Optimal execution with regime-switchingmarket resilience

Monash CQFIS working paper

2018 – 5

Chi Chung SiuSchool of Mathematics and AppliedStatistics, University of Wollongong

Ivan GuoSchool of Mathematical Sciences,Monash University and Centre for

Quantitative Finance and InvestmentStrategies

Song-Ping ZhuSchool of Mathematics and AppliedStatistics, University of Wollongong,

Robert J. ElliottSchool of Commerce, University of

South Australia,Haskayne School of Business,

University of Calgary

Abstract

In this paper, we study the optimal placement of market ordersin a limit order book (LOB) market when the market resilience rate,which is the rate at which market replenishes itself after each trade,is stochastic. More specifically, we extend the optimal executionmodel under the discrete-time framework of Obizhaeva and Wang(2013) by modelling the dynamics of the resilience rate to be drivenby a Markov chain. When the LOB replenishes itself stochasticallythrough time, we show that the optimal execution strategy becomesstate-dependent, and is driven linearly by the current remainingposition and the current temporary price impact, with their lin-ear dependence based on the expectation of the dynamics of futureresilience rate. A trader would optimally place more aggressive (re-spectively, conservative) market orders when the limit order bookswitches from a low to a high resilience state, (respectively, froma high to a low resilience state). Our cost saving analysis showsthat state-independent strategies, such as a deterministic execu-tion strategy associated with a constant resilience rate and the nave strategy are sub-optimal, and a substantial cost reduction canbe achieved by choosing the optimal execution strategy over theaforementioned state-independent strategies.

Centre for Quantitative Finance and Investment Strategies

Optimal Execution with Regime-switching MarketResilience

Chi Chung Siu∗,a, Ivan Guob, Song-Ping Zhua, and Robert J. Elliott c,d

aSchool of Mathematics and Applied Statistics, University of Wollongong, AustraliabSchool of Mathematical Sciences, Monash University, AustraliacSchool of Commerce, University of South Australia, AustraliadHaskayne School of Business, University of Calgary, Canada

July 16, 2018

Abstract

In this paper, we study the optimal placement of market orders in a limit orderbook (LOB) market when the market resilience rate, which is the rate at which marketreplenishes itself after each trade, is stochastic. More specifically, we extend the optimalexecution model under the discrete-time framework of Obizhaeva and Wang (2013) bymodelling the dynamics of the resilience rate to be driven by a Markov chain. Whenthe LOB replenishes itself stochastically through time, we show that the optimal execu-tion strategy becomes state-dependent, and is driven linearly by the current remainingposition and the current temporary price impact, with their linear dependence basedon the expectation of the dynamics of future resilience rate. A trader would optimallyplace more aggressive (respectively, conservative) market orders when the limit orderbook switches from a low to a high resilience state, (respectively, from a high to alow resilience state). Our cost saving analysis shows that state-independent strategies,such as a deterministic execution strategy associated with a constant resilience rate andthe naıve strategy are sub-optimal, and a substantial cost reduction can be achievedby choosing the optimal execution strategy over the aforementioned state-independentstrategies.

Key words: Optimal Execution Problem, Limit Order Book, Stochastic Market Resilience,Permanent Price Impact, Temporary Price Impact; Markov chains

1 Introduction

An institutional trader often buys or sells a large block of shares in the market. The largeplacement of a buy/sell order induces an adverse price impact for the institutional trader: a

∗Corresponding Author. Email: ccsiu@uow.edu.au

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large buy (respectively, sell) order would have an immediate upward (respectively, downward)pressure on the price, thereby increasing (respectively, decreasing) the cost (respectively,revenue) of the trader. To reduce the overall price impact, an institutional trader shouldsplit a larger block of shares into smaller ones and execute them over time. An optimalexecution problem faced by an institutional trader is then to find an optimal way to split alarge order into smaller ones in order to minimize the overall price impact incurred over atrading horizon. Earlier works focused on characterizing the optimal execution problem asan optimal dynamic control problems for a single asset, in which various price impacts areparsimoniously incorporated into the asset price dynamics.

Bertsimas and Lo (1998) initiated the study of the optimal execution problem of a risk-neutraltrader which minimizes the expected cost of buying a large block of a stock over a fixed timehorizon. Motivated by empirical studies, they separated the price impact into two compo-nents: the permanent and temporary price impacts. The permanent price impact accountsfor an indefinite shift of the asset price, whereas the temporary price impact only influencesthe asset price locally as it dissipates instantaneously. When trading occurs discretely, theoptimal trading strategy of the trader is to split the order into small trades of equal sizesand execute them at each period over the trading horizon . Almgren and Chriss (2000) en-riched the model of Bertsimas and Lo (1998) to the case of a risk-averse trader by minimizingvolatility risks, in addition to minimizing the expected execution cost in a continuous-timesetting. Under the continuous-time setting, only the permanent price impact would affectthe return of the asset; the temporary price impact incurs an execution cost for a trader buthas no influence on the asset price dynamics. To maintain tractability, Almgren and Chriss(2000) considered absolutely continuous strategies to be the admissible strategies and, hence,the optimal trading strategy has a natural interpretation as optimal trading speed. For otherimportant works on price impact since Bertsimas and Lo (1998) and Almgren and Chriss(2000), see, for example, He and Mamaysky (2005), Schied and Schoeneborn (2008), Forsyth(2011), Forsyth et al. (2012), and Almgren (2012).

According to Parlour and Seppi (2008), most equity and derivatives exchanges across theglobe are either pure electronic limit order book markets, or allow electronic limit ordersin addition to on-exchange market making. It is then essential to study the price impactof trading under a limit order book market. Although seminal works of Bertsimas and Lo(1998) and Almgren and Chriss (2000) provided valuable insights into the price impact oftrading, their works did not explicitly model the dynamics of the a limit order book ofsingle asset. Moreover, extensive empirical literature (e.g. Biais et al. (1995), Hamao andHasbrouck (1995), Ranaldo (2004), Degryse et al. (2005), Kempf et al. (2009)) has shownthat the market resilience rate, which is the rate at which market replenishes after eachtrade, is finite. Yet, both Bertsimas and Lo (1998) and Almgren and Chriss (2000) made afundamental assumption of an infinite market resilience rate.

Obizhaeva and Wang (2013) incorporated the concept of a finite market resilience rate intoan optimal execution problem under the limit order book framework. More specifically,Obizhaeva and Wang (2013) distinguished the price impact of each trade into permanent andtemporary price impacts in which permanent price impact affects the stationary levels of thebest-ask and best-bid prices, whereas the temporary price impact dissipates at the constantmarket resilience rate of the limit order book. In the discrete-time framework, the optimaltrading strategy under their limit order book framework admits a closed-form expression andis shown to be a function of the remaining position and the cumulative temporary price impact

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at each time period. Their numerical examples indicated that an optimal trading strategy intheir framework has an interesting feature concerning large orders placed in both the initialand final trading periods, with smaller orders of constant sizes placed in the intermediateperiods. Moreover, the size of each market order is sensitive to the market resilience rate:As the market resilience rate decreases, the initial and final market orders increase, whileintermediate market orders decrease. In addition, Obizhaeva and Wang (2013) showed thatthe absolutely continuous strategy can become sub-optimal in continuous time and thatoptimal trading strategy can consist of both lump-sum and absolutely continuous strategies.Finally, they showed that the optimal execution strategy in continuous time does not dependon the bid-ask spread and market depth, but depends on the resilience rate.

The work of Obizhaeva and Wang (2013) has attracted some important extensions. Alfonsiet al. (2010) relaxed the block-shaped market depth assumption in Obizhaeva and Wang(2013) and showed that the main qualitative characteristics of the optimal strategies inthe OW model still hold in the discrete-time setting. Predoiu et al. (2011) generalizedthe continuous-time optimal execution problem in Obizhaeva and Wang (2013) under moregeneral price dynamics. Fruth et al. (2014) introduced the time-dependent market depth andmarket resilience functions in the continuous-time framework of Obizhaeva and Wang (2013)and demonstrated that the optimal strategy employs a wait region/buy region. Tsoukalaset al. (2017) generalized Obizhaeva and Wang (2013) with multiple assets and a risk-aversetrader and showed that cross price impact plays a critical role in the optimal trading strategy.

In addition to the finite market resilience rate, it is well documented that trading volume,order flows, and transaction costs all exhibit typical U-shaped intra-day patterns, (see, e.g.Biais et al. (1995), Handa and Schwartz (1996), and Ahn et al. (2001)). High volumes oflarge orders are placed at the start of the trading day, but the speed and size of the ordersdecline gradually toward the mid-day, before regaining momentum as the market closes.In this respect, limit order book models with deterministic liquidity characteristics cannotcapture such complex intraday patterns.

The development of optimal execution with stochastic liquidity in the limit order book frame-work is relatively recent. Chen, et al. (2017) modelled the market depth of the limit orderbook model in Obizhaeva and Wang (2013) as Markov chains and developed a partitioningalgorithm to solve the corresponding optimal execution problem. Their numerical resultsindicated that the optimal execution strategies under the stochastic market depth outper-formed those under the deterministic market depth. Fruth et al. (2017) extended Obizhaevaand Wang (2013) by assuming that the market depth follows a general stochastic process.They showed that, when the market depth follows a diffusion process under some appropriateconditions, the optimal execution strategy still follows a single wait region/buy region struc-ture, as in the case of the time-varying market depth (Fruth et al. (2014)) . For the generalmarket depth processes, the optimal execution strategy is proven to take a more general formin which multiple waiting regions can occur and it does not vary monotonically with the sizeof the remaining position. Becherer et al. (2018) studied the optimal liquidation problem foran infinite time horizon with multiplicative stochastic price impact. The main objective oftheir paper is to find an optimal strategy to maximize the discounted liquidation proceedsover an infinite time horizon in which the dynamics of the stock price are modelled as theproduct of the fundamental value process, (i.e. a value unaffected by trades), and a functionof a stochastic volume-effect process. Graewe and Horst (2017) generalized the optimal ex-ecution problem of Obizhaeva and Wang (2013) with stochastic resilience and a stochastic

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risk aversion parameter. By allowing only an absolutely-continuous strategy, they reformu-lated the optimal execution problem into a three dimensional system of backward stochasticdifferential equations (BSDEs) with a singular terminal condition in one component.

Motivated by empirical studies (e.g. Large (2007), Lo and Hall (2015)) which show thatresilience rate of the limit order is stochastic, we proposed a tractable generalization ofthe discrete-time optimal execution problem of Obizhaeva and Wang (2013) and includea stochastic market resilience rate. Indeed, Obizhaeva and Wang (2013) showed that thevalue of the resilience rate has a profound impact on the optimal execution cost, for whichthe cost difference between the resilience rates of 0.001 and 1000 can reach nearly 50% intheir numerical examples. In other words, modelling market resilience based on a single cali-brated resilience parameter becomes too restrictive, implying that richer resilience modellingbecomes necessary.

Unlike Graewe and Horst (2017), who modelled stochastic resilience rate as a diffusion pro-cess, we model the stochastic market resilience as a continuous-time Markov chain. Modellingthe stochastic market resilience rate as function of a Markov chain maintains high tractabil-ity. Indeed, we show that the optimal execution strategy is structurally similar to that inObizhaeva and Wang (2013), with the key exception that the optimal execution strategyis state-dependent when resilience rate is stochastic. In particular, for the case of low andhigh resilience states, we show that optimal execution strategy in our framework alleviatesthe problem of placing a large initial market order at the beginning of the trading horizon,even when the initial resilience rate is low, as long as there is a non-zero probability thatthe market will move to a high resilience rate in later trading periods. For each of the inter-mediate periods, the optimal execution strategy updates the expected resilience rate in thefuture periods, and will place larger market orders when the resilience state switches fromlow to high resilience states, and vice versa. Our numerical studies also indicate that the costsaving achieved by choosing an optimal execution strategy over other state-independent exe-cution strategies can be substantial, showing that execution strategies under the assumptionof constant resilience rates are sub-optimal.

We conclude this introduction with a comparison between Chen et al. (2017) and this paper.Although Chen et al. (2017) and this paper model the stochastic liquidity of the limit orderbook dynamics using a Markov chain, our focus is different. In Chen et al. (2017), stochasticmarket depth with a Markov chain captures the essential feature that the arrival of the quote-non-improving orders, i.e. cancellation of submitted limit orders and the limit orders thatare placed outside of the bid-ask spread, is stochastic. In this paper, stochastic resiliencewith Markov chain captures the feature that the arrival of the quote-improving limit orders,i.e. the limit orders that are placed within the bid-ask spread, is stochastic. Placementof each market order attracts liquidity as it provides the market the signal of additionaldemand/supply of the asset. However, as the intensity of trading differs across a tradingday, our model assumes that the limit order book replenishes stochastically itself after eachmarket order. Finally, we consider both temporary and permanent price impacts, whereasChen et al. (2017) only consider permanent price impacts.

The rest of the paper is organized as follows. Section 2 characterizes the optimal executionproblem and the limit order book dynamics in this paper. Section 3 solves the optimalexecution problem proposed in Section 2. Section 4 provides detailed numerical studieson the impact of the stochastic market resilience rate to the optimal execution strategies.

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Section 5 concludes the paper with few comments for future research. Proofs of all resultsare relegated to Appendix A.

2 Model and Problem Formulation

Suppose (Ω,F ,P) is a complete probability space which models all uncertainties. More specif-ically, the filtration F = (Ft)t≥0 is generated by stochastic processes Ftt≥0 and Mtt≥0 tobe defined below.

Consider an optimal execution problem of a trader under the framework of Obizhaeva andWang (2013), (hereafter, the OW model). Suppose a trader wishes to purchase X0 ≥ 0units of one security over a time horizon [0, T ]. There are N + 1 equally-spaced tradingopportunities over [0, T ], tn , nτ for n = 0, ..., N and τ , T

N+1. The trader can split the

order X0 into series of smaller orders xtn over the period [tn, tn+1), for n = 0, 1, ..., N . Inother words,

∑Nn=0 xtn = X0.

Let Θ be the set of feasible execution strategy strategies, i.e.

Θ ,

x = (xt0 , ..., xtN ) : xtn is Ftn measurable; xtn ≥ 0, for all n,

N∑n=0

xtn = X0

. (1)

The main objective of the trader is to choose an execution strategy x ∈ Θ which minimizesthe expected execution cost over time horizon [0, T ], i.e.

minx∈Θ

E

[N∑n=0

P tn(xtn)xtn

], (2)

where P t(x) denotes the average execution price of a market order x at time t ≥ 0. Asdiscussed in the OW model, the average execution price P tn(xtn) depends on the price impactof both current market xtn and past trades xtkn−1

k=0 . To characterize price impact of trades,we now specify the dynamics of the limit order book (LOB) for the security.

2.1 Dynamics of Limit Order Book

We begin with a stylized description of a LOB of a security. As discussed in Parlour andSeppi (2008), supply and demand of a security is reflected in the electronic LOB for thesecurity at any given time. Indeed, in a typical LOB market, each trader has a choice ofthree basic strategies: limit orders, market orders, and cancellation. A limit order is an orderthat specifies a number of shares to be purchased or sold at a pre-specified price, called thelimit price. The LOB of a security is then a collection of the limit orders listed with respectto the limit price: limit sell (respectively, buy) orders of a security are typically listed inthe descending (respectively, ascending) order of the limit ask prices (respectively, limit buyprices). The lowest limit sell price (respectively, highest limit buy price) is referred to as bestask price (respectively, best bid price) of the security. The difference between the best askand best bid prices of a security is known as the bid-ask spread of the security. We denoteAt and Bt to be the best ask and the best bid prices of a security at time t ≥ 0, respectively.

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At any given time, the LOB of the security reflects the supply and demand of the securityby indicating total shares available at each limit price to traders. Traders who submit thelimit orders are said to be liquidity providers. A market sell (respectively, buy) order is anorder which specifies the number of shares to be sold, (respectively, purchased), at the bestbid, (respectively, ask), price of the security. Trade is said to occur when a market order isexecuted immediately against the opposite side of the existing limit orders of the security.For example, each market sell order would first consume the existing limit buy orders of thesecurity at the best bid price before rolling up to higher bid prices, until the market sellorder is completely exhausted, and the bid price that completed the market sell order thenbecomes the new best bid price. Traders who submit market orders are said to be liquidityconsumers. Finally, traders can cancel their previously submitted limit orders before thoselimit orders are consumed by market orders. As our primary focus of this paper is to studythe problem of optimal execution of market buy orders over a fixed time horizon, we shallhereafter assume our trader to be a liquidity consumer, i.e. the trader only submits marketbuy orders to consume the existing limit sell orders of the security.

To maintain the tractability of the optimal execution problem in this paper, we shall follow thesimplification in the OW model by assuming that both prices and order sizes are continuousinstead of discrete. Consequently, the LOB of the security with respect to limit price canbe modelled by its density function. To understand this, suppose qAt (P ) and qBt (P ) are thedensities of ask and bid sides of a security at price P , respectively, and let dP be the smallprice differential from P . In terms of qAt (P ) (respectively qBt (P )), the total size of limit sell(respectively buy) orders in the price range of [P, P + dP ) (respectively (P − dP, P ]) at timet ≥ 0 is qAt (P )dP (respectively qBt (P )dP ). Moreover, as we consider only market buy orders,we shall focus on qA exclusively.

To facilitate subsequent analysis, we shall also make the following assumption on the shapeof the density function qAt , i.e.

Assumption 1. The limit sell order book is assumed to have a block-shape density:

qAt (P ) , qA1P≥At, for qA > 0 and t ≥ 0, (3)

and qA is referred to as market depth of the limit sell order book of the security. Here, 1x≥ydenotes the indicator function that yields 1 if x ≥ y and 0 if x < y, for x, y ∈ R.

Next, we specify the dynamic nature of the limit order book of the security. We assume thatthe evolution of the best ask price At of the security is comprised of two components: thefundamental value of the security and the trading impact on the security. We first discussthe fundamental value of the security. The fundamental value of a security is the value ofthe security in absence of any trading activities. Suppose Ft is the fundamental value of thesecurity at time t ≥ 0. Following the OW model, (see also Chen et al. (2017), and Tsoukalaset al. (2017)), we shall assume that Ft follows a random walk. More specifically,

Assumption 2. Let ηtnNn=1 be the sequence of independent and identically distributed nor-mal random variables, i.e. ηtn ∼ N(0, σ2τ) and E[ηtiηtj ] = 0, for i 6= j and i, j = 1, .., N .The fundamental value of the security Ftn is assumed to have the following dynamics

Ftn+1 = Ftn + ηtn+1 , Ft0 > 0, (4)

with E[Ftn+1|Ftn ] = Ftn, for n = 0, ..., N − 1.

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In absence of any trades, the best ask price At can be expressed in terms of fundamentalvalue Ft as follows:

Atn = Ftn +s

2, for n ≥ 0, (5)

where s denotes as the bid-ask spread in steady-state, i.e. this is the bid-ask spread of thesecurity when the effects of all trades have been dissipated in the long run.

Remark 2.1. As shown in the OW model (see also Chen et al. (2017), Tsoukalas et al.(2017)), when the fundamental value of the security is the only source of uncertainty inthe model, the optimal execution problem under Assumption 2 is an example of an optimaldeterministic control problem, i.e. the optimal execution strategy is deterministic. This isdue to the linear relationship between the fundamental value of the security and the randomwalk such that zero mean and zero serial correlation imply that random walk plays no role inthe optimal execution problem of the trader under expectation. However, when we introducea stochastic resilience rate, the optimal execution problem considered in this paper remainsstochastic even when Assumption 2 is in force. That is, the optimal trading strategy undera stochastic resilience rate becomes state-dependent. In this respect, Assumption 2 allowsus to isolate the effect of stochastic resilience rate on the optimal execution strategy and thecorresponding optimal execution cost.

In addition to its fundamental value, the second component that drives the evolution of thebest ask price of the security is the impact of market buy orders. Consider an arrival ofmarket order xtn at time tn, and denote Atn to be the best ask price of the security at timetn before the arrival of market order xtn . In terms of the density function qA of the LOB, wehave ∫ Atn+

Atn

qAtn(P )dP = xtn ,

where Atn+ is the best ask price immediately after a placement of the market order xtn . Theblock-shaped density assumption of qA in Assumption 1 allows a more transparent expressionof Atn+ in terms of Atn and the market order xtn :

Atn+ = Atn +xtnqA. (6)

In other words, the immediate impact of the market buy order xtn is given as Atn+−Atn = xtnqA

.

Note that greater the market depth qA, the less immediate impact the market buy order xtnwould have on the best ask price Atn+ . For this reason, market depth is one of the liquiditymeasures of the LOB market.

Next, we assume that the immediate impact of market buy order xtn on the best ask priceAt can be decomposed into permanent and temporary price impacts. We begin with thepermanent price impact. Let Vtn be the steady-state mid-price of a security before the arrivalof market order xtn at time tn, for n = 0, .., N . Assume that the steady-state mid-price of thesecurity follows its fundamental value at time t0, i.e. Vt0 = Ft0 . The following assumptionexpresses the steady-state mid-price of the security Vtn as the sum of the fundamental valueFtn of the security and the permanent price impact of past trades xtkn−1

k=0 :

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Assumption 3. Each market order xtn, for n = 0, ..., N , affects the steady-state mid-priceVtn, linearly, i.e.

Vtn = Vtn−1 + λxtn−1 + ηtn = Ftn + λ

n−1∑i=0

xti , (7)

where λ ∈ [0, 1qA

) denotes the permanent price impact parameter of a market order; qA > 0 isthe market depth from Assumption 1; and ηtn is the normal random variable in Assumption2.

From Assumption 3, we see that the steady-state mid-price Vt is the sum of the fundamentalvalue Ft and the accumulated permanent price impact of past trades before time t. All pasttrades xti , for ti < t, affect the steady-state mid-price Vt equally, and their permanent priceimpact is measured using parameter λ.

In addition, Assumption 3 also leads to the assumption that a steady-state best ask priceexists. To see this, assume that no market order arrives after time tn−1. In this case, thesteady-state mid-price Vt would become Ft + λ

∑n−1i=0 xti . Since s denotes the steady-state

bid-ask spread, it follows that, in absence of any additional market order after tn−1, the bestask price At would eventually converge to Vt + s

2= Ft + λ

∑n−1i=0 xti + s

2for sufficiently large

t ≥ tn. Following the OW model, we shall denote A∞ to be the steady-state best ask price,i.e.

A∞tn , Vtn +s

2, (8)

where Vtn is the steady-state mid-price given in (7). Note that the best ask price Atn andits steady-state counterpart A∞tn are generally different. We shall call the difference between

them as the displacement and denote it by Dtn , Atn − A∞tn , with Dt0 , 0.

Consider now time tn+, i.e. the time immediately after the arrival of the market buy orderxtn . The steady-state mid-price V and the steady-state best ask price A∞ now become

Vtn+ = Vtn + λxtn ,

A∞tn+ = A∞tn + λxtn .

Using the expression for the best ask price Atn+ immediately after the arrival of market orderxtn in (6), the displacement at time tn+ now becomes

Dtn+ = Atn+ − A∞tn+

= Atn +xtnqA−(A∞tn + λxtn

)= Dtn + κxtn ,

where κ , 1qA− λ.

After the arrival of each market buy order, the replenishment of the LOB begins. The tempo-rary price impact of past trades is closely linked with the specification of this replenishmentprocess, as we now describe. The replenishment of the limit order is due to the fact thatmarket order xtn at time tn sends a signal to other market participants that there is a demand

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for the security, as they see the best ask price jumps from Atn to Atn+ immediately after theplacement of market buy order xtn . This signal attracts some market participants to submitadditional limit sell orders which cause the narrowing of the bid-ask spread of the security.In other words, a market order xtn attracts quote-improving limit sell orders which eventuallylead to the decrease in the best ask price Atn+ . The arrival rate of these quote-improvinglimit sell orders is known as the market resilience rate. In the OW model (see also, Chen etal. (2017) and Tsoukalas et al. (2017)), the market resilience rate is assumed to be constantand is denoted as ρ. A constant market resilience rate implies that the quote-improving limitsell orders always arrive at a constant rate, i.e. other market participants always react thesame way after the placement of each market order at any time during a trading day. Whenthe market resilience rate ρ is low, the market is relatively inert toward the arrival of a newmarket order xtn , and consequently the best ask price would remain close to Atn+ in latertrading periods. On the other hand, when the market resilience rate ρ is high, the market isvery reactive towards the market order xtn , with a higher volume of quote-improving limitsell orders to lower the best ask price. For the ease of exposition, we shall hereafter refer tomarket resilience rate as the resilience rate.

As discussed in the introduction, trading volume, order flows, and transaction costs all exhibitthe typical U-shaped intra-day patterns (see, e.g. Biais et al. (1995), Handa and Schwartz(1996), and Ahn et al. (2001)): high trading activities and large trades typically appearat the starting and closing times of a trading day, whereas the market tends to slow downduring the mid-day. In this respect, a LOB with a constant resilience rate appears to beinconsistent with the U-shaped intra-day pattern. Obizhaeva and Wang (2013) offered aremedy by suggesting the replacement of the constant resilience rate with a deterministicresilience rate. However, Large (2007) showed that the impact of large trades on the resilienceof the limit order book fitted well with a mutually-exciting ten-variate Hawkes point process,suggesting the stochastic nature of market resilience. Indeed, using high-frequency data fromthe London Stock Exchange, Large showed that fitted Hawkes process offer an insightfulexplanation how randomly the market recovers after large trade shocks. See also Lo and Hall(2015) for another econometric model of stochastic resilience rate using high-frequency vectorautoregression (VAR) model. In order to reconcile with this financial econometric literaturewhile maintaining tractability of the problem, we propose to model the stochastic resiliencerate using a continuous-time Markov chain.

To this end, we follow the standard description of a finite-state, continuous-time Markovchain (see Elliott et al. (1994)). Write E , e1, ..., ed ⊂ Rd for the set of standard unitvectors in which the kth component of ej is a Kronecker delta δjk, for 1 ≤ j, k ≤ d. LetMtt≥0 be the continuous-time Markov chain on a state space E with a d×d-transition ratematrix Ψ , (ψjk)1≤j,k≤d under measure P with ψjj = −

∑k 6=j ψjk and ψjk ≥ 0. In addition,

we assume that Markov chain M is independent to ηtnNn=1. Write ρ , (ρ1, .., ρd)> ∈ Rd

for the vector of resilience rates of the LOB, satisfying 0 < ρ1 < ... < ρd < ∞, where ρ>

denotes the transpose of the vector ρ ∈ Rd. The stochastic resilience rate of the limit orderbook is defined as ρ(Mt) , 〈ρ,Mt〉, where 〈x,y〉 denotes the usual inner product of vectorsx,y ∈ Rd. In terms of the stochastic resilience rate ρ(Mt), we can now characterize theevolution of displacement process Dt:

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Assumption 4. For n = 0, .., N−1, the displacement process Dtn has the following dynamics

Dtn+1 , (Dtn + κxtn) e−∫ tn+1tn

ρ(Ms)ds = κ

n∑i=0

xtie−

∫ tn+1ti

ρ(Ms)ds. (9)

Furthermore, the displacement process Dtn measures temporary price impact.

When there is only one state in ρ, i.e. ρ = (ρ, ..., ρ)>, our model reduces to the OWmodel. When there is more than one state, Assumption 4 shows that the displacement Dt

is stochastic, with the resilience rate depending on the state of the Markov chain Mtt≥0.To see this, consider the case when there are only two resilience states, a low resilience stateρL and a high resilience state ρH . When the market is very reactive toward new marketbuy orders over a short period (t, s), for t ≤ s, the quote-improving limit sell orders arriveat a high resilience rate ρ(Mu) , ρH for t ≤ u ≤ s. During the period (t, s), the fastreplenishment of the LOB leads to a drop in the best ask price at time s, i.e. As. The oppositeeffect occurs when the market is relatively inert to the new market buy orders. From thenumerical examples in Section 4, we shall see that the optimal execution strategies behavevery differently to those in the OW model when a stochastic resilience rate is introduced.

To understand how the displacement process Dtn measures the temporary price impact, letus now put Assumptions 2-4 together. For n = 0, ..., N − 1, the the best ask price A nowbecomes

Atn+1 = A∞tn+1+Dtn+1

= Ftn+1 +s

2+ λ

n∑i=0

xti︸ ︷︷ ︸permanent price impact

+ κn∑i=0

xtie−

∫ tn+1ti

ρ(Ms)ds

︸ ︷︷ ︸temporary price impact

. (10)

Assume that no market order arrives after tn. The best ask price A in (10) indicates thatthe displacement process Dt decreases at a stochastic resilience rate ρ(Mt) as time increases.When time t is sufficiently large, Dt would then converge to zero and the best ask price Awould then converge to its steady-state A∞t , or Vt + s

2. This is consistent with our definition

of A∞t . In this respect, it is now natural to see that displacement Dt directly measures thetemporary price impact, and the stochastic resilience rate can be interpreted as the speed atwhich the temporary price impact of past trades dissipate. Unlike the case of the permanentprice impact, the displacement Dt in (10) indicates that past market buy orders xtk , fortk < t, have unequal temporary price impacts on the best ask price At. More specifically,(10) shows that the temporary price impact of each past market buy order xti is multiplied

by e−∫ tn+1ti

ρ(Ms)ds, indicating that an older executed market buy order has less temporaryprice impact on the current best ask price than a more recently executed market buy order.This is intuitive as the temporary price impact of the recently market order xtn has less timeto dissipate than that of other past market orders xtkn−1

k=0 , and hence the temporary priceimpact of the xtn has the greatest effect on the best-ask price Atn+1 , as shown in (10).

2.2 Optimal execution problem

With the representation of best ask price Atn , for n = 0, ...N , in place, we now return tospecify the average execution price P tn of a market order xtn . Denote c to be the cost of

10

market order xtn , i.e.

c(xtn) ,∫ xtn

0

Ptn(x)dx, (11)

where Pt(x) satisfies x =∫ Pt(x)

AtqAt (P )dP . From Assumption 1, Pt(x) admits a closed-form

representation,

Pt(x) = At +x

qA,

where At is given in (10). The cost of the market order xtn now becomes

c(xtn) =

∫ xtn

0

(Atn +

x

qA

)dx =

(Atn +

xtn2qA

)︸ ︷︷ ︸

,P tn (xtn )

xtn , (12)

from which we specify the average execution price P t of market buy order x as P t(x) ,At + x

2qA. Using the representation of best-ask price Atn in (10), we have

P tn(x) = Atn +xtn2qA

= Ftn +s

2+ λ

n−1∑i=0

xti + κn−1∑i=0

xtie−

∫ tnti

ρ(Ms)ds +xtn2qA

.

This implies that the average execution price P tn depends not only on the current marketorder xtn , but also past market orders xtkn−1

k=0 .

Write Xtn+1 , with n = 0, ..., N − 1, for the remaining order to be executed at time tn. Wethen have

Xtn+1 , X0 −n∑i=0

xtn . (13)

Summing trading costs in (12) over N +1 market buy orders xtn , for n = 0, ..., N , and takingthe expectation at time t0, we now obtain a more transparent formulation of the the optimalexecution problem with stochastic market resilience,

Problem 1. We wish to find

J(t0, Ft0 ,Mt0 , Xt0 , Dt0) , minx∈Θ

E

[N∑n=0

(Atn +

xtn2qA

)xtn

], (14)

subject to Ftn+1 = Ftn + ηtn+1 ,

Atn+1 = Ftn+1 + s2

+ λ(Xt0 −Xtn+1) +Dtn+1 ,

Xtn+1 = X0 −∑n

k=0 xtk ,

Dtn+1 = (Dtn + κxtn) e−∫ tn+1tn

ρ(Ms)ds,

(15)

with n = 0, ..., N − 1.

11

Before providing the solution to this optimal execution problem for a trader (Problem 1), weneed to impose a sufficient condition to ensure that Problem 1 is jointly convex in (xti , ..., xtN ),for i = 0, ...N . To this end, for s ≥ t, let

g(t, s) , κe−∫ st ρ(Mu)du + λ. (16)

Note that for each realized sample path ρ(Ms)0≤s≤T , the realized execution cost can bere-expressed as follows

N∑n=0

(Atn +

xtn2qA

)xtn =

N∑n=0

(Ftn +

s

2

)xtn +

N∑n=0

(n−1∑i=0

xtig(ti, tn)

)xtn +

1

2qA

N∑n=0

x2tn ,

where the equality follows from κ = 1qA− λ and the definition of g in (16). In matrix form,

the realized execution cost is

N∑n=0

(Atn +

xtn2qA

)xtn = x>

(F +

s

21N+1

)+ x>Gx, (17)

where

x , (xt0 , ..., xtN )> ∈ RN+1

F , (Ft0 , ..., FtN )> ∈ RN+1, 1N+1 , (1, ..., 1)> ∈ RN+1,

G , (Gij)0≤i,j≤N with Gij =

g(tj, ti), if i > j,12g(ti, ti), if i = j,

0, if i < j,

Write G , G+G>

2, and we then have the following result.

Lemma 2.2. For each realized sample path of ρ(Ms)0≤s≤T , G is positive definite.

Let

V (tn, xtn ,Mtn , Ftn , Xtn , Dtn) ,(Ftn +

s

2

)xtn + xtn

n−1∑i=0

xtig(ti, tn) +x2tn

2qA

+Etn[J(tn+1, Ftn+1 ,Mtn+1 , Xtn+1 , Dtn+1)

], (18)

where J is defined in (14). We then have the following lemma,

Lemma 2.3. V defined in (18) is jointly convex in xtkNk=n, for n = 0, ..., N .

3 Optimal execution strategy and the optimal execu-

tion cost

In this section, we show that the optimal execution problem (Problem 1) admits an explicitsolution under Assumptions 2-4. Suppose ydiag is a d × d diagonal matrix with y on thediagonal. We now state the main result of the paper.

12

Theorem 3.1. Under Assumptions 2-4, the optimal execution strategy x∗ , (x∗t0 , ..., x∗tN

) ofProblem 1 admits the following form,

x∗tn =

12δ(tn,Mtn)

[A(tn,Mtn)Xtn −B(tn,Mtn)Dtn

], for n = 0, ...N − 1,

XtN , for n = N ;

(19)

and the corresponding optimal execution cost at time tn, for n = 0, ..., N , is

J(tn, Ftn ,Mtn , Xtn , Dtn) =(Ftn +

s

2

)Xtn + λX0Xtn + α(tn,Mtn)X2

tn + β(tn,Mtn)DtnXtn

+γ(tn,Mtn)D2tn , (20)

where coefficients α, β and γ are recursively determined as follows:For n = N ,

α(tN ,MtN ) = 12qA− λ,

β(tN ,MtN ) = 1,

γ(tN ,MtN ) = 0.

(21)

For n = 0, ..., N − 1,α(tn,Mtn) = α(tn,Mtn)− 1

4δ(tn,Mtn)A2(tn,Mtn),

β(tn,Mtn) = β(tn,Mtn) + 12δ(tn,Mtn)A(tn,Mtn)B(tn,Mtn),

γ(tn,Mtn) = γ(tn,Mtn)− 14δ(tn,Mtn)B2(tn,Mtn),

(22)

where

α(tn,Mtn) , Etn[α(tn+1,Mtn+1)

](23)

=⟨exp (Ψ(tn+1 − tn))αtn+1 ,Mtn

⟩,

β(tn,Mtn) , Etn[e−

∫ tn+1tn

ρ(Ms)dsβ(tn+1,Mtn+1)]

(24)

=⟨

exp((

Ψ− ρdiag

)(tn+1 − tn)

)βtn+1

,Mtn

⟩,

γ(tn,Mtn) , Etn[e−2

∫ tn+1tn

ρ(Ms)dsγ(tn+1,Mtn+1)]

(25)

=⟨

exp((

Ψ−(2ρdiag

)(tn+1 − tn)

)γtn+1

,Mtn

⟩;

with αtn+1 = (α(tn+1, e1), ..., α(tn+1, ed))> ∈ Rd, βtn+1

= (β(tn+1, e1), ..., β(tn+1, ed))> ∈ Rd,

and γtn+1= (γ(tn+1, e1), ..., γ(tn+1, ed))

> ∈ Rd;A(tn,Mtn) ,(λ+ 2α(tn,Mtn)− κβ(tn,Mtn)

)B(tn,Mtn) ,

(1− β(tn,Mtn) + 2κγ(tn,Mtn)

),

(26)

and

δ(tn,Mtn) =

[1

2qA+ α(tn,Mtn)− κβ(tn,Mtn) + κ2γ(tn,Mtn)

]−1

. (27)

13

Theorem 3.1 shows that the optimal execution strategy x∗tnNn=0 and the optimal execu-

tion cost J admit analogous forms to those in the OW model, except that our results arestate-dependent whereas the results in OW are deterministic. More specifically, the optimalexecution strategy x∗tn

Nn=0 (19) indicates that x∗tn is influenced by the remaining position Xtn

and the displacement Dtn at time tn, for n = 0, ..., N , and the magnitudes of their influencesare described by the coefficients A(tn,Mtn) and B(tn,Mtn), respectively. The remaining po-sition Xtn provides a benchmark on how close the trader is from meeting the ultimate targetX0 while the displacement Dtn updates the temporary price impact from the past marketorders x∗ti , where 0 ≤ i < n. Both the remaining position Xtn and the displacement Dtn areforward processes, as their values are determined by the past market buy orders x∗tk

n−1k=0

and the realized path of the resilience rate ρ(Mt), for 0 ≤ t < tn. A large displacement Dtn

indicates that the temporary price impact of the past trades still persists, reflecting a highbest-ask price Atn at time tn. As a result, the trader should place a less aggressive, (i.e.smaller), market buy order x∗tn and wait for the temporary price impact of past trades topartially dissipate in subsequent periods.

On the other hand, the coefficients A(tn,Mtn) and B(tn,Mtn) in (19) are backwardly deter-mined, as they are functions of α(tn,Mtn), β(tn,Mtn), and γ(tn,Mtn) in (23)–(25). Indeed,from (23)–(25), we see that α(tn,Mtn), β(tn,Mtn), and γ(tn,Mtn) are the respective expectedvalues of α(tn+1,Mtn+1), β(tn+1,Mtn+1), and γ(tn+1,Mtn+1) evaluated at time tn. Given thecurrent resilience rate ρ(Mtn), coefficients A(tn,Mtn) and B(tn,Mtn) are determined by thetrader’s expectation of the future resilience rate, which is stochastic, for the remaining timeperiod. As we shall see in the numerical examples in Section 4.2, a high resilience environ-ment typically results in a smaller B(tn,Mtn), indicating the displacement Dtn would haveless impact when the resilience rate is currently high. This results in a larger x∗tn , i.e. a largermarket order at time tn. The opposite result would appear for the case of a low resiliencestate.

To understand the optimal execution cost, let us focus exclusively on the execution costat time t0, for the optimal execution cost at other time tn can be interpreted analogously.Assume that Mt0 = ej with a probability πj ∈ [0, 1], with

∑dj=1 πj = 1. As Xt0 = X0 and

Dt0 = 0, the optimal execution cost J in (20) at time t0 becomes

J(t0, Ft0 ,Mt0 , Xt0 , Dt0) =(Ft0 +

s

2

)X0︸ ︷︷ ︸

Fundamental cost

+ λX20 +

d∑j=1

πjα(t0, ej)X20︸ ︷︷ ︸

Cost due to price impact

. (28)

From (28), the optimal execution cost at time t0 is decomposed into two components: thefundamental cost of the trade and the price impact of the trade. The fundamental cost of thetrade,

(Ft0 + s

2

)X0 = Ft0At0 in (28), is the execution cost the trader would pay if there were

no price impact. The remaining terms in (28) constitute the expected price impact resultsfrom the trader pursuing the optimal trading strategy x∗tn

Nn=0 in (19). In Section 4.3, we

shall show that the magnitude of cost saving obtained by pursuing the optimal executionstrategy x∗tn

Nn=0 in (19) can be substantial when comparing the optimal execution cost at

time t0 in (28) against the execution costs associated with other state-independent executionstrategies, (see Section 4.3).

14

4 Numerical Analysis

In this section, we shall provide several numerical studies on the impact of the stochasticresilience rate to the optimal execution problem. For simplicity, we shall assume there areonly two states, i.e. E = eL, eH, in which eL , (1, 0)> and eH , (0, 1)> denotes low andhigh resilience states, respectively. Correspondingly, we have ρ = (ρL, ρH)>, in which ρLand ρH denote the low and high resilience rates, respectively. Unless otherwise stated, thefollowing numerical studies are performed with the parameters in Table 1.

X0 A0 T N qA λ ρL ρH ρAvg π100, 000 $100 1 10 5000 1

10,0000.05 20 10.025 0.5

Table 1: Model Parameters

In Table 1, size of the total order X0 is 100, 000 shares. The initial best ask price A0 is set as$100; the trading horizon is set to be T = 1 day, and we assume that there are N + 1 = 11trading periods of equal size T

N+1within the one trading day, which corresponds to 6.5 hours

(or 390 minutes) of trading in a given trading day. The market depth q is set to be 5000 units,and the permanent price impact coefficient is λ = 1

2qA= 1

10,000. We set the low resilience

rate ρL = 0.05, the high resilience rate ρH = 20, and denote ρAvg = ρL+ρH2

to be the averageof high and low resilience rates. Following the half-life interpretation from the OW model,the low resilience rate ρL = 0.05 corresponds to the case when the half-life for the LOB toreplenish itself after being hit by a trade is 1

ρLln(2) = 13.86 days (or 5.41× 103 minutes); the

half-life for the LOB in the high resilience rate ρH is 1ρH

ln(2) = 0.035 day (or 13.52 minutes);

and the half-life for the LOB in the average resilience rate ρAvg is 1ρAvg

ln(2) = 0.069 day

(or 26.91 minutes). Finally, we assume that the initial distribution of the Markov chain is(π, 1− π) = (1

2, 1

2).

Consider the following two intensity matrices of the Markov chain Mst≥0: For ε > 0,

• Case 1: Asymmetric intensity matrix

Ψ1(ε) ,

(−10× ε 10× ε10× 1

ε−10× 1

ε

), (29)

• Case 2: Symmetric intensity matrix

Ψ2(ε) , ε×(−10 1010 −10

), (30)

where ε denotes the resilience switching speed.

4.1 Expected half-life

As a constant resilience rate has a natural interpretation as the half-life of the LOB afterbeing hit by a trade, let us now introduce the concept of the expected half-life of the LOBafter being hit by a trade under a stochastic resilience rate. To this end, suppose Mt = ej,

15

ε ∆HalfL (ε) ∆Half

H (ε)

10−4 13.863 13.85910−3 13.857 13.81810−2 13.341 12.9610.1 3.2278 2.38880.5 0.31353 0.0690751 0.14894 0.0435802 0.089763 0.03796010 0.045809 0.034944100 0.035668 0.0346611000 0.034757 0.03465710000 0.034667 0.034657

ε ∆HalfL (ε) ∆Half

H (ε)

10−4 13.592 0.03465810−3 11.562 0.03466510−2 4.6698 0.0347340.1 0.73944 0.0354360.5 0.21980 0.0388281 0.14894 0.0435802 0.11069 0.05286610 0.077579 0.067128100 0.069983 0.0689831000 0.069226 0.06912610000 0.069150 0.069140

Table 2: Expected half-life for Case 1 (left); Expected half-life for Case 2 (right)

for j = L,H. Define the expected half-life for the limit order book to be time ∆Halfj (ε) > 0

satisfying

Et

[e−

∫ t+∆Halfj

(ε)

t ρ(Ms)ds

]=

1

2. (31)

Applying Lemma A.1 with ν = 1 and f = 12 , (1, 1)> to the left-hand side of (31), we obtainan implicit function of the expected half-life ∆Half

j (ε) > 0, i.e. given Mt = ej, for j = L,H,

e>j exp

((Ψl(ερ)− ρdiag

)∆Halfj (ε)

)12 =

1

2. (32)

Table 2 displays the half-life of the LOB after being hit by a trade under different valuesof the resilience switching speed ε. In Case 1, we see that ∆Half

L (ε) and ∆HalfH (ε) decrease

toward the half-life for the LOB under a constant resilience rate ρH as ε increases. It sufficesto discuss ∆Half

L (ε) for ∆HalfH (ε) follows similarly. Indeed, for ε = 100, Table 2 indicates the

expected half-life for the LOB ∆HalfL (100) is 0.036 day (or 14.04 minutes). On the other

hand, when ε = 1100

, the expected half-life for the limit order book ∆HalfL ( 1

100) is 13.34 days

(or 5.20× 103 minutes).

To understand this, when Mt = eL, the time that the market would remain in the lowresilience state eL before leaving for the high resilience state eH is an exponentially distributedrandom variable with mean µ = 1

10εday. This implies that when ε = 100, the duration of

remaining in the low resilience state is exponentially distributed with mean µ = 11000

day (or0.39 minute), whereas the duration of remaining in the low resilience state is exponentiallydistributed with mean µ = 10 days (or 3900 minutes) for ε = 1

100. In other words, it is more

likely that the market would be in the high resilience state eH than in the low resilience stateeL over a given trading day when ε = 100. This results in the expected half-life for the LOBafter hit by a trade to be close to that of constant resilience rate ρH , which is 0.035 day (or13.52 minutes). The same arguments can be applied to show that ∆Half

L (ε) and ∆HalfH (ε) are

close to the half-life for the LOB under the constant resilience rate ρL when ε = 1100

.

16

In Case 2, Table 2 shows that ∆HalfL (ε) and ∆Half

H (ε) converge toward the half-life for theLOB under the constant resilience rate ρAvg as ε increases. In Case 2, a higher value of εwould imply the market switches between the low and high resilience states frequently. As εapproaches infinity, standard Markov chain theory indicates that the market resilience rateρ(Mt) would approach the average of states ρL and ρH , i.e. ρAvg. Indeed, consider the case

when ε = 100. If ρ(Mt) = ρL (respectively, ρ(Mt) = ρH), the expected half-life ∆HalfL (100)

(respectively, ∆HalfH (100)) is 0.070 day (or 27.30 minutes) (respectively, 0.069 day (or 26.91

minutes)), which is almost indistinguishable to the half-life for the LOB under the constantresilience rate ρAvg, which is 0.069 day (or 26.91 minutes). Finally, note that when ε = 1,Cases 1 and 2 coincide, i.e. Ψ1(1) = Ψ2(1). In light of the discussion of the expected half-lifefor the LOB, we shall illustrate the impact of resilience switching speed ε to the optimalexecution strategies and the optimal execution costs in Sections 4.2 and 4.3.

4.2 Optimal execution strategies

Optimal Execution Strategy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

2

4

6

8

10

12

14

16

18

20Sample path of

Remaining Position

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

1

2

3

4

5

6

7

8

9

10

Re

ma

inin

g P

ositio

n

104 Displacement (Temporary Price Impact)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.2

0.4

0.6

0.8

1

1.2

Dis

pla

ce

me

nt

Figure 1: Optimal execution strategy, x∗tn10n=0; Sample path of ρ, ρ(Mtn)10

n=0; Remainingposition Xtn10

n=0; and temporary price impact (Displacement) Dtn10n=0, for ε = 1.

Figure 1 displays the optimal execution strategy, x∗tn10n=0, a sample path of ρ(Mt), the

remaining position Xtn10n=0, and the displacement Dtn10

n=0 when ε = 1. As shown inTheorem 3.1, the trader’s optimal execution strategy x∗tn at time tn, for n = 0, ..., 10, is

17

a linear function of the current remaining position Xtn minus the current temporary priceimpact (displacement) Dtn of the past trades before tn, i.e. x∗tk

n−1k=0 . While Xtn generally

decreases in time, the dynamics of Dtn is more sensitive to the fluctuation of the resiliencerate, as shown in Figure 1. More specifically, when the Markov chain Mt spends more timein the low resilience state eL, the resulting displacement Dtn remains high. This indicatesthat there are not enough quote-improving limit sells orders to absorb the price impact ofpast trades. As a result, the high Dtn keeps the best ask price Atn (10) from returning to thelower level, widening the bid-ask spread. This certainly increases the trader’s execution costsfor any remaining orders. In this case, the trader would optimally choose to execute a smallermarket order at time tn, and wait until the market becomes more resilient, i.e. Mt = eH ,before placing larger market buy orders.

In addition, Figure 1 also indicates that the optimal execution strategy x∗tn is mainly influ-enced by the last trade x∗tn−1

. To understand this, recall from the discussion in Section 2 thatthe temporary price impact Dtn (9) is the weighted sum of all past trades x∗t0 , ..., x

∗tn−1

, withthe last trade x∗tn−1

having the largest weight. These weights are expressed in terms of therealized path of ρ(Mt) up to time tn. Hence, as shown in Figure 1, the size of the optimalmarket buy order at time tn would be significantly smaller than the size of the optimal mar-ket buy order at time tn−1, when Mt spends more time in state eL than in state eH during(tn−1, tn]. Finally, the final executed order x∗tN must be equal to XtN in order to achieve thetarget of X0 = 100, 000 shares at the end of the trading day. Therefore, x∗tN can be sizeablewhen past trades were not large enough to meet the final target X0.

Figure 2 compares the optimal execution strategy x∗tn10n=0 against the deterministic execu-

tion strategies, denoted by xρtn10n=0, which correspond to constant resilience rates ρ = ρL

and ρ = ρH in Case 1. When the resilience switching speed ε is 10−3, the optimal executionstrategy x∗tn

10n=0 is indistinguishable from the deterministic execution strategy correspond-

ing to the constant resilience rate ρL. This is consistent with our previous observations thatthe expected half-lives ∆Half

L ( 11000

) and ∆HalfH ( 1

1000) are close to that under constant resilience

rate ρL. As a result, the trader’s optimal execution strategy would behave as in the case ofconstant resilience ρL, as shown in Figure 2. Similar reasoning can be applied for the casewhen ε = 103 to deduce that the optimal execution strategy coincides with the deterministicexecution strategy corresponding to the constant resilience rate ρH .

For intermediate values of ε, e.g. ε = 12, Figure 2 also shows that the optimal execution strat-

egy x∗tn10n=0 becomes distinct from the deterministic execution strategies corresponding to

constant resilience rates ρL and ρH . Deterministic execution strategies xρtn10n=0 correspond-

ing to constant resilience rates ρ = ρL and ρ = ρH have the general feature of large marketbuy orders at time t0 and tN with smaller market buy orders of equal sizes at times t1, ...tN−1,as shown in Figure 2. Moreover, deterministic execution strategies xρtn10

n=0 are sensitive tothe values of ρ. As the resilience rate ρ decreases, the market buy orders xt0 and xtN increasewhile market buy orders at times t1, ...tN−1 decrease, as first shown in the OW model. Whenthe resilience rate is stochastic, the initial market buy order xt0 drops substantially from thedeterministic initial market buy order xρLt0 . This can be explained using the concept of the

expected half-life ∆Halfj (1

2) in (32), for j = L,H. Consider the case when j = L. When ε = 1

2,

the expected half-life for the LOB after a hit by a trade is ∆HalfL (1

2) = 0.31 day, whereas the

half-life for the LOB after a hit by a trade under constant resilience rate ρL is 13.86 days. Alower expected half-life indicates that there is a high probability of high resilience market in

18

Optimal Execution Strategy , =10-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

H=20

Optimal Execution Strategy , =1/2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

H=20

Optimal Execution Strategy, =2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

H=20

Optimal Execution Strategy , =103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

H=20

Figure 2: Comparisons of optimal execution strategy, x∗tn10n=0, for ε = 10−3, 1

2, 2, 103, with

the deterministic execution strategies (Case 1, Ψ1(ε)). Optimal execution strategy is denotedas “2 Regime” and is displayed as the average of the optimal execution strategies over 10, 000sample paths of ρ. Deterministic execution strategy under the constant resilience rates ρLand ρH are displayed as “ρL = 0.05” and “ρH = 20”, respectively.

later trading periods. Consequently, a trader would not optimally choose to submit a largeinitial market buy order x∗t0 and then wait for a high resilience market before submittinglarger market orders later.

Figure 3 compares the optimal execution strategy x∗tn10n=0 against the deterministic execu-

tion strategies xρtn10n=0, for ρ = 0.05, 10.025, 20, under Case 2. As discussed at the beginning

of this section, Case 2 represents the case when the probabilities of the Markov chain Mt

switching between states eL and eH are identical, as Ψ2(ε) is then a symmetric intensitymatrix for each ε. Moreover, as the resilience switching speed ε increases, the Markov chainMt switches between the states eL and eH more frequently. The optimal execution strategywould then converge to the optimal deterministic execution strategy corresponding to theconstant resilience rate ρAvg = 10.025, i.e. xρAvgtn 10

n=0. This is again consistent with our un-derstanding that, when ε is high, the expected half-life for the LOB after a hit by a trade isindistinguishable from that under the constant resilience rate ρAvg, regardless the initial stateρ(M0). For example, Figure 3 shows that the optimal strategy x∗tn

10n=0 and the deterministic

execution strategy xρAvgtn 10n=0 under constant resilience rate ρAvg become indistinguishable

19

Optimal Execution Strategy , =10-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

Avg=10.025

H=20

Optimal Execution Strategy , =1/2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

Avg=10.025

H=20

Optimal Execution Strategy, =2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

Avg=10.025

H=20

Optimal Execution Strategy , =103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

trade time

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Op

tim

al E

xe

cu

tio

n S

tra

teg

y

104

L=0.05

2 Regimes

Avg=10.025

H=20

Figure 3: Comparisons of optimal execution strategy, x∗tn10n=0, for ε = 10−3, 1

2, 2, 103, with

the deterministic execution strategies (Case 2, Ψ2(ε)). Optimal execution strategy is denotedas “2 Regime” and is displayed as the average of the optimal execution strategies over 10, 000sample paths of ρ.. Deterministic execution strategy under the constant resilience rates ρL,ρH , and ρAvg are displayed as “ρL = 0.05”, “ρH = 20”,“ρH = 10.025” respectively.

when ε = 103.

4.3 Cost Saving Comparisons

The numerical studies in Section 4.2 indicate that, under a stochastic resilience rate, the op-timal execution strategy can differ significantly from its deterministic counterparts. However,such a difference has yet to confirm the superiority of choosing the optimal execution strategyover other deterministic execution strategies. To unify the following discussion, let us denotexSItn

Nn=0 to be the state-independent (SI) execution strategies, where SI = ρL, ρH , ρAvg,N .

When SI = ρL, ρH , ρAvg, these SI execution strategies are the execution strategies under theassumption of constant resilience rates ρL, ρH ,and ρAvg. That is, they admit the determin-istic form as given in Proposition 1 of Obizhaeva and Wang (2013). When SI = N , theSI execution strategy is referred to as a naıve execution strategy in which trades are splitequally over the entire trading horizon, i.e. xNtn , X0

N+1, for n = 0, ...N . When the resilience

rate is infinite, ρ = ∞, Bertsimas and Lo (1998) have shown that xNtn , X0

N+1constitutes an

20

optimal execution strategy.

We first give the following lemma on the execution cost associated with a state-independentexecution strategy xSItn

Nn=0.

Lemma 4.1. Suppose xSItn Nn=0 is a state-independent execution strategy and JSI is the

corresponding expected execution cost at time t0. Under Assumptions 1-4, JSI admits thefollowing explicit form:

JSI(tn, Ftn ,Mtn , XSItn , Dtn) =

(Ftn +

s

2

)XSItn + λX0X

SItn − λ

N∑j=n

XSItjxSItj +

N∑j=n

(xSItj )2

2qA

+Dtn

N∑j=n

M>tn exp ((Ψ− ρdiag) (tj − tn)) 1d

+N∑j=n

xSItj

(j−1∑k=n

κxSItk M>tn exp (Ψ(tk − tn))

× exp ((Ψ− ρdiag) (tj − tk)) 1d

), (33)

where XSItj+1

= XSItj− xSItj , for j = n, ..., N − 1.

For simplicity, we shall write Jtn , J(tn, Ftn ,Mtn , Xtn , Dtn) in (20). Correspondingly, weshall write JSItn , JSI(tn, Ftn ,Mtn , X

SItn , Dtn) in (33). In addition, our subsequent discussion

exclusively focuses on the cost comparisons between Jt0 , (see (28)), and JSIt0 . Structurallysimilar to the optimal execution cost Jt0 (28) at time t0, the execution cost JSIt0 associatedwith a state-independent strategy in (33) consists of two terms: a fundamental value oftrading

(Ft0 + s

2

)X0 = Ft0At0 plus an extra cost due to the price impact induced by the

state-independent execution strategy. As first discussed in the OW model, the fundamentalvalue is unrelated to the execution strategy. We shall now only consider the net execution costJt0 , which is defined as Jt0 , Jt0−X0At0 . Correspondingly, the net execution cost associatedwith the state-independent strategy xSItn

Nn=0 is denoted as JSIt0 , JSIt0 −X0At0 . The relative

cost saving is then defined as the relative difference between the net optimal execution costand the net state-independent execution cost, i.e.

Relative Cost Saving(SI) ,JSIt0 − Jt0JSIt0

× 100%. (34)

In the remaining section, we shall first discuss the relative cost saving over the state-independentstrategies xSItn

10n=0, for SI = ρL, ρH , ρAvg (see Tables 3 and 4), followed by the cost saving

over the naive strategy xSItn 10n=0, (see Tables 5 and 6).

Table 3 displays the relative cost saving x∗tnNn=0 over the state-independent execution strat-

egy xSItn Nn=0 in (34) when SI = ρL and SI = ρH in Case 1, (see Ψ1(ε) in (29)). For a

fixed value of the resilience switching speed ε, each row of Table 3 shows that the cost savingincreases as the permanent price impact coefficient λ decreases. From the decompositionof the best ask price Atn in (10), for n = 0, ..., N , we see that the permanent price impactterm is the sum of past trades x∗tk

n−1k=0 multiplied by the coefficient λ, while the temporary

21

HHHH

HHελ 1

2qA1

10qA1

50qA1

100qA0

0.001 1.0555× 10−6 1.9193× 10−6 2.0942× 10−6 2.1161× 10−6 2.1380× 10−6

0.01 0.00011460 0.00020883 0.00022795 0.00023035 0.000232750.1 0.068007 0.12961 0.14282 0.14449 0.146170.5 8.1641 19.522 22.752 23.187 23.6311 18.084 45.067 53.161 54.269 55.4002 22.880 57.269 67.643 69.064 70.51710 24.480 61.344 72.482 74.008 75.569100 24.543 61.504 72.672 74.203 75.7681000 24.544 61.509 72.678 74.210 75.774

HHHHHHελ 1

2qA1

10qA1

50qA1

100qA0

0.001 0.32259 0.58502 0.63798 0.64461 0.651250.01 0.46588 0.84567 0.92241 0.93202 0.941630.1 2.2218 4.1535 4.5578 4.6088 4.65990.5 4.2671 10.844 12.867 13.146 13.4311 2.1286 7.4779 10.057 10.467 10.9032 0.48852 2.1695 3.3435 3.5626 3.807010 0.0030847 0.015100 0.025059 0.027090 0.029426100 9.1084× 10−6 4.4742× 10−5 7.4472× 10−5 8.0554× 10−5 8.7563× 10−5

1000 7.8890× 10−8 3.8759× 10−7 6.4519× 10−7 6.9790× 10−7 7.5863× 10−7

Table 3: Relative Cost Saving (ρL) (top); Relative Cost Saving (ρH) (bottom) under Case 1(see Ψ1(ε)) in (29)

price impact term is the weighted sum of past trades multiplied by κ = 1qA− λ. When the

permanent price impact coefficient λ diminishes, the temporary price impact becomes thedominant liquidity component which influences the best ask price Atn . As the optimal ex-ecution strategy x∗tn

Nn=0 incorporates the stochastic nature of the resilience rate while the

deterministic execution strategy xSItn Nn=0, for SI = ρL, ρH does not, the relative cost saving

becomes more pronounced as the permanent price impact coefficient λ decreases. Indeed,for ε = 1, Table 3 shows that the cost saving achieved by choosing the optimal executionstrategy x∗tk

n−1k=0 over xSItk

n−1k=0 , for SI = ρL (respectively, SI = ρH) can increase from 18%

(respectively, from 2.12%) to 55.40%, (respectively, to 10.90%), as λ decreases from 12qA

to 0.

For a fixed linear price impact coefficient λ, the varying resilience switching speed ε canalso result in a substantial cost saving by choosing the optimal execution strategy over itsdeterministic counterparts, as shown in each column of Table 3. For illustration, let us fixthe linear price impact coefficient λ to be 1

2qAin Table 3. We first discuss the relative cost

saving achieved over pursuing xρLtn Nn=0 under different values of ε. When ε = 0.001, therelative cost saving over the state-independent execution strategies xρLtn Nn=0 is negligiblewith a cost saving equal to 1.0555 × 10−6%, whereas for ε = 2, the cost saving over thestate-independent execution strategies xρLtn Nn=0 is 22.88%. When the resilience switchingspeed ε is extremely small, we know from Figure 2 that the optimal execution strategyx∗tn

Nn=0 resembles the state-independent strategy xρLtn Nn=0, yielding negligible cost saving.

22

However, when the resilience switching speed ε = 2, the expected half-life ∆HalfL (2) for the

LOB after a hit by a trade is 0.0897 day (see Table 2), as opposed to the half-life for the LOBafter a hit by a trade under constant resilience state ρL, which is 13.86 days. Consequently,the trader can embrace a higher probability that the LOB recovers at a higher resiliencerate in the later periods by choosing an optimal execution strategy x∗tn

Nn=0. As a result,

the trader can achieve a significant cost reduction over the alternative, state-independentstrategy xρLtn Nn=0. Applying a similar line of reasoning, we can deduce that the cost savingachieved by choosing an optimal execution strategy x∗tn

Nn=0 over the state-independent

strategy xρHtn Nn=0 is negligible when the resilience switching speed ε is high (i.e. ε = 1000),but it becomes substantial for intermediate values of ε (i.e. ε = 2), which is again consistentwith Figure 2.

An interesting phenomenon occurs when inspecting the relationship between cost saving andthe switching resilience speed ε for SI = ρL, ρH in Table 3 (under a fixed λ). While theRelative Cost Saving(ρL) increases as ε increases, no such monotonic relationship exists be-tween Relative Cost Saving(ρH) and ε. Indeed, the Relative Cost Saving(ρH) decreases asε decreases from 0.5 to 0.001. In Case 1, when the resilience switching speed ε decreasesfrom 0.5 to 0.001, it is more likely that the Markov chain Mt is in the low resilience stateeL. Although the optimal execution strategy x∗tn

Nn=0 looks very different from the execu-

tion strategy xρHtn Nn=0 under the constant resilience rate ρH (see Figure 2), we know fromObizhaeva and Wang (2013) that the execution cost becomes strategy independent as theconstant resilience rate ρ approaches 0, (see also Table 2 of Obizhaeva and Wang (2013)).Consequently, the Relative Cost Saving(ρH) becomes minuscule when ε decreases from 0.5to 0.001. In all scenarios, varying the permanent price impact coefficient λ and the resilienceswitching intensity ε, we see that the optimal execution strategy x∗tn

Nn=0 always outperforms

its deterministic counterparts xSItn Nn=0, for SI = ρL, ρH , in achieving lower execution cost.

HHHHHHελ 1

2qA1

10qA1

50qA1

100qA0

0.001 0.24107 0.56620 0.65657 0.66869 0.681020.01 0.47500 1.1283 1.3125 1.3373 1.36260.1 3.0397 7.9527 9.5233 9.7420 9.96620.5 3.0263 9.8015 12.704 13.146 13.6101 1.7539 6.2214 8.4064 8.7559 9.12772 0.84542 3.1935 4.4564 4.6661 4.891710 0.13075 0.52479 0.75784 0.79812 0.84198100 0.011384 0.046358 0.067529 0.071225 0.0752621000 0.0011198 0.0045668 0.0066582 0.0070237 0.0074231

Table 4: Relative Cost Saving (ρAvg) under Case 2 (see Ψ2(ε) in (30))

To investigate the cost saving obtained by choosing an optimal execution strategy x∗tnNn=0

over the state-independent strategy xρAvgtn Nn=0, we now turn to Table 4 under Case 2 (seeΨ2(ε) in (30)). First, as in Table 3, we also see that cost saving achieved by choosing anoptimal execution strategy x∗tn

Nn=0 over the state-independent strategy xρAvgtn Nn=0 increases

as λ increases for a fixed ε. For low values of ε, Table 4 shows that substantial cost savings

23

can be achieved by pursuing the optimal execution strategy, as it captures a cost reductionwhen the market switches to a high resilience state, whereas xρAvgtn Nn=0 only assumes theconstant resilience rate ρAvg. For example, when ε = 0.5 and λ = 0, the cost saving achievedreaches 13.61%. When ε increases, ρ(Mt) converges to ρAvg and hence the cost saving becomesnegligible. Indeed, when ε = 1000, the cost savings achieved over xρAvgtn Nn=0 are all less than0.01% across different values of λ. This observation is again consistent with the optimalexecution strategy in Figure 3.

Finally, we discuss the cost savings achieved by pursuing an optimal execution strategy overthe naıve strategy xNtn

Nn=0. Bertsimas and Lo (1998) showed that when the resilience rate

is infinite, ρ = ∞, the optimal execution strategy constitutes the naıve strategy. Numericalexamples in Obizhaeva and Wang (2013) indicate that the cost saving using the optimalexecution strategy over that of the nıave strategy can be substantial when the deterministicresilience rate is finite. We now show that the degree of cost saving can be even morepronounced in the case of the stochastic resilience rate.

HHHHHHρ

λ 12qA

110qA

150qA

1100qA

0

0.05 0.32570 0.59054 0.64397 0.65066 0.6573610.025 0.30324 1.2254 1.7768 1.8726 1.9770

20 0.027852 0.13668 0.22731 0.24584 0.26718

Table 5: Cost Saving under different ρ (No Regime-Switch)

HHHHHHε

λ 12qA

110qA

150qA

1100qA

0

0.001 0.34168 0.61955 0.67562 0.68264 0.689660.01 0.49346 0.89554 0.97675 0.98692 0.997090.1 2.3608 4.4078 4.8357 4.8896 4.94360.5 4.6513 11.747 13.913 14.211 14.5151 2.3578 8.2348 11.044 11.489 11.9622 0.56081 2.4843 3.8220 4.0711 4.348910 0.029762 0.14553 0.24132 0.26082 0.28327100 0.027652 0.13569 0.22564 0.24403 0.265201000 0.027830 0.13657 0.22714 0.24565 0.26697

Table 6: Relative Cost Saving (N ) under Case 1 (see Ψ1(ε)) in (29)

To facilitate comparisons, let us first reproduce the cost saving of choosing the deterministicexecution strategy over the naıve strategy in Obizhaeva and Wang (2013) within our modelsetting in Table 5. By varying the permanent price impact coefficient λ in Table 5, we see thatthe cost savings of choosing the deterministic execution strategy xSItn

Nn=0, for SI = ρL, ρH ,

over the naıve strategy are all less than 1%. However, when ρ(Mt) can switch randomlybetween ρL and ρH , Table 6 shows that for the intermediate values of ε, the cost saving over

24

the naıve strategy can reach almost over 12% in the case ε = 1 and λ = 0. As long as theresilience switching speed ε is not zero, there is a non-zero probability of the LOB recoveringat the high resilience rate ρH . The optimal execution strategy x∗tn

Nn=0 incorporates this

feature by making larger purchases at times when the resilience rate is high and reducespurchase size when the resilience is low to minimize the execution cost. On the other hand,the naıve strategy places an equal size of market order regardless of the state of the resiliencerate at any time. Consequently, when the resilience rate moves stochastically between ρLand ρH , a trader who follows the naıve strategy incurs substantial execution costs over thatof the optimal execution strategy. In other words, our stochastic resilience model enrichesthe OW model by showing that the resilience switching speed ε plays a critical role in givinga significant cost saving over the state-independent strategies considered in this paper.

5 Conclusion

Motivated by the existing empirical studies (e.g. Large (2007) and Lo and Hall (2015))which show that the resilience rate of the limit order book (LOB) is inherently stochastic,we revisit the optimal execution problem of Obizhaeva and Wang (2013) by modelling theresilience rate as a function of a Markov chain. Incorporating stochastic resilience does notdestroy a salient feature of the tractable Obizhaeva and Wang (2013) model, as we showthat the optimal execution strategy and the corresponding optimal execution cost retainclosed-form expressions. When the LOB replenishes itself stochastically through time, weshow that the optimal execution strategy becomes state-dependent and is driven linearlyby the trader’s current remaining position and the current temporary price impact, withtheir linear dependence based on the future expectation of the dynamics of the resiliencerate. Our cost saving analysis shows that state-independent strategies, such as deterministicexecution strategy associated with a constant resilience rate and the naıve strategy, are sub-optimal and a substantial cost reduction can be achieved by choosing the optimal executionstrategy. Moreover, the resilience switching speed of the Markov chain serves a crucial rolein capturing the expected half-life for the LOB of the security after a hit by a trade. Forextreme values of the resilience switching speed, the optimal execution strategy converges toits deterministic counterpart, whereas the optimal execution strategy departs significantlyfrom its deterministic execution strategies for intermediate values of the resilience switchingspeed.

The results in this paper also suggest several future investigations. For solution transparency,we have chosen to study the optimal execution problem in a discrete time setting. It will beinteresting to see the structure of the optimal execution strategy with stochastic resilienceunder a continuous time framework. As discussed in Fruth et al. (2014, 2017), the optimalexecution strategy in the continuous time framework would have at least one critical buy/waitregion. It would then be imperative to characterize of the critical buy/wait regions when thestochastic resilience rate is introduced. Secondly, as the main objective of this paper isto highlight the impact of the stochastic resilience rate to the optimal execution problem,we have not changed other features in the OW model. However, as indicated by Chen etal. (2017), stochastic market depth provides a logical characterization behind submittingand cancelling limit orders, and the partitioning algorithm in their paper offers a versatilenumerical scheme to obtain optimal execution strategies and optimal execution costs when

25

closed-form solutions are not available. In this respect, it is interesting to further enrich theoptimal execution problem in this paper by including stochastic market depth.

Another important consideration in the optimal execution problem is information asymmetryin a multi-agent framework. This paper only considers the impact of stochastic marketresilience in a single-agent optimal execution problem, in which information on the LOBdynamics is assumed to be visible to the trader. However, it is quite commonplace thatagents trade in the LOB market under asymmetric information. It is, therefore, importantto study the aggregate impact of traders’ decisions on the LOB dynamics. To this end,Chiarella et al. (2017) incorporated behavioral sentiments in a multi-agent LOB model bymodelling each trader’s belief on the mean-growth rate of the fundamental price driven by atwo-state hidden Markov chain. They showed that such a model can reconcile many stylizedfacts in a typical LOB market such as fat-tails in return distribution and long-memory intrading volume. Yang et al. (2018) proposed an event-triggered regime-switching jump-diffusion model to incorporate the impact of asymmetric information on optimal tradingdecisions and inventory controls of three groups of traders: informed, partially informed,and uninformed traders. They showed that traders that have more superior event-triggeredinformation such as market turmoil would engage in more aggressive trading strategies. Inlight of these studies, it would be insightful to extend the optimal execution model of thispaper to a multi-agent setting by modelling information asymmetry on the price impact andmarket resilience using a hidden Markov chain. To endogenize the trading decisions of allagents within a trading system, a high-dimensional simulation scheme becomes necessary,and the agent-based genetic algorithm proposed by Chiarella et al. (2015) could shed somelight on the subsequent numerical implementations. We leave these issues for future research.

6 Acknowledgement

The authors acknowledge the financial support from the Australian Research Council (ARC)Discovery Grant “The role of liquidity in financial markets” (Grant ID: DP170101227).

A Proofs

Proof. (Proof of Lemma 2.2)We first re-express the matrix G+G> as κL+λ1N+1, where 1N+1 is a (N+1)×(N+1) matrix

with all entries equal to 1; L = (Lij)0≤i,j≤N is the symmetric matrix with Lij = e−∫ tjtiρ(Ms)ds,

for 0 ≤ i ≤ j ≤ N . Note that the indices run from 0 to N . Since 1N+1 is positive-semidefinite,it suffices to show that L is positive definite.

To this end, note first that Lij =∏j−1

k=i ak, where ak , e−∫ tk+1tk

ρ(Ms)ds are random processesvalued strictly between 0 and 1. Since the leading principal submatrices of L have the sameform as L, it suffices to check that L has a positive determinant.

Denote Li , (Li0, ...LiN), for i = 0, ..., N . Perform the following row operations on the matrixL: for i = N,N − 1, . . . , 1, (Lj − ai−1Li−1) 7→ Lj. Observe that L0 remains unchanged underthese row operations. After these row operations, matrix L is then transformed into an upper

26

triangular matrix, i.e., for i, j = 1, ..., N ,

Lij =

0, if i > j,

(1− a2i−1), if i = j,

(1− a2i−1)

∏j−1k=i ak, if i < j,

from which its determinant can be easily computed to be∏N

k=1(1− a2k), which is positive as

required.

Proof. (Proof of Lemma 2.3)

Write x(1)tn Nk=n and x(2)

tn Nk=n for the respective feasible execution strategies which minimize

the expected execution cost from time tn+1 to tN after x(l)tn is chosen at time tn, for l = 1, 2,

i.e.

x(l)tn

Nk=n+1 = arg minEtn

[N∑

k=n+1

(Ftk +

s

2

)x

(l)tk

+N∑

k=n+1

(k−1∑i=0

x(l)ti g(ti, tk)

)x

(l)tn +

1

2qA

N∑k=n+1

(x(l)tk

)2

],

where g is defined in (16). Let φ ∈ (0, 1) and consider an execution strategy xφtnNk=n ,

φx(1)tn +(1−φ)x

(2)tn Nk=n. It is easy to see that xφtnNk=n is also a feasiable strategy. Abbreviating

V (tn, xtn ,Mtn , Ftn , Xtn , Dtn) as V (tn, xtn), it suffices to show that

φV (tn, xφtn) ≤ φV (tn, x

(1)tn ) + (1− φ)V (tn, x

(2)tn ). (35)

By Lemma 2.2, we know that for each realized sample path of ρ(Ms)tn+1≤s≤tN , the realized

execution cost∑N

k=n+1

(Atk +

xtk2q

)xtk is jointly convex in (xtn+1 , ..., xtN ). Taking expectation

over all possible realizations of ρ(Ms)tn+1≤s≤tN preserves this joint convexity. Therefore, itfollows that(

Ftn +s

2

)xφtn +

(n−1∑i=0

xtig(ti, tn)

)xφtn +

(xφtn)2

2qA

+Etn

[N∑

k=n+1

(Ftk +

s

2

)x

(φ)tk

+N∑

k=n+1

(k−1∑i=0

xφtig(ti, tk)

)xφtn +

1

2qA

N∑k=n+1

(xφtk)2

](36)

≤ φ

(Ftn +

s

2

)x

(1)tn +

(n−1∑i=0

xtig(ti, tn)

)x

(1)tn +

(x(1)tn )2

2qA

+Etn

[N∑

k=n+1

(Ftk +

s

2

)x

(1)tk

+N∑

k=n+1

(k−1∑i=0

x(1)ti g(ti, tk)

)x

(1)tk

+1

2qA

N∑k=n+1

(x(1)tk

)2

]

+ (1− φ)

(Ftn +

s

2

)x

(2)tn +

(n−1∑i=0

xtig(ti, tn)

)x

(2)tn +

(x(2)tn )2

2qA

+Etn

[N∑

k=n+1

(Ftk +

s

2

)x

(2)tk

+N∑

k=n+1

(k−1∑i=0

x(2)ti g(ti, tk)

)x

(2)tk

+1

2qA

N∑k=n+1

(x(2)tk

)2

]= φV (tn, x

(1)tn ) + (1− φ)V (tn, x

(2)tn )

27

where the last equality follows from the definitions of execution strategies x(l)tn Nk=n, for

l = 1, 2. From the Bellman equation V (tn, xφtn), the execution strategy which yields V (tn, x

φtn)

must be the one that minimizes the expected execution cost from tn+1 to tN after xφtn is

chosen at tn. Therefore, V (tn, xφtn) must be less than or equal to (36), from which the result

follows.

Before proving Theorem 3.1, we first state the following lemma:

Lemma A.1. Fix u ≥ 0 and for ν ∈ N, 0 ≤ t ≤ u < ∞, and Mt = ei, ei ∈ E, letf , (f1, ..., fd)

> ∈ Rd. Then

Et[e−

∫ ut νρ(Ms)dsf(Mu)

]= e>i exp

((Ψ− νρdiag

)(u− t)

)f , (37)

where f(x) , 〈f ,x〉.

Proof. Define

h(ν)ij (t, u) , Et

[e−

∫ ut νρ(Ms)ds1Mt=ei1Mu=ejfj

].

For 0 < ε ≤ u− t and ν ∈ N, we have

h(ν)ij (t, u) = Et

[e−

∫ t+εt νρ(Ms)ds1Mt=ei1Mt+ε=eie

−∫ ut+ε νρ(Ms)ds1Mu=ejfj

]+∑k 6=i

Et[e−

∫ t+εt νρ(Ms)ds1Mt=ei1Mt+ε=eke

−∫ ut+ε νρ(Ms)ds1Mu=ejfj

]= Et

[e−

∫ t+εt νρ(Ms)ds1Mt=ei1Mt+ε=ei

]Et+ε

[e−

∫ ut+ε νρ(Ms)ds1Mt+ε=ei1Mu=ejfj

]+∑k 6=i

Et[e−

∫ t+εt νρ(Ms)ds1Mt=ei1Mt+ε=ek

]Et+ε

[e−

∫ ut+ε νρ(Ms)ds1Mt+ε=ek1Mu=ejfj

]= (1 + ψiiε) e−νρ(ei)εEt+ε

[e−

∫ ut+ε νρ(Ms)ds1Mt+ε=ei1Mu=ejfj

]+∑k 6=i

ψikεe−νρ(ek)εEt+ε

[e−

∫ ut+ε νρ(Ms)ds1Mt+ε=ek1Mu=ejfj

]+ o(ε)

= (1 + ψiiε) e−νρ(ei)εh(ν)ij (t+ ε, u) +

∑k 6=i

ψikεe−νρ(ek)εh

(ν)kj (t+ ε, u) + o(ε),

where r(ε) = o(ε) implies that r(ε)ε→ 0 as ε ↓ 0. We then have

h(ν)ij (t+ ε, u)− h(ν)

ij (t, u)

ε=

1

ε

(1− e−νρ(ei)ε

)h

(ν)ij (t+ ε, u) +

d∑k=1

e−νρ(ek)εψikh(ν)kj (t+ ε, u) +

1

εo(ε),

from which it follows that

limε→0

h(ν)ij (t+ ε, u)− h(ν)

ij (t, u)

ε= νρ(ei)h

(ν)ij (t, u)−

d∑k=1

ψikh(ν)kj (t, u).

28

Since h(ν)(t, u) =(h(ν)(t, u)ij

)1≤i,j≤d and h

(ν)ii (u, u) = fi, we have, in matrix form,

dh(ν)(t,u)dt

= − (Ψ− νρdiag) h(ν)(t, u),

h(ν)(u, u) = fdiag.

Since Mt = ei, we therefore have

Et[e−

∫ ut νρ(Ms)dsf(Mu)

]= e>i

(h(ν)(t, u)

)1d = e>i exp

((Ψ− νρdiag

)(u− t)

)fdiag1d

= e>i exp((

Ψ− νρdiag

)(u− t)

)f .

Proof. (Theorem 3.1)We shall solve the optimal execution problem (Problem 1) using the dynamic programmingprinciple. Consider the following Antsaz of J ,

J(tn, Ftn ,Mtn , Xtn , Dtn) =(Ftn +

s

2

)Xtn + λX0Xtn + α(tn,Mtn)X2

tn + β(tn,Mtn)DtnXtn

+γ(tn,Mtn)D2tn . (38)

We begin with tN = T and proceed backwards in time. For tn = T , it is clear that x∗tN = XtN

as XtN is the remaining order at time tN in order to achieve XtN+= XT+ = 0. Hence, we

have

J(tN , FtN ,MtN , XtN , Dtn) =

[FtN +

s

2+ λ(X0 −XtN ) +DtN +

XtN

2qA

]XtN .

Using the Antsaz of J in (38) for n = N , it now becomes clear thatα(tN ,MtN ) = 1

2qA− λ,

β(tN ,MtN ) = 1,

γ(tN ,MtN ) = 0,

yielding (21).

For n = 0, ..., N − 1, write Et[.] , E[.|Ft]. Using the dynamic programming principle yields

J(tn, Ftn ,Mtn , Xtn , Dtn) = minxtn

[Ftn +

s

2+ λ(X0 −Xtn) +Dtn +

xtn2qA

]xtn

+Etn[J(tn+1, Ftn+1 ,Mtn+1 , Xtn+1 , Dtn+1)

]. (39)

To simplify the notation, we recall the abbreviation Jtn , J(tn, Ftn ,Mtn , Xtn , Dtn). Since

Xtn+1 = Xtn − xtn , Dtn+1 = (Dtn + κxtn)e−∫ tn+1tn

ρ(Ms)ds, and using the Antsaz of J in (38) for

29

n, we have:

Jtn = minxtn

[Ftn +

s

2+ λ(X0 −Xtn) +Dtn +

xtn2qA

]xtn + Etn

[(Ftn+1 +

s

2

)(Xtn − xtn)

+λX0 (Xtn − xtn) + α(tn+1,Mtn+1) (Xtn − xtn)2

+β(tn+1,Mtn+1) (Dtn + κxtn) e−∫ tn+1tn

ρ(Ms)ds (Xtn − xtn)

+γ(tn+1,Mtn+1) (Dtn + κxtn)2 e−2∫ tn+1tn

ρ(Ms)ds

].

By Assumption 2, Etn [Ftn+1 ] = Ftn . Note that xtn , Dtn in (9), and Xtn in (13) are Ftn-measurable. Hence, for all n = 0, ..., N , we have

Jtn = minxtn

[Ftn +

s

2+ λ(X0 −Xtn) +Dtn +

xtn2qA

]xtn +

(Ftn +

s

2

)(Xtn − xtn)

+λX0 (Xtn − xtn) + Etn[α(tn+1,Mtn+1)

](Xtn − xtn)2

+Etn[e−

∫ tn+1tn

ρ(Ms)dsβ(tn+1,Mtn+1)]

(Dtn + κxtn) (Xtn − xtn)

+Etn[e−2

∫ tn+1tn

ρ(Ms)dsγ(tn+1,Mtn+1)]

(Dtn + κxtn)2

= minxtn

[−λXtn +Dtn +

xtn2qA

]xtn +

(Ftn +

s

2

)Xtn + λX0Xtn

+α(tn,Mtn) (Xtn − xtn)2 + β(tn,Mtn) (Dtn + κxtn) (Xtn − xtn)

+γ(tn+1,Mtn+1) (Dtn + κxtn)2

=

(Ftn +

s

2

)Xtn + λX0Xtn + α(tn,Mtn)X2

tn + β(tn,Mtn)DtnXtn + γ(tn,Mtn)D2tn

+ minxtn

[1

2qA+ α(tn,Mtn)− κβ(tn,Mtn) + κ2γ(tn,Mtn)

]x2tn

−

[ (λ+ 2α(tn,Mtn)− κβ(tn,Mtn)

)Xtn −

(1− β(tn,Mtn) + 2κγ(tn,Mtn)

)Dtn

]xtn

.

Here the second equality follows from the definitions of α, β, and γ in (23)–(25). By Lemma2.3,

1

2qA+ α(tn,Mtn)− κβ(tn,Mtn) + κ2γ(tn,Mtn) > 0,

and hence δ(tn) in (27), for n = 0, ..., N − 1, is well-defined. Therefore, from the definitions

30

of A and B in (26), it follows that

Jtn =(Ftn +

s

2

)Xtn + λX0Xtn + α(tn,Mtn)X2

tn + β(tn,Mtn)DtnXtn + γ(tn,Mtn)D2tn

+ minxtn

(δ(tn,Mtn))−1 x2

tn − (A(tn,Mtn)Xtn −B(tn,Mtn)Dtn)xtn

=

(Ftn +

s

2

)Xtn + λX0Xtn + α(tn,Mtn)X2

tn + β(tn,Mtn)DtnXtn + γ(tn,Mtn)D2tn

−1

4δ(tn,Mtn) (A(tn,Mtn)Xtn −B(tn,Mtn)Dtn)2

+ minxtn

(δ(tn,Mtn))−1

[xtn −

1

2δ(tn,Mtn) [A(tn,Mtn)Xtn −B(tn,Mtn)Dtn ]

]2

=(Ftn +

s

2

)Xtn + λX0Xtn +

[α(tn,Mtn)− 1

4δ(tn,Mtn)A2(tn,Mtn)

]X2tn

+

[β(tn,Mtn) +

1

2δ(tn,Mtn)A(tn,Mtn)B(tn,Mtn)

]DtnXtn

+

[γ(tn,Mtn)− 1

4δ(tn,Mtn)B2(tn,Mtn)

]D2tn , (40)

where the last equality follows from Jtn and is minimized when xtn = x∗tn in (19).

Since Jtn admits the form (38), collecting the coefficients of X2tn , D2

tn , and XtnDtn in (40),we have

α(tn,Mtn) = α(tn,Mtn)− 14δ(tn,Mtn)A2(tn,Mtn),

β(tn,Mtn) = β(tn,Mtn) + 12δ(tn,Mtn)A(tn,Mtn)B(tn,Mtn),

γ(tn,Mtn) = γ(tn,Mtn)− 14δ(tn,Mtn)B2(tn,Mtn),

yielding (22).

It remains to obtain the explicit forms of α(tn,Mtn), β(tn,Mtn), and γ(tn,Mtn), in (23)–(25),for tn, n = 0, ..., N . By Lemma A.1 with ν = 0, ν = 1, and ν = 2, respectively, we have

α(tn,Mtn) = Etn[α(tn+1,Mtn+1)

]=

⟨exp (Ψ(tn+1 − tn))αtn+1 ,Mtn

⟩;

β(tn,Mtn) = Etn[e−

∫ tn+1tn

ρ(Ms)dsβ(tn+1,Mtn+1)]

=⟨

exp((

Ψ− ρdiag

)(tn+1 − tn)

)βtn+1

,Mtn

⟩;

γ(tn,Mtn) = Etn[e−

∫ tn+1tn

2ρ(Ms)dsγ(tn+1,Mtn+1)]

=⟨

exp((

Ψ− 2ρdiag

)(tn+1 − tn)

)γtn+1

,Mtn

⟩,

yielding the second equalities in (23)–(25). The result now follows.

31

Proof. (Proof of Lemma 4.1)Since the execution strategy xSItn

Nn=0 is state-independent, the remaining position XSI

tn Nn=0

is also state-independent. Recall the abbreviation JSItn , JSI(tn, Ftn ,Mtn , XSItn , Dtn). Direct

computation shows that

JSItn =N∑j=n

Etn

[(Atj +

xSItj2qA

)xSItj

]=

(Ftn +

s

2

)XSItn + λX0X

SItn − λ

N∑j=n

XSItjxSItj

+N∑j=n

(xSItj )2

2qA+Dtn

N∑j=n

Etn[e−

∫ tjtnρ(Ms)ds

]+

N∑j=n

xSItj

(j−1∑k=n

κxSItk Etn[e−

∫ tjtkρ(Ms)ds

]).

It remains to compute Etn[e−

∫ tjtnρ(Ms)ds

]and Etn

[e−

∫ tjtkρ(Ms)ds

]. Applying Lemma A.1 gives

Etn[e−

∫ tjtnρ(Ms)ds

]= M>

tn exp ((Ψ− ρdiag) (tj − tn)) 1d.

Etn[e−

∫ tjtkρ(Ms)ds

]= M>

tn exp (Ψ(tk − tn)) exp ((Ψ− ρdiag) (tj − tk)) 1d.

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