 (r) α(r, Z(r))), reflecing barrier ˆX 1 and erminal condiion f ( ˆX 3 (T ) α(t, Z(T ))). Similarly, he process R 1,2 P 2, Q 2, R 2,2, κ 2ˆη 2 R 3,2 is a soluion o he refleced BSDE wih driver κ 2 ρ 2 P 2 (r) + h ( ˆX 3 (r) α(r, Z(r))), reflecing barrier ˆX 2 and erminal condiion f ( ˆX 3 (T ) α(t, Z(T ))). Proof. We only check he firs asserion. The firs wo iems of Definiion are clear. Iem (3) follows from he dynamics of he adjoin process by direc compuaion. Specifically, we have for s, T P 1 (s) P 1 ( ) = κ 1 (P 1 (s) P 1 ( )) (P 3 (s) P 3 ( )) s ( ) = κ 1 ρ 1 P 1 (r) h ˆX3 (r) α(r, Z(r)) dr s + s κ 1 Q 1 (r) Q 3 (r)dw (r) + κ 1 R 1,1 (r) R 1,3 (r)ñ1(dr) + κ 1 R 2,1 (r) R 2,3 (r)ñ2(dr) s + κ 1 R 3,1 (r, θ) R 3,3 (r, θ) M(dr, dθ) d(κ 1ˆη 1 (r)), R k,s P ( ) 1 (T ) = κ 1 P 1 (T ) P 3 (T ) = h ˆX3 (T ) α(t, Z(T )). Iem (4) follows from equaion (2.17) in Theorem In order o verify iem (5) we apply Proposiion o ge ( P 1 (r) + ˆX 1 (r))d(κ 1ˆη 1 (r)) (2.24),T =κ 1,T ( κ 1 P 1 (r) P 3 (r) + ˆX 1 (r))1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)=0} dˆη 1(r) = 0. The second asserion follows from similar argumens. s As our main focus is on a soluion o he curve following problem and no on refleced BSDEs, we shall no pursue his furher and insead refer he ineresed reader o Øksendal and Sulem 2010, El Karoui e al. 1997a as well as Cvianic and Ma

105 2.7. Examples 2.7. Examples In his example secion we shall apply he general resuls on curve following o specific subproblems. We firs consider porfolio liquidaion, where an invesor wans o unwind a large posiion of sock shares in a shor period of ime, wih as lile adverse price impac as possible. Models and soluions have been proposed among ohers by Almgren and Chriss 2001 and Schied and Schöneborn Our framework is inspired by Obizhaeva and Wang In Subsecion we show how our characerisaion of opimaliy can be used o recover heir 3 soluion. We exend he model o passive orders in Subsecion In he concluding Subsecion we analyse an opimisaion problem where i is opimal never o rade Porfolio Liquidaion wih Singular Marke Orders The invesor sars wih sock holdings X 3 (0 ) = x 3 > 0 and wans o sell hem such ha X 3 (T ) = 0. (2.25) In he presen subsecion, we remove passive orders and only consider conrols from he following se U η 0 {η : 0, T R 2 + η i(0 ) = 0, η i (T ) 2 <, } η i is nondecreasing and càdlàg for i = 1, 2. The soluion wih passive orders will be given in Subsecion Heurisically, i should be opimal o use only marke sell and no buy orders, however we allow for boh ypes of orders and hen prove ha buying is no opimal. The consrain (2.25) ensures ha he porfolio is liquidaed by mauriy. Thus we do no need o penalise deviaion and may choose h = f = α = 0. The porfolio liquidaion problem is Problem Minimise J(η) X 1 (r ) + κ 1 0,T 2 η 1(r) dη 1 (r) + X 2 (r ) + κ 2 0,T 2 η 2(r) dη 2 (r) over conrols η U η 0 such ha ˆX 3 (T ) = 0. Noe ha his is a deerminisic problem, so ha P, X and η are deerminisic funcions. We give he explici soluion in he following proposiion. While he resul is known from Obizhaeva and Wang 2005 Proposiion 3, who use an ad hoc maringale opimaliy principle, we show here how he resul can be derived from he more gen- 3 To be precise, hey consider he equivalen ask of acquiring a large number of sock shares. 95

106 2. When o Cross he Spread: Curve Following wih Singular Conrol eral maximum principle given in Theorem As in Obizhaeva and Wang 2005 we assume ha he iniial spreads are zero, i.e. x i = X i (0 ) = 0 for i = 1, 2. As in he proof of Proposiion 1.7.3, we do no solve he above consrained problem direcly. Insead we inroduce a sequence of auxiliary conrol problems wihou consrains, bu wih a penaly for sock holdings a mauriy. For n N we define he relaxed opimisaion problem wih addiional erminal coss nx 3 (T ) 2 by Problem Minimise J n (η) X 1 (r ) + κ 1 0,T 2 η 1(r) dη 1 (r) + X 2 (r ) + κ 2 0,T 2 η 2(r) dη 2 (r) + nx 3 (T ) 2 over conrols η U η 0. We firs solve he auxiliary conrol problem. Proposiion The soluion o Problem is given by discree marke sell orders a imes s = 0 and s = T of size ˆη n 2 (0) = ˆη n 2 (T ) = x 3 ρ 2 T κ 2 n and a consan rae of marke sell orders in (0, T ) given by dˆη n 2 (s) = ρ 2 ˆη n 2 (0)ds = Marke buy orders are no used, i.e. ˆη n 1 0. ρ 2 x 3 ρ 2 T κ 2 n ds. Proof. We firs noe ha Assumpions and are saisfied in he presen framework, so ha he resuls from he previous secions can be applied. For fixed n N, he sae dynamics are given for s 0, T by X ˆηn 1 (s) = 0, X ˆηn 2 (s) = s 0 ρ 2X ˆηn 2 (r)dr + 0,s κ 2dˆη 2 n(r), X ˆηn 3 (s) = x 3 (2.26) 0,s dˆηn 2 (r), and he adjoin equaion is now for 0 s T P 1 (s) P 1 ( ) = s ρ 1P 1 (r)dr +,s dˆηn 1 (r), P 1(T ) = 0, P 2 (s) P 2 ( ) = s ρ 2P 2 (r)dr +,s dˆηn 2 (r), P 2(T ) = 0, P 3 (s) P 3 ( ) = 0, P 3 (T ) = 2n ˆX 3 (T ). (2.27) I can be verified by direc compuaion ha if ˆη n is chosen as in he asserion of he 96

107 2.7. Examples proposiion, hen he sae dynamics saisfy ˆX 1 (s) =0, ˆX 2 (s) =κ 2 ˆη n 2 (0)1 {s<t } + 2κ 2 ˆη n 2 (0)1 {s=t }, ˆX 3 (s) = ρ 2(T s) κ 2 κ 2 n ρ 2 T κ x {s<t } + n n ρ 2 T κ x {s=t }. n We claim ha he soluion o he adjoin equaion is hen given on 0, T by P 1 (s) =0, P 2 (s) = ˆη n 2 (0)1 {s<t }, P 3 (s) = 2n ˆX 3 (T ). Indeed, P 1 and P 3 are consan on 0, T and P 2 is consan on 0, T ) and jumps o zero a T. We noe ha for s 0, T ) we have s ρ 2 P 2 (r)dr + dˆη 2 n (r) =,s s ρ2 ˆη n 2 (0) + ρ 2 ˆη n 2 (0) dr = 0, so ha (2.27) is saisfied. A furher compuaion shows ha for all s 0, T we hen have { ˆX1 (s) κ 1 P 1 (s) P 3 (s) = P 3 (s) = 2n ˆX 3 (T ) 0, ˆX 2 (s) κ 2 P 2 (s) + P 3 (s) = 0. In paricular, condiions (2.17) and (2.18) of Theorem are saisfied, and hus ˆη n is he unique soluion of Problem Now ha he soluion o he unconsrained opimisaion problem is known, we ake he limi n o ge he soluion o he problem wih erminal consrain. Proposiion The soluion o Problem is given by discree marke sell orders a imes s = 0 and s = T of size ˆη 2 (0) = ˆη 2 (T ) = x 3 ρ 2 T + 2 and a consan rae of marke sell orders in (0, T ) given by dˆη 2 (s) = ρ 2 ˆη 2 (0)ds = ρ 2x 3 ρ 2 T + 2 ds. Marke buy orders are no used, i.e. ˆη 1 0. Proof. We rewrie he performance funcional in he following way: J(η) =E X 1 (r ) + κ 1 0,T 2 η 1(r) dη 1 (r) 97

108 2. When o Cross he Spread: Curve Following wih Singular Conrol + X 2 (r ) + κ 2 0,T 2 η 2(r) dη 2 (r) + δ {R\{0}} (X 3 (T )). where δ {R\{0}} is he indicaor funcion in he sense of convex analysis. We hen have for each η U η 0 J n (η) J(η). (2.28) Moreover, one can check by direc calculaion ha he sraegy ˆη saisfies he liquidaion consrain (2.25), i.e. we have ˆX 3 (T ) = 0 and hus ˆη is admissible. Before we prove he opimaliy, le us esablish some convergence resuls. We firs noe ha he opimal sraegies converge in he sense ha lim n ˆη n (s) = ˆη(s) for all s 0, T a.s. We now show ha he associaed rading coss also converge. Indeed, using he known form of X ˆηn 3 (T ) from (2.26) implies ha he erminal coss saisfy { { lim nx ˆη n n 3 (T )2} = lim n x3 ρ 2T } n ρ 2 T κ x3 2 n 2 } = lim n { 1 n κ 2 ρ 2 T κ 2 n x3 The inegrand of he singular cos erm defined in Problem converges poinwise in he sense { lim X ˆηn n 2 (r ) + κ } 2 2 ˆηn 2 (r) dˆη 2 n (r) = lim n = 0. { X ˆηn 2 (r )ρ 2 ˆη n 2 (0)dr + κ 2 2 ˆηn 2 (0) 2 + κ 2 2 ˆηn 2 (T ) 2 = ˆX 2 (r )ρ 2 ˆη 2 (0)dr + κ 2 2 ˆη 2(0) 2 + κ 2 2 ˆη 2(T ) 2 = X ˆη 2 (r ) + κ 2 2 ˆη 2(r) dˆη 2 (r). We now apply Faou s Lemma ogeher wih (2.28) o ge for each η U 0 J(ˆη) =E ˆX 2 (r ) + κ 2 0,T lim inf n N E 0,T 2 ˆη 2(r) dˆη 2 (r) X ˆηn 2 (r ) + κ 2 2 ˆηn 2 (r) = lim inf n N J n (ˆη n ) lim inf n N J n (η) J(η). This proves ha ˆη is indeed he soluion o Problem } dˆη n 2 (r) + nx ˆηn 3 (T )2 Remark The opimal liquidaion( sraegy has he following srucure: An iniial discree rade is chosen such ha 0, ˆX(0), ) P (0) is on he boundary of he sell region. Aferwards, a consan rae of sell orders and a final discree rade are 98

109 2.7. Examples used such ha ( s, ˆX(s), P (s) ) is on he boundary of he sell region for all s (0, T. While Obizhaeva and Wang 2005 work in a one sided model and only consider marke sell orders, we consider a larger class of conrols and allow for boh marke buy and sell orders. I is a consequence of Proposiion ha marke buy orders are never used. The soluion given above only holds for iniial spread zero. If we sar wih a larger spread, i migh be opimal no o use marke orders for a cerain period of ime and wai for he spread o grow back Porfolio Liquidaion wih Singular and Passive Orders In his secion, we exend he model described in Subsecion such ha i also allows for passive orders. The porfolio liquidaion problem wih passive orders is Problem Minimise J(η, u) E X 1 (r ) + κ 1 0,T 2 η 1(r) dη 1 (r) + X 2 (r ) + κ 2 0,T 2 η 2(r) dη 2 (r) over conrols (η, u) U 0 such ha ˆX 3 (T ) = 0. Noe ha in conras o Problem his is now a sochasic problem, because passive order execuion is random. We shall see ha he opimal conrol is no longer deerminisic, bu adaped o he jumps of he Poisson process N 2. Again we inroduce a sequence of auxiliary conrol problems wihou consrains, bu wih a penaly for sock holdings a mauriy. For n N we define Problem Minimise J n (η, u) E over conrols (η, u) U ,T 0,T X 1 (r ) + κ 1 2 η 1(r) X 2 (r ) + κ 2 2 η 2(r) Again we firs solve he auxiliary conrol problem. dη 1 (r) dη 2 (r) + nx 3 (T ) 2 Proposiion The soluion o Problem is given ds dp a.e. on 0, T Ω by a passive sell order of size an iniial discree marke sell order of size û n 2 (s) = ˆX 3 (s ), ˆη n 2 (0) = 2nλ 2 ρ 2 2ne λ2t (λ 2 + ρ 2 ) 2 + e λ2t λ 2 κ 2 (λ 2 + 2ρ 2 ) 2nρ 2 x 3, 2 99

110 2. When o Cross he Spread: Curve Following wih Singular Conrol Figure 2.2.: Sock holdings and rading rae wih (red, λ 2 = 1) and wihou (black, λ 2 = 0) passive orders. If here are no passive orders, here are equally sized iniial and erminal discree rades and a consan rading rae in beween. If passive orders are allowed, he iniial rade is smaller and he rading rae is increasing in ime. If he passive order is execued, he sock holdings jump o zero. The parameers in his simulaion are T = 2, x 3 = 1, ρ 2 = 1 and κ 2 = a erminal discree marke sell order of size ˆη n 2 (T ) = λ 2 + ρ 2 ρ 2 e λ 2T ˆη n 2 (0)1 {T <τ2 } and he following rae of marke sell orders in (0, T ), dˆη n 2 (s) = (λ 2 + ρ 2 )e λ 2s ˆη n 2 (0)1 {s<τ2 }ds, where τ 2 denoes he firs jump ime of he Poisson process N 2. Marke and passive buy orders are no used, i.e. a.s. ˆη 1 n(s) = 0 for each s 0, T and ûn 1 = 0 ds dp a.e. on 0, T Ω. Proof. The proof proceeds as follows: Taking he candidae opimal conrol (ˆη n, û n ) as given, we firs compue he associaed sae process and hen he adjoin equaion. This provides a soluion o he forward backward sysem and i hen only remains o check he opimaliy condiions from Theorem The sae rajecory associaed o he conrol (ˆη n, û n ) is given on 0, T by ˆX 1 (s) = x 1 e ρ1s, κ 2 e λ2s ˆη 2 n(0), if s τ 2 and s < T, ˆX 2 (s) = ˆX 2 (τ 2 )e ρ 2(s τ 2 ), if τ 2 < s, κ 2 ρ 2 (λ 2 + 2ρ 2 )e λ2t ˆη 2 n(0), if s = T < τ 2, x 3 λ 2+ρ 2 λ 2 (e λ2s 1) ˆη 2 n(0), if s < τ 2 and s < T, 1 κ ˆX 3 (s) = 2 2n ρ 2 (λ 2 + 2ρ 2 )e λ2t ˆη 2 n (0), if s = T < τ 2, 0, if s τ 2. (2.29) 100

111 2.7. Examples Noe ha he sock holdings ˆX 3 are sricly posiive on 0, τ 2 ) and jump o zero a τ 2, i.e. if N 2 jumps and he passive order is execued. A his insan, he invesor sops rading. Aferwards, he sell spread ˆX 2 recovers exponenially due o resilience. We will now use he represenaion (2.14) o consruc he adjoin process. Firs noe ha ˆη n 1 0 implies P 1 = 0 ds dp a.e. We now compue P 3. For s 0, T we have using (2.14) P 3 (s) = E s,x 2n ˆX3 (T ). We know from (2.29) ha ˆX 3 = 0 on he sochasic inerval τ 2, T, so ha P 3 (s)1 {s τ2 } = 0. We also have P 3 (T ) = 2n ˆX 3 (T ). I remains o consider s 0, τ 2 T ) and for such s we compue using he exponenial densiy of τ 2 P 3 (s) = E s,x 2n ˆX 3 (T ) = E s,x 2n 1 κ 2 (λ 2 + 2ρ 2 )e λ2t ˆη 2 n (0)1 2n ρ {T <τ2 } 2 = κ 2 (λ 2 + 2ρ 2 )e λ2t ˆη 2 n (0) λ 2 e λ2(z s) dz ρ 2 T = κ 2 (λ 2 + 2ρ 2 )e λ2t ˆη 2 n (0)e λ 2(T s) ρ 2 = κ 2 ρ 2 (λ 2 + 2ρ 2 )e λ 2s ˆη n 2 (0). We now urn o P 2. A calculaion based on he known form of ˆη 2 n, he represenaion (2.14) and he densiy of τ 2 shows ha P 2 (s) =E s,x T = s T (s,t λ 2 e λ 2(z s) λ 2 e λ 2(z s) e ρ 2(r s) dˆη n 2 (r) z s { T = λ 2 + ρ 2 ρ 2 e λ 2s ˆη n 2 (0). To sum up, he adjoin process is given explicily as P 1 (s) = 0, e ρ 2s e (λ 2 ρ 2 )r (λ 2 + ρ 2 ) ˆη n 2 (0)drdz s e ρ 2s e (λ 2 ρ 2 )r (λ 2 + ρ 2 ) ˆη n 2 (0)dr + e ρ2s e (λ 2 ρ 2 )T 1 } (λ 2 + ρ 2 ) ˆη 2 n (0) dz ρ 2 101

112 2. When o Cross he Spread: Curve Following wih Singular Conrol P 2 (s) = P 3 (s) = { λ 2 +ρ 2 ρ 2 e λ2s ˆη 2 n(0), if s < τ 2 and s < T, 0, else, κ 2 ρ 2 (λ 2 + 2ρ 2 )e λ2s ˆη 2 n(0), if s < τ 2 and s < T, 2n ˆX 3 (T ), if s = T < τ 2 0, else. In paricular, P i is zero on he sochasic inerval τ 2, T for i = 2, 3. Having consruced a soluion o he forward backward sysem, we will now use Theorem o show ha he conrol (û n, ˆξ n ) is indeed opimal. Using he known form of ˆX i and P i for i = 1, 2, 3, we check he opimaliy condiions and compue ha a.s. ˆX 1 (s) P 3 (s) κ 1 P 1 (s) = P 3 (s) 0, s 0, T ˆX 2 (s) + P 3 (s) κ 2 P 2 (s) = 0, s 0, τ 2 T, ˆX 2 (s) + P 3 (s) κ 2 P 2 (s) = ˆX 2 (s) 0, s (τ 2 T, T, so ha condiion (2.17) is saisfied. In order o check (2.18), we firs noe ha ˆη n 1 (r) = 0 for each r 0, T a.s. so ha ( ) P 1 { ˆX1 (r) κ 1 P 1 (r) P 3 (r)>0} dˆηn 1 (r) = 0 = 1. 0,T In addiion, we have ˆX 2 κ 2 P 2 + P 3 = 0 on 0, τ 2 T and ˆη 2 n is consan on τ 2 T, T so ha 1 { ˆX2 (r) κ 2 P 2 (r)+p 3 (r)>0} dˆηn 2 (r) = 0,T =0. 0,τ 2 T 1 { ˆX2 (r) κ 2 P 2 (r)+p 3 (r)>0} dˆηn 2 (r) + 1 { ˆX2 (r) κ 2 P 2 (r)+p 3 (r)>0} dˆηn 2 (r) (τ 2 T,T Finally, le us check condiion (2.19). A consequence of P 1 = 0 is ha R 1,3 = 0 ds dp a.e. and we have R 1,3 (s) + P 3 (s ) = P 3 (s ) 0 and û 1 (s) = 0. If he Poisson process N 2 jumps, hen P 3 jumps o zero, so we have ds dp a.e. on 0, T Ω R 2,3 (s) + P 3 (s ) = 0. An applicaion of Theorem now yields ha (û n, ˆη n ) is opimal. We now proceed o he porfolio liquidaion problem wih passive orders and erminal consrain. 102

113 2.7. Examples Proposiion The soluion o Problem is given ds dp a.e. on 0, T Ω by a passive sell order of size an iniial discree marke sell order of size û 2 (s) = ˆX 3 (s ), ˆη 2 (0) = λ 2 ρ 2 e λ2t (λ 2 + ρ 2 ) 2 ρ 2 x 3, 2 a erminal discree marke sell order of size ˆη 2 (T ) = λ 2 + ρ 2 e λ2t ˆη 2 (0)1 ρ {T <τ2 } = λ 2(λ 2 + ρ 2 )e λ2t 2 e λ2t (λ 2 + ρ 2 ) 2 ρ 2 x 3 1 {T <τ2 }, 2 and he following rae of marke sell orders in (0, T ), dˆη 2 (s) = (λ 2 + ρ 2 )e λ 2s ˆη 2 (0)1 {s<τ2 }ds = λ 2 ρ 2 (λ 2 + ρ 2 ) e λ2t (λ 2 + ρ 2 ) 2 ρ 2 e λ2s x 3 1 {s<τ2 }ds, 2 where τ 2 denoes he firs jump ime of he Poisson process N 2. Marke and passive buy orders are no used, i.e. a.s. ˆη 1 (s) = 0 for each s 0, T and û 1 = 0 ds dp a.e. on 0, T Ω. Proof. The argumen is he same as in he proof of Proposiion We conclude wih some remarks on he srucure of he opimal conrol. Remark I is opimal o offer all ousanding shares as a passive order, and simulaneously rade using marke orders. Le us compare he soluions wih and wihou passive orders. Proposiion shows ha in he laer, here are equally sized iniial and erminal discree rades and a consan rading rae in beween. If passive orders are allowed, i follows from Proposiion ha he iniial discree rade is small and he invesor sars wih a small rading rae, which increases as mauriy approaches. The inerpreaion is ha he is relucan o use marke orders and raher wais for passive order execuion. See Figure 2.2 for an illusraion. The sell region is in his case { R sell = (s, x, p) 0, T R 3 R 3 } x 2 + p 3 κ 2 p 2 < 0. The iniial discree rade is chosen such ha he conrolled sysem jumps o he boundary of he sell region. Then a rae of marke sell orders is chosen such ha he sae process remains on his boundary unil he passive order is execued. The opimal sraegy does no depend on he inverse order book heigh κ 2 and is linear in he iniial porfolio size x 3 = X 3 (0 ). 103

114 2. When o Cross he Spread: Curve Following wih Singular Conrol The soluion o he porfolio liquidaion problem wih passive orders given in Proposiion is similar o he one obained in Kraz and Schöneborn 2009 Proposiion 4.2; wha hey call dark pool can be inerpreed as a passive order in our seup. Noe however ha hey work in discree ime in a model wihou spread and resilience. Our soluion is also similar o he one obained in Proposiion 1.7.3, where he porfolio liquidaion problem is solved in coninuous ime using passive and marke, bu no discree orders and wihou resilience. Remark As he jump inensiy λ 2 ends o zero, he soluion given in Proposiion for he model wih passive orders converges o he soluion given in Proposiion for he model wihou passive orders. Specifically we have for s (0, T ) lim ˆη 2(0) = lim λ 2 0 λ 2 0 lim ˆη 2(T ) = lim λ 2 0 λ 2 0 lim dˆη 2(s) = lim λ 2 0 λ 2 0 λ 2 ρ 2 e λ2t (λ 2 + ρ 2 ) 2 ρ 2 x 3 = 2 λ 2 (λ 2 + ρ 2 )e λ 2T e λ2t (λ 2 + ρ 2 ) 2 ρ 2 x 3 = 2 λ 2 ρ 2 (λ 2 + ρ 2 ) e λ2t (λ 2 + ρ 2 ) 2 ρ 2 2 x 3 ρ 2 T + 2, x 3 ρ 2 T + 2, e λ 2s x 3 ds = ρ 2x 3 ρ 2 T + 2 ds. This shows ha Proposiion is a special case of Proposiion An Example Where I Is Opimal Never To Trade We know from Proposiion ha he conrolled sysem never leaves (he closure of) he no-rade region. In he preceding subsecions we solved he problem of curve following wih and wihou passive orders and found ha in his case he conrolled sysem remains on he boundary beween he sell and he no-rade region a all imes. We shall now presen anoher example where he conrolled sysem always remains inside he no-rade region, and hus i is opimal never o use marke orders. In he presen subsecion we remove passive orders and only consider deerminisic marke orders from he se U η 0 {η : 0, T R 2 + η i(0 ) = 0, η i (T ) 2 <, } η i is nondecreasing and càdlàg for i = 1, 2. Le α be a deerminisic bounded arge funcion and le f, h be penaly funcions saisfying Assumpions and We consider he following opimisaion problem Problem Minimise J(η) X 1 (r ) + κ 1 2 η 1(r) 0,T T + 0 h(x 3 (r) α(r))dr + f(x 3 (T ) α(t )). dη 1 (r) + X 2 (r ) + κ 2 0,T 2 η 2(r) dη 2 (r) 104

115 2.7. Examples over conrols η U η 0. This is again a deerminisic opimisaion problem since passive order execuion as well as he signal process Z do no play a role here. We wan o consruc he problem such ha i is opimal never o rade. To his end, we choose iniial values for he buy and sell spreads which are so high ha marke orders are never beneficial. Specifically, we assume for i = 1, 2 { x i = X i (0 ) > e ρ it sup r 0,T h (x 3 α(r)) T + f (x 3 α(t )) }. (2.30) Proposiion The soluion o Problem is given by ˆη i (s) = 0 for i = 1, 2 and all s 0, T. Proof. A consequence of (2.7) is ha he sae dynamics associaed o he conrol ˆη 0 are given for s 0, T by ˆX 1 (s) =e ρ 1s x 1, ˆX 2 (s) =e ρ 2s x 2, ˆX 3 (s) =x 3. The adjoin processes can now be represened using (2.14) by P 1 (s) =0, P 2 (s) =0, P 3 (s) = T s h (x 3 α(r))dr f (x 3 α(t )). To confirm ha ˆη 0 is indeed opimal, by Theorem i is enough o check ha for each s 0, T we have { ˆX1 (s) κ 1 P 1 (s) P 3 (s) 0, ˆX 2 (s) κ 2 P 2 (s) + P 3 (s) 0. (2.31) We only check he firs inequaliy, he second follows from similar argumens. We have for s 0, T using (2.30) κ 1 P 1 (s) + P 3 (s) =P 3 (s) T = h (x 3 α(r))dr f (x 3 α(t )) s sup h (x 3 α(r)) T + f (x 3 α(t )) r 0,T <x 1 e ρ 1T x 1 e ρ 1s = ˆX 1 (s). This proves (2.31) even wih sric inequaliy and complees he proof. 105

116 2. When o Cross he Spread: Curve Following wih Singular Conrol Remark The proof of he preceding proposiion shows ha for each s 0, T we have he sric inequaliies { ˆX1 (s) κ 1 P 1 (s) P 3 (s) > 0, ˆX 2 (s) κ 2 P 2 (s) + P 3 (s) > 0, so ha he conrolled sysem is always inside he no-rade region and never his he boundary. 106

117 3. On Marke Manipulaion in Illiquid Markes 3.1. Inroducion A key feaure of illiquid markes is ha large ransacions move prices. This is a clear disadvanage for raders ha need o liquidae large porfolios or keep heir sock holdings close o a prespecified arge as discussed in he previous chapers. Typically, marke impac should be avoided, bu here are siuaions where invesors may benefi from moving prices. Specifically, a rader ha holds a large number of opions may have an incenive o uilise his impac on he dynamics of he underlying and o move he opion value in a favorable direcion if he increase in he opion value ouweighs he rading coss in he underlying. Gallmeyer and Seppi 2000 provide some empirical evidence ha in illiquid markes opion raders are in fac able o increase a derivaive s value by moving he price of he underlying. Pirrong, 2001, p.222f wries ha a rader wih a large long posiion in a cash-seled conrac can drive up is selemen value by buying excessive quaniies of he underlying. Kumar and Seppi 1992 call such rading behavior punching he close. In his chaper we provide a coninuous ime framework o model he ineracion beween several invesors which have an incenive o punch he close. We se his up as a sochasic differenial game and esablish exisence and uniqueness of Markov equilibria for risk neural invesors and hose wih exponenial uiliy and consan absolue risk aversion (CARA). For cerain cases we have explici soluions which allow o discuss some ideas how manipulaion in he sense of punching he close could poenially be reduced. Our work builds on previous research in a leas hree differen fields. The firs is he mahemaical modeling of illiquid financial markes. The role of liquidiy as a source of financial risk has been exensively invesigaed in boh he mahemaical finance and financial economics lieraure over he las couple of years. Much of he lieraure focusses on eiher opimal hedging and porfolio liquidaion sraegies for a single large invesor under marke impac (Çein e al. 2004, Alfonsi e al. 2010, Rogers and Singh 2010), predaory rading (Brunnermeier and Pedersen 2005, Carlin e al. 2007, Schied and Schöneborn 2007a) or he role of derivaive securiies including he problem of marke manipulaion using opions (Jarrow 1994, Kumar and Seppi 1992). I has been shown by Jarrow 1994, for insance, ha by inroducing derivaives ino an oherwise complee and arbirage-free marke, cerain manipulaion sraegies for a large rader may appear, such as marke corners and fron runs. Schönbucher and Wilmo 2000 discuss an illiquid marke model where a large rader can influence he sock price wih vanishing coss and risk. They argue ha he risk of manipulaion on he par 107

118 3. On Marke Manipulaion in Illiquid Markes of he large rader makes he small raders unwilling o rade derivaives any more. In paricular, hey predic ha he opion marke breaks down. Our analysis indicaes ha markes do no necessarily break down when sock price manipulaion is cosly as i is in our model. Kraf and Kühn 2009 analyse he behaviour of an invesor in a Black Scholes ype marke, where rading has a linear permanen impac on he sock s drif. They consruc he hedging sraegy and he indifference price of a European payoff for a CARA invesor, and show ha he opimal sraegy is a combinaion of hedging and manipulaion. In order o exploi her marke impac, he invesor over- or underhedges he opion, depending on his endowmen and he sign of he impac erm. The second line of research our work is conneced o are sochasic differenial games, see e.g. Fleming and Soner 1993 chaper XI and Hamadène and Lepelier 1995 for zero sum games, Friedman 1972 and Buckdahn e al for nonzero sum games or Nisio 1988 and Buckdahn and Li 2008 for viscosiy soluions of Hamilon-Jacobi- Bellman (henceforh HJB) equaions. The sraegic ineracion beween large invesors and is implicaions for marke microsrucure are discussed in Kyle 1985, Foser and Viswanahan 1996, Back e al. 2000, and Chau and Vayanos 2008, for insance. Brunnermeier and Pedersen 2005, Carlin e al and Schied and Schöneborn 2007a consider predaory rading, where liquidiy providers ry o benefi from he liquidiy demand ha comes from some large invesor. Vanden 2005 considers a pricing game in coninuous ime where he opion issuer conrols he volailiy of he underlying bu does no incur liquidiy coss. He derives a Nash equilibrium in he wo player, risk neural case and shows ha seemingly harmless derivaives, such as ordinary bull spreads, offer incenives for manipulaion ha are idenical o hose offered by digial opions (p.1892). Closes o our seup is he paper by Gallmeyer and Seppi They consider a binomial model wih hree periods and finiely many risk neural agens holding call opions on an illiquid underlying. Assuming a linear permanen price impac and linear ransacion coss, and assuming ha all agens are iniially endowed wih he same derivaive hey prove he exisence of a Nash equilibrium rading sraegy and indicae how marke manipulaion can be reduced. A hird line of research we build on is marke manipulaion. Differen noions of marke manipulaions have been discussed in he lieraure including shor squeezes, he use of privae informaion or false rumours, cf. Kyle 1985, Back 1992, Jarrow 1994, Allen and Gale 1992, Pirrong 2001, Du and Harris 2005, Kyle and Viswanahan Mos of hese aricles are se up in discree ime. We sugges a general mahemaical framework in coninuous ime wihin which o value derivaive securiies in illiquid markes under sraegic ineracions. Specifically, we consider a sochasic differenial game beween a finie number of large invesors ( players ) holding European claims wrien on an illiquid sock. Their goal is o maximise he expeced porfolio value a mauriy, composed of rading coss and he opion payoff, which depends on he rading sraegies of all he oher players hrough heir impac on he dynamics of he underlying. Following Almgren and Chriss 2001 we assume ha he players have a permanen impac on sock prices and ha all rades are seled a he prevailing marke price plus a liquidiy premium. The liquidiy premium can be viewed as an insananeous price impac ha affecs ransacion prices bu no he value of he players invenory. This form 108

119 3.1. Inroducion of marke impac modeling is analyically more racable han ha of Obizhaeva and Wang 2005 which also allows for emporary price impacs and resilience effecs. I has also been adoped by, e.g. Carlin e al and Schied and Schöneborn 2007a and some praciioners from he financial indusry, as poined ou by Schied and Schöneborn Our framework is flexible enough o allow for raher general liquidiy coss including he linear cos funcion of Almgren and Chriss 2001 and some form of bid ask spread, cf. Example We show ha when he marke paricipans are risk neural or have CARA uiliy funcions he pricing game has a unique Nash equilibrium. We solve he problem of equilibrium pricing using echniques from he heory of sochasic opimal conrol and sochasic differenial games. We show ha he family of he players value funcions can be characerised as he soluion o a coupled sysem of non-linear PDEs. Coupled sysems of non-linear PDEs arise naurally in differenial sochasic games. Since general exisence and uniqueness of soluion resuls for sysems of nonlinear PDEs on unbounded sae spaces are hard o prove much of he lieraure on sochasic differenial games is confined o bounded sae spaces; see e.g. he seminal paper of Friedman We prove an a priori esimae for Nash equilibria. More precisely we prove ha under raher mild condiions any equilibrium rading sraegy is uniformly bounded. This allows us o prove ha he PDE sysem ha describes he equilibrium dynamics has a unique classical soluion. The equilibrium problem can be solved in closed form for a specific marke environmen, namely he linear cos srucure and risk neural agens. I is imporan o know which measures may reduce marke manipulaion. For insance, Du and Harris 2005 propose posiion limis; Pirrong 2001 suggess efficien conrac designs. We use he explici soluion for risk neural invesors o show when punching he close is no beneficial. For insance, no manipulaion occurs in zero sum games, i.e. in a game beween an opion wrier and an opion issuer. In our model manipulaion decreases wih he number of informed liquidiy providers and wih he number of compeiors, if he produc is spli beween hem. Furhermore, we find ha he bid ask spread is imporan deerminan of marke manipulaion. I urns ou ha he higher he spread, he less beneficial marke manipulaion: high spread crossing coss make rading more cosly and hence discourage frequen re-balancing of porfolio posiions. The remainder of his chaper is organised as follows: We presen he marke model as well as he opimisaion problem and some a priori esimaes in Secion 3.2. The soluion for risk neural and CARA invesors are given in Secions 3.3 and 3.4, respecively. We use he explici soluion for he risk neural case in Secion 3.5 o show how marke manipulaion can be reduced. Pars of his chaper are published in Hors and Naujoka

120 3. On Marke Manipulaion in Illiquid Markes 3.2. The Model We adop he marke impac model of Schied and Schöneborn 2007a wih a finie se of agens, or players, rading a single sock whose price process depends on he agens rading sraegies. I is a muliplayer exension of he model inroduced in Chaper 1, wih an addiional permanen price impac as in Almgren and Chriss All rades are seled a prevailing marke prices plus a liquidiy premium which depends on he change in he players porfolios. In order o be able o capure changes in porfolio posiions in an analyically racable way, we assume ha he sock holdings of player j {1,..., N} are governed by he following SDE, dx j (s) = u j (s)ds, X j (0) = 0, where he rading speed u j = Ẋj is chosen from he following se of admissible conrols, for 0, T : U {u :, T Ω R progressively measurable}. There is a an array of large invesor models which assume ha sock holdings are absoluely coninuous and ha he price dynamics depend on he change of he invesors posiions, e.g. Almgren e al. 2005, Almgren and Lorenz 2007, Schied and Schöneborn (2007, 2008), Carlin e al and Rogers and Singh In all hese papers he assumpion of absolue coninuiy is made merely for analyical convenience. We also remark ha he dynamics specified above are similar o he seup in Chaper 1, bu we do no consider passive orders here since heir marke impac is ypically negligible Price dynamics and he liquidiy premium Our focus is on opimal manipulaion sraegies (in he sense of punching he close ) for derivaives wih shor mauriies under sraegic marke ineracions. For shor rading periods we deem i appropriae o model he fundamenal sock price, i.e. he value of he sock in he absence of any marke impac, as a Brownian Moion wih volailiy σ > 0. Marke impac is accouned for by assuming ha he invesors accumulaed sock holdings N i=1 X i have a linear permanen impac on he sock process P so ha for s 0, T N P (s) = P (0) + σw (s) + λ X i (s) (3.1) i=1 wih an impac parameer λ > 0. The linear permanen impac is consisen wih he work of Huberman and Sanzl 2004 who argue ha lineariy of he permanen price impac is imporan o exclude quasi-arbirage. We remark ha in Chapers 1 and 2 our focus was on he radeoff beween he liquidiy coss of rading and he penaly for deviaing from a arge funcion. In ha case he liquidiy coss were deermined by he rading sraegy and he bid ask spread. I was herefore no necessary o model 110

121 3.2. The Model he sock price explicily and we only considered insananeous and emporary (bu no permanen) price impac. However in he presen chaper we invesigae how o increase he payoff of a given opion wih illiquid underlying. We do model he price of he underlying explicily and we now also include a permanen price impac. A rade a ime s 0, T is seled a a ransacion price P (s) ha includes an addiional insananeous price impac, or liquidiy premium. Specifically, ( N ) P (s) = P (s) + g u i (s) i=1 (3.2) wih a cos funcion g ha depends on he insananeous change N i=1 u i in he agens posiion in a possibly non-linear manner, jus as in he firs chaper. The liquidiy premium accouns for limied available liquidiy, ransacion coss, fees or spread crossing coss, cf. Example Remark In he single player case discussed in Chaper 1 we considered a liquidiy cos funcion g(u, Z) which was driven by a sochasic facor Z. In he presen case wih several agens we remove Z for racabiliy. In our model he liquidiy coss are he same for all raders and depend only on he aggregae demand hroughou he enire se of agens. This capures siuaions where he agens rade hrough a marke maker or clearing house ha reduces he rading coss by collecing all orders and maching incoming demand and supply prior o seling he ousanding balance N i=1 u i (s) a marke prices. We assume ha g is normalised, g(0) = 0 and smooh. The following addiional mild assumpions on g will guaranee ha he equilibrium pricing problem has a soluion for risk neural and CARA invesors. Assumpion The derivaive g is bounded away from zero, ha is g > ɛ > 0. The mapping z g(z) + zg (z) is sricly increasing. The firs assumpion is a echnical condiion needed in he proof of Proposiion I appears no oo resricive for a cos funcion. Since he liquidiy coss associaed wih a ne change in he overall posiion z is given by zg(z), he second assumpion saes ha he agens face increasing marginal coss of rading. Example Among he cos funcions which saisfy Assumpion are he linear cos funcion g(z) = κz wih κ > 0 and cos funcions of he form g(z) = κz + c 2 arcan(cz) wih c, C > 0. π The former is he cos funcion associaed wih a block-shaped limi order book. The laer can be viewed as a smooh approximaion of he map z κz + c sign(z) which is he cos funcion associaed wih a block-shaped limi order book and bid ask spread c >

122 3. On Marke Manipulaion in Illiquid Markes The Opimisaion Problem Each agen is iniially endowed wih a coningen claim H j = H j (P (T )), whose payoff depends on he sock price a mauriy. Our focus is on opimal rading sraegies in he sock, given an iniial endowmen. As in Gallmeyer and Seppi 2000 and Kraf and Kühn 2009, we assume ha he agens do no rade he opion in 0, T. A consisen model for rading an illiquid opion wih illiquid underlying in a muliplayer framework in coninuous ime is no available, o he bes of our knowledge. Our work migh be considered a sep in his direcion. We assume ha he funcions H j are smooh and bounded wih bounded derivaives H j p. This is needed in he a priori esimaes as well as in he proof of exisence of a smooh soluion o he HJB equaion. Remark We only consider opions wih cash selemen. This assumpion is key. While cash selemen is suscepible o marke manipulaion, we show in Proposiion below ha when deals are seled physically, i.e. when he opion issuer delivers he underlying, marke manipulaion is no beneficial: Any price increase is ouweighed by he liquidiy coss of subsequen liquidaion. We noice ha his only applies o punching he close. There are oher ypes of marke manipulaion, such as corners and shor squeezes, which migh be beneficial when deals are seled physically, bu which are no capured by our model, cf. Jarrow 1994 or Kyle and Viswanahan We shall now give a heurisic derivaion of he opimisaion problem. Consider a single risk neural invesor who builds up a posiion in sock holdings X(T ) using he rading sraegy u in 0, T and aferwards liquidaes his sock holdings using a consan rae of liquidaion η, so ha a ime T T + X(T ) η he porfolio is liquidaed. In view of (3.2), he proceeds from such a round rip sraegy are = T 0 T 0 u(s) P T (s)ds + σw (s)dx(s) λ T T 0 η P (s)ds T X(s)dX(s) u(s)g(u(s))ds X(T )g(η). 0 Using inegraion by pars and X(0) = X(T ) = 0 we see ha he firs erm in he second line has zero expecaion 1 and he second erm also vanishes. The las erm describes he liquidiy coss of he consan liquidaion rae η and goes o zero if η goes o zero since g(0) = 0. In his sense, infiniely slow liquidaion incurs no coss. I follows ha he round rip sraegy described above incurs expeced liquidiy coss of T 0 u(s)g(u(s))ds. Taking ino accoun he opion payoff, he opimisaion problem for a single risk neural invesor becomes sup E u U 0 T 0 u(s)g (u(s)) ds + H(P (T )). (3.3) 1 We will prove an a priori esimae in Proposiion and hen only consider bounded sraegies, so ha he sochasic inegral T 0 X()dW () is indeed a maringale. 112

123 3.2. The Model This reflecs he radeoff beween liquidiy coss (he coss of punching he close ) and an increased opion payoff. In our model, he only purpose of rading is an increased opion payoff and no, for insance, hedging. For a sudy on he inerplay of hedging and manipulaion we refer he reader o Kraf and Kühn Unforunaely, he heurisic derivaion given above has no direc counerpar in he muliplayer case. As one prerequisie one would need he opimal liquidaion sraegies (and corresponding liquidaion value) of several agens in a marke wih general liquidiy srucure. Defining a noion of liquidaion value under sraegic ineracion is sill an open quesion (Carlin e al and Schied and Schöneborn 2007a derived soluions in special cases) and i is no he focus of he presen work. Our focus is on he radeoff beween increased opion payoff and liquidiy coss in a muliplayer framework. Specifically, we assume ha he preferences of player j a ime 0, T are described by a preference funcional Ψ j (condiional expeced value or condiional enropic risk measure) and ha his goal a ime = 0 is o maximise he uiliy from he opion payoff minus he cos of rading (given he oher players sraegies). We hence consider he following opimisaion problem: Problem Given he sraegies u i U 0 for all he players i j he opimisaion problem of player j N is sup u j U 0 Ψ j 0 ( ( T N ) ) u j (s)g u i (s) ds + H j (P (T )). 0 i=1 Remark We remark ha in he preceding chapers he dynamics of he sae were degenerae, so he dynamic programming approach was no direcly applicable and we based our characerisaion of opimaliy on a suiable version of he sochasic maximum principle insead. In he presen case he sae variable P is given by a nondegenerae diffusion. As a resul, we shall see ha he HJB equaion is uniformly parabolic and we proceed via he dynamic programming approach. In conras, he mehods from he previous chapers do no apply here since he above opimisaion problem is no necessarily convex in he conrol. The case where all invesors are risk neural, Ψ j (Z) = EZ F, is sudied in Secion 3.3. The case of condiional expeced exponenial uiliy maximising invesors is sudied in Secion 3.4. In ha case we may choose Ψ j (Z) = 1 log E exp( α j Z) F α j where α j > 0 denoes he risk aversion of player j. Boh preference funcionals are ranslaion invarian 2. This means ha Ψ j (Z + Y ) = Ψj (Z) + Y for any random variable Y ha is measurable wih respec o he informaion available a ime 0, T. As a resul, he rading coss incurred up o ime do no affec he opimal rading sraegy a laer imes. This propery is key and will allow us o esablish he exisence of Nash equilibria in our financial marke model. ( ) Definiion We say ha a vecor of sraegies u 1,..., u N is a Nash equilibrium if for each agen j N his rading sraegy u j is a bes response agains he behavior of 2 Translaion invarian preferences have recenly araced much aenion in he mahemaical finance lieraure in he conex of opimal risk sharing and equilibrium pricing in dynamically incomplee markes. We refer o Cheridio e al for furher deails. 113

124 3. On Marke Manipulaion in Illiquid Markes all he oher players, i.e. if u j solves Problem 3.2.5, given he oher players aggregae rading u j i j ui. Remark Our resuls hinge on wo key assumpions: he resricion o absoluely coninuous rading sraegies and he focus on he radeoff beween rading coss and marke manipulaion. Boh resricions may be considered undesirable. On he oher hand, if singular conrols are considered, he dynamic programming approach would lead o a sysem of quasi-variaional inequaliies, which is beyond he scope of our work. Also, i is no obvious how he maximum principle approach we derived in Chaper 2 can be exended o a muliplayer framework. Insead we work in a raher simple framework in he spiri of Almgren and Chriss 2001 and our model should be viewed as a firs benchmark o more sophisicaed models. Despie is many simplificaions, i allows for explici soluions and hus yields some insigh ino he qualiaive behaviour of opimal manipulaion sraegies as well as rules of humb for raders or regulaors. Moreover, he closed-form soluions will be used in Secion 3.5 o indicae how manipulaion can be reduced A Priori Esimaes In he sequel we show ha Problem admis a unique soluion for risk neural and CARA invesors. The proof uses he following a priori esimaes for he opimal rading sraegies. I saes ha, if an equilibrium exiss, hen each player s rading speed is bounded. The reason is ha he derivaives Hp j of he payoff funcions H j are assumed o be bounded, so each invesor benefis a mos linearly from fas rading. However, rading coss grow more han linearly, and hus very fas rading is no beneficial. Noe ha his resul does no depend on he preference funcional. ( ) Proposiion Le u 1,..., u N be a Nash equilibrium for Problem Then each sraegy u j saisfies ds dp a.e. u j (s) N λ ɛ where ɛ is aken from Assumpion Proof. Le j N, h max i H i p and ( max i N ) Hp i + 1, { } N A (s, ω) 0, T Ω : u i (s, ω) 0 i=1 be he se where he aggregae rading speed is nonnegaive. Le us fix he sum of he compeiors sraegies u j. On he se A he bes response u j (s) is bounded from above by K λ ε (h + 1). Oherwise he runcaed sraegy ūj (s) u j (s) K 1 A + u j (s) 1 A c would ouperform u j (s). To see his, le us compare he payoffs associaed wih u j and ū j. We denoe by P ūj (T ) and P uj (T ) he sock price under he sraegies ū j and u j, 114

125 3.3. Soluion for Risk Neural Invesors respecively. The payoff associaed wih ū j minus he payoff associaed wih u j can be esimaed from below as T + 0 T 0 T 0 T + ( ) ū j (s)g ū j (s) + u j (s) ds + H j (P ūj (T )) ( ) u j (s)g u j (s) + u j (s) ds H j (P uj (T )) ū j (s) 0 ( ( ) ( )) g u j (s) + u j (s) g ū j (s) + u j (s) ds ( ) ( ) u j (s) ū j (s) g u j (s) + u j (s) ds λ(x j (T ) Y j (T )) H p. Noe ha u j (s) + u j (s) 0 on A and hus g ( u j (s) + u j (s) ) 0 due o Assumpion Furhermore, g ( u j (s) + u j (s) ) g ( ū j (s) + u j (s) ) ɛ ( u j (s) ū j (s) ), again by Assumpion The difference in he payoffs is herefore larger han = T 0 ( ) T ū j (s)ɛ u j (s) ū j (s) ds λh u j (s)>ū j (s) ( ) ( ) ɛū j (s) λh u j (s) ū j (s) ds 0 ( ) u j (s) ū j (s) ds On he se { u j (s) > ū j (s) } we have ū j (s) = K = λ ε (h + 1) and he above expression is sricly posiive, a conradicion. This shows ha u j (s) is bounded above by K on he se A for each j N. Sill on he se A, we ge he following lower bound: N u j (s) = u i (s) + u i (s) 0 (N 1)K. (3.4) i=1 i j com- A symmeric argumen on he se B plees he proof. { (s, ω) 0, T Ω : } N i=1 u i (s, ω) Soluion for Risk Neural Invesors In his secion we use dynamic programming o show ha Problem admis a unique soluion (in a cerain class) for risk neural agens. Here he preference funcional is Ψ j (Z) = EZ F for each j N. We also show ha he soluion can be given in closed form for he special case of a linear cos funcion. The idea is o consider he value funcion associaed o Problem for player j, where his compeiors sraegies are fixed, and o characerise i as he soluion of he HJB PDE. Solving he resuling coupled sysem of PDEs for all players simulaneously hen provides an equilibrium poin of he sochasic differenial game, cf. Friedman To begin wih, we fix he sraegies (u i ) i j and define he value funcion for 115

126 3. On Marke Manipulaion in Illiquid Markes player j N as V j (, p) = sup u j U E,p subjec o he sae dynamics ( T N ) u j (s)g u i (s) ds + H j (P (T )), i=1 N dp (s) = σdw (s) + λ u i (s)ds, P () = p. i=1 Here we use he noaion E,p E P = p. Given ime 0, T and sock price p R he value funcion represens he condiional expeced porfolio value a mauriy ha player j can achieve by rading opimally, given he oher players sraegies. The associaed HJB equaion is, cf. Fleming and Soner 1993 Theorem IV.3.1, 0 = v j + 1 ( 2 σ2 vpp j + sup λ c j + u j) ( vp j c j g c j + u j), (3.5) c j R wih erminal condiion v j (T, p) = H j (p), where v and v p denoe ime and spaial derivaives, respecively. The HJB equaion is formulaed in erms of he candidae value funcions v 1,..., v N insead of he acual value funcions V 1,..., V N. We firs need o show exisence and uniqueness of a smooh soluion o (3.5) before we can idenify v i wih V i. Given he aggregae rading sraegy u j of all he oher agens, a candidae for he maximiser c j = u j in (3.5) should saisfy ( 0 = λvp j g c j + u j) ( c j g c j + u j). (3.6) We have one equaion of his ype for each player j N. Summing hem up and defining he aggregae rading speed as yields he following condiion N u ag u i i=1 ( N N ) ( 0 = λ vp i N ) ( Ng u i N ) (s) u i (s) g u i (s) i=1 i=1 i=1 i=1 N = λ vp i Ng (u ag (s)) u ag (s)g (u ag (s)). (3.7) i=1 In view of Assumpion he map z Ng(z) + zg (z) is sricly increasing. Hence condiion (3.7) admis a unique soluion u ag which depends on N i=1 v i p. Plugging u ag 116

127 3.3. Soluion for Risk Neural Invesors back ino (3.6) allows o compue he candidae opimal conrol for player j N as c j = u j = λvj p g(u ag ) g (u ag. (3.8) ) This expression is well defined since g > 0 again by Assumpion Plugging his candidae opimal conrol ino he HJB equaion, we see ha he sysem of HJB PDEs now akes he form 0 = v j + 1 ( ) 2 σ2 vpp j + λ u ag g(uag ) g (u ag v jp + g(uag ) 2 ) g (u ag (3.9) ) wih erminal condiion v j (T, p) = H j (p) for j N. Noe ha he coupling sems from he aggregae rading speed u ag via condiion (3.7). Remark Looking back, we have urned he individual HJB equaions (3.5) ino he sysem of coupled PDEs (3.9). Sysems of his form appear naurally in he heory of differenial games, bu we did no find a reference which covers his paricular case. Theorem 1 of Friedman 1972 for insance is valid only on a bounded sae space. We shall use our a priori esimaes of Proposiion in order o prove exisence of a unique soluion o (3.9). The following heorem, whose proof is given in Appendix A.3.1, shows ha he sysem of PDEs (3.9) has a unique classical soluion if H j C 2 b, i.e. Hj is wice coninuously differeniable and is derivaives up o order 2 are bounded, for each j. Similarly, C 1,2 is he space of funcions which are coninuously differeniable in ime and wice coninuously differeniable in space. Theorem Le H Cb 2. Then he Cauchy problem (3.9) admis a unique classical soluion in C 1,2, which is he vecor of value funcions. Remark An alernaive way of solving he sysem (3.9) is he following: If we sum up he N equaions, we ge a Cauchy problem for he aggregae value funcion v N i=1 v i, namely 0 = v σ2 v pp + u ag λv p g (u ag ) (3.10) wih erminal condiion v(t, p) = N i=1 H i (p). Exisence and uniqueness of a soluion o his one-dimensional problem can be shown using Theorem V.8.1 in Ladyzenskaja e al Once he soluion is known, we can plug i back ino (3.9) and ge N decoupled equaions. This echnique is applied in he following secion where we consruc an explici soluion for linear cos funcions. I is hard o find a closed form soluion for he coupled PDE (3.9). However, for he paricular choice g(z) = κz wih a liquidiy parameer κ > 0 he soluion o (3.9) can 117

128 3. On Marke Manipulaion in Illiquid Markes be given explicily. Here and hroughou, we denoe by ( ) 1 f µ,σ 2(z) exp (z µ)2 2πσ 2 2σ 2 he normal densiy wih mean µ and variance σ 2. Proposiion Le g(z) = κz. Then he soluion of (3.9) can be given in closed form as he soluion o a nonhomogeneous hea equaion. Proof. The opimal rading speed from (3.8) and he aggregae rading speed from (3.7) are u j = λ ( vp j 1 ) N vp i (3.11) κ N + 1 N u ag = u i = i=1 λ κ(n + 1) i=1 N i=1 v i p = Equaion (3.9) for player j s value funcion now becomes 0 = v j σ2 v j pp + κ(u ag ) 2. λ κ(n + 1) v p. (3.12) Combining his wih (3.12) and summing up for j = 1,..., N yields he following PDE for he aggregae value funcion v = N i=1 v i : 0 = v σ2 v pp + λ 2 N κ(n + 1) 2 v2 p (3.13) wih erminal condiion v(t, p) = N i=1 H i (p). This PDE is a varian of Burgers equaion, cf. Rosencrans I allows for an explici soluion, which we cie in Lemma A.3.3 in he appendix. Wih his soluion a hand, we can solve for each single invesor s value funcion. We plug he soluion v back ino he equaions (3.11) and (3.12) for he rading speeds, and hose ino he PDE (3.9). This yields 0 = v j σ2 v j pp + λ 2 κ(n + 1) 2 v2 p wih erminal condiion v j (T, p) = H j (p). This is now a PDE in he unknown funcion v j wih known funcion v p. We see ha i is a nonhomogeneous hea equaion wih soluion given by v j (T, p) = R H j (z)f p,σ 2 (z)dz + λ 2 κ(n + 1) 2 0 R v 2 p(s, z)f p,σ 2 ( s)(z)dzds where v is known from Lemma A.3.3 (in paricular i is bounded and inegrable). 118

129 3.3. Soluion for Risk Neural Invesors Figure 3.1.: Trading speed and surplus for one risk neural invesor holding a European Call opion. Le us conclude his secion wih some numerical illusraions. For risk neural players and a linear cos srucure, we reduced he sysem of PDEs o he one-dimensional PDE (3.13) for he aggregae value funcion. This can be inerpreed as he value funcion of he represenaive agen. Such reducion o a represenaive agen is no always possible for more general uiliy funcions. In he sequel we illusrae he opimal rading speed u(s, p) and surplus of a represenaive agen as funcions of ime and spo prices for a European call opion H(P (T )) = (P (T ) K) + and digial opion H(P (T )) = 1 {P (T ) K}, respecively. 3 By surplus, we mean he difference beween he represenaive agen s opimal expeced porfolio value v(, p) and he condiional expeced payoff E,p H(P (T )) in he absence of any marke impac. I represens he expeced ne benefi due o price manipulaion. We choose a linear cos funcion, srike K = 100, mauriy T = 1, volailiy σ = 1 and liquidiy parameers λ = κ = We see from Figure 3.1 ha for he case of a call opion boh he opimal rading speed and he surplus increases wih he spo; he laer also increases wih he ime o mauriy. Furhermore, he increase in he rading speed is maximal when he opion is a he money. For digial opions (figure 2) he rading speed is highes for a he money opions close o mauriy as he rader ries o push he spo above he srike. If he spo is far away from he srike, he rading speed is very small as i is unlikely ha he rader can push he spo above he srike before expiry. For boh opion ypes a high spread renders manipulaion unaracive. Figures 3 and 4 show he opimal rading speed and he surplus a ime = 0 for he Call and Digial opion for a represenaive agen. We used he cos funcion g(z) = κz + c sign(z) for differen spreads c {0, 0.001, 0.002, 0.003, 0.004} (3.14) wih he remaining parameers as above. We see ha he higher he spread, he smaller 3 Noe ha he cos funcion in (3.14) is no smooh, and he Call and Digial opions are no smooh and bounded, so Theorem does no apply direcly. There are wo ways o overcome his difficuly: We could eiher approximae g and H by smooh and bounded funcions. Or we could inerpre v no as a classical, bu only as a viscosiy soluion of (3.5), cf. Fleming and Soner 1993 chaper V. 119