Analysis of optimal liquidation in limit order books

Transcription

1 Analysis of opimal liquidaion in limi order books James W. Blair, Paul V. Johnson, & Peer W. Duck Absrac In his paper we sudy he opimal rading sraegy of a passive rader who is rading in he limi order book. Using a combined approach of accurae numerical mehods and asympoical analysis we examine he problem using differen sochasic processes o model he asse price, as well as inroducing a proporional resilience for he limi order book. This resuls in more complex equaions o solve han when examined under he case of sandard Brownian moion, allowing us o perform ineresing analyical asympoic analysis which adds insigh ino he soluion space. Under Geomeric Brownian Moion, we reduce he resuling four-dimensional Hamilon-Jacobi-Bellman parial differenial equaion PDE o a novel hree-dimensional non-linear PDE, as well as rescaling he variables o reduce he number of inpu parameers by wo. We use numerical mehods o solve he PDE before asympoically examining i in several limis, wih each approach informing and confirming he oher. We find he ransiion from a ime-varying soluion o a perpeual-ype soluion resuls in he developmen of singular behaviour, and his ransiion is examined in some deail. Finally we emphasise he adapabiliy of our proposed mehodologies by implemening he same mehods on a mean-revering process for he asse price. Throughou he paper we also analyse he resuling rading sraegies from a financial perspecive. The rading sraegies we develop are asse-price dependen, which o our knowledge is a unique concep in he passive opimal rading lieraure, and is arguably more realisic. Opimal liquidaion; Asympoic analysis; Sochasic opimal conrol; Algorihmic rading; High-frequency rading 1 Inroducion Opimal liquidaion execuion consiss of selling buying a large amoun of an asse before a specified ime, while obaining he bes price under some specified risk crieria. Generally speaking, liquidaion selling and execuion buying are inerchangeable for mos mehods by simply changing iniial condiions and consrains. In his paper we focus on he opimal liquidaion of a porfolio of asses, alhough adaping he mehod o fill a porfolio would be sraighforward. Algorihmic rading has exploded in recen years, wih repors of 5-77% of rading volume in he US coming from compuerised algorihms, see SEC 21. Various algorihms have been invesigaed exensively by boh academics and insiuional raders; hese algorihms focus on being he mos profiable, he leas risky, or a rade-off beween he wo. The main focus of previous lieraure is spli beween he modelling of aggressive and passive opimal rading sraegies. Aggressive rading focuses on finding he opimal rae o rade by filling orders while passive rading focuses on placing orders ino he Limi Order Book LOB and waiing for an aggressive rader o fill hose orders. Opimal rading of large orders o conrol he rading coss was originally proposed for aggressive rading o find an opimal balance beween reducing marke impac by rading slow and reducing marke risk by rading fas. The Previously: Opimal liquidaion in limi order books under general uncerainies School of Mahemaics, The Universiy of Mancheser, M13 9PL Mancheser, UK jblair@ma.man.ac.uk, pjohnson@ma.man.ac.uk, duck@ma.man.ac.uk. 1

2 firs models were developed around he urn of he millennium by Bersimas e al and Almgren and Chriss 21. These models focused on rading in he marke book, in which he objecive was o maximise he efficien fronier of he invesor s wealh. Boh models used linear impac funcions for discree ime rading and developed saic rading sraegies, meaning he sraegies were known before rading had begun and hus were independen of he asse price, which was driven by a sandard Brownian moion wih drif. Exensions o his framework include: Almgren 23 who considered coninuous ime rading wih non-linear impac funcions, Almgren and Lorenz 27 who allow one updae of he asse price a a fixed ime and from his creaed dynamical sraegies based on his updae, Lorenz and Almgren 211 who allow coninuous, bu discree, updaes of he share price o develop dynamical rading sraegies, Schied and Schöneborn 29 who maximise general uiliy funcions, raher han a mean-variance rade-off, o develop price dependen sraegies, Schied e al. 21 who maximise he invesor s uiliy using CARA uiliy funcions o find deerminisic rading sraegies and Forsyh e al. 212 who used numerical mehods o solve he problem under Geomeric Brownian Moion GBM. More recenly, he concep of dark pools have been inroduced ino he lieraure. Kraz and Schöneborn 213 consider a rader who can rade in he marke book wih price impac and in a dark pool no price impac, wih execuion occuring randomly in he dark pool according o a Poisson process. The framework of Almgren and Chriss 21 was furher developed for rading in he LOB, which is a beer replicaion of real-life rading han rading in he marke book. In he laer case, asses can be bough or sold a he same price, ha being he fair asse price. However, in realiy here is a difference beween he price ha asses can be bough and sold. The difference beween he highes price ha a buyer is willing o pay for an asse and he lowes price for which a seller is willing o sell i is known as he bid-ask spread. The LOB includes he highes bid price, he lowes ask price as well as all offers lower han he bes bid price and higher han he bes ask price. Obizhaeva and Wang 213, which has been a preprin paper since 24, inroduced aggressive rading in he LOB in which a block shaped densiy for he LOB was assumed. The emporary marke impac funcion used in marke book models, in which only one rade was affeced, was replaced by a ransien marke impac funcion, a decreasing funcion of ime represening he LOB refilling pos rade. Alfonsi e al. 21 considered general shape funcions for he LOB and ackled he problem hrough he use of Lagrange mulipliers, raher han hrough he use of dynamical programming, as carried ou by Obizhaeva and Wang 213. Kharroubi and Pham 21 considered a coninuous ime seing bu, o beer replicae real life, allowed only for discree ime rading by inroducing a lag variable racking he ime inerval beween successive rades, under a GBM diffusion process. This paper was followed up by Guilbaud e al. 21 who discussed numerical finie difference mehods used o solve his ype of problem. Bouchard e al. 211 examined he use of differen algorihms depending on he preferences of a rader. For a full review of aggressive rading mehods he auhors recommend he review paper of Schied and Slynko 211. The dynamics of he LOB have also been examined and mehods have been suggesed o make i mahemaically racable, see for example Con e al. 21 and Con and De Larrard 213. Passive rading on he oher hand does no involve he rade-off beween marke impac and marke risk. Marke impac is replaced by a new ype of risk: he risk of non-execuion. In passive rading, orders are placed ino he LOB and are only filled when me by an aggressive rader s order. The furher ino he LOB he rades are placed, he higher he payoff for he rader bu wih a lower probabiliy of he order being filled. Given his rading scenario, he filling of an order is no guaraneed prior o he erminal ime, and hence here is a risk of non-execuion. As he rader can coninuously amend his asking price, he price a which he asse is sold is conrolled by he rader. However, he rader canno conrol he fair price of he asse, which is driven by a sochasic process, and hus marke risk is sill presen. This ype of model was firs suggesed by Ho and Soll 1981 bu lay dorman in he lieraure for some 2

3 ime unil Avellaneda and Soikov 28 revisied his problem wih some modificaions. The mehods used by hese laer auhors do no explicily consider he LOB bu insead consider he saisical properies of is liquidiy, reducing he dimension of he problem o a level ha is more amenable o work wih. Under he objecive of maximising he invesor s uiliy, wih a general Brownian Moion driving he asse price, hey derive a Hamilon-Jacobi-Bellman HJB parial differenial equaion PDE and examine is asympoics. Guéan e al. 212a reduce his HJB PDE o a sysem of firs-order ordinary differenial equaions ODEs by inroducing a consrain on he invenory ha he invesor can hold. In a similar framework, Guéan e al. 212b examined he problem for a single sided LOB, one in which asses could only be sold, while inroducing a erminal ime penaly for any remaining invenory, before examining a seady-sae soluion. Bayrakar and Ludkovski 212 who developed a framework for porfolio liquidaion for a risk-neural invesor in which he objecive is o maximise he expeced revenues of sales. These auhors considered a one-sided LOB, similar o Guéan e al. 212b, bu inroduced a power-law inensiy, as opposed o he negaive exponenial inensiy which has dominaed he lieraure. Guéan and Lehalle 213 developed he framework of he previous paper Guéan e al. 212b by considering general inensiy shapes for he Poisson process. Carea and Jaimungal 213b used a similar se-up o Avellaneda and Soikov 28 bu under a high frequency rading HFT framework. Their objecive was o maximise erminal wealh while penalising invenory deviaions from zero and erminal invenory holdings, properies which comply wih insiuional HFT as oulined by Brunei e al In his paper we expand on he above framework, in which he asse price was modelled as a sandard Brownian moion wih drif, by inroducing a novel and perhaps more realisic concep o he opimal rading lieraure, ha being he use of more general diffusion processes for he asse price. I was over a decade afer Almgren and Chriss 21 published heir seminal paper, which examines rading in he marke book raher han he limi order book wih a similar ime frame in mind, ha Forsyh e al. 212 examined he same problem under GBM. I has been argued hrough empirical evidence ha GBM is favourable in modelling asse prices over arihmeic Brownian moion see Osborne, GBM can be used for boh long and shor rading horizons, and avoids he fallacy of negaive price scenarios ha can appear in sandard Brownian moion. Alhough i is argued ha under small ime horizons sandard Brownian moion approximaes GBM, many of he problems in opimal execuion consider infinie horizon problems see for example Guéan e al. 212b and Schied and Schöneborn 29 o which his approximaion is no longer valid. Addiionally, he rescaling of variables we use in secion 2 makes our model applicable for various ime scales. As opposed o Guéan e al. 212b and Carea and Jaimungal 213b, in which hey reduced he HJB PDE o a sysem of ODEs, using GBM and mean-revering processes will disable us from using similar mehods. Alhough he equaions are more complex o solve, his could be a reason GBM and more general diffusion processes was avoided in he iniial framework. Having more complex equaions allows us o do wha we consider o be ineresing analyical asympoic analysis, finding analyical soluions in various limis, and discovering he developmen of a singulariy when a seady-sae framework is examined. We have hus aken a novel combined approach in solving he problem which involves boh heoreically analyically/asympoically examining he problem and using numerical mehods o obain soluions, wih each approach informing and confirming he oher. Using more general diffusion processes will resul in assedependen rading sraegies, as opposed o he asse-independen sraegies found by Guéan e al. 212b and Carea and Jaimungal 213b, among ohers. Asse-dependen sraegies appear more realisic from a financial perspecive, as he addiional amoun you ask for he asse should dependen on he price of he asse iself. This also resuls in no selling he asse for a negaive price effecively paying someone o ake he asse, which can occur in he work of Guéan e al. 212b and Carea and Jaimungal 213b, amongs ohers. Addiionally, we model he conrol parameer, he addiional amoun we ask for he asse also referred o 3

4 as he opimal rading sraegy, as a proporional quaniy of he asse price, raher han an absolue amoun as modeled by Guéan e al. 212b. This no only complicaes he model bu produces ineresing and arguably more realisic resuls while doing so. Having an assedependen conrol parameer will induce some ransparency in our resuls, given he opimal rading sraegy will be expressed as a percenage of he asse. The use of a proporional opimal conrol resuls in a proporional resilience parameer for he limi order book, a novel concep in he lieraure. As we will see shorly, he mehods we inroduce in his paper are no consrained o he use of GBM as he driving process for he asse price. Conrasing frameworks for obaining opimal rading sraegies have also been developed. To name a few, Carea and Jaimungal 213a developed a hidden Markov model o undersand he key behaviour of sock dynamics resuling in an opimal ick-by-ick rading sraegy ha an invesor who uses limi orders o profi from he bid-ask spread should follow. Hul and Kiessling 21 modeled he enire LOB as a high-dimensional Markov chain. Opimal rading sraegies were discussed in he heory of Markov decision processes, and a value ieraion procedure was presened which enables opimal sraegies o be found numerically. Finally, combinaions of limi order book and marke book rading have been invesigaed in he lieraure. Huiema 213 derive a model where a rader, whose asses follow sandard Brownian moion, can rade in boh he marke book wih he rae of rading impacing he price and he limi order book wih a Poisson process wih conrolled inensiy driving he execuion of orders, and solves he resuling PIDE numerically. Guilbaud and Pham 213b develop a model where passive and aggressive rading is possible, in which limi orders can be placed a bes quoe or bes quoe improved by one ick and occur according o a Cox process, while aggressive rading can only occur a discree imes. This is furher developed in Guilbaud and Pham 213a by examining a pro-raa micro-srucure, raher han he usual price/ime prioriy srucure. The layou of his paper is as follows: in secion 2 we inroduce he problem, suggesing an ansaz soluion which reduces he HJB PDE o a non-linear PDE before deriving boundary condiions and providing numerical soluions. In secion 3 we discuss a small-ime-o-erminaion soluion for he problem, examined using perurbaion heory. Secion 4 considers a perpeualype soluion for he problem. In secion 5 we examine he same problem bu wih he use of a mean-revering process, ha being he Cox e al CIR process, as he diffusion process for he asse price, replacing he GBM. This no only emphasises he adapabiliy of our mehods bu also produces ineresing resuls given ha he CIR process has previously been quie exensively suggesed for he diffusion process of an asse price, mos noably commodiies see Linesky, 24. We conclude in secion 6. 2 Problem Formulaion We consider an invesor who wishes o maximise his expeced uiliy, given a porfolio of asses o liquidae, before a specified erminal ime, T. We assume a ime = ha he invesor sars wih an iniial invenory of q asses, in which q akes posiive ineger values, and an iniial wealh X. Le Ω, F, P be a probabiliy space wih a filraion, F, [, T ]. We assume he fair asse price S follows a GBM, and so he diffusion process is defined as ds = µsd + σsdw. 1 wih µ as he relaive drif, σ as he relaive volailiy and W as a Wiener process which is F measurable. The invesor will coninuously pos orders ino he ask side of he LOB for price S a which is δ = δ, X, q, S percen greaer han he fair asse value S, i.e.: S a = S 1 + δ. 2 4

5 This is anoher disinc aspec o our approach as Guéan e al. 212b, and ohers, do no model he conrol as a percen of he asse price bu insead have asking price S a = S + δ. Asse sales follow a ime-dependen Poisson process, N, which is F measurable and independen of W : dq = dn. 3 for q >, hus assuming he rader becomes inacive afer liquidaing and does no shor sell. For each occurrence of a jump sale, he invesor s wealh increases by he amoun ha asse was sold for, i.e. S 1 + δ. The dynamics of he wealh is given by dx = S 1 + δ dn, 4 where N is he same Poisson process as before. Therefore, when a jump occurs, he values of q and X change simulaneously, according o 3 and 4 respecively. We are hus assuming sales are of uniary size, consisen wih ha of previous lieraure. N has inensiy Λδ which akes he form: Λδ = λe l S a S S = λe lδ 5 for some posiive consans λ and l. The liquidiy of he marke is described by he inensiy of he Poisson process. If no addiional amoun is added o he fair asse value hen he rae a which he asses are sold is Λ, which for he case of 5 is equal o λ. Given he negaive exponenial form of 5 here is less liquidiy for asses sold for prices higher han heir fair value and as such he probabiliy of execuion is lower. The parameer l can be inerpreed as he exponenial decay facor for he fill rae of orders placed away from he fair price see Carea and Jaimungal, 213b, i.e. how quickly or slowly he demand changes as we move furher ino he LOB and hus how quickly or slowly he probabiliy of execuion decreases, which is proporional o he price of he asse. This is differen han he sandard, absolue decay used in previous lieraure. For ease of reference we shall denoe l as he resilience of he LOB. Jusificaion for using he form of 5 for he inensiy of he Poisson processes is described horoughly in Avellaneda and Soikov 28 which is suppored by empirical evidence by Gopikrishnan e al. 2 and Maslov and Mills 21 for he disribuion of he size of he marke orders and by Gabaix e al. 26, Weber and Rosenow 25 and Poers and Bouchaud 23 for he change in price following a marke order. The objecive is o liquidae his porfolio before some final ime T. Asses ha are no liquidaed before his ime will be sold in he marke for heir fair value. The uiliy, Φ, we seek o maximise, akes he form of a negaive exponenial funcion and as such he invesor has consan absolue risk aversion CARA defined by AW = Φ W W W Φ W W, 6 which for he case of he exponenial uiliy family is consan and equal o he risk aversion parameer, γ, noing he subscrip in 6 represens he derivaive and W represens he invesor s wealh. This form of he uiliy funcion is consisen wih he previous lieraure of Avellaneda and Soikov 28 and Guéan e al. 212b. We define our value funcion, u, X, q, S, as he maximum expeced uiliy a ime [ u, X, q, S = sup E e γxt +qt ST ], 7 δ [,T ] A where γ > is he risk aversion characerising he invesor and A 1, is he se of admissible rading sraegies. Assuming he rader sars wih some non-negaive wealh, x, and wih some posiive quaniy of invenory, q, he erm γ X + qs is sricly posiive. Therefore he objecive funcion, and by definiion he value funcion, are bound, wih u 1,. We 5

6 should noe ha he lower bound of he admissable sraegy occurs naurally due o he invesor never waning o sell his asse for a negaive price, implying he would be paying someone o ake he asse from him, and hence i does no have o be explicily implemened. Using he form of he asking price S a = S + δ, under a sandard Brownian moion, rading sraegies are independen of he asse price and hence selling for a negaive price can occur, which is a fundamenal flaw of previous models. Given he opimisaion problem of 7, a HJB equaion can be derived by applying he Bellman 1957 principle of opimaliy and using Iō s lemma: u, X, q, S + µsu S, X, q, S σ2 S 2 u SS, X, q, S [ ] +sup δ λe lδ u, X + S 1 + δ, q 1, S u, X, q, S =, 8 wih condiions: 2 ut, X, q, S = ΦX, q, S = e γx+qs 9 and u, X,, S = ΦX,, S = e γx. 1 A derivaion of he HJB PDE 8 can be found in he appendix A.1 along wih a verificaion heorem in appendix A.2 verifying ha he soluion of he HJB PDE 8 is in fac he soluion of he original opimisaion problem Reducion of he problem The problem as saed in 8 is a four-dimensional HJB PDE. Guéan e al. 212b and Carea and Jaimungal 213b boh reduced heir similar problems o a sysem of ODEs by suggesing an ansaz form of he soluion which assumes he soluion can be separaed ino wo funcions, one involving variables X, S and q he oher involving variables q and, wih he PDE being reduced o a -dependen ODE, hus arriving wih rading sraegies independen of X and S. Due o he use of GBM, as opposed o he arihmeic Brownian moion used by Guéan e al. 212b and Carea and Jaimungal 213b, reducing he problem o a sysem of ODEs is no longer possible. However, we can sill make use of he form of he uiliy funcion o make a significan reducion o he complexiy of he problem. We begin by assuming an ansaz soluion u, X, q, S = e γx f τ, q, S, 11 where we also use a change in he ime variable τ = T so ha we are now solving forward in τ raher han backward in. Using his form of he soluion we obain: e γx f τ τ, q, S + µse γx f S τ, q, S σ2 S 2 e γx f SS τ, q, S [ ] +sup λe lδ e γx+s1+δ f τ, q 1, S e γx f τ, q, S =. 12 δ We can solve for he opimal conrol by differeniaing he supremum wih respec o he opimal conrol δ and seing he resul equal o zero which locaes he saionary poin. 2 In some of he lieraure, such as Guéan e al. 212b, a erminal penaly is included such he asses are sold a a discoun of here acual price a he erminal price, while ohers, such as Avellaneda and Soikov 28 and Guéan e al. 212a, neglec inclusion of his erminal penaly. Inclusion of he penaly is quie rivial bu in erms of boh solving his problem numerically and invesigaing i asympoically he erminal penaly would make lile difference o boh he difficuly of he mehods used and he resuls obained. We have herefore chosen o neglec i. 6

7 Solving his we obain: δ τ, q, S = 1 Sγ ln γs + l f τ, q 1, S lf τ, q, S 1, 13 which we noice is independen of X, hence confirming he use of our ansaz soluion. We also noice he opimal sraegy is a funcion of S which is a unique addiion o he lieraure for his general class of problem. Using he form of 13 we find he asking price of he asse o be S a τ = 1 γ ln γs + l f τ, q 1, S lf τ, q, S. 14 To confirm wih previous lieraure, noably Guéan e al. 212b, we will keep our focus on he opimal conrol δ when examining our resuls. Subsiuing 13 ino 12 and cancelling common facors we obain he following non-linear PDE wih and f τ τ, q, S + µsf S τ, q, S σ2 S 2 f SS τ, q, S λel γsf τ, q, S Sγ + l l lf τ, q, S Sγ =, 15 Sγ + l f τ, q 1, S f, q, S = e γqs, 16 f τ,, S = When discussing resuls we shall now refer o f τ, q, S as he value funcion Rescaling of he PDE We shall now perform a rescaling effecively a non-dimensionalisaion of his PDE o eliminae several of he parameers. We use he following change of variables: τ = λτ, = Sγ, µ = µ λ, σ = σ λ If we were o follow he framework of Guéan e al. 212b and use an absolue conrol parameer, so our asking price would be S a = S + δ, we would sill have an asse-dependen rading sraegy, given by δ τ, q, S = 1 «γs + l f τ, q 1, S γ ln S, lf τ, q, S and would sill resul arrive wih 12 and 14 if we used a proporional resilience, so 5 would ake he form If we weren o use a proporional resilence, i.e. he rading sraegies we obain would ake he form Λδ = λe l S a S S = λe lδ S. Λδ = λe lsa S = λe lδ δ τ, q, S = 1 γ ln γ + l f τ, q 1, S lf τ, q, S «S, which would leave o a non-linear PDE analogous o 15 wih rading sraegies ha are asse price-dependen. 7

8 All he new variables of 18 are hen dimensionless. This resuls in he non-linear PDE: f τ τ, q, + µ f τ, q, σ2 2 f τ, q, l e l f τ, q, + l lf + l f τ, q, τ, q 1, =, 19 wih f, q, q = e 2 f τ,, = is now dimensionless as are he condiions 2 and 21. The opimal rading sraegy in dimensionless form is hen δ τ, q, = 1 + l f τ, q 1, ln lf τ, q, To make his problem well-posed we mus consider boundary condiions for = and. For = we ake he limi of 19 which resuls in Given 23 and 2 he = boundary condiion we arrive a is f τ τ, q, =. 23 f τ, q, = 1, 24 and so δ τ, q, = f τ, q, f τ, q 1, l For, f τ, q, alhough for numerical purposes we assume a sofer Neumann boundary condiion f. 26 which, from invesigaion, we found o be saisfacory. From a financial perspecive we can inerpre his as he invesor no being able o become any more saisfied when he has an asse worh an infinie amoun of money. To solve 19 we use a finie difference scheme. We have esed boh implemening implici differences on he derivaive erms while aking he non-linear erm as an explici erm, hus negaing he need o use an ieraive scheme, and using an ieraive Crank-Nicolson scheme, wih each mehod confirming he oher. For he former we expeced our mehod o exhibi O τ, 2 convergence which we found was he case; for he laer we expeced O τ 2, 2 convergence which we also confirmed, where τ and are he grid sizes in τ and respecively. A number of calculaions were performed on a ransformed grid Y = ln, which did have some advanages for cerain calculaions, bu also exhibied some disadvanages including he need for wo, raher han one, domain runcaion parameers. However he resuls hus obained did provide a useful check of he resuls from he unransformed grid. 8

9 τ τ 1 τ 2 Opimal Sraegy: δ Opimal Sraegy: δ τ a Opimal sraegy for various values of b Opimal sraegy for various values of τ Figure 1: Opimal sraegy for invesor wih one asse remaining, wih µ =.4, σ =.4 and l = Numerical Examples We shall focus on he behaviour of he opimal rading sraegy, δ, given ha i is a ransformaion of he value funcion and, from a financial perspecive, is more ransparen han examining he value funcion per se. The parameer values used are: µ =.4, σ =.4, l = 25 and T = 2 which comply wih Avellaneda and Soikov 28, and he empirical discussions wihin. The resuls can be seen in figure 1 for q = 1. Looking firs a he opimal sraegy over ime, figure 1a, for various values of we noice ha he general behaviour is decreasing for increasing. This is due o he CARA characerisic of he uiliy funcion. Given he invesor exhibis consan absolue risk aversion, he invesor will dislike he higher absolue volailiy presen as increases. He will hus op o sell he asse quicker, so as o lock in he curren price and avoid risk. Wha is paricularly vivid for larger values of bu also rue for smaller values, is ha as τ increases he opimal rading sraegy ends o a perpeual ime-independen ype soluion. This can be seen in he do-dash line ending o a consan value around τ.3. I can also be seen in figure 1b ha he soluion is ending o a perpeual ype soluion as we can see he values for τ = 1 are close o hose a τ = 2, wih he wo lines are almos idenical for > 1. We noice in boh figure 1a and figure 1b ha as we perurb away from he erminal ime soluion here is ineresing behaviour and as such a small- τ soluion is also of some ineres. Given he opimal sraegy δ is a ransformaion of f τ, q, he funcion f exhibis similar behaviour as he opimal sraegy δ does in figure 1. Therefore in he following secions we shall invesigae boh he small τ soluion and perpeual soluion for f. The remainder of his secion consiss of examining rading sraegies for varying amouns of invenory, q, and varying parameer values. We examine hese a a ime close o he erminal ime, τ =.1, and a an earlier ime, τ = 2. The reason for doing so is ha he soluion varies rapidly as τ iniially increases before ending o a perpeual soluion. Therefore, showing he soluions a boh imes give a good insigh ino he overall behaviour of he soluion. The properies of he opimal rading sraegy for q > 1 are similar o hose of q = 1. Figure 2 gives an indicaion of how he opimal rading sraegy behaves for various values of q for a single ime sep around he erminal ime τ =.1, figure 2a and a an earlier ime τ = 2, figure 2b. The opimal rading sraegy is q independen a τ =, which can be confirmed 9

10 Opimal Sraegy: δ q=1 q=2 q=3 τ = Opimal Sraegy: δ q=1 q=2 q= a Opimal sraegy a τ = b Opimal sraegy a τ = 2 Figure 2: Opimal sraegy a various imes for numerous asses remaining. In figure 2a in which τ is close o he erminal ime we have included he soluion of he opimal sraegy a τ = o signify he impac of a sligh deviaion from zero for various q. The parameer values used are µ =.4, σ =.4 and l = 25. by subsiuing 2 ino 22 which resuls in: δ τ =, q, = 1 ln + l l. 27 We have included a plo of 27 in figure 2a o highligh he significan difference in soluions from a small deviaion in ime, and as q is increased. To conclude he discussion on he numerical examples we shall briefly describe how he opimal rading sraegy changes in relaion o changes o he various parameers in he model. In figure 3 he opimal sraegy for various imes has been ploed wih each parameer, µ, σ and l being varied, as well as a base case which is calculaed using he same parameers as saed above. As can be seen he same properies hold a boh ends of he ime specrum for a given parameer variaion. As he drif, µ, increases he invesor will be more happy o hold on o he asse as he expecs he price o rise and hus his wealh o rise. He herefore asks for a higher price for he asse over asses wih lower drif. As he volailiy, σ, increases he invesor s asking price decreases. Undersandably, an invesor who is risk-averse dislikes higher volailiy as i brings a level of risk in sock price movemen and hus will wish o sell quicker han if volailiy was lower. As we decrease l, he resilience parameer for he LOB, we see he invesor asks for a higher price when his asse price is low and a lower price when his asse price is high. This swich is cenered around wheher he invesor is selling he asse above par or a a discoun, i.e. δ greaer han zero or less han zero, which occurs as is increased. When selling above par a lower l signifies ha he invesor can place his asse furher ino he LOB wihou significanly reducing his probabiliy of sale. When he invesor is eager o sell he will sell he asse for a discoun. In order o do so under lower l he invesor mus sell a a high discoun in order o increase his probabiliy of sale by a significan amoun. The opposie is rue for larger l; when selling above par he probabiliy of selling significanly decreases while he probabiliy of selling significanly increases when selling a a discoun; his can be seen in figure 3b. However if we were o consider τ =.1 for higher values of we would see he curve represening lower l cross he curve represening he base case. 1

11 .15.1 Base Case µ doubled σ doubled l halved.2 Base Case µ doubled σ doubled l halved Opimal Sraegy: δ Opimal Sraegy: δ a Opimal sraegies a τ = b Opimal sraegies a τ = 2 Figure 3: Opimal sraegy under varying parameers. The base case has parameer values: µ =.4, σ =.4 and l = Small-ime-o-erminaion soluion In he ligh of he resuls from secion 2.3 we now visi he concep of a small- τ soluion. From he numerical resuls we see ha here is a lo of aciviy which mirrors financial ineres aking place around τ =. This is ofen he case in applied mahemaics wih financial applicaions, as, i is well known ha opions priced using he Black-Scholes-Meron framework close o expiry see Evans e al., 22 can also exhibi rapid changes in value. In his regime he invesor is more worried abou no being able o sell he asse for any price higher han is curren price or, when he asse is already a a high price, he invesor is worried ha volailiy could cause a decrease in his asse price before he erminal ime. This is a propery of he invesor s risk-aversion. This behaviour around his regime presens an ineresing asympoic problem. The behaviour shape of he opimal rading sraegy is similar o ha of he value funcion if examined much more closely. From 13 we see ha he opimal sraegy is derived by a log ransformaion of he value funcion, hence he similariy in shape. Given his informaion, he same ineresing small-ime-o-erminaion soluion is presen in he value funcion as is in he opimal rading sraegy. To examine he small-ime-o-erminaion soluion we shall expand f τ, q, as follows f τ, q, = f q, + τf 1 q, + τ 2 f 2 q, + O τ The O τ erm, f q,, is merely equal o he erminal condiion given by 2. By subsiuing 28 ino 19 we can find an analyical soluion for f 1 q, by collecing he O τ erms f 1 q, = µ q σ2 2 q 2 l l f q, + l, 29 + l 11

12 .5.45 Full Numerical 2 erm Asympoic 3 erm Asympoic.3.2 Full Numerical 2 erm Asympoic 3 erm Asympoic Opimal Sraegy: δ Opimal Sraegy: δ τ a = τ b = 1 Figure 4: Comparison of exac opimal rading sraegy and asympoic expansion for near zero and near 1 using a wo-erm and hree-erm model, wih parameers µ =.4, σ =.4 and l = 5. and from he O τ 1 erms f 2 q, = 1 µ 2 f 1 q, σ2 2 f 1 q, 1 l l + l + l l + f 1 q, lf 1 q 1, e. 3 f 1 and f 1 are he firs and second derivaive of f 1 respecively which can boh be easily calculaed analyically; herefore 29 and 3 have fully analyical soluions. We shall now discuss he accuracy of his asympoic expansion, along wih is limiaions. The values we use for our parameers are he same as ha of secion 2.3 bu wih l = 5, for reasons ha will be explained laer. In figure 4 we have ploed he soluion of he opimal rading sraegy along wih is asympoic approximaions for wo values of which was calculaed by subsiuing 28 ino 22. We can see ha in boh figures he hree-erm approximaion is a good approximaion for he soluion for τ of O 1 while he wo-erm approximaion is valid for smaller τ. We have esed his for various q values and hese observaions sill hold rue. This approximaion is surprisingly accurae for a range of parameer values. However, here are resricions o how well he model holds depending on he parameer values used. For higher values of and l he soluion of f τ, q, diverges faser from he small-ime-o-erminaion soluion due o he sronger presence of he non-linear erm, hence he use of he smaller l in our numerical example. In figure 5 we have ploed he opimal sraegy agains for increasing τ. We can see ha as τ increases he range of for which he approximaion is accurae decreases. I can also be seen ha for τ =.1 he difference beween he full numerical and hree-erm asympoic approximaion is negligible. If l were o be reduced even furher he asympoic approximaion remains valid for a larger range of τ values. We shall now move on o examine wha happens when τ, i.e. he perpeual sae. 12

13 Opimal Sraegy: δ Full Numeircal 3 erm Asympoic τ increasing Figure 5: Comparison of exac opimal rading sraegy and hree-erm asympoic expansion for increasing τ, wih τ = {.1,.5, 1} and parameer values µ =.4, σ =.4 and l = 5. 4 Perpeual Soluion We saw from figure 1 ha i is possible ha perpeual ime-independen soluions exis. Indeed, Guéan e al. 212b also found such soluions in cerain parameer regimes. In his secion we shall sudy he perpeual soluion in some deail. For he perpeual soluion we require f τ τ, q, as τ. In making his assumpion, we are implicily assuming ha he erminal ime in which he asses mus be sold is sufficienly disan for us o neglec he ime variaion. Then 19 becomes µ f q, σ2 2 f q, e l f q, + l + l f lf q, q 1, l =, 31 wih f, = To solve his non-linear ODE 31 we mus consider wo boundary condiions, one for = and one for. We know from he non-seady-sae problem ha f τ, q, = = 1 and hus f q, = = 1 for he perpeual case. For we use he same boundary condiion as used in secion 2.2 for he full problem, ha being a Neumann boundary condiion as given by 26. We solve 31 using a Newon ieraive mehod, see Press e al. 29. The opimal rading sraegy akes he form: δ q, = 1 + l f q 1, ln lf q, We plo he value funcion figure 6a and opimal rading sraegy figure 6b for q {1, 2, 3, 4} wih he same parameer values as used in secion 2.3. This soluion was esed agains he full numerical soluion for he non-perpeual ime-varying case. We noice ha he opimal sraegy for he perpeual case is larger for small han for he full problem. This is for wo reasons. The firs being ha δ τ > for small as can be seen in figure 4a which is due o he prospec of he drif increasing he value of he asse dominaing over he prospec 13

14 fq, q=1.8 q=2 q=3 1 q= a Value funcion Opimal Sraegy: δ q=1 q=2 q=3 q= b Opimal sraegy Figure 6: Value funcion and opimal sraegy for perpeual soluion under GBM for various q, wih parameer values: µ =.4, σ =.4 and l = 25. of he volailiy decreasing he value. The second is due o he presence of a singulariy abou =, which we will now discuss. We found ha here are convergence issues for cerain parameer values. Examining f for small i appeared ha a singulariy was presen for 1. On closer inspecion using asympoic balancing suggess ha in his limi, he soluion behaves as f q, 1 + cq β 34 for some posiive consan β and cq, and so he soluion is bounded, provided which gives us he requiremen β = 1 2 µ >, 35 σ 2 µ < σ is necessary for a soluion o exis, given β >. We have verified his consrain numerically laer in his secion we address he issue of parameers ouside his region. This consrain is analogous o ha found by Guéan e al. 212b in which hey derived in heir noaion µ < γσ2 2 o be a necessary condiion for a soluion o exis for he perpeual case under sandard Brownian moion. We noice here is a lack of he γ parameer in our consrain in comparison o Guéan e al. 212b. Alhough we have scaled ou γ in our model, even wihou his rescaling we sill arrive a a consrain ha is independen of γ. As can be seen in figure 6b we noice a singulariy is apparenly presen in he opimal rading sraegy. We can make an asympoic approximaion for he opimal rading sraegy for 1 given as: δ q, Kq β 1 38 which is found by aking he limi of 33 as +, using he form 34 for he value funcion, and noing ha cq > cq 1. Given he laer poin we expec Kq o be a decreasing funcion of q, ending o zero as q, which can be seen in figure

15 Kq q Figure 7: Kq variaion wih q for S wih parameer values: µ =.4, σ =.4 and l = 1. We se Y min = 2. In he following we have se l = 1; larger values of l lead o a general dominance of he non-linear erm, and rapid valuaion variaions close o =, boh of which conceal valuaion behaviours, and hereby make insighs hereof more difficul o discern. I is of some ineres o inspec how quickly he ime-varying soluion approaches he perpeual sae assuming 35 is saisfied. Figure 8 shows sample resuls for he difference beween he former and laer saes, indicaing a quie rapid asympoic approach in his case which, indeed, generally was a canonical observaion found in mos calculaions. However, alhough his difference is diminishing, quie rapidly, i is clear ha here is a maximum deviaion a decreasing values of as τ increases. I is clear ha he O β in he perpeual sae as leads o singular behaviour, whils he numerical resuls of he ime-varying soluion very srongly poin o he non-linear erm in 19 being negligible in his regime, and so a series soluion of he linearised form of his PDE 4 akes he form f τ, q, = 1 n+1 n= n! q n e n µ+ 1 2 nn 1 σ2 τ. 39 Alhough his is clearly a divergen series for sufficienly large n, noneheless i does highligh he likely occurrence of very shor scales as τ increases and suiable runcaions of his series does lead o resuls consisen wih he full numerical sysem. The n = 1 erm leads o he resul ha f τ, q, = qe µ τ, 4 which clearly suggess he developmen of an exponenially hin region close o =, ogeher wih an increasing magniude for his derivaive as opposed o a diminishing value for f, which we surmise leads ulimaely wih a connecion o 34. Ouside of his region, he valuaion is effecively ha prediced by he perpeual soluion. A naural quesion ha arises is wha is he siuaion if he values of µ and σ do no comply wih 36? Figure 9 shows such a case corresponding o β =.92, and i is very clear paricularly when compared wih previous resuls, ha a ime-independen sae is no 4 I is possible o find an exac soluion o his linear sysem, saisfying he appropriae condiions a τ =, namely wih Y = ln. f τ, q, Y = 1 σ 2π τ Z exp qe Y p ` µ σ2! 1 τ 2 + p 1 p2 2 σ 2 2 µ σ 2 2 σ 2 τ dp, 15

16 .2 f τ,q, fq, Increasing τ Figure 8: Soluion of 19, f.14 {5, 1, 15, 2, 25} wih µ =.4, σ =.4 and l = τ, q,, minus he soluion of he 31, f q,, for τ = being approached, raher ha he non-negligible valuaion is confined o a diminishingly small regime, close o =. 4.1 Perpeual soluion for large q To conclude our examinaion of he perpeual problem we shall look a he case of large q. Examinaion of figure 6 suggess ha as q is increased he soluions end o collapse on o a single disribuion a conclusion suppored by a number of numerical experimens conduced by he auhors. Furher o ha we have seen in figure 7 ha K q as q. Examining he non-linear erm we find lim f q, q f q 1, Using 41 we can reduce 31 o a linear ODE which is now independen of q, µ f + 1 e l f l l 2 σ2 2 f =, 42 + l + l in which we shall refer f as he q soluion, wih boundary condiions f = = 1 and f, as used on he non-linear ODE previously. We have been unable o find an analyical soluion for 42 bu i is sraighforward o solve numerically using finie difference mehods for example. Remembering he explici form of he opimal conrol given by 33, we can see ha under he assumpion of 41 he opimal rading sraegy is given by δ = 1 ln + l l The soluion is shown in figure 1 for he same parameer values as used for figure 6. In figure 1a we can see ha as q is increased he soluions end o he q soluion. The opimal rading sraegy is given by 43 and can be seen in figure 1b. We noice how, as wih he 16

17 f τ,q, Increasing τ.8 Figure 9: Soluion of 19, f β =.92 and l = τ, q,, for τ = {2, 4,..., 2} wih q = 1, µ =.6, σ =.25 value funcion, he soluions of he opimal rading sraegies converge o he q opimal rading sraegy as q is increased. I can be seen ha he soluion for δ a S + is near zero for he q soluion while singular for he soluions for finie q. If we were o increase q significanly, he coefficien of he singulariy in his limi decreases owards zero, in line wih our observaions above regarding Kq. 5 Opimal liquidaion of mean-revering asses In his secion we consider using a mean-revering diffusion process for he asse price, so as o emphasise he adapabiliy and advanages of using numerical mehods o solve his class of problem. The process we shall use is he CIR process, which was firs suggesed in 1985 by Cox e al as an exension o he Vasicek 1977 model. The araciveness of he CIR model is is avoidance of he possibiliy of negaive values. I was originally suggesed for he modelling of ineres raes bu has since been suggesed for oher models such as sochasic volailiy models and, of paricular ineres o us for his framework, commodiies see Linesky, 24. An asse S following he CIR process solves he follow sochasic differenial equaion ds = κ ζ S d + σ SdW 44 wih κ as he mean-reversion speed, ζ is he long-erm mean asse price and σ S is he absolue volailiy. Following he same problem formulaion as we have in secion 2 bu wih he CIR process for he asse price, we can derive a similar HJB PDE. Furhermore we use he change of variables τ = λ T, = Sγ, ζ = ζγ, κ = κ λ, o derive he dimensionless non-linear PDE: f τ τ, q, + κ ζ f e l f τ, q, + l γ σ = σ λ τ, q, σ2 f τ, q, l lf + l f τ, q, τ, q 1, 45 =, 46 17

18 2 fq, Opimal Sraegy: δ a Value funcion b Opimal rading sraegy Figure 1: Perpeual soluion for large q. In figure 1a he solid line represens he soluion of 42 while he broken lines represen he soluions of 31 for q = {1, 2, 3, 4, 5}, beginning wih q = 1 which is furhes from he q soluion. We can see ha as q is increased he soluions converge o he q soluion. The same can be seen for he opimal sraegy in figure 1b. The parameer values used: µ =.4, σ =.4 and l = 25. wih f, q, = e q, 47 f τ,, = Noe ha he opimal rading sraegy akes he same form as 22. For he CIR model i is necessary o be careful when esablishing boundary condiions as he concep of posiiviy and non-negaiviy come ino play, he difference being he inclusion of zero o he asse-price domain. Under cerain condiions he CIR model is defined on a sricly posiive domain, while when hese condiion are no saisfied he CIR model has a posiive probabiliy of reaching zero. The boundary behaviour of 44 has been sudied in grea deail by Feller 1951 in which he found ha he CIR model is defined on a posiive domain if in our noaion 2 κ ζ σ is saisfied for mos realisic parameer values applicable o finance, and as such we shall focus only on his regime in his paper. For he boundary condiion a = we implemen he degenerae form of he PDE 46 resuling in f τ τ, q, + κ ζf τ, q, =, 5 which could be seen as he boundary behaviour, as never reaches zero. Indeed, 5 implies ha fτ, q >, = approaches zero as τ increases since f τ >, q, >, hen as, δ = O 1. For he case of we found from numerical invesigaion ha he same boundary condiion as used for he GBM case, ha being a Neumann condiion given by 26, is consisen and saisfacory. 5.1 Numerical resuls Several of he properies found in he resuls for he GBM model are mirrored in he CIR model and hus we shall no repea hem. However a unique propery for he mean-revering process 18

19 x f τ,q, a Value funcion Opimal Sraegy: δ b Opimal rading sraegy Figure 11: Value funcion and opimal rading sraegy under CIR process a ime τ = 2. The parameers ake he values: q = 1, ζ = 1, κ = 2.5, σ =.8 and l = 25. Noe he value funcion is no concave over he whole domain and as such he invesor swiches from being risk-averse o risk-seeking. no found in he GBM case was ha of an invesor becoming risk-seeking. If we examine he graph of he value funcion given in figure 11a we see ha he funcion is no sricly concave. Ineresingly his occurs when he asse price increases pas he long-erm mean, ζ. In his case he asse price is expeced o reurn back o ζ, wih some volailiy of course. As he asse price is due o decrease he invesor will ry o sell he asse before i does so. His rading sraegy is hus o change he asking price drasically in comparison o he GBM model where he asse value was expeced o increase. This resuls in a negaive δ which implies selling he asse a a discoun, bu a a price higher han ζ. This change in asking price resuls in he risk-seeking behaviour we observe around he long-erm mean; his rading sraegy can be seen in figure 11b. We can observe ha he opimal rading sraegy grows significanly as approaches zero in line wih our commens above. This is due o he mean-revering characerisic of he asse price. When is near zero he invesor expecs he fuure value of he asse o rever o he long-erm mean, ζ. He will hus ask for a large muliple of he curren asse price. Figure 12, which is he soluion of 46 for =, reinforces he explanaion of why he opimal rading sraegy increases in his region and also as τ increases. The risk-seeking behaviour observed in figure 11a is more pronounced for higher values of he resilience, l, and he speed of reversion, κ. This is due o he invesor decreasing his asking price more predominanly as he probabiliy of execuion decreases more rapidly for larger l and he asse price decreases faser o he long-erm mean ζ for large values of κ. Similar o he case under GBM, he CIR model has ineresing characerisics close o he erminal ime which we shall invesigae in he nex secion. However, unlike he GBM model, he CIR model does no end o a perpeual ype soluion as τ, bu raher he value diminishes owards zero, as evidenced by figure 12. Therefore we canno consruc a non-rivial perpeual soluion in he same sense as we could for he GBM case. From a financial perspecive, when is small and less han ζ he invesor asks for a relaively high price for he asse given i is expeced o increase and rever back o he is long-erm mean, ζ. The invesor s value funcion hus approaches zero as τ increases. 19

20 .2 f τ,q, τ Figure 12: Value funcion a =. The parameers ake he values: q = 1, ζ = 1, κ = 2.5, σ =.8 and l = 25. Noice he approach of f τ, q, = o zero as τ increases, which reinforces he growh in δ as as seen in figure 11b. 5.2 Small-ime-o-erminaion soluion To examine he small- τ soluion we use he same perurbaion mehod as used in secion 3. Subsiuing 28 ino 46 we collec he O τ and O τ 1 erms in which we find he values for f 1 q, and f 2 q, o be f 1 q, = κ ζ q σ2 q 2 l l f q, + l + l 51 and f 2 q, = 1 κ ζ f 2 1 q, σ2 f1 q, 1 + l l + f 1 q, lf 1 q 1, e l l + l 52 A comparison of he full numerical soluion agains a wo-erm and hree-erm asympoic can be found in figure 13. The same parameer values as in secion 5.1 are used. We see he asympoic approximaion is of O 1. Noe ha in his case, boh f 1 and f 2 are boh numerically large, which explains he iniial rapid variaion in values close o he erminal ime, especially as. 6 Conclusion In his paper we have developed a novel approach o he opimal rading in he LOB problem. We sugges using GBM as he driving process for he asse along wih an asse proporional conrol parameer rading sraegy and asse proporional resilience. This is disinc from he sandard Brownian moion and non-proporional conrol and resilience used by Guéan e al. 212b. The rading sraegies we found were variable wih respec o he asse price, as opposed o he rading sraegies found by Guéan e al. 212b which were found o be consan for all asse values. The former is a characerisic we feel would be more realisic in he finance indusry. 2

21 f τ, q, x Full Numerical 7 2 erm Asympoic 3 erm Asympoic τ a Value funcion Opimal Sraegy: δ Full Numerical 2 erm Asympoic 3 erm Asympoic τ b Opimal rading sraegy Figure 13: Comparison of exac value funcion and opimal rading sraegy agains a woerm and hree-erm asympoic expansion for = ζ for he CIR model. The parameers ake he values: ζ = 1, κ = 2.5, σ =.8 and l = 5. Focusing on he problem, we reduced he four-dimensional HJB PDE o a hree-dimensional non-linear PDE by finding an explici form of he opimal conrol in erms of he reduced variable funcion. We also used a change of variables non-dimensionalisaion which eliminaed wo parameers from he model. We used numerical mehods o solve his problem, noing ha he hree-dimensional PDE approach is much less compuaional expensive han solving he full HJB PDE and is hus much more aracive from an algorihmic rading perspecive. We invesigaed boh he small-ime-o-erminaion soluion and he perpeual soluion, which provided us wih a deeper undersanding of he soluion opology. The small- τ soluion provided rading sraegies which could be calculaed exremely quickly and hus would be especially useful for a high-frequency rading framework in which lile ime is lef o liquidae he porfolio and a quick soluion is needed. The perpeual soluion on he oher hand provided rading sraegies which could be implemened for an invesor wih a long ime remaining before expiry, as well as an ineresing asympoical problem. We found a consrain for he perpeual case similar bu differen from ha found by Guéan e al. 212b under sandard Brownian moion. Solving his problem numerically has led o he developmen of a mehodology which can easily be generalised. This was highlighed in secion 5 by changing he diffusion process for he asse price from a GBM o a mean-revering process, ha being he CIR process. Implemening oher such mean-revering processes such as he Uhlenbeck and Ornsein 193 process, he Dixi and Pindyck 1994 process or he Schwarz 1997 process would also be sraighforward using our numerical mehods. As well as changing he sochasic process driving he asse price, numerical mehods allow us o expand on his problem in many forms o make i more mahemaically ineresing and financially realisic. Examples include inroducing anoher form of sochasiciy such as including sochasic resilience, as suggesed bu no implemened by Carea e al. 211, or allow rading of muliple correlaed asses, he laer which will be he opic of a fuure paper. 21