Optimal trade execution and price manipulation in order books with time-varying liquidity
aa r X i v : . [ q -f i n . T R ] S e p Optimal trade execution and price manipulation in order bookswith time-varying liquidity ∗ Antje Fruth † Torsten Schöneborn ‡ Mikhail Urusov § October 8, 2018
Abstract
In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. Inthis article we consider a limit order book model that allows for time-dependent, deterministic depthand resilience of the book and determine optimal portfolio liquidation strategies. In a first modelvariant, we propose a trading dependent spread that increases when market orders are matchedagainst the order book. In this model no price manipulation occurs and the optimal strategy is ofthe wait region - buy region type often encountered in singular control problems. In a second model,we assume that there is no spread in the order book. Under this assumption we find that pricemanipulation can occur, depending on the model parameters. Even in the absence of classical pricemanipulation there may be transaction triggered price manipulation. In specific cases, we can statethe optimal strategy in closed form.
KEYWORDS: Market impact model, optimal order execution, limit order book, resilience, time-varyingliquidity, price manipulation, transaction-triggered price manipulation
Empirical investigations have demonstrated that liquidity varies over time. In particular deterministictime-of-day and day-of-week liquidity patterns have been found in most markets, see, e.g., Chordia, Roll,and Subrahmanyam (2001), Kempf and Mayston (2008) and Lorenz and Osterrieder (2009). In spiteof these findings the academic literature on optimal trade execution usually assumes constant liquidityduring the trading time horizon. In this paper we relax this assumption and analyze the effects ofdeterministically varying liquidity on optimal trade execution for a risk-neutral investor. We characterizeoptimal strategies in terms of a trade region and a wait region and find that optimal trading strategiesdepend on the expected pattern of time-dependent liquidity. In the case of extreme changes in liquidity, ∗ We would like to thank Peter Bank for valuable suggestions. † Technische Universität Berlin, Germany, [email protected] ‡ Deutsche Bank AG, London, UK, [email protected] § Ulm University, Germany, [email protected] Not all changes in liquidity are deterministic; an additional stochastic component has been investigated empirically by,e.g., Esser and Mönch (2003) and Steinmann (2005). See, e.g., Fruth (2011) for an analysis of the implications of suchstochastic liquidity on optimal trade execution.
1t can even be optimal to entirely refrain from trading in periods of low liquidity. Incorporating suchpatterns in trade execution models can hence lower transaction costs.Time-dependent liquidity can potentially lead to price manipulation. In periods of low liquidity, a tradercould buy the asset and push market prices up significantly; in a subsequent period of higher liquidity,he might be able to unwind this long position without depressing market prices to their original level,leaving the trader with a profit after such a round trip trade. In reality such round trip trades are often notprofitable due to the bid-ask spread: once the trader starts buying the asset in large quantities, the spreadwidens, resulting in a large cost for the trader when unwinding the position. We propose a model withtrading-dependent spread and demonstrate that price manipulation does not exist in this model in spiteof time-dependent liquidity. In a similar model with fixed zero spread we find that price manipulation ortransaction-triggered price manipulation (a term recently coined by Alfonsi, Schied, and Slynko (2011)and Gatheral, Schied, and Slynko (2011b)) can be a consequence of time-dependent liquidity. Phenomenaof such type, i.e. existence of “illusory arbitrages”, which disappear when bid-ask spread is taken intoaccount, are also observed in different modelling approaches (see e.g. Section 5.1 in Madan and Schoutens(2011)).Our liquidity model is based on the limit order book market model of Obizhaeva and Wang (2006), whichmodels both depth and resilience of the order book explicitly. The instantaneously available liquidity inthe order book is described by the depth. Market orders issued by the large investor are matched withthis liquidity, which increases the spread. Over time, incoming limit orders replenish the order book andreduce the spread; the speed of this process is determined by the resilience. In our model both depth andresilience can be independently time dependent. We show that there is a time dependent optimal ratio ofremaining order size to bid-ask spread: If the actual ratio is larger than the optimal ratio, then the traderis in the “trade region” and it is optimal to reduce the ratio by executing a part of the total order. If theactual ratio is smaller than the optimal ratio, then the trader is in the “wait region” and it is optimal towait for the spread to be reduced by future incoming limit orders before continuing to trade.Building on empirical investigations of the market impact of large transactions, a number of theoreticalmodels of illiquid markets have emerged. One part of these market microstructure models focuses on theunderlying mechanisms for illiquidity effects, e.g., Kyle (1985) and Easley and O’Hara (1987). We followa second line that takes the liquidity effects as given and derives optimal trading strategies within sucha stylized model market. Two broad types of market models have been proposed for this purpose. First,several models assume an instantaneous price impact, e.g., Bertsimas and Lo (1998), Almgren and Chriss(2001) and Almgren (2003). The instantaneous price impact typically combines depth and resilience ofthe market into one stylized quantity. Time-dependent liquidity in this setting then leads to executingthe constant liquidity strategy in volume time or liquidity time, and no qualitatively new features occur.In a second group of models resilience is finite and depth and resilience are separately modelled, e.g.,Bouchaud, Gefen, Potters, and Wyart (2004), Obizhaeva and Wang (2006), Alfonsi, Fruth, and Schied(2010) and Predoiu, Shaikhet, and Shreve (2011). Our model falls into this last group. Allowing forindependently time-dependent depth and resilience leads to higher technical complexity, but allows us tocapture a wider range of real world phenomena.The remainder of this paper is structured as follows. In the next section, we introduce the market modeland formulate an optimization problem. In Section 3, we show that this model is free of price manipulation,which allows us to simplify the model setup and the optimization problem in Section 4. Before we state ourmain results on existence, uniqueness and characterization of the optimal trading strategy in Sections 6to 7, we first provide some elementary properties, like the dimension reduction of our control problem, inSection 5. Section 6 discusses the case where trading is constrained to discrete time and Section 7 containsthe continuous time case. In Section 8 we investigate under which conditions price manipulation occurs2n a zero spread model. In some special cases, we can calculate optimal strategies in closed form for ourmain model as well as for the zero spread model of Section 8; we provide some examples in Section 9.Section 10 concludes.
In order to attack the problem of optimal trade execution under time-varying liquidity, we first need tospecify a price impact model in Section 2.1. Our model is based on the work of Obizhaeva and Wang(2006), but allows for time-varying order book depth and resilience. Furthermore we explicitly modelboth sides of the limit order book and thus can allow for strategies that buy and sell at different pointsin time. After having introduced the limit order book model, we specify the trader’s objectives in Section2.2.
Trading at most public exchanges is executed through a limit order book, which is a collection of thelimit orders of all market participants in an electronic market. Each limit order has the number of shares,that the market participant wants to trade, and a price per share attached to it. The price represents aminimal price in case of a sell and a maximal price in case of a buy order. Compared to a limit order,a market order does not have an attached price per share, but instead is executed immediately againstthe best limit orders waiting in the book. Thus, there is a tradeoff between price saving and immediacywhen using limit and market orders. We refer the reader to Cont, Stoikov, and Talreja (2010) for a morecomprehensive introduction to limit order books.In this paper we consider a one-asset model that derives its price dynamics from a limit order book thatis exposed to repeated market orders of a large investor (sometimes referred to as the trader). The goalof the investor is to use market orders in order to purchase a large amount x of shares within a certaintime period [0 , T ] , where T typically ranges from a few hours up to a few trading days. Without lossof generality we assume that the investor needs to purchase the asset (the sell case is symmetrical) andhence first describe how buy market orders interact with the ask side of the order book (i.e., with the selllimit orders contained in the limit order book). Subsequently we turn to the impact of buy market orderson the bid side and of sell market orders on both sides of the limit order book.Suppose first that the trader is not active. We assume that the corresponding unaffected best ask price A u (i.e. the lowest ask price in the limit order book) is a càdlàg martingale on a given filtered probabilityspace (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) satisfying the usual conditions. This unaffected price process is capturingall external price changes including those due to news as well as due to trading of noise traders andinformed traders. Our model includes in particular the case of the Bachelier model A ut = A u + σW At witha ( F t ) -Brownian motion W A , as considered in Obizhaeva and Wang (2006). It also includes the driftlessgeometric Brownian motion A ut = A u exp( σW At − σ t ) , which avoids the counterintuitive negative pricesof the Bachelier model. Moreover, we can allow for jumps in the dynamics of A u .We now describe the shape of the limit order book, i.e. the pattern of ask prices in the order book. Wefollow Obizhaeva and Wang (2006) and assume a block-shaped order book: The number of shares offered On this macroscopic time scale, the restriction to market orders is not severe. A subsequent consideration of small timewindows including limit order trading is common practice in banks. See Naujokat and Westray (2011) for a discussion of alarge investor execution problem where both market and limit orders are allowed. umber of shares Price per shareB t A ut t A =A +D t+ t t+u
A =A +D t t tu
Resiliencemarket buyorder:shares x t extralimit buy orders spread spread limit sell orders Figure 1:
Snapshot of the block-shaped order book model at time t . at prices in the interval [ A ut , A ut + ∆ A ] is given by q t · ∆ A with q t > being the order book height (seeFigure 1 for a graphical illustration). Alfonsi, Fruth, and Schied (2010) and Predoiu, Shaikhet, and Shreve(2011) consider order books which are not block shaped and conclude that the optimal execution strategyof the investor is robust with respect to the order book shape. In our model, we allow the order bookdepth q t to be time dependent. As mentioned above, various empirical studies have demonstrated thetime-varying features of liquidity, including order book depth. In theoretical models however, liquidity isstill usually assumed to be constant in time. To our knowledge first attempts to non-constant liquidityin portfolio liquidation problems has only been considered so far in extensions of the Almgren and Chriss(2001) model such as Kim and Boyd (2008) and Almgren (2009). In this modelling framework, priceimpact is purely temporary and several of the aspects of this paper do not surface.Let us now turn to the interaction of the investor’s trading with the order book. At time t , the best ask A t might differ from the unaffected best ask A ut due to previous trades of the investor. Define D t := A t − A ut as the price impact or extra spread caused by the past actions of the trader. Suppose that the traderplaces a buy market order of ξ t > shares. This market order consumes all the shares offered at pricesbetween the ask price A t just prior to order execution and A t + immediately after order execution. A t + isgiven by ( A t + − A t ) · q t = ξ t and we obtain D t + = D t + ξ t /q t . See Figure 1 for a graphical illustration.It is a well established empirical fact that the price impact D exhibits resilience over time. We assumethat the immediate impact ξ t /q t can be split into a temporary impact component K t ξ t which decays tozero and a permanent impact component γξ t with γ + K t = q − t . We assume that the temporary impact decays exponentially with a fixed time-dependent, deterministicrecovery rate ρ t > . The price impact at time s ≥ t of a buy market order ξ t > placed at time t isassumed to be γξ t + K t e − R st ρ u du ξ t . Notice that this temporary impact model is different to the one which is used, e.g., in Almgren and Chriss(2001) and Almgren (2003). It slowly decays to zero instead of vanishing immediately and thus pricesdepend on previous trades. Obizhaeva and Wang (2006) limit their analysis to a constant decay rate4 t ≡ ρ , but suggest the extension to time dependent ρ t . Weiss (2010) considers exponential resilienceand shows that the results of Alfonsi, Fruth, and Schied (2010) and in particular Obizhaeva and Wang(2006) can be adapted when the recovery rate depends on the extra spread D caused by the large investor.Gatheral (2010) considers more general deterministic decay functions than the exponential one in a modelwith a potentially non-linear price impact and discusses which combinations of decay function and priceimpact yield ’no arbitrage’, i.e. non-negative expected costs of a round trip. Alfonsi, Schied, and Slynko(2011) study the optimal execution problem for more general deterministic decay functions than theexponential one in a model with constant order book height. For the calibration of resilience see Large(2007) and for a discussion of a stochastic recovery rate ρ we refer to Fruth (2011).Let us now discuss the impact of market buy orders on the bid side of the limit order book. According tothe mechanics of the limit order book, a single market buy order ξ t directly influences the best ask A t + ,but does not influence the best bid price B t + = B t immediately. The best ask A t + recovers over time (inthe absence of any other trading from the investor) on average to A t + γξ t . In reality market orders onlylead to a temporary widening of the spread. In order to close the spread, B t needs to move up by γξ t over time and converge to B t + γξ t , i.e. the buy market order ξ t influences the future evolution of B . Weassume that B converges to this new level exponentially with the same rate ρ t . The price impact on thebest bid B s at time s ≥ t of a buy market order ξ t > placed at time t is hence γ (cid:16) − e − R st ρ u du (cid:17) ξ t . We assume that the impact of sell market orders is symmetrical to that of buy market orders. It shouldbe noted that our model deviates from the existing literature by explicitly modelling both sides of theorder book with a trading dependent spread. For example Obizhaeva and Wang (2006) only model oneside of the order book and restrict trading to this side of the book. Alfonsi, Schied, and Slynko (2011),Gatheral, Schied, and Slynko (2011b) and Gatheral, Schied, and Slynko (2011a) on the other hand allowfor trading on both sides of the order book, but assume that there is no spread, i.e. they assume A u = B u for unaffected best ask and best bid prices, and that the best bid moves up instantaneously when a marketbuy order is matched with the ask side of the book. They find that under this assumption the modelparameters (for example the decay kernel) need to fulfill certain conditions, otherwise price manipulationarises. We will revisit this topic in Sections 3 and 8.We can now summarize the dynamics of the best ask A t and best bid B t for general trading strategiesin continuous time. Let Θ and ˜Θ be increasing processes that describe the number of shares which theinvestor bought respectively sold from time until time t . We then have A t = A ut + D t ,B t = B ut − E t , where D t = D e − R t ρ s ds + Z [0 ,t ) (cid:16) γ + K s e − R ts ρ u du (cid:17) d Θ s − Z [0 ,t ) γ (cid:16) − e − R ts ρ u du (cid:17) d ˜Θ s , t ∈ [0 , T +] , (1) E t = E e − R t ρ s ds + Z [0 ,t ) (cid:16) γ + K s e − R ts ρ u du (cid:17) d ˜Θ s − Z [0 ,t ) γ (cid:16) − e − R ts ρ u du (cid:17) d Θ s , t ∈ [0 , T +] , (2)with some given nonnegative initial price impacts D ≥ and E ≥ . Assumption 2.1 (Basic assumptions on Θ , ˜Θ , A u , B u , K , and ρ ) . Throughout this paper, we assume the following. 5
The set of admissible strategies is given as ˜ A := (cid:8) (Θ , ˜Θ) : Ω × [0 , T +] → [0 , ∞ ) | Θ and ˜Θ are ( F t ) -adapted nondecreasingbounded càglàd processes with (Θ , ˜Θ ) = (0 , (cid:9) . Note that (Θ , ˜Θ) may have jumps. In particular, trading in rates and impulse trades are allowed. • The unaffected best ask price process A u is a càdlàg H -martingale with a deterministic startingpoint A u , i.e. E p [ A u , A u ] T < ∞ , or, equivalently, E sup t ∈ [0 ,T ] | A ut | < ∞ . The same condition holds for the unaffected best bid price B u . Furthermore, B ut ≤ A ut for all t ∈ [0 , T ] . • The price impact coefficient K : [0 , T ] → (0 , ∞ ) is a deterministic strictly positive bounded Borelfunction. • The resilience speed ρ : [0 , T ] → (0 , ∞ ) is a deterministic strictly positive Lebesgue integrable func-tion. Remark 2.2. i) The purchasing component Θ of a strategy from ˜ A consists of a left-continuous nondecreasing process (Θ t ) t ∈ [0 ,T ] and an additional random variable Θ T + with ∆Θ T := Θ T + − Θ T ≥ being the lastpurchase of the strategy. Similarly, for t ∈ [0 , T ] , we use the notation ∆Θ t := Θ t + − Θ t . The sameconventions apply for the selling component ˜Θ .ii) The processes D and E depend on (Θ , ˜Θ) , although this is not explicitly marked in their notation.iii) As it is often done in the literature on optimal portfolio execution, Θ , ˜Θ , D and E are assumed tobe càglàd processes. In (1), the possibility t = T + is by convention understood as D T + = D e − R T ρ s ds + Z [0 ,T ] (cid:16) γ + K s e − R Ts ρ u du (cid:17) d Θ s − Z [0 ,T ] γe − R Ts ρ u du d ˜Θ s . A similar convention applies to all other formulas of such type. Furthermore, the integrals of theform Z [0 ,t ) K s d Θ s or Z [0 ,t ] K s d Θ s , are understood as pathwise Lebesgue-Stieltjes integrals, i.e. Lebesgue integrals with respect to themeasure with the distribution function s Θ s + .iv) In the sequel, we need to apply stochastic analysis (e.g. integration by parts or Ito’s formula) to càglàdprocesses of finite variation and/or standard semimartingales. This will always be done as follows:if U is a càglàd process of finite variation, we first consider the process U + defined by U + t := U t + andthen apply standard formulas from stochastic analysis to it. An example (which will be often usedin proofs) is provided in Appendix A. 6 .2 Optimization problem Let us go ahead by describing the cost minimization problem of the trader. When placing a single buymarket order of size ξ t ≥ at time t , he purchases at prices A ut + d , with d ranging from D t to D t + , seeFigure 1. Due to the block-shaped limit order book, the total costs of the buy market order amount to ( A ut + D t ) ξ t + D t + − D t ξ t = ( A ut + D t ) ξ t + ξ t q t = ξ t (cid:18) A t + ξ t q t (cid:19) . Thus, the total costs of the buy market order are the number of shares ξ t times the average price pershare ( A t + ξ t q t ) . More generally, the total costs of a strategy (Θ , ˜Θ) ∈ ˜ A are given by the formula C (Θ , ˜Θ) := Z [0 ,T ] (cid:18) A t + ∆Θ t q t (cid:19) d Θ t − Z [0 ,T ] B t − ∆ ˜Θ t q t ! d ˜Θ t . We now collect all admissible strategies that build up a position of x ∈ [0 , ∞ ) shares until time T in theset ˜ A ( x ) := n (Θ , ˜Θ) ∈ ˜ A | Θ T + − ˜Θ T + = x a.s. o . Our aim is to minimize the expected execution costs inf (Θ , ˜Θ) ∈ ˜ A ( x ) E C (Θ , ˜Θ) . (3)We hence consider the large investor to be risk-neutral and explicitly allow for his optimal strategy toconsist of both buy and sell orders. In the next section, we will see that in our model it is never optimalto submit sell orders when the overall goal is the purchase of x > shares.Let us finally note that problem (3) with x ∈ ( −∞ , is the problem of maximizing the expected proceedsfrom liquidation of | x | shares and, due to symmetry in modelling ask and bid sides, can be consideredsimilarly to problem (3) with x ∈ [0 , ∞ ) . Market manipulation has been a concern for price impact models for some time. We now define thecounterparts in our model of the notions of price manipulation in the sense of Huberman and Stanzl(2004) and of transaction-triggered price manipulation in the sense of Alfonsi, Schied, and Slynko (2011)and Gatheral, Schied, and Slynko (2011b). Note that in defining these notions in our model we explicitlyaccount for the possibility of D and E being nonzero. Definition 3.1. A round trip is a strategy from ˜ A (0) . A price manipulation strategy is a round trip (Θ , ˜Θ) ∈ ˜ A (0) with strictly negative expected execution costs E C (Θ , ˜Θ) < . A market impact model(represented by A u , B u , K , and ρ ) admits price manipulation if there exist D ≥ , E ≥ and (Θ , ˜Θ) ∈ ˜ A (0) with E C (Θ , ˜Θ) < . Definition 3.2.
A market impact model (represented by A u , B u , K , and ρ ) admits transaction-triggeredprice manipulation if the expected execution costs of a buy (or sell) program can be decreased by inter-mediate sell (resp. buy) trades. More precisely, this means that there exist x ∈ [0 , ∞ ) , D ≥ , E ≥ and (Θ , ˜Θ ) ∈ ˜ A ( x ) with E C (Θ , ˜Θ ) < inf { E C (Θ , | (Θ , ∈ ˜ A ( x ) } (4)7r there exist x ∈ ( −∞ , , D ≥ , E ≥ and (Θ , ˜Θ ) ∈ ˜ A ( x ) with E C (Θ , ˜Θ ) < inf { E C (0 , ˜Θ) | (0 , ˜Θ) ∈ ˜ A ( x ) } . (5)Clearly, if a model admits price manipulation, then it admits transaction-triggered price manipulation.But transaction-triggered price manipulation can be present even if price manipulation does not exist ina model. This situation has been demonstrated in limit order book models with zero bid-ask spread bySchöneborn (2008) (Chapter 9) in a multi-agent setting and by Alfonsi, Schied, and Slynko (2011) in asetting with non-exponential decay of price impact. In this section, we will show that the limit order bookmodel introduced in Section 2 is free from both classical and transaction-triggered price manipulation. InSection 8 we will revisit this topic for a different (but related) limit order book model.Before attacking the main question of price manipulation in Proposition 3.4, we consider the expectedexecution costs of a pure purchasing strategy and verify in Proposition 3.3 that the costs resulting fromchanges in the unaffected best ask price are zero and that the costs due to permanent impact are the samefor all strategies. Proposition 3.3 (Only temporary impact has to be considered) . Let (Θ , ˜Θ) ∈ ˜ A ( x ) with x ∈ [0 , ∞ ) and ˜Θ ≡ . Then E "Z [0 ,T ] (cid:18) A t + ∆Θ t q t (cid:19) d Θ t = A u x + γ x + E "Z [0 ,T ] (cid:18) D γ =0 t + K t t (cid:19) d Θ t (6) with D γ =0 t := D e − R t ρ s ds + Z [0 ,t ) K s e − R ts ρ u du d Θ s , t ∈ [0 , T +] . (7) Proof.
We start by looking at the expected costs caused by the unaffected best ask price martingale.Using (48) with U := Θ , Z := A u , the facts that Θ is bounded and that A u is an H -martingale yield E "Z [0 ,T ] A ut d Θ t = E [ A uT Θ T + − A u Θ ] = A u x. (8)Let us now turn to the simplification of our optimization problem due to permanent impact. To this end,we differentiate between the temporary price impact D γ =0 t and the total price impact D t = D γ =0 t + γ Θ t that we get by adding the permanent impact. Notice that ˜Θ ≡ . We can then write E "Z [0 ,T ] (cid:18) A ut + D t + ∆Θ t q t (cid:19) d Θ t = A u x + E "Z [0 ,T ] (cid:18) D γ =0 t + γ Θ t + γ + K t t (cid:19) d Θ t = A u x + E "Z [0 ,T ] (cid:18) D γ =0 t + K t t (cid:19) d Θ t + γ E "Z [0 ,T ] (cid:18) Θ t + ∆Θ t (cid:19) d Θ t . The assertion follows, since integration by parts for càglàd processes (see (49) with U = V := Θ ) and Θ =0 , Θ T + = x yield Z [0 ,T ] (cid:18) Θ t + ∆Θ t (cid:19) d Θ t = Θ T + − Θ x . (9)8e can now proceed to prove that our model is free of price manipulation and transaction-triggered pricemanipulation. Proposition 3.4 (Absence of transaction-triggered price manipulation) . In the model of Section 2, there is no transaction-triggered price manipulation. In particular, there is noprice manipulation.Proof.
Consider x ∈ [0 , ∞ ) and (Θ , ˜Θ) ∈ ˜ A ( x ) . Making use of B t = B ut − E t ≤ A ut − E t ≤ A ut + γ (cid:16) Θ t − ˜Θ t (cid:17) yields E "Z [0 ,T ] (cid:18) A t + ∆Θ t q t (cid:19) d Θ t − E "Z [0 ,T ] B t − ∆ ˜Θ t q t ! d ˜Θ t ≥ E "Z [0 ,T ] (cid:18) A ut + γ Θ t + D γ =0 t − γ ˜Θ t + γ t + K t t (cid:19) d Θ t − E "Z [0 ,T ] (cid:18) A ut + γ Θ t − γ ˜Θ t − γ t − K t t (cid:19) d ˜Θ t ≥ E "Z [0 ,T ] A ut d (Θ t − ˜Θ t ) + γ E "Z [0 ,T ] (cid:18) Θ t − ˜Θ t + ∆Θ t (cid:19) d Θ t + Z [0 ,T ] ˜Θ t − Θ t + ∆ ˜Θ t ! d ˜Θ t + E "Z [0 ,T ] (cid:18) D γ =0 t + K t t (cid:19) d Θ t . Analogously to (8), the first of these terms equals A u x since Θ , ˜Θ are bounded and A u is an H -martingale.For the second one, we do integration by parts (use (49) three times) to deduce Z [0 ,T ] (2Θ t + ∆Θ t ) d Θ t + Z [0 ,T ] (cid:16) t + ∆ ˜Θ t (cid:17) d ˜Θ t − Z [0 ,T ] ˜Θ t d Θ t − Z [0 ,T ] Θ t d ˜Θ t = Θ T + + ˜Θ T + − T + ˜Θ T + − Z [0 ,T ] ˜Θ t d Θ t + 2 Z [0 ,T ] ˜Θ t + d Θ t ≥ (cid:16) Θ T + − ˜Θ T + (cid:17) = x . That is we have shown E "Z [0 ,T ] (cid:18) A t + ∆Θ t q t (cid:19) d Θ t − Z [0 ,T ] B t − ∆ ˜Θ t q t ! d ˜Θ t ≥ A u x + γ x + E "Z [0 ,T ] (cid:18) D γ =0 t + K t t (cid:19) d Θ t and thanks to Proposition 3.3 the right-hand side is larger or equal to the expected execution costs of thestrategy ( ˇΘ , ∈ ˜ A ( x ) with ˇΘ t := (cid:26) Θ t if Θ t ≤ xx otherwise (cid:27) . (Θ , ˜Θ) ∈ ˜ A ( x ) containing a selling component isalways greater or equal to the costs of the modified strategy ( ˇΘ , ∈ ˜ A ( x ) without a selling component.Thus, (4) does not occur. By a similar reasoning, (5) does not occur as well.The central economic insight captured in the previous proposition is that price manipulation strategiescan be severely penalized by a widening spread. This idea can easily be applied to different variations ofour model, for example to non-exponential decay kernels as in Gatheral, Schied, and Slynko (2011b). Due to Propositions 3.3 and 3.4, we can significantly simplify the optimization problem (3). Let us fix x ∈ [0 , ∞ ) . Then it is enough to minimize the expectation in the right-hand side of (6) over the purepurchasing strategies that build up the position of x shares until time T . That is to say, the problemin general reduces to that with A u ≡ , γ = 0 , ˜Θ ≡ . Moreover, due to (6), (7) and the fact that K and ρ are deterministic functions, it is enough to minimize over deterministic purchasing strategies. Weare going to formulate the simplified optimization problem, where we now consider a general initial time t ∈ [0 , T ] because we will use dynamic programming afterwards.Let us define the following simplified control sets only containing deterministic purchasing strategies: A t := (cid:8) Θ : [ t, T +] → [0 , ∞ ) | Θ is a deterministicnondecreasing càglàd function with Θ t = 0 (cid:9) , A t ( x ) := { Θ ∈ A t | Θ T + = x } . As above, a strategy from A t consists of a left-continuous nondecreasing function (Θ s ) s ∈ [ t,T ] and anadditional value Θ T + ∈ [0 , ∞ ) with ∆Θ T := Θ T + − Θ T ≥ being the last purchase of the strategy. Forany fixed t ∈ [0 , T ] and δ ∈ [0 , ∞ ) , we define the cost function J ( t, δ, · ) : A t → [0 , ∞ ) as J (Θ) := J ( t, δ, Θ) := Z [ t,T ] (cid:18) D s + K s s (cid:19) d Θ s , (10)where D s := δe − R st ρ u du + Z [ t,s ) K u e − R su ρ r dr d Θ u , s ∈ [ t, T +] . (11)The cost function J represents the total temporary impact costs of the strategy Θ on the time interval [ t, T ] when the initial price impact D t = δ . Observe that J is well-defined and finite due to Assumption 2.1.Let us now define the value function for continuous trading time U : [0 , T ] × [0 , ∞ ) → [0 , ∞ ) as U ( t, δ, x ) := inf Θ ∈A t ( x ) J ( t, δ, Θ) . (12)We also want to discuss discrete trading time , i.e. when trading is only allowed at given times t < t < ... < t N = T. Define ˜ n ( t ) := inf { n = 0 , ..., N | t n ≥ t } . We then have to constrain our strategy sets to A Nt := (cid:8) Θ ∈ A t | Θ s = 0 on [ t, t ˜ n ( t ) ] , Θ s = Θ t n + on ( t n , t n +1 ] for n = ˜ n ( t ) , ..., N − (cid:9) ⊂ A t , A Nt ( x ) := (cid:8) Θ ∈ A Nt | Θ T + = x (cid:9) ⊂ A t ( x ) , value function for discrete trading time becomes U N ( t, δ, x ) := inf Θ ∈A Nt ( x ) J ( t, δ, Θ) ≥ U ( t, δ, x ) . (13)Note that the optimization problems in continuous time (12) and in discrete time (13) only refer to theask side of the limit order book. The results for optimal trading strategies that we derive in the followingsections are hence applicable not only to the specific limit order book model introduced in Section 2,but also to any model which excludes transaction-triggered price manipulation and where the ask priceevolution for pure buying strategies is identical to the ask price evolution in our model. This includesfor example models with different depth of the bid and ask sides of the limit order book, or differentresiliences of the two sides of the book.We close this section with the following simple result, which shows that our problem is economicallysensible. Lemma 4.1 (Splitting argument) . Doing two separate trades ξ α , ξ β > at the same time s has the same effect as trading at once ξ := ξ α + ξ β ,i.e., both alternatives incur the same impact costs and the same impact D s + .Proof. The impact costs are in both cases (cid:18) D s + K s ξ (cid:19) ξ = D s ( ξ α + ξ β ) + K s (cid:0) ξ α + 2 ξ α ξ β + ξ β (cid:1) = (cid:18) D s + K s ξ α (cid:19) ξ α + (cid:18) D s + K s ξ α + K s ξ β (cid:19) ξ β and the impact D s + = D s + K s ( ξ α + ξ β ) after the trade is the same in both cases as well. In this section, we first show that in our model optimal strategies are linear in ( δ, x ) , which allows usto reduce the dimensionality of our problem from three dimensions to two dimensions. Thereafter, weintroduce the concept of WR-BR structure in Section 5.2, which appropriately describes the value functionand optimal execution strategies in our model as we will see in Sections 6 and 7. Finally, we establishsome elementary properties of the value function and optimal strategies in Section 5.3.In this entire section, we usually refer only to the continuous time setting, for example, to the valuefunction U . We refer to the discrete time setting only when there is something there to be added explicitly.But all of the statements in this section hold both in continuous time (i.e. for U ) and in discrete time(i.e. for U N ), and we will later use them in both situations. In this section, we prove a scaling property of the value function which helps us to reduce the dimensionof our optimization problem. Our approach exploits both the block shape of the limit order book and theexponential decay of price impact and hence does not generalize easily to more general dynamics of D as,11.g., in Predoiu, Shaikhet, and Shreve (2011). We formulate the result for continuous time, although italso holds for discrete time. Lemma 5.1 (Optimal strategies scale linearly) . For all a ∈ [0 , ∞ ) we have U ( t, aδ, ax ) = a U ( t, δ, x ) . (14) Furthermore, if Θ ∗ ∈ A t ( x ) is optimal for U ( t, δ, x ) , then a Θ ∗ ∈ A t ( ax ) is optimal for U ( t, aδ, ax ) .Proof. The assertion is clear for a = 0 . For any a ∈ (0 , ∞ ) and Θ ∈ A t , we get from (10) and (11) that J ( t, aδ, a Θ) = a J ( t, δ, Θ) . (15)Let Θ ∗ ∈ A t ( x ) be optimal for U ( t, δ, x ) and ¯Θ ∈ A t ( ax ) be optimal for U ( t, aδ, ax ) . If no such optimalstrategies exist, the same arguments can be performed with minimizing sequences of strategies. Using (15)two times and the optimality of Θ ∗ , ¯Θ , we get J ( t, aδ, ¯Θ) ≤ J ( t, aδ, a Θ ∗ ) = a J ( t, δ, Θ ∗ ) ≤ a J (cid:18) t, δ, a ¯Θ (cid:19) = J ( t, aδ, ¯Θ) . Hence, all inequalities are equalities. Therefore, a Θ ∗ is optimal for U ( t, aδ, ax ) and (14) holds.For δ > , we can take a = δ and apply Lemma 5.1 to get U ( t, δ, x ) = δ U (cid:16) t, , xδ (cid:17) = δ V ( t, y ) with (16) y := xδ ,V ( t, y ) := U ( t, , y ) , V ( T, y ) = y + K T y , V ( t, ≡ . In this way we are able to reduce our three-dimensional value function U defined in (12) to a two-dimensional function V . That is U ( t, δ fix , x ) for some δ fix > or U ( t, δ, x fix ) for some x fix > alreadydetermines the entire value function . Instead of keeping track of the values x and δ separately, only theratio of them is important. It should be noted however that the function V itself is not necessarily thevalue function of a modified optimization problem. In a similar way we define the function V N throughthe function U N . Let us consider an investor who at time t needs to purchase a position of x > in the remaining timeuntil T and is facing a limit order book dislocated by D t = δ ≥ . Any trade ξ t at time t is decreasingthe number of shares that are still to be bought, but is increasing D at the same time (see Figure 2 fora graphical representation). In the δ - x -plane, the investor can hence move downwards and to the right.Note that due to the absence of transaction-triggered price manipulation (as shown in Proposition 3.4)any intermediate sell orders are suboptimal and hence will not be considered. In the following, we will often analyze the function V in order to derive properties of U . Technically this does notdirectly allow us to draw conclusions for U ( t, , x ) , where δ = 0 , since in this case y = x/δ is not defined. The extensionof our proofs to the possibility δ = 0 however is straightforward using continuity arguments (see Proposition 5.5 below) oralternatively by analyzing ˜ V ( t, ˜ y ) := U ( t, ˜ y, . Barrierc(t)=x/ d Buy Wait d ( +K ,x- ) d x t t x t ( ,x) d Trade x t Figure 2:
The δ - x -plane for fixed time t . Intuitively one might expect the large investor to behave as follows: If there are many shares x left to bebought and the price deviation δ is small, then the large investor would buy some shares immediately. Inthe opposite situation, i.e. small x and large δ , he would defer trading and wait for a decrease of the pricedeviation due to resilience. We might hence conjecture that the δ - x -plane is divided by a time-dependentbarrier into one buy region above and one wait region below the barrier. Based on the linear scalingof optimal strategies (Lemma 5.1), we know that if ( δ, x ) is in the buy region at time t , then, for any a > , ( aδ, ax ) is also in the buy region. The barrier between the buy and wait regions therefore hasto be a straight line through the origin and the buy and sell region can be characterized in terms of theratio y = xδ . In this section, we formally introduce the buy and wait regions and the barrier function. InSections 6 and 7, we prove that such a barrier exists for discrete and continuous trading time respectively.In contrast to the case of a time-varying but deterministic illiquidity K considered in this paper, for stochastic K , this barrier conjecture holds true in many, but not all cases, see Fruth (2011).We first define the buy and wait regions and subsequently define the barrier function. Based on the abovescaling argument, we can limit our attention to points (1 , y ) where δ = 1 , since for a point ( δ, x ) with δ > we can instead consider the point (1 , x/δ ) . Definition 5.2 (Buy and wait region) . For any t ∈ [0 , T ] , we define the inner buy region as Br t := (cid:26) y ∈ (0 , ∞ ) | ∃ ξ ∈ (0 , y ) : U ( t, , y ) = U ( t, K t ξ, y − ξ ) + (cid:18) K t ξ (cid:19) ξ (cid:27) , and call the following sets the buy region and wait region at time t : BR t := Br t , W R t := [0 , ∞ ) \ Br t (the bar means closure in R ).The inner buy region at time t hence consists of all values y such that immediate buying at the state (1 , y ) is value preserving. The wait region on the other hand contains all values y such that any non-zeropurchase at (1 , y ) destroys value. Let us note that Br T = (0 , ∞ ) , BR T = [0 , ∞ ) and W R T = { } .Regarding Definition 5.2, the following comment is in order. We do not claim in this definition that Br t is an open set. A priori one might imagine, say, the set (10 , as the inner buy region at some timepoint. But what we can say from the outset is that, due to the splitting argument (see Lemma 4.1), Br t is in any case a union of (not necessarily open) intervals or the empty set.13he wait-region/buy-region conjecture can now be formalized as follows. Definition 5.3 (WR-BR structure) . The value function U has WR-BR structure if there exists a barrier function c : [0 , T ] → [0 , ∞ ] such that for all t ∈ [0 , T ] , Br t = ( c ( t ) , ∞ ) with the convention ( ∞ , ∞ ) := ∅ . For the value function U N in discrete time to have WR-BR structure,we only consider t ∈ { t , ..., t N } and set c N ( t ) = ∞ for t / ∈ { t , ..., t N } .Let us note that we always have c ( T ) = 0 . Below we will see that it is indeed possible to have c ( t ) = ∞ ,i.e. at time t any strictly positive trade is suboptimal no matter at which state we start. For c ( t ) < ∞ ,having WR-BR structure means that BR t ∩ W R t = { c ( t ) } . Figure 3 illustrates the situation in continuoustime. y t0 Barrierc(t)Buy BRWR T Figure 3:
Schematic illustration of the buy and wait regions in continuous time.
Thus, up to now we have the following intuition. An optimal strategy is suggested by the barrier functionwhenever the value function has WR-BR structure. If the position of the large investor at time t satis-fies xδ > c ( t ) , then the portfolio is in the buy region. We then expect that it is optimal to execute thelargest discrete trade ξ ∈ (0 , x ) such that the new ratio of remaining shares over price deviation x − ξδ + K t ξ isstill in the buy region, i.e. the optimal trade is ξ ∗ = x − c ( t ) δ K t c ( t ) , which is equivalent to c ( t ) = x − ξ ∗ δ + K t ξ ∗ . Notice that the ratio term x − ξδ + K t ξ is strictly decreasing in ξ . Consequently, trades have the effect ofreducing the ratio as indicated in Figure 3, while the resilience effect increases it. That is one trades justenough shares to keep the ratio y below the barrier. In Figure 3 we demonstrate an intuitive case where the barrier decreases over time, i.e. buying becomesmore aggressive as the investor runs out of time. This intuitive feature however does not need to hold forall possible evolutions of K and ρ as we will see e.g. in Figure 4. Intuitively, this implies that apart from a possible initial and final impulse trade, optimal buying occurs in infinitesimalamounts provided that c is continuous in t on [0 , T ) . For diffusive K as in Fruth (2011), this would lead to singular optimalcontrols. U N always has WR-BR structure, there exists a unique optimal strategy, which is of the type“trade to the barrier when the ratio is in the buy region, do not trade when it is in the wait region” (seeSection 6). In continuous time the situation is more delicate. It may happen, for example, that the valuefunction U has WR-BR structure, but the strategy consisting in trading towards the barrier is not optimal(see the example in the beginning of Section 7, where an optimal strategy does not exist). However, if theilliquidity K is continuous, there exists an optimal strategy, and, under additional technical assumptions,it is unique (see Section 7). Moreover, if K and ρ are smooth and satisfy some further technical conditions,we have explicit formulas for the barrier and for the optimal strategy (see Section 9). We first state comparative statics satisfied by both the continuous and the discrete time value function.The value function is increasing in t, δ, x and the price impact coefficient K as well as decreasing withrespect to the resilience speed function. Proposition 5.4 (Comparative statics for the value function) . a) The value function is nondecreasing in t, δ, x .b) Fix t ∈ [0 , T ] . Assume that < ˇ K s ≤ ˆ K s for all s ∈ [ t, T ] . Then the value function correspondingto ˇ K is less than or equal to the one corresponding to ˆ K .c) Fix t ∈ [0 , T ] . Assume that < ˇ ρ s ≤ ˆ ρ s for all s ∈ [ t, T ] . Then the value function corresponding to ˆ ρ is less than or equal to the one corresponding to ˇ ρ . The proof is straightforward.
Proposition 5.5 (Continuity of the value function) . For each t ∈ [0 , T ] , the functions U ( t, · , · ) : [0 , ∞ ) → [0 , ∞ ) and V ( t, · ) : [0 , ∞ ) → [0 , ∞ ) are continuous.Proof. Due to Lemma 5.1 it is enough to prove that the function U ( t, · , · ) is continuous. Let us fix t ∈ [0 , T ] , x ≥ , ≤ δ < δ , ǫ > and take a strategy Θ ǫ ∈ A t ( x ) such that J ( t, δ , Θ ǫ ) < U ( t, δ , x ) + ǫ. For i = 1 , , we define D is := δ i e − R st ρ u du + Z [ t,s ) K u e − R su ρ r dr d Θ ǫu , s ∈ [ t, T +] . Using Proposition 5.4, we get U ( t, δ , x ) ≤ U ( t, δ , x ) ≤ Z [ t,T ] (cid:18) D s + K s ǫs (cid:19) d Θ ǫs ≤ Z [ t,T ] (cid:18) D s + K s ǫs (cid:19) d Θ ǫs + ( δ − δ ) x = J ( t, δ , Θ ǫ ) + ( δ − δ ) x < U ( t, δ , x ) + ǫ + ( δ − δ ) x. t ∈ [0 , T ] and x ≥ , the function U ( t, · , x ) is continuous on [0 , ∞ ) . For t ∈ [0 , T ] , δ ≥ and x > , by Lemma 5.1, we have U ( t, δ, x ) = x U ( t, δ/x, , hence the function U ( t, · , · ) is continuous on [0 , ∞ ) × (0 , ∞ ) . Considering the strategy of buying the wholeposition x at time t , we get U ( t, δ, x ) ≤ (cid:18) δ + K t x (cid:19) x −−−→ x ց U ( t, δ, , i.e. the function U ( t, · , · ) is also continuous on [0 , ∞ ) × { } . This concludes the proof. Proposition 5.6 (Trading never completes early) . For all t ∈ [0 , T ) , δ ∈ [0 , ∞ ) and x ∈ (0 , ∞ ) , the value function satisfies U ( t, δ, x ) < (cid:18) δ + K t x (cid:19) x, i.e. it is never optimal to buy the whole remaining position at any time t ∈ [0 , T ) .Proof. For ǫ ∈ [0 , x ] , define the strategies Θ ǫ ∈ A t ( x ) that buy ( x − ǫ ) shares at t and ǫ shares at T . Thecorresponding costs are J ( t, δ, Θ ǫ ) = (cid:18) δ + K t x − ǫ ) (cid:19) ( x − ǫ ) + (cid:18) ( δ + K t [ x − ǫ ]) e − R Tt ρ s ds + K T ǫ (cid:19) ǫ. Clearly, U ( t, δ, x ) ≤ J (cid:0) t, δ, Θ (cid:1) = (cid:18) δ + K t x (cid:19) x, but we never have equality since ∂∂ǫ J ( t, δ, Θ ǫ ) (cid:12)(cid:12)(cid:12) ǫ =0 = − (cid:16) − e − R Tt ρ s ds (cid:17) ( K t x + δ ) < . As discussed above, we always have Br T = (0 , ∞ ) and W R T = { } . In two following propositions wediscuss Br t (equivalently, W R t ) for t ∈ [0 , T ) . Proposition 5.7 (Wait region near ) . Assume that the value function U has WR-BR structure with the barrier c . Then for any t ∈ [0 , T ) , c ( t ) ∈ (0 , ∞ ] (equivalently, there exists ǫ > such that [0 , ǫ ) ⊂ W R t ).Proof. We need to exclude the possibility c ( t ) = 0 , i.e. Br t = (0 , ∞ ) . But if Br t = (0 , ∞ ) , we get byProposition 5.5 that for any y > , V ( t, y ) = (cid:18) K t y (cid:19) y, which contradicts Proposition 5.6.The following result illustrates that the barrier can be infinite.16 roposition 5.8 (Infinite barrier) . Assume there exist ≤ t < t ≤ T such that K s e − R t s ρ u du > K t for all s ∈ [ t , t ) . Then Br s = ∅ for s ∈ [ t , t ) . In particular, if the assumption of Proposition 5.8 holds with t = 0 and t = T , then the value functionhas WR-BR structure and the barrier is infinite except at terminal time T . Proof.
For any s ∈ [ t , t ) , δ ∈ [0 , ∞ ) , x ∈ (0 , ∞ ) and Θ ∈ A s ( x ) with Θ t > , we get the followingby applying (11), the assumption of the proposition, monotonicity of J in δ , and integration by parts asin (9) J ( s, δ, Θ) = Z [ s,t ) (cid:18) D u + K u u (cid:19) d Θ u + J (cid:0) t , D t , (Θ u − Θ t ) u ∈ [ t ,T +] (cid:1) ≥ Z [ s,t ) δe − R us ρ r dr + Z [ s,u ) K r e − R ur ρ w dw d Θ r + K u u ! d Θ u + J (cid:16) t , δe − R t s ρ u du + K t Θ t , (Θ u − Θ t ) u ∈ [ t ,T +] (cid:17) > (cid:18) δe − R t s ρ u du + K t t (cid:19) Θ t + J (cid:16) t , δe − R t s ρ u du + K t Θ t , (Θ u − Θ t ) u ∈ [ t ,T +] (cid:17) . That is it is strictly suboptimal to trade on [ s, t ) . In particular, Br s = ∅ .Proposition 5.8 can be extended in the following way. Proposition 5.9 (Infinite barrier, extended version) . Let K be continuous and assume there exist ≤ t < t ≤ T such that K t e − R t t ρ u du > K t . Then Br t = ∅ .Proof. Define ˜ t as the minimal value of the set argmin t ∈ [ t ,t ] K t e R t ρ u du with ˜ t being well-defined due to the continuity of K . Then we know that ˜ t > t . By definition of ˜ t , wehave that for all t ∈ [ t , ˜ t ) K t e R t ρ u du > K ˜ t e R ˜ t ρ u du and hence K t e − R ˜ tt ρ u du > K ˜ t . By Proposition 5.8, we can conclude that Br t = ∅ for all t ∈ [ t , ˜ t ) and hence in particular for t = t .17 Discrete time
In this section we show that the optimal execution problem in discrete time has WR-BR structure. Let usfirst rephrase the problem in the discrete time setting and define K n := K t n , D n := D t n and ξ n := ∆Θ t n for n = 0 , ..., N . The optimization problem (12) can then be expressed as U N ( t n , δ, x ) = inf ξj ∈ [0 ,x ] P ξ j = x N X j = n (cid:18) D j + K j ξ j (cid:19) ξ j . (17)with D n = δ and D j +1 = ( D j + K j ξ j ) a j , where a j := exp − Z t j +1 t j ρ s ds ! . (18)Recall the dimension reduction from Lemma 5.1 U N ( t n , δ, x ) = δ V N (cid:16) t n , xδ (cid:17) with V N ( t n , y ) := U N ( t n , , y ) . The following theorem establishes the WR-BR structure in discrete time.
Theorem 6.1 (Discrete time: WR-BR structure) . The discrete time value function U N has WR-BR structure with some barrier function c N . There existsa unique optimal strategy, which corresponds to the barrier c N as described in Section 5.2. Furthermore, V N ( t n , · ) : [0 , ∞ ) → [0 , ∞ ) has the following properties for n = 0 , ..., N .(i) It is continuously differentiable .(ii) It is piecewise quadratic , i.e., there exists M ∈ N , constants ( α i , β i , γ i ) i =1 ,...,M and < y
Let a ∈ (0 , , κ > and let the function v : [0 , ∞ ) → [0 , ∞ ) satisfy (i), (ii), (iii) given inTheorem 6.1. Then the following statements hold true.(a) There exists c ∗ ∈ [0 , ∞ ] such that L ( y ) := 1 + 2 κa v ( ya − )(1 + κy ) , y ∈ [0 , ∞ ) , is strictly decreasing for y ∈ [0 , c ∗ ) and strictly increasing for y ∈ ( c ∗ , ∞ ) .(b) The function ˜ v ( y ) := (cid:26) κ (cid:2) (1 + κy ) L ( c ∗ ) − (cid:3) if y > c ∗ a v ( ya − ) otherwise (cid:27) again satisfies (i), (ii), (iii) with possibly different coefficients.Proof of Theorem 6.1. We proceed by backward induction. Notice that V N ( t N , y ) = (cid:0) K N y (cid:1) y ful-fills (i), (ii), (iii) with M = 1 , α = K N , β = 1 , γ = 0 . Let us consider the induction step from t n +1 to t n .We are going to use Lemma 6.2 for a = a n , κ = K n , v = V N ( t n +1 , · ) . We then have that L = L N ( t n , · ) and we obtain c ∗ as the unique minimum of L N ( t n , · ) from Lemma 6.2 (a). From (21) we see that theunique optimal value for η is given by η ∗ := argmin η ∈ [0 ,y ] K n (cid:2) (1 + K n y ) L N ( t n , η ) − (cid:3) = min { y, c ∗ } and accordingly that the unique optimal trade is given by ξ ∗ := ξ ( η ∗ ) = max (cid:26) , y − c n K n c n (cid:27) . Therefore we have a unique optimal strategy and the value function has WR-BR structure with c N ( t n ) := c ∗ . Plugging ξ ∗ into (20) and applying the definition of V N yields V N ( t n , y ) = ˜ v ( y ) . Lemma 6.2 (b) nowconcludes the induction step. Proof of Lemma 6.2. (a) The function L is continuously differentiable with L ′ ( y ) = 2 κ (1 + κy ) l ( y ) , (23) l ( y ) := y (cid:0) α m ( ya − ) − κβ m ( ya − ) a (cid:1) + (cid:0) β m ( ya − ) a − κγ m ( ya − ) a − (cid:1) . L is constant. Assume there would be an intervalwhere l is zero, i.e., there exists i ∈ { , ..., M } such that (2 α i − κβ i a ) = 0 and ( β i a − κγ i a −
1) = 0 .Solving these equations for α respectively γ yields α i γ i + a − β i − β i = 0 . This is a contradiction to (19).Let us assume l (ˇ y ) > for some ˇ y ∈ [0 , ∞ ) with j := m (ˇ ya − ) . We are done if we can conclude l (ˆ y ) > for all ˆ y ∈ [ˇ y, ∞ ) . Because of the continuity of l , it is sufficient to show that L keeps increasingon [ˇ y, y j ] , i.e., we need to show l (ˆ y ) > for all ˆ y ∈ [ˇ y, y j ] . Due to the form of l , this is guaranteedwhen α j − κβ j a > . Let us suppose that this term would be negative which is equivalent to α j β − j a − ≤ κ . Together with the inequalities from (19) one gets al (ˇ y ) = − κ a (cid:0) ˇ ya − β j + 2 γ j (cid:1) + (cid:0) ya − α j + β j − a − (cid:1) ≤ − α j β − j (cid:0) ˇ ya − β j + 2 γ j (cid:1) + (cid:0) ya − α j + β j − a − (cid:1) = − β j (cid:0) α j γ j + β j a − − β j (cid:1) < . This is a contradiction to l (ˇ y ) > .(b) If c ∗ is finite, the function ˜ v is continuously differentiable at c ∗ since a brief calculation showsthat ˜ v ′ ( c ∗ − ) = ˜ v ′ ( c ∗ +) is equivalent to l ( c ∗ ) = 0 . We have ˜ v ( y ) = ˜ α ˜ m ( y ) y + ˜ β ˜ m ( y ) y + ˜ γ ˜ m ( y ) , i.e. ˜ v is piecewise quadratic with ˜ M = 1 + m ( c ∗ a − ) , ˜ y ˜ M − := c ∗ , ˜ y i := y i a for i = 1 , ..., ˜ M − and ˜ α ˜ M = κ L ( c ∗ ) > , ˜ β ˜ M = L ( c ∗ ) > , ˜ γ ˜ M = L ( c ∗ ) − κ , (24) ˜ α i = α i > , ˜ β i = aβ i > , ˜ γ i = a γ i for i = 1 , ..., ˜ M − . We therefore get α i ˜ γ i + ˜ β i − ˜ β i = (cid:26) if i = ˜ Ma (cid:0) α i γ i + a − β i − β i (cid:1) otherwise (cid:27) ≥ . It remains to show that ˜ v also inherits the last inequality in (19) from v . For y ≤ c ∗ , y ˜ β ˜ m ( y ) + 2˜ γ ˜ m ( y ) = a (cid:0) ya − β m ( ya − ) + 2 γ m ( ya − ) (cid:1) ≥ . Due to ˜ v being continuously differentiable in c ∗ , we get ˜ α ˜ M ( c ∗ ) + ˜ β ˜ M c ∗ + ˜ γ ˜ M = ˜ α ˜ M − ( c ∗ ) + ˜ β ˜ M − c ∗ + ˜ γ ˜ M − , α ˜ M c ∗ + ˜ β ˜ M = 2 ˜ α ˜ M − c ∗ + ˜ β ˜ M − . Taking two times the first equation and subtracting c ∗ times the second equation yields c ∗ ˜ β ˜ M + 2˜ γ ˜ M = c ∗ ˜ β ˜ M − + 2˜ γ ˜ M − . Since we already know that the right-hand side is positive, also y ˜ β ˜ M + 2˜ γ ˜ M ≥ for all y > c ∗ .20e need the following lemma as a preparation for the WR-BR proof in continuous time. Lemma 6.3.
Let K be continuous. Then at least one of two following statements is true: • The function y L N (0 , y ) is convex on (cid:2) , c N (0) (cid:1) ; • The continuous time buy region is simply Br = ∅ , i.e. c (0) = ∞ . We stress that the first statement in this lemma concerns discrete time, while the second one concernsthe continuous time optimization problem.
Proof.
Recall that the definition of L N (0 , · ) from (22) contains V N ( t , · ) which is continuously differen-tiable and piecewise quadratic with coefficients ( α i , β i , γ i ) . Analogously to (23), it turns out that ∂∂y L N (0 , y ) = 2 K (1 + K y ) " y α m (cid:18) ye R t ρsds (cid:19) − K β m (cid:18) ye R t ρsds (cid:19) e − R t ρ s ds ! + β m (cid:18) ye R t ρsds (cid:19) e − R t ρ s ds + 2 K γ m (cid:18) ye R t ρsds (cid:19) e − R t ρ s ds − ! . We distinguish between two cases. First assume that all i satisfy (2 α i − K β i e − R t ρ s ds ) ≥ . Then ∂∂y L N (0 , · ) must be increasing on [0 , c N (0)) as desired, since L N (0 , · ) is decreasing on this interval as we know fromLemma 6.2.Assume to the contrary that there exists i such that (2 α i − K β i e − R t ρ s ds ) < . Recall how α i and β i are actually computed in the backward induction of Theorem 6.1. In each induction step, Lemma 6.2 isused and the coefficients ˜ α ˜ M , ˜ β ˜ M get updated in (24). It gets clear that there exists n ∈ { , ..., N } suchthat α i − K β i e − R t ρ s ds = (cid:16) K t n − K e − R tn ρ s ds (cid:17) L N (cid:0) t n , c N ( t n ) (cid:1) . We get the resilience multiplier e − R tn ρ s ds thanks to the adjustment ˜ β i = aβ i from the second line of (24).Due to L N being positive, it follows that K t n < K e − R tn ρ s ds . That is for this choice of K , it cannot be optimal to trade at t = 0 as we see from Proposition 5.9. Hence,the buy region at t = 0 is the empty set for both discrete and continuous time.The proof of Theorem 6.1 is constructive. It not only establishes the existence of a unique barrier, butalso provides means to calculate the barrier numerically through the following recursive algorithm.Initialize value function V N ( t N , y ) = (cid:0) K N y (cid:1) y For n = N − , ..., Set L N ( t n , y ) := K n a n V N ( t n +1 ,ya − n ) (1+ K n y ) Compute c N ( t n ) := c n := argmin y ≥ L N ( t n , y ) Set V N ( t n , y ) := (cid:26) K n (cid:2) (1 + K n y ) L N ( t n , c n ) − (cid:3) if y > c n a n V N ( t n +1 , ya − n ) otherwise (cid:27)
21e close this section with a numerical example. Figure 4 was generated using the above numerical schemeand illustrates the optimal barrier and trading strategy for several example definitions of K and ρ . Forconstant K , we recover the Obizhaeva and Wang (2006) “bathtub” strategy with impulse trades of thesame size at the beginning and end of the trading horizon and trading with constant speed in between.The corresponding barrier is a decreasing straight line as we will explicitly see for continuous time inExample 9.5. For high values of the resilience ρ , the barriers have the typical decreasing shape, i.e. thebuy region increases if less time to maturity remains. For low values of the resilience ρ , the barrier mustnot be decreasing and can even be infinite, i.e. the buy region is the empty set, as illustated for K withless liquidity in the middle than in the beginning and the end of the trading horizon. We now turn to the continuous time setting. In Section 7.1 we discuss existence of optimal strategies usingHelly’s compactness theorem and a uniqueness result using convexity of the value function. Thereafter inSection 7.2 we prove that the WR-BR result from Section 6 carries over to continuous time.
In continuous time existence of an optimal strategy is not guaranteed in general. For instance, considera constant resilience ρ t ≡ ρ > and the price impact parameter K following the Dirichlet-type function K t = (cid:26) for t rational for t irrational (cid:27) . (25)In order to analyze model (25), let us first recall that in the model with a constant price impact K t ≡ κ > there exists a unique optimal strategy, which has a nontrivial absolutely continuous component(see Obizhaeva and Wang (2006) or Example 9.5 below for explicit formulas). Approximating this strategyby strategies trading only at rational time points we get that the value function in model (25) coincideswith the value function for the price impact K t ≡ . But there is no strategy in model (25) attaining thisvalue because the nontrivial absolutely continuous component of the unique optimal strategy for K t ≡ will count with price impact instead of in the total costs. Thus, there is no optimal strategy inmodel (25). We can therefore hope to prove existence of optimal strategies only under additional conditions on themodel parameters. In all of Section 7 we will assume that K is continuous; the following theorem assertsthat this is a sufficient condition for existence of an optimal strategy. Theorem 7.1. (Continuous time: Existence).
Let K : [0 , T ] → (0 , ∞ ) be continuous. Then there exists an optimal strategy Θ ∗ ∈ A t ( x ) , i.e. J ( t, δ, Θ ∗ ) = inf Θ ∈A t ( x ) J ( t, δ, Θ) . In the proof we construct an optimal strategy as the limit of a sequence of (possibly suboptimal) strategies.Before we can turn to the proof itself, we need to establish that strategy convergence leads to costconvergence. Let us, however, note that the value function here has WR-BR structure with the barrier from Example 9.5 with κ = 1 . æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì K Low Resilience æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì K High Resilience æ æ æ æ æ æ æ æ æ æà à à à à à à à à à ì ì ì ì ì ì ì ì ì ì æ æ æ æ æ æ æ æ æ æà à à à à à à à à à ì ì ì ì ì ì ì ì ì ì æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì
Figure 4:
Illustration of the numerically computed barrier ( c N ( t n )) n =0 ,...,N and the corresponding optimalstrategy (∆Θ t n ) n =0 ,...,N in discrete time for T = 1 , N = 10 , x = 100 , δ = 0 , ρ = 2 (left-hand side) and ρ = 10 (right-hand side). We used K t ≡ . , K t = 1 − . t and K t = 1 − . t − . as the given evolution of theilliquidity. roposition 7.2. (Costs are continuous in the strategy, K continuous). Let K : [0 , T ] → (0 , ∞ ) be continuous and let ¯Θ , (Θ n ) be strategies in A t ( x ) with Θ n w → ¯Θ , i.e., lim n →∞ Θ ns = ¯Θ s for every point s ∈ [ t, T ] of continuity of ¯Θ (i.e. Θ n converges weakly to ¯Θ ). Then (cid:12)(cid:12) J ( t, δ, ¯Θ) − J ( t, δ, Θ n ) (cid:12)(cid:12) −−−−→ n →∞ . Note that Proposition 7.2 does not hold when K has a jump. To prove Proposition 7.2, we first showin Lemma 7.3 that the convergence of the price impact processes follows from the weak convergence ofthe corresponding strategies. We then conclude in Lemma 7.5 that Proposition 7.2 holds for absolutelycontinuous K . This finally leads to Proposition 7.2 covering all continuous K . Lemma 7.3. (Price impact process is continuous in the strategy).
Let K : [0 , T ] → (0 , ∞ ) be continuous and let ¯Θ , (Θ n ) be strategies in A t ( x ) with Θ n w → ¯Θ .Then lim n →∞ D ns = ¯ D s for s = T + and for every point s ∈ [ t, T ] of continuity of ¯Θ .Proof. Recall equation (11) D s = Z [ t,s ) K u e − R su ρ r dr d Θ u + δe − R st ρ u du , which holds for s = T + and s ∈ [ t, T ] . Due to the weak convergence (note that the total mass is preserved,i.e. ¯Θ T + = Θ nT + = x , since ¯Θ , Θ n ∈ A t ( x ) ) and the integrand being continuous in u , the assertionfollows for s = T + . Due to the weak convergence we also have that for all s ∈ [ t, T ] with ∆ ¯Θ s = 0 and f s ( u ) := K u e − R su ρ r dr I [ t,s ) ( u ) (i.e. f s is continuous d ¯Θ -a.e.) D ns = Z [ t,T ] f s ( u ) d Θ nu + δe − R st ρ u du −−−−→ n →∞ Z [ t,T ] f s ( u ) d ¯Θ u + δe − R st ρ u du = ¯ D s . Lemma 7.4. (Costs rewritten in terms of the price impact process).
Let K : [0 , T ] → (0 , ∞ ) be absolutely continuous, i.e. K s = K + R s µ u du . Then J ( t, δ, Θ) = 12 " D T + K T − δ K t + Z [ t,T ] (cid:18) ρ s K s + µ s K s (cid:19) D s ds . (26) Proof.
Applying d Θ s = dD s + ρ s D s dsK s , ∆Θ s = ∆ D s K s yields J ( t, δ, Θ) = Z [ t,T ] (cid:18) D s + K s s (cid:19) d Θ s = Z [ t,T ] D s + ∆ D s K s dD s + Z [ t,T ] ρ s D s K s ds + Z [ t,T ] 12 ∆ D s ρ s D s K s ds. In this expression, the last term is zero since D has only countably many jumps. Using integration byparts for càglàd processes, namely (49) with U := D, V := DK , and d (cid:16) D s K s (cid:17) = K s dD s + D s d (cid:16) K s (cid:17) , we canwrite Z [ t,T ] D s K s dD s = 12 D T + K T − δ K t − Z [ t,T ] D s d (cid:18) K s (cid:19) − X s ∈ [ t,T ] (∆ D s ) K s . d (cid:16) K s (cid:17) = − µ s K s ds yields (26) as desired.The following result is a direct consequence of Lemma 7.3 and Lemma 7.4. Lemma 7.5. (Costs are continuous in the strategy, K absolutely continuous). Let K : [0 , T ] → (0 , ∞ ) be absolutely continuous and ¯Θ , (Θ n ) be strategies in A t ( x ) with Θ n w → ¯Θ . Then (cid:12)(cid:12) J ( t, δ, ¯Θ) − J ( t, δ, Θ n ) (cid:12)(cid:12) −−−−→ n →∞ . Proof of Proposition 7.2.
We use a proof by contradiction and suppose there exists a subsequence ( n j ) ⊂ N such that lim j →∞ Z [ t,T ] (cid:18) D n j s + K s n j s (cid:19) d Θ n j s = Z [ t,T ] (cid:18) ¯ D s + K s s (cid:19) d ¯Θ s , where the limit on the left-hand side exists. Without loss of generality assume lim j →∞ Z [ t,T ] (cid:18) D n j s + K s n j s (cid:19) d Θ n j s < Z [ t,T ] (cid:18) ¯ D s + K s s (cid:19) d ¯Θ s . (27)We now want to bring Lemma 7.5 into play. For ǫ > , we denote by K ǫ : [ t, T ] → (0 , ∞ ) an absolutelycontinuous function such that max s ∈ [ t,T ] | K ǫs − K s | ≤ ǫ . For Θ ∈ A t ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [ t,T ] (cid:18) D ǫs + K ǫs s (cid:19) d Θ s − Z [ t,T ] (cid:18) D s + K s s (cid:19) d Θ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z [ t,T ] (cid:18) | D ǫs − D s | + 12 | K ǫs − K s | ∆Θ s (cid:19) d Θ s ≤ x ǫ. We therefore get from (27) that there exists ǫ > such that lim sup j →∞ Z [ t,T ] (cid:18) D n j ,ǫs + K ǫs n j s (cid:19) d Θ n j s < Z [ t,T ] (cid:18) ¯ D ǫs + K ǫs s (cid:19) d ¯Θ s . This is a contradiction to Lemma 7.5.We can now conclude the proof of the existence Theorem 7.1.
Proof of Theorem 7.1.
Let (Θ n ) ⊂ A t ( x ) be a minimizing sequence. Due to the monotonicity of theconsidered strategies, we can use Helly’s Theorem in the form of Theorem 2, §2, Chapter III of Shiryaev(1995), which also holds for left-continuous processes and on [ t, T ] instead of ( −∞ , ∞ ) . It guarantees theexistence of a deterministic ¯Θ ∈ A t ( x ) and a subsequence ( n j ) ⊂ N such that (Θ n j ) converges weaklyto ¯Θ . Note that we can always force ¯Θ T + to be x , since weak convergence does not require that Θ n j T converges to ¯Θ T whenever ¯Θ has a jump at T . Thanks to Proposition 7.2, we can conclude that U ( t, δ, x ) = lim j →∞ J ( t, δ, Θ n j ) = J ( t, δ, ¯Θ) . The price impact process D is affine in the corresponding strategy Θ . That is in the case when K is notdecreasing too quickly, Lemma 7.4 guarantees that the cost term J is strictly convex in the strategy Θ .Therefore, we get the following uniqueness result. 25 heorem 7.6. (Continuous time: Uniqueness). Let K : [0 , T ] → (0 , ∞ ) be absolutely continuous, i.e. K s = K + R s µ u du , and additionally µ s + 2 ρ s K s > a.e. on [0 , T ] with respect to the Lebesgue measure.Then there exists a unique optimal strategy. For continuous K , we have now established existence and (under additional conditions) uniqueness of theoptimal strategy. Let us now turn to the value function in continuous time and demonstrate that it hasWR-BR structure, consistent with our findings in discrete time. Theorem 7.7. (Continuous time: WR-BR structure).
Let K : [0 , T ] → (0 , ∞ ) be continuous. Then the value function has WR-BR structure. We are going to deduce the structural result for the continuous time setting by using our discrete timeresult. First, we show that the discrete time value function converges to the continuous time value function.Without loss of generality, we set t = 0 . Lemma 7.8. (The discrete time value function converges to the continuous time one).
Let K : [0 , T ] → (0 , ∞ ) be continuous and consider an equidistant time grid with N trading intervals.Then lim N →∞ V N (0 , y ) = V (0 , y ) . Proof.
Thanks to Theorem 7.1, there exists a continuous time optimal strategy Θ ∗ ∈ A ( y ) . Approximateit suitably via step functions Θ N ∈ A N ( y ) . Then V (0 , y ) = J (0 , , Θ ∗ ) = lim N →∞ J (0 , , Θ N ) ≥ lim sup N →∞ V N (0 , y ) . The inequality V (0 , y ) ≤ lim inf N →∞ V N (0 , y ) is immediate. Proof of Theorem 7.7.
By the same change of variable from ξ to η that was used in Section 6, we cantransform the optimal trade equation V (0 , y ) = min ξ ∈ [0 ,y ] (cid:26)(cid:18) K ξ (cid:19) ξ + (1 + K ξ ) V (cid:18) , y − ξ K ξ (cid:19)(cid:27) into the optimal barrier equation V (0 , y ) = 12 K (cid:20) (1 + K y ) min η ∈ [0 ,y ] L (0 , η ) − (cid:21) , (28)where L (0 , y ) := L ( y ) := 1 + 2 K V (0 , y )(1 + K y ) . (29)Now it follows from (28) and (29) that min η ∈ [0 ,y ] L ( η ) = L ( y ) ,
26n particular the function L is nonincreasing in y . Define ˜ L N ( y ) := min η ∈ [0 ,y ] L N (0 , η ) , which is a nonincreasing positive function. If for some N the function y L N (0 , y ) is not convex on [0 , c N (0)) , then the second alternative in Lemma 6.3 holds, i.e. we have WR-BR structure with c (0) = ∞ .Thus, below we assume that for any N the function y L N (0 , y ) is convex on [0 , c N (0)) , hence, byLemma 6.2 (a) and Theorem 6.1, the function ˜ L N is convex on [0 , ∞ ) . Moreover, by rearranging (21) weobtain that ˜ L N ( y ) = 1 + 2 K V N (0 , y )(1 + K y ) . Hence ˜ L N converges pointwise to L as N → ∞ by Lemma 7.8 and (29). Therefore, L is also convex.Due to L being nonincreasing and convex, there exists a unique c ∗ ∈ [0 , ∞ ] such that L is strictly decreasingfor y ∈ [0 , c ∗ ) and constant for y ∈ ( c ∗ , ∞ ) . One can now conclude that for all y > c ∗ and η ∈ ( c ∗ , y ) ,setting ξ := y − η K η , i.e. η = y − ξ K ξ , and using (29) and L ( y ) = L ( η ) , we have V (0 , y ) = 12 K (cid:2) (1 + K y ) L ( y ) − (cid:3) = 12 K (cid:2) (1 + K y ) L ( η ) − (cid:3) = 12 K (cid:20) (1 + K y ) L (cid:18) y − ξ K ξ (cid:19) − (cid:21) . We now use the definition of L from (29) once again to get V (0 , y ) = (cid:18) K ξ (cid:19) ξ + (1 + K ξ ) V (cid:18) , y − ξ K ξ (cid:19) . (30)Therefore ( c ∗ , ∞ ) ⊂ Br . In case of c ∗ > , consider y ≤ c ∗ , take any η ∈ [0 , y ) , and set ξ := y − η K η . Thena similar calculation using that L ( y ) < L ( η ) shows that V (0 , y ) is strictly smaller than the right-hand sideof (30). Hence Br = ( c ∗ , ∞ ) . Thus, we get WR-BR structure with c (0) = c ∗ ∈ [0 , ∞ ] as desired.In Section 9 we will investigate the value function, barrier function and optimal trading strategies forseveral example specifications of K and ρ . In the model introduced in Section 2, we assumed a trading dependent spread between the best ask A t and best bid B t . This has allowed us to exclude both forms of price manipulation in Section 3. Analternative assumption that is often made in limit order book models is to disregard the bid-ask spreadand to assume A t = B t , see, for example, Huberman and Stanzl (2004), Gatheral (2010), Alfonsi, Schied,and Slynko (2011) and Gatheral, Schied, and Slynko (2011b). The canonical extension of these models toour framework including time-varying liquidity is the following. Assumption 8.1.
In the zero spread model , we have the unaffected price S u , which is a càdlàg H -martingale with a deterministic starting point S u , and assume that the best bid and ask are equal and27iven by A l t = B l t = S ut + D l t with D l t = D l e − R t ρ s ds + Z [0 ,t ) K s e − R ts ρ u du ( d Θ s − d ˜Θ s ) , t ∈ [0 , T +] , (31)where D l ∈ R is the initial value for the price impact. For convenience, we will furthermore assume that K : [0 , T ] → (0 , ∞ ) is twice continuously differentiable and ρ : [0 , T ] → (0 , ∞ ) is continuously differentiable.As opposed to this zero spread model, the model introduced in Section 2 will be referred to as the dynamicspread model in the sequel. In this section we study price manipulation and optimal execution in the zerospread model. In particular, we provide explicit formulas for optimal strategies. This in turn will be usedin the next section to study explicitly several examples both in the dynamic spread and in the zero spreadmodel.We have excluded permanent impact from the definition above ( γ = 0 ). It can easily be included, but,like in the dynamic spread model, proves to be irrelevant for optimal strategies and price manipulation.Note that for pure buying strategies ( ˜Θ ≡ ) the zero spread model is identical to the model introducedin Section 2. The difference between the two models is that if sell orders occur, then they are executedat the same price as the ask price. Furthermore, buy and sell orders impact this price symmetrically.We can hence consider the net trading strategy Θ l := Θ − ˜Θ instead of buy orders Θ and sell orders ˜Θ separately. The simplification of the stochastic optimization problem of Section 2.2 to a deterministicproblem in Section 4 applies similarly to the zero spread model defined in Assumption 8.1. Thus, for anyfixed x ∈ R , we define the sets of strategies A l := (cid:8) Θ l : [0 , T +] → R | Θ l is a deterministic càglàdfunction of finite variation with Θ l = 0 (cid:9) , A l ( x ) := n Θ l ∈ A l | Θ l T + = x o . Strategies from A l ( x ) allow buying and selling and build up the position of x shares until time T . Wefurther define the cost function J l : R × A l → R as J l (Θ l ) := J ( δ, Θ l ) := Z [0 ,T ] (cid:18) D l s + K s l s (cid:19) d Θ l s , where D l is given by (31) with D l = δ . The function J l represents the total temporary impact costs in the zero spread model of the strategy Θ l on the time interval [0 , T ] when the initial price impact D l = δ . Observe that J l is well-defined and finite because K is bounded, which in turn follows fromAssumption 8.1. The value function U l : R → R is then given as U l ( δ, x ) := inf Θ l ∈A l ( x ) J l ( δ, Θ l ) . (32)The zero spread model admits price manipulation if, for D l = 0 , there is a profitable round trip, i.e. thereis Θ l ∈ A l (0) with J l (0 , Θ l ) < . The zero spread model admits transaction-triggered price manipulation if, for D l = 0 , the execution costs of a buy (or sell) program can be decreased by intermediate sell(resp. buy) trades (more precisely, this should be formulated like in Definition 3.2). In the case of liquidation of shares (i.e. Θ l T + < ) the word “costs” should be understood as “minus proceeds from theliquidation”. emark 8.2. The conceptual difference with Section 3 is that we require D l = 0 in these definitions. Thereason is that even in “sensible” zero spread models (that do not admit both types of price manipulationaccording to definitions above), we typically have profitable round trips whenever D l = 0 . In the zerospread model, the case D l = 0 can be interpreted as that the market price is not in its equilibrium statein the beginning. In the absence of trading the process ( D l t ) approaches zero due to the resilience, henceboth best ask and best bid price processes ( A l t ) and ( B l t ) (which are equal) approach their evolution inthe equilibrium ( S ut ) . The knowledge of this “direction of deviation from S u ” plus the fact that both buyand sell orders are executed at the same price clearly allow us to construct profitable round trips. Forinstance, in the Obizhaeva–Wang-type model with a constant price impact K t ≡ κ > , the strategy Θ l s := − D l κ I (0 ,ǫ ] ( s ) , s ∈ [0 , T +] , where ǫ ∈ (0 , T ] , is a profitable round trip whenever D l = 0 , as can be checked by a straightforwardcalculation.Let us first discuss classical price manipulation in the zero spread model. If the liquidity in the orderbook rises too fast ( K falls too quickly), then a simple pump and dump strategy becomes attractive. Inthe initial low liquidity regime (high K ), buying a large amount of shares increases the price significantly.Quickly thereafter liquidity increases. Then the position can be liquidated with little impact at thiselevated price, leaving the trader with a profit. The following result formalizes this line of thought. Proposition 8.3. (Price manipulation in the zero spread model).
Assume the zero spread model of Assumption 8.1 and that K ′ t + 2 ρ t K t < for some t ∈ [0 , T ) . Then price manipulation occurs and, for any δ, x ∈ R , there is no optimal strategy in problem (32) .Proof. By the assumption of the theorem, K ′ t = lim ǫ ց K t + ǫ − K t ǫ < − ρ t K t = lim ǫ ց K t (cid:16) e − R t + ǫt ρ u du − (cid:17) − K t ǫ , hence for a sufficiently small ǫ > we have K t + ǫ < K t (cid:16) e − R t + ǫt ρ u du − (cid:17) . (33)Let us consider the round trip Θ l m ∈ A l (0) , which buys share at time t and sells it at time t + ǫ , i.e. Θ l ms := I ( t,t + ǫ ] ( s ) , s ∈ [0 , T +] . A straightforward computation shows that, for D l = 0 , the cost of such a round trip is J l (0 , Θ l m ) = K t + K t + ǫ − K t e − R t + ǫt ρ u du . Due to (33), J l (0 , Θ l m ) < . Thus, price manipulation occurs. Let us observe that this does not apply to the dynamic spread model of Section 2, where we have different processes D and E for the deviations of the best ask and best bid prices from the unaffected ones due to the previous trades. δ, x ∈ R , consider a strategy Θ l ∈ A l ( x ) , and, for any z ∈ R , define Θ l z := Θ l + z Θ l m . Then Θ l z ∈ A l ( x ) and we have J l ( δ, Θ l z ) = c z + c z + c with c = J l (0 , Θ l m ) < and some constants c and c . Since z is arbitrary, we get U l ( δ, x ) = −∞ .An optimal strategy is this situation would be a strategy from A l ( x ) with the cost −∞ . But for anystrategy Θ l , its cost J l ( δ, Θ l ) is finite as discussed above, hence there is no optimal strategy in prob-lem (32).Interestingly, the condition K ′ t +2 ρ t K t < for some t in Proposition 8.3 is not symmetric; quickly falling K leads to price manipulation, but quickly rising K does not.If K ′ t + 2 ρ t K t ≥ holds at all points in time, then the situation remains unclear so far. In their model,Alfonsi, Schied, and Slynko (2011) and Gatheral, Schied, and Slynko (2011b) have shown that even in theabsence of profitable round trip trades, we might still be facing transaction-triggered price manipulation.This can happen also in our zero spread model. The following theorem provides explicit formulas foroptimal strategies and leads to a characterization of transaction-triggered price manipulation. Theorem 8.4. (Optimal strategies in the zero spread model).
Assume the zero spread model of Assumption 8.1 and that K ′ t + 2 ρ t K t > on [0 , T ] . Define f t := K ′ t + ρ t K t K ′ t + 2 ρ t K t , t ∈ [0 , T ] . (34) Then, for any δ, x ∈ R , the strategy Θ l∗ given by the formulas ∆Θ l∗ = δ l f K − δK , d Θ l∗ t = δ l f ′ t + ρ t f t K t dt, ∆Θ l∗ T = δ l − f T K T (35) with δ l := c (cid:16) x + δK (cid:17) and c := Z T f ′ t + ρ t f t K t dt + f K + 1 − f T K T > , (36) is the unique optimal strategy in problem (32) . Furthermore, we have U l ( δ, x ) = J l ( δ, Θ l∗ ) = ( δ l ) Z T ( K ′ t + 2 ρ t K t ) f t K t dt + 12 K T ! − δ K . (37) Corollary 8.5 (Transaction-triggered price manipulation in the zero spread model) . Under the assumptions of Theorem 8.4 price manipulation does not occur. Furthermore, transaction-triggered price manipulation occurs if and only if f < or f ′ t + ρ t f t < for some t ∈ [0 , T ] .Proof. Using (35) with δ = 0 , we immediately get that price manipulation does not occur. Noting furtherthat f T < , we obtain that transaction-triggered price manipulation occurs if and only if either f < or f ′ t + ρ t f t < for some t ∈ [0 , T ] . 30e can summarize Proposition 8.3 and Corollary 8.5 as follows. If K ′ t + 2 ρ t K t < for some t , i.e. liquiditygrows very rapidly, then price manipulation (and hence transaction-triggered price manipulation) occurs.If K ′ t + 2 ρ t K t > everywhere, but f < or f ′ t + ρ t f t < for some t , i.e. liquidity grows fast but not quiteas fast, then price manipulation does not occur, but transaction-triggered price manipulation occurs. If K ′ t + 2 ρ t K t > everywhere, f ≥ and f ′ t + ρ t f t ≥ everywhere, i.e. liquidity never grows too fast, thenneither form of price manipulation occurs and an investor wishing to purchase should only submit buyorders to the market. Figure 5 illustrates optimal transaction-triggered price manipulation strategies. InExample 1, the liquidity K is slightly growing at the end of the trading horizon, which makes the optimalstrategy Θ l∗ non-monotonic. As we see in Example 2, the number of shares hold by the large investorcan become negative although the overall goal is to buy a positive amount of shares. KK'+2pK f '+pf0.2 0.4 0.6 0.8 1.0 Time - - Example 1
KK'+2pK f '+pf0.2 0.4 0.6 0.8 1.0 Time - Example 2
Transaction - triggered price manipulation Transaction - triggered price manipulation Figure 5:
In Example 1, we consider K t = sin(2 . t ) + 0 . and K t = sin(10 t ) + 4 in Example 2. The otherparameters are T = 1 , ρ = 2 , x = 100 , δ = 0 . The plots at the bottom illustrate the corresponding optimalstrategies Θ l∗ from (35). In the proof of Theorem 8.4, we are going to exploit the fact that there is a one-to-one correspondencebetween Θ l and D l . We rewrite the cost term, which is essentially R T D l t d Θ l t , in terms of the deviationprocess D l by applying d Θ l t = dD l t + ρ t D l t dtK t , (38)and get the following result. Lemma 8.6 (Costs rewritten in terms of the price impact process) . nder Assumption 8.1, for any δ ∈ R and Θ l ∈ A l , we have J l ( δ, Θ l ) = 12 (cid:16) D l T + (cid:17) K T − δ K + Z [0 ,T ] ( K ′ t + 2 ρ t K t ) (cid:16) D l t (cid:17) K t dt . (39)The formal proof, where one needs to take into account possible jumps of Θ l , is similar to that ofLemma 7.4.Similar to Bank and Becherer (2009) and as explained in Gregory and Lin (1996), we can then use theEuler-Lagrange formalism to find necessary conditions on the optimal D l . Under our assumptions, theseconditions turn out to be sufficient and the optimal D l directly gives us an optimal Θ l . Unfortunately,we cannot use the Euler-Lagrange approach directly in the full generality of all strategies in A l , but needto impose a continuity property. Motivated by the WR-BR structure established in previous sections aswell as the optimal strategy in the case of constant K from Obizhaeva and Wang (2006), we introduce,for x ∈ R , the set of strategies A l c ( x ) ⊂ A l ( x ) with impulse trades at t = 0 and t = T only: A l c ( x ) := (cid:8) Θ l ∈ A l ( x ) | Θ l is continuous on (0 , T ) (cid:9) . We will also need a notation for a similar set of monotonic strategies, i.e., for y ∈ [0 , ∞ ) , we define A c ( y ) := (cid:8) Θ ∈ A ( y ) | Θ is continuous on (0 , T ) (cid:9) . Lemma 8.7. (Approximation by continuous strategies).
Assume the zero spread model of Assumption 8.1. Then, for any δ, x ∈ R , U l ( δ, x ) := inf Θ l ∈A l ( x ) J l ( δ, Θ l ) = inf Θ l ∈A l c ( x ) J l ( δ, Θ l ) . (40) Proof.
Let us take any Θ l ∈ A l ( x ) and find Θ , ˜Θ ∈ A such that Θ l = Θ − ˜Θ . We set y := Θ T + ∈ [0 , ∞ ) , ˜ y := ˜Θ T + ∈ [0 , ∞ ) , so that x = y − ˜ y . Below we will show that ∃ Θ n ∈ A c ( y ) , ˜Θ n ∈ A c (˜ y ) such that Θ n w −→ Θ , ˜Θ n w −→ ˜Θ . (41)Let us define Θ l n := Θ n − ˜Θ n ∈ A l c ( x ) . It follows from (31) and the weak convergence of the strategiesthat the price impact D l nt corresponding to Θ l n converges to the price impact D l t corresponding to Θ l for t = T + and for every point t ∈ [0 , T ] , where both Θ and ˜Θ are continuous (i.e. the convergence ofprice impact functions holds at T + and everywhere on [0 , T ] except at most a countable set). By (39),we get J l ( δ, Θ l n ) → J l ( δ, Θ l ) as n → ∞ . Since Θ l ∈ A l ( x ) was arbitrary, we obtain (40).It remains to prove (41). Clearly, it is enough to consider some Θ ∈ A ( y ) and to construct Θ n ∈ A c ( y ) weakly convergent to Θ . Let P denote the class of all probability measures P on ([0 , T ] , B ([0 , T ])) and P c = (cid:8) P ∈ P (cid:12)(cid:12) P ( { s } ) = 0 for all s ∈ (0 , T ) (cid:9) . The formula P ([0 , s )) := Θ s y , s ∈ [0 , T ] , with Θ ∈ A ( y ) , provides a one-to-one correspondence be-tween A ( y ) and P , where A c ( y ) is mapped on P c . Thus, it is enough to show that any probabilitymeasure P ∈ P can be weakly approximated by probability measures from P c . To this end, let usconsider independent random variables ψ and ζ such that Law ( ψ ) = P and Law ( ζ ) is continuous. Forany n ∈ N , we define ψ n := (cid:18)(cid:18) ψ + ζn (cid:19) ∨ (cid:19) ∧ T. Q n := Law ( ψ n ) ∈ P c and Q n w → P as n → ∞ because ψ n → ψ a.s. This concludes the proof. Lemma 8.8.
Assume K ′ t + 2 ρ t K t > on [0 , T ] and define χ ( t ) := Z t f ′ s + ρ s f s K s dt + f K + 1 − f t K t . Then χ ( t ) > for all t ∈ [0 , T ] . In particular, c = χ ( T ) > .Proof. We have χ (0) = 1 K > . Furthermore, χ ′ ( t ) = f ′ t + ρ t f t K t + − f ′ t K t − (1 − f t ) K ′ t K t = ρ t K ′ t + 2 ρ t K t > . Proof of Theorem 8.4.
We first note that c from (36) is strictly positive by Lemma 8.8. Also note that ifan optimal strategy in (32) exists, then it is unique in the class A l ( x ) because the function Θ l J l ( δ, Θ l ) is strictly convex on A l (this is due to (39) and the assumption K ′ t + 2 ρ t K t > on [0 , T ] ).For the strategy Θ l∗ given in (35), we have Θ l∗ T + = x as desired. This follows from the formula for δ l .Let us further observe that Θ l∗ corresponds to the deviation process D l∗ = δ, D l∗ t = δ l f t on (0 , T ] , D l∗ T + = δ l , (42)which immediately follows from (38) (direct computation using (31) is somewhat longer). A straightfor-ward calculation gives J l ( δ, Θ l∗ ) = ( δ l ) Z T f t f ′ t + ρ t f t K t dt + f K + 1 − f T K T ! − δ K . (43)Using integration by parts we get Z T f t f ′ t K t dt = 12 " f T K T − f K + Z T f t K t K ′ t dt . Substituting this into (43) we get that J l ( δ, Θ l∗ ) equals the right-hand side of (37).It remains to prove optimality of Θ l∗ . Due to Lemma 8.7 it is enough to prove that Θ l∗ is optimal in theclass A l c ( x ) , which we do below. In terms of D l∗ , the corresponding trading costs are J l ( δ, Θ l∗ ) = Z (0 ,T ) D l∗ t d Θ l∗ t + (cid:18) δ + K l∗ (cid:19) ∆Θ l∗ + (cid:18) D l∗ T + K T l∗ T (cid:19) ∆Θ l∗ T = Z (0 ,T ) D l∗ t K t dD l∗ t + Z (0 ,T ) ρ t ( D l∗ t ) K t dt + ( D l∗ ) − δ K + ( D l∗ T + ) − ( D l∗ T ) K T . ˆΘ ∈ A l c ( x ) with corresponding ˆ D = D l∗ + h and show thatthese alternative strategies cause higher trading costs than Θ l∗ . That is in the following, we work withfunctions h : [0 , T +] → R , which are of bounded variation and continuous on (0 , T ) with h = 0 , h T =lim t ր T h t and a finite limit h (so that there are possibly jumps ( h − h ) , ( h T + − h T ) ∈ R ). Using ∆ ˆΘ = ∆Θ l∗ + h K , d ˆΘ t = d Θ l∗ t + dh t + ρ t h t dtK t , ∆ ˆΘ T = ∆Θ l∗ T + h T + − h T K T , a straightforward calculation yields J l ( δ, ˆΘ) = Z (0 ,T ) ˆ D t d ˆΘ t + (cid:18) δ + K (cid:19) ∆ ˆΘ + (cid:18) ˆ D T + K T T (cid:19) ∆ ˆΘ T = J l ( δ, Θ l∗ ) + ∆ J + ∆ J , ∆ J := Z (0 ,T ) ρ t D l∗ t h t K t dt + Z (0 ,T ) h t K t dD l∗ t + Z (0 ,T ) D l∗ t K t dh t + D l∗ h K + D l∗ T + h T + − D l∗ T h T K T , ∆ J := Z (0 ,T ) ρ t h t K t dt + Z (0 ,T ) h t K t dh t + h K + h T + − h T K T . Notice that we collect all terms containing D l∗ in ∆ J . We are now going to finish the proof by showingthat ∆ J = 0 and ∆ J > if h does not vanish.Let us first rewrite ∆ J exploiting the fact that D l∗ t = δ l f t , use integration by parts, the definition of f and again integration by parts to get ∆ J = δ l (Z (0 ,T ) ρ t f t h t K t dt + Z (0 ,T ) h t K t df t + Z (0 ,T ) f t K t dh t + f h K + h T + − f T h T K T ) = δ l (Z (0 ,T ) ρ t K t + K ′ t K t f t h t dt + h T + K T ) = δ l (Z (0 ,T ) ρ t h t K t dt + h T + K T + Z (0 ,T ) K ′ t K t h t dt ) = δ l (Z (0 ,T ) ρ t h t K t dt + Z (0 ,T ) K t dh t + h K + h T + − h T K T ) . Clearly, ∆ J = 0 whenever δ l = 0 . If δ l = 0 , we have x = Z (0 ,T ) d ˆΘ t + ∆ ˆΘ + ∆ ˆΘ T = Z (0 ,T ) d Θ l∗ t + ∆Θ l∗ + ∆Θ l∗ T ! + Z (0 ,T ) ρ t h t K t dt + Z (0 ,T ) K t dh t + h K + h T + − h T K T ! = x + ∆ J δ l . Therefore ∆ J = 0 . Hence, J l ( δ, ˆΘ) − J l ( δ, Θ l∗ ) = ∆ J . Applying integration by parts to the dh t integralyields ∆ J = Z (0 ,T ) h t K t (cid:18) ρ t + K ′ t K t (cid:19) dt + h T + K T . K ′ t + 2 ρ t K t > on [0 , T ] , we get that ∆ J is positive as desired. Let us now turn to explicit examples of dynamics of the price impact parameter K and the resilience ρ . Wecan use the formulas derived in the previous section to calculate optimal trading strategies in problem (32)in the zero spread model. We also want to investigate optimal strategies in problem (12) in the dynamicspread model introduced in Section 2. In (12) we considered a general initial time t ∈ [0 , T ] . Without lossof generality below we will consider initial time for both models, e.g. we will mean the function U (0 , · , · ) when speaking about the value function in the dynamic spread model. Further, in the dynamic spreadmodel we had a nonnegative initial value δ for the deviation of the best ask price from its unaffected leveland considered strategies with the overall goal to buy a nonnegative number of shares x . That is, wewill consider δ, x ∈ [0 , ∞ ) in this section when speaking about either model. It is clear that strategy (35)is optimal also in the dynamic spread model whenever it does not contain selling. Thus, Theorem 8.4,applied with δ, x ∈ [0 , ∞ ) , provides us with formulas for the value function and optimal strategy alsoin the dynamic spread model whenever there is no transaction-triggered price manipulation in the zerospread model (see Corollary 8.5) and δ is sufficiently close to (so that ∆Θ l∗ given by the first formulain (35) is still nonnegative). Furthermore, in this case we get an explicit formula for the barrier functionof Definition 5.3. Proposition 9.1. (Closed form optimal barrier in the dynamic spread model).
Assume the dynamic spread model of Section 2 and that K : [0 , T ] → (0 , ∞ ) is twice continuously differ-entiable and ρ : [0 , T ] → (0 , ∞ ) is continuously differentiable. Let K ′ t + 2 ρ t K t > on [0 , T ] , f ≥ and f ′ t + ρ t f t ≥ on [0 , T ] , (44) where f is defined in (34) . Then the barrier function of Definition 5.3 is explicitly given by c ( t ) = 1 f t Z Tt f ′ s + ρ s f s K s ds + 1 − f T K T ! , t ∈ [0 , T ) , c ( T ) = 0 . (45) Furthermore, for any x ∈ [0 , ∞ ) and δ ∈ h , xc (0) i , there is a unique optimal strategy in the prob-lem U (0 , δ, x ) (see (12) ) and it is given by formula (35) in Theorem 8.4, and the value function U (0 , δ, x ) equals the right-hand side of (37) . Remark 9.2 (Comments to (45)) . i) First let us note that (44) implies f t ≥ on [0 , T ] (see Lemma 9.3 below). Hence the right-hand sideof (45) has the form a/b with a > (note that f T < ) and b ≥ , i.e. c ( t ) ∈ (0 , ∞ ] for t ∈ [0 , T ) .The case c ( t ) = ∞ can occur (see e.g. Example 9.6 with ν = − ).ii) Let us further observe that lim t ր T c ( t ) = 1 − f T f T K T ∈ (0 , ∞ ] , i.e. the barrier always jumps at T . Proof of Proposition 9.1.
Let us first notice that c from (36) is strictly positive by Lemma 8.8, so thatTheorem 8.4 applies. Further, it follows from (44) that in the zero spread model with such functions K ρ there is no transaction-triggered price manipulation. Hence, for any x > , the optimal strategy Θ l∗ from (35) with δ = 0 in the problem U l (0 , x ) will also be optimal in the problem U (0 , , x ) . Let us recallthat the value c (0) of the barrier is the ratio x − ∆Θ l∗ D l∗ for the optimal strategy Θ l∗ in the problem U (0 , , x ) and the corresponding D l∗ (with D l∗ = 0 ). Thus, we get c (0) = x − ∆Θ l∗ K ∆Θ l∗ = 1 f Z T f ′ s + ρ s f s K s ds + 1 − f T K T ! . A similar reasoning applies to an arbitrary t ∈ [0 , T ) . Recall that we always have c ( T ) = 0 . Finally,for δ > , under condition (44), formula (35) for the zero spread model will give the optimal strategy inthe problem U (0 , δ, x ) (i.e. for the dynamic spread model) if and only if ∆Θ l∗ ≥ . Solving this inequalitywith respect to δ we get δ ≤ xc (0) (note that δ l from (35) also depends on δ ).Condition (44) ensures the applicability of Theorem 8.4 and additionally excludes transaction-triggeredprice manipulation in the zero spread model (see Corollary 8.5). The following result provides an equivalentform for this condition, which we will use below when studying specific examples. Lemma 9.3 (An equivalent form for condition (44)) . Assume that K : [0 , T ] → (0 , ∞ ) is twice continuously differentiable and ρ : [0 , T ] → (0 , ∞ ) is continuouslydifferentiable. Then condition (44) is equivalent to K ′ t + ρ t K t ≥ on [0 , T ] and f ′ t + ρ t f t ≥ on [0 , T ] . (46) Proof.
Clearly, (46) implies (44). Let us prove the converse. Suppose (44) is satisfied and K ′ s + ρ s K s < for some s ∈ [0 , T ] . Then there exists [ u, v ] ⊂ [0 , T ] such that u < v , f u = 0 and f t < on ( u, v ] . By themean value theorem, there exists w ∈ ( u, v ) such that f ′ w = ( f v − f u ) / ( v − u ) . Thus, we get f ′ w < and f w < , which contradicts the condition f ′ t + ρ t f t ≥ on [0 , T ] .When we have transaction-triggered price manipulation in the zero spread model, optimal strategies inthe dynamic spread model are different from the ones given in Theorem 8.4. The following propositiondeals with the case of K ′ t + ρ t K t < for some t (cf. with (46)). Proposition 9.4. (Wait if decrease of K outweighs resilience). Assume the dynamic spread model of Section 2. Let, for some t ∈ [0 , T ) , K be continuously differentiableat t and ρ continuous at t with K ′ t + ρ t K t < . Then Br t = ∅ , i.e., c ( t ) = ∞ .Proof. Since K ′ + ρK is continuous at t , we have K ′ s + ρ s K s < on an interval around t . Then thereexists ǫ > such that K s e − R t + ǫs ρ u du > K t + ǫ for all s ∈ [ t, t + ǫ ) . By Proposition 5.8, it is not optimal totrade at t .Let us finally illustrate our results by discussing several examples. For simplicity, take constant re-silience ρ > . Then condition (46) takes the form K ′ t + ρK t ≥ on [0 , T ] and K ′′ t + 3 ρK ′ t + 2 ρ K t ≥ on [0 , T ] . (47)A sufficient condition for (47), which is sometimes convenient (e.g. in Example 9.6 below), is K ′ t + ρK t ≥ on [0 , T ] and K ′′ t + ρK ′ t ≥ on [0 , T ] . In all examples below we consider δ = 0 and x ∈ [0 , ∞ ) .36 xample 9.5. (Constant price impact K t ≡ κ > ).Assume that the price impact K t ≡ κ > is constant. Clearly, condition (47) is satisfied, so we can useformula (35) to get the optimal strategy in both models. We have f t ≡ and δ l = κxρT +2 . The optimalstrategy in both the dynamic and zero spread models is given by the formula ∆Θ = ∆Θ T = xρT + 2 , d Θ t = xρρT + 2 dt, which recovers the results from Obizhaeva and Wang (2006). The large investor trades with constantspeed on (0 , T ) and consumes all fresh limit sell orders entering the book due to resilience in such a waythat the corresponding deviation process D t is constant on (0 , T ] (see (42) and note that f t is constant).The barrier is linearly decreasing in time (see (45)): c ( t ) = 1 + ρ ( T − t ) κ , t ∈ [0 , T ) , c ( T ) = 0 . Let us finally note that the optimal strategy does not depend on κ , while the barrier depends on κ . SeeFigure 6 for an illustration. WRBR0.2 0.4 0.6 0.8 1.0 Time0.51.01.52.02.53.0Barrier 0.2 0.4 0.6 0.8 1.0 Time20406080100Optimal strategy
Figure 6:
Constant price impact ( T = 1 , ρ = 2 , κ = 1 , x = 100 , δ = 0 ). Example 9.6. (Exponential price impact K t = κe νρt , κ > , ν ∈ R \ { } ).Assume that the price impact K t = κe νρt is growing or falling exponentially with ν ∈ R \ { } beingthe slope of the exponential price impact relative to the resilience. The case ν = 0 was studied in theprevious example. We exclude this case here because some expressions below will take the form / when ν = 0 (however, the limits of these expressions as ν → will recover the corresponding formulas from theprevious example). Condition (47) is satisfied if and only if ν ≥ − . We first consider the case ν ≥ − .We have f t ≡ ν + 1 ν + 2 and δ l = xκν ( ν + 2)( ν + 1) − e − νρT . In particular, like in the previous example, the large investor trades in such a way that the deviationprocess D t is constant on (0 , T ] . The optimal strategy in both the dynamic and zero spread models isgiven by the formula ∆Θ = xν ( ν + 1)( ν + 1) − e − νρT , d Θ t = xν ( ν + 1)( ν + 1) − e − νρT ρe − νρt dt, ∆Θ T = xν ( ν + 1) − e − νρT e − νρT . We see that, for ν = − , it is optimal to buy the entire order at T . Vice versa, the initial trade ∆Θ approaches x as ν ր ∞ . The barrier is given by the formula c ( t ) = ( ν + 1) e − νρt − e − νρT κν ( ν + 1) , t ∈ [0 , T ) , c ( T ) = 0 c ( t ) = ∞ for t ∈ [0 , T ) if ν = − and the barrier is finite everywhere if ν > − ). Foreach ν > − , the barrier is decreasing in t , i.e. buying becomes more aggressive as the investor runs outof time. Furthermore, one can check that, for each t ∈ [0 , T ) , the barrier is decreasing in ν . That is, thegreater is ν , the larger is the buy region since it is less attractive to wait. Like in the previous example,the optimal strategy does not depend on κ , while the barrier depends on κ .Let us now consider the case ν < − . In the zero spread model, transaction-triggered price manipulationoccurs for ν ∈ ( − , − (one checks that the assumptions of Theorem 8.4 are satisfied) and classical pricemanipulation occurs for ν < − (see Proposition 8.3). In the dynamic spread model, for ν < − , it isoptimal to trade the entire order at T because K t e − ρ ( T − t ) > K T for all t ∈ [0 , T ) (see Proposition 5.8).Thus, in the case ν < − , we have c ( t ) = ∞ for t ∈ [0 , T ) .See Figure 7 for an illustration. K - Figure 7:
Exponential price impact ( T = 1 , ρ = 2 , κ = 1 , x = 100 , δ = 0 , ν = 0 . and − . (dashed)). Example 9.7. (Straight-line price impact K t = κ + mt , κ > , m > − κT ).Assume that the price impact K t = κ + mt changes linearly over time. The condition m > − κT ensuresthat K is everywhere strictly positive. Condition (47) is satisfied if and only if m ≥ − ρκ ρT . Note that − ρκ ρT > − κT . Let us first assume that m ≥ − ρκ ρT . In this case, the optimal strategy in both thedynamic and zero spread models is given by the formulas ∆Θ = 2 m ( m + κρ ) x ( m + 2 κρ ) ˜ m , d Θ t = 2 mκρ (2 κρ + m (3 + 2 ρt )) x ( m + 2 κρ + 2 mρt ) ˜ m dt, ∆Θ T = 2 mκρx ( m + 2 κρ + 2 mρT ) ˜ m with ˜ m := 2 m + κρ log (cid:18) m + 2 κρ + 2 mρTm + 2 κρ (cid:19) . The barrier is given by the formula c ( t ) = ρ m − ( m + 2 κρ + 2 mρt ) log (cid:16) m +2 κρ +2 mρtm +2 κρ +2 mρT (cid:17) m ( m + κρ + mρt ) . In the zero spread model, transaction-triggered price manipulation occurs for m ∈ ( − ρκ ρT , − ρκ ρT ) (seeTheorem 8.4) and classical price manipulation occurs for m ∈ ( − κT , − ρκ ρT ) (see Proposition 8.3). In thedynamic spread model, we can check by Proposition 5.8 that it is optimal to trade the entire order at T for m ∈ (cid:16) − κT , − κT (cid:0) − e − ρT (cid:1)(cid:17) (see Lemma B.1). We observe that − κT (cid:0) − e − ρT (cid:1) < − ρκ ρT (see Lemma B.2). Let us finally note thatthe presented methods do not allow us to calculate the optimal strategy in closed form in the dynamic38pread model for m ∈ h − κT (cid:0) − e − ρT (cid:1) , − ρκ ρT (cid:17) , but we can approximate it numerically in discrete time(see e.g. the case with K t = 1 − . t , ρ = 2 , T = 1 in Figure 4).See Figure 8 for an illustration. K Figure 8:
Straight-line price impact ( T = 1 , ρ = 2 , κ = 1 , x = 100 , δ = 0 , m = 0 . and − . (dashed)).
10 Conclusion
Time-varying liquidity is a fundamental property of financial markets. Its implications for optimal liq-uidation in limit order book markets is the focus of this paper. We find that a model with a dynamic,trading influenced spread is very robust and free of two types of price manipulation. We prove that valuefunctions and optimal liquidation strategies in this model are of wait-region/buy-region type, which isoften encountered in problems of singular control. In the literature on optimal trade execution in limitorder books, the spread is often assumed to be zero. Under this assumption we show that time-varying liq-uidity can lead to classical as well as transaction-triggered price manipulation. For both dynamic and zerospread assumptions we derive closed form solutions for optimal strategies and provide several examples.
A Integration by parts for càglàd processes
In various proofs in this paper we need to apply stochastic analysis (e.g. integration by parts or Ito’sformula) to càglàd processes of finite variation and/or standard semimartingales. As noted in Section 2,this is always done as follows: if U is a càglàd process of finite variation, we first consider the process U + defined by U + t := U t + and then apply standard formulas from stochastic analysis to it. As an example ofsuch a procedure we provide the following lemma, which is often applied in the proofs in this paper. Lemma A.1. (Integration by parts).
Let U = ( U t ) t ∈ [0 ,T +] and V = ( V t ) t ∈ [0 ,T +] be càglàd processes of finite variation and Z a semimartingale(in particular càdlàg), which may have a jump at . For t ∈ [0 , T ] , we have U t + Z t = U Z − + Z [0 ,t ] U s dZ s + Z [0 ,t ] Z s dU s , (48) U t + V t + = U V + Z [0 ,t ] U s dV s + Z [0 ,t ] V s + dU s . (49)39 roof. Let X and Y be càdlàg processes (possibly having a jump at ) with X being a semimartingaleand Y a finite variation process. By Proposition I.4.49 a) in Jacod and Shiryaev (2003), which is a variantof integration by parts for the case where one of the semimartingales is of finite variation, X t Y t = X − Y − + Z [0 ,t ] Y s − dX s + Z [0 ,t ] X s dY s , t ∈ [0 , T ] . (50)Equation (48) is a particular case of (50) applied to X := Z, Y := U + and equation (49) is a particularcase of (50) applied to X := V + , Y := U + , where U + t := U t + and V + t := V t + . B Technical lemmas used in Example 9.7
Below we use the notation of Example 9.7.
Lemma B.1.
For m ∈ (cid:0) − κT , − κT (cid:0) − e − ρT (cid:1)(cid:1) we have ( κ + mt ) e − ρ ( T − t ) > κ + mT, t ∈ [0 , T ) , (51) i.e. Proposition 5.8 applies.Proof. Inequality (51) is equivalent to m < − κ (cid:0) − e − ρ ( T − t ) (cid:1) T − te − ρ ( T − t ) . The assertion now follows from our assumption on m . To see this, we need to show that − κT (cid:0) − e − ρT (cid:1) ≤ − κ (cid:0) − e − ρ ( T − t ) (cid:1) T − te − ρ ( T − t ) , which in turn is equivalent to g ( t ) := 1 − e − ρ ( T − t ) T − te − ρ ( T − t ) ≤ − e − ρT T = g (0) . This is a true statement since − x ≤ e − x for all x ∈ R and therefore g ′ ( t ) = 1 − ρ ( T − t ) − e − ρ ( T − t ) e ρ ( T − t ) ( T − te − ρ ( T − t ) ) ≤ . Lemma B.2.
We have − κT (cid:0) − e − ρT (cid:1) < − ρκ ρT .Proof. The statement reduces to proving that ρT ρT < − e − ρT . Setting x := ρT > we see that it isenough to establish that e − x < x , which is true as, clearly, e x > x .40 eferences Alfonsi, A., A. Fruth, and A. Schied, 2010, Optimal execution strategies in limit order books with generalshape functions,
Quantitative Finance
10, 143–157.Alfonsi, A., A. Schied, and A. Slynko, 2011, Order book resilience, price manipulation, and the positiveportfolio problem,
Preprint .Almgren, R.F., 2003, Optimal execution with nonlinear impact functions and trading-enhanced risk,
Applied Mathematical Finance
10, 1–18., 2009, Optimal trading in a dynamic market,
Preprint ., and N. Chriss, 2001, Optimal execution of portfolio transactions,
Journal of Risk
3, 5–40.Bank, P., and D. Becherer, 2009, Talk: Optimal portfolio liquidation with resilient asset prices,
Liquidity- Modelling Conference, Oxford .Bertsimas, D., and A. Lo, 1998, Optimal control of execution costs,
Journal of Financial Markets
1, 1–50.Bouchaud, J. P., Y. Gefen, M. Potters, and M. Wyart, 2004, Fluctuations and response in financialmarkets: The subtle nature of ‘random’ price changes,
Quantitative Finance
4, 176–190.Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2001, Market liquidity and trading activity,
Journal of Finance
56, 501–530.Cont, R., S. Stoikov, and R. Talreja, 2010, A stochastic model for order book dynamics,
OperationsResearch
58, 217–224.Easley, D., and M. O’Hara, 1987, Price, trade size, and information in securities markets,
Journal ofFinancial Economics
19, 69–90.Esser, A., and B. Mönch, 2003, Modeling feedback effects with stochastic liquidity,
Preprint .Fruth, A., 2011, Optimal order execution with stochastic liquidity, PhD Thesis, TU Berlin.Gatheral, J., 2010, No-dynamic-arbitrage and market impact,
Quantitative Finance
10, 749–759., A. Schied, and A. Slynko, 2011a, Exponential resilience and decay of market impact,
Econophysicsof Order-driven Markets pp. 225–236., 2011b, Transient linear price impact and Fredholm integral equations,
To appear in MathematicalFinance .Gregory, J., and C. Lin, 1996,
Constrained Optimization in the Calculus of Variations and Optimal ControlTheory (Springer).Huberman, G., and W. Stanzl, 2004, Price manipulation and quasi-arbitrage,
Econometrica
72, 1247–1275.Jacod, J., and A. Shiryaev, 2003,
Limit Theorems for Stochastic Processes, 2nd edition (Springer).Kempf, A., and D. Mayston, 2008, Commonalities in the liquidity of a limit order book,
Journal ofFinancial Research
31, 25–40.Kim, S.J., and S. Boyd, 2008, Optimal execution under time-inhomogeneous price impact and volatility,
Preprint . 41yle, A.S., 1985, Continuous auctions and insider trading,
Econometrica
53, 1315–1335.Large, J., 2007, Measuring the resiliency of an electronic limit order book,
Journal of Financial Markets
10, 1–25.Lorenz, J., and J. Osterrieder, 2009, Simulation of a limit order driven market,
The Journal Of Trading
4, 23–30.Madan, D. B., and W. Schoutens, 2011, Tenor specific pricing,
Preprint .Naujokat, F., and N. Westray, 2011, Curve following in illiquid markets,
Mathematics and FinancialEconomics
4, 1–37.Obizhaeva, A., and J. Wang, 2006, Optimal trading strategy and supply/demand dynamics, Preprint.Predoiu, S., G. Shaikhet, and S.E. Shreve, 2011, Optimal execution in a general one-sided limit-orderbook,
SIAM Journal on Financial Mathematics
2, 183–212.Schöneborn, T., 2008, Trade execution in illiquid markets, PhD Thesis, TU Berlin.Shiryaev, A., 1995,
Probability, 2nd edition (Springer).Steinmann, Georges, 2005, Order book dynamics and stochastic liquidity in risk-management, Master’sthesis ETH Zurich and University of Zurich.Weiss, A., 2010, Executing large orders in a microscopic market model,