General Intensity Shapes in Optimal Liquidation

Transcription

1 General Intensity Shapes in Optimal Liquidation Olivier Guéant, Charles-Albert Lehalle First draft: November his version: July 2012 Abstract We study the optimal liquidation problem using limit orders. If the seminal literature on optimal liquidation, rooted to Almgren-Chriss models, tackles the optimal liquidation problem using a trade-off between market impact and price risk, it only answers the general question of the liquidation rhythm. he very question of the actual way to proceed is indeed rarely dealt with since most classical models use market orders only. Our model, that incorporates both price risk and non-execution risk, answers this question using optimal posting of limit orders. he very general framework we propose regarding the shape of the intensity generalizes both the risk-neutral model presented of Bayraktar and Ludkovski [13] and the model developed in [25], restricted to exponential intensity. Introduction Since the late nineties and the first papers on the impact of execution costs on trading strategies e.g. [14], an important literature has appeared to tackle the problem of optimal liquidation. his literature, often rooted to the seminal papers by Almgren and Chriss [7, 8], has long been characterized by a trade-off between, on one hand, market impact that incites to trade slowly and, on the other hand, price risk that provides incentive to trade fast. he first family of models, following Almgren and Chriss, considered general instantaneous price impact sometimes called execution cost and linear permanent price impact. Several generalizations have been proposed such as an extension to random execution costs [10], or stochastic volatility and stochastic liquidity [5, 6]. Also, many objective functions to design optimal strategies have been proposed and discussed in order to understand the assumptions under which optimal strategies are deterministic as opposed to adaptive. he initial mean-variance framework has been expressed in the more rigorous framework of a CARA utility function [45], a mean-quadratic-variation framework has been considered [19, 47], initial-time mean-variance criterion has been discussed [9, 40] and the case of a his research was partially supported by the Research Initiative Microstructure des marchés financiers under the aegis of the Europlace Institute of Finance. he authors also wish to acknowledge the helpful conversations with Yves Achdou Université Paris-Diderot, Guy Barles Université de ours, Erhan Bayraktar University of Michigan, Bruno Bouchard Université Paris-Dauphine, Jean-Michel Lasry Université Paris-Dauphine, Mike Ludkovski UC Santa Barbara and Nizar ouzi Ecole Polytechnique. UFR de Mathématiques, Laboratoire Jacques-Louis Lions, Université Paris-Diderot. 175, rue du Chevaleret, Paris, France. olivier.gueant@ann.jussieu.fr Head of Quantitative Research. Crédit Agricole Cheuvreux. 9, Quai du Président Paul Doumer, Courbevoie, France. clehalle@cheuvreux.com 1

2 general utility function has recently been considered [44] to justify aggressive-in-the-money or passive-in-the-money strategies. Slightly different models have been proposed in this first generation of models see e.g. [30] and [32] and they all derive from the initial models by Almgren and Chriss since market impact is either permanent or instantaneous and they do not take into account the resilience of the underlying order book. Another family of models appeared following a paper by Obizhaeva and Wang [41] that directly models the limit order book and considers its resilient dynamic after each trade. his second generation of optimal liquidation models, that incorporate transient market impact, has developed in recent years [1], [2], [3] and [42] and raises the theoretical question of the functional form for the transient market impact that are compatible with the absence of price manipulation see [4], [21] and [23]. All these models only make use of market orders and hence only consider strategies that consume available liquidity. Hence, they do not consider the possible use of limit orders that provide liquidity to the market nor the post-mifid possible use of dark pools. Notwithstanding the preceding criticism, models à la Almgren-Chriss provide a rather acceptable answer to the macroscopic question of the optimal rhythm at which liquidation must be proceeded at least once the instantaneous market impact function has been replaced by an execution cost function modeling the ability to trade over short periods of time with all possible means including limit orders, dark pools and market orders. However, they do not answer the question of the optimal way to proceed in practice and the methods currently used in the industry are seldom based on optimal control models at the microscopic level. his paper provides such a model of optimal liquidation using limit orders 1 and can be used either to liquidate a portfolio as a whole over a few hours or on shorter periods of time to follow a trading curve, be it a WAP curve, a VWAP curve or an Almgren-Chriss trading curve. In our approach, a trader sends limit orders to the market thus providing liquidity instead of consuming it and does not know when his orders are going to be executed, if at all. As a consequence, the classical trade-off between market impact or execution cost and price risk is not at the core of our model 2. In our setting, a new risk is in fact borne by the trader because execution is now a random process and this non-execution risk is very different, in its nature, from price risk. his new risk characterizes the recent literature on optimal liquidation, focusing rather on the optimal way to liquidate than on the optimal liquidation rhythm. he recent literature on optimal liquidation focuses indeed on alternatives to the use of market orders. Kratz and Schoneborn [36] proposed an approach inspired from models of the first family, but with both market orders and access to dark pools. Although they considered no risk-aversion with respect to the new risk borne by the trader, their model is one of the first in this new family of models. he optimal split of large orders across liquidity pools has then been studied by Laruelle, Lehalle and Pagès in [37] but the literature focuses perhaps more today on limit orders than on dark pools. Liquidation with limit orders has indeed been developed by Bayraktar and Ludkovski [13] for general intensity functions but only in a risk-neutral framework. Guéant, Lehalle and Fernandez-apia [25] considered in parallel the specific case of an exponential intensity for a risk-adverse agent. Very recently, Huitema [33] considered liquidation involving market orders and limit orders, and Guilbaud and Pham [29] also proposed a liquidation model in a pro-rata microstructure. 1 and also market orders as it will be discussed in section 5. 2 Execution costs only appear to liquidate the shares which may be remaining at the end of the period we consider. 2

3 One should also note that many models dealing with high-frequency market making have been developed and can be adapted to deal with optimal liquidation. Building on the model proposed by Ho and Stoll [31] and then modified by Avellaneda and Stoikov [11] 3, Cartea, Jaimungal and Ricci [18] consider a model with exponential intensity, market impact on the limit order book, adverse selection effects and predictable α. Cartea and Jaimungal [17] recently used a similar model to introduce risk measures for high-frequency trading. Earlier, the same authors proposed a model [16] in which the reference price is modeled by a Hidden Markov Model. Eventually, Guilbaud and Pham [28] also used a model including both market orders and limit orders at best and next to best bid and ask together with stochastic spreads. In this paper, we generalize both [13] and [25]. We indeed consider both general shapes for the intensity functions, and an investor with a CARA utility function. Also, we present a limiting case in which the size of the orders tends to 0 and show that this limiting case approximation is intrinsically linked to the usual continuous framework of Almgren and Chriss although the latter framework only considers market orders. In the first section, we present the setting of the model and the main hypotheses on execution. he second section is devoted to the resolution of the partial differential equations arising from the control problem. hen, in section 3 we provide illustrations of the model and we exhibit the asymptotic behavior of the quotes, generalizing therefore a result presented in [25]. Section 4 is then dedicated to the study of a limit regime that corresponds to orders of small size. his fourth section leads to results linked to those obtained for the fluid limit in [13], here in a risk-adverse setting. his result is exploited in section 5 that draws parallels between our model and the usual Almgren-Chriss framework. We eventually discuss the choice of the intensity and the use of the model in practice. 1 Optimal execution with limit orders: the model 1.1 Setup of the model Let us fix a probability space Ω, F, P equipped with a filtration F t t 0 satisfying the usual conditions. We assume that all random variables and stochastic processes are defined on Ω, F, F t t 0, P. We consider a trader who has to liquidate a portfolio containing a quantity q 0 of a given stock. We suppose that the reference price of the stock that can be considered the first bid quote for example follows a Brownian motion 4 : ds t = σdw t he trader under consideration will continuously propose an ask quote denoted St a and will hence sell shares according to the rate of arrival of liquidity-consuming orders at the prices he quotes. His inventory q, that is the quantity he holds, is given by q t = q 0 N t where N is a point process giving the number of executed orders, each order being of size and entirely 3 See also [26]. 4 he model can be generalized with the adjunction of a drift. 3

4 filled 5 we suppose obviously that is a fraction of q 0. Arrival rates depend on the ask price St a quoted by the trader at date t and we posit that the intensity λ associated to N is of the following form, that depends on : λs a, s, = Λ s a s where Λ : R R + satisfies the following assumptions 6 : Λ is strictly decreasing the cheaper the order price, the faster it will be executed, lim δ + Λ δ = 0, Λ C 2, Λ δλ δ 2Λ δ 2. As a consequence of his trades, the trader has an amount of cash whose dynamics is given by: dx t = S a t dn t. Now, coming to the liquidation problem, the trader has a time horizon to liquidate his shares and his goal is to optimize the expected utility of his P&L at time. We focus on CARA utility functions so that the trader maximizes: E [ exp γ X + q S lq ] where γ > 0 is the absolute risk aversion parameter characterizing the trader, where X is the amount of cash at time and where q is the remaining quantity of shares in the inventory at time. In this setting, the trader can sell the shares remaining at time in his portfolio at a price below the reference price, namely S lq, the function l being a positive and increasing function measuring execution cost. 1.2 From Hamilton-Jacobi to a system of ODEs he optimization problem set up in the preceding section can be solved using classical Bellman tools. o this purpose, we introduce the Hamilton-Jacobi-Bellman equation associated to the optimization problem, where the unknown u is going to be the value function of the problem: with the final condition: and the boundary condition: HJB 0 = t u t, x, q, s σ2 2 ssu t, x, q, s + sup Λ s a s [u t, x + s a, q, s u t, x, q, s] s a u, x, q, s = exp γ x + qs lq u t, x, 0, s = exp γx 5 here is no partial fill in this model. 6 We assume that Λ is defined on the entire real line and not only on R +. We shall discuss in section 5 the interpretation of quotes below the reference price hereafter called negative quotes. 4

5 Since we use a CARA function, we can factor out the Mark-to-Market MtM value of the portfolio and we consider the change of variables u t, x, q, s = exp γx + qs + θ t, q. In that case, the above HJB equation with 4 variables is formally reduced to the following system of ODEs indexed by q: with where HJ θ 0 = γ t θ t, q 1 2 γ2 σ 2 q 2 + H θ t, q θ t, q θ, q = lqq, θ t, 0 = 0 H p = sup Λ δ 1 e γδ p δ 2 Solving the optimal control problem his section aims at solving the optimal control problem set up in the preceding section. We first concentrate on the equation HJ θ and we then provide a verification theorem that indeed provides a solution to the control problem and characterizes in a simple way the optimal quotes. 2.1 A solution to HJ θ We start with a lemma about the hamiltonian function H. Lemma 1. Let us define L p, δ = Λ δ 1 e γδ p. p R, δ L p, δ reaches its maximum at δ p uniquely characterized by: δ p 1 γ log 1 γ Λ δ p Λ δ p = p Moreover, p δ p is a C1 function. Subsequently H is a C 1 function with: Proof: Λ δ H p = γ p2 γλ δ p Λ δ p First, let us notice that L p, p = 0 and that lim δ + L p, δ = 0. Now, if we differentiate, we get: δ L p, δ = Λ δ 1 e γδ p + γλ δe γδ p Hence, δ L p, p = γλ p > 0 and there is at least one δ δ L p, δ = 0 and corresponding to a maximum of L p,. p, + such that Such a δ must satisfy: δ 1 γ log 1 γ Λ δ Λ δ 5 = p

6 Now fx = x 1 γ log 1 γ Λ x Λ x defines a strictly increasing function since f x = 1 + Λ x Λ x 1 γ Λ x Λ x = 1 + Λ x2 Λ xλ x Λ x2 γλ xλ x = γλ xλ x Λ x2 γλ xλ x + 2Λ x2 Λ xλ x Λ x2 γλ xλ x and this expression is strictly positive because of the hypotheses on Λ. Hence L p, reaches its unique maximum at δ p uniquely characterized by: δ p 1 γ log 1 γ Λ δ p Λ δ p = p Using the implicit function theorem, this also gives that p δ p is a C1 function. Plugging the relation for δ p in the definition of H then gives the last part of the lemma. Now, we are going to prove a comparison principle for the system of ODEs. his result is useful in two ways. First, it gives a priori bounds that will allow us to prove the existence of a solution to HJ θ. Second, it will provide bounds to θ along with Lipschitz-type regularity properties, both being independent of. Proposition 1 Comparison principle. Let τ [0,, k {0, 1,..., q 0 }. and Let θ : [τ, ] {0,,..., k} R be a C 1 function with respect to time with q {0,,..., k}, θ, q lqq t [τ, ], θ t, γ t θ t, q 1 2 γ2 σ 2 q 2 θ t, q θ + H t, q Let θ : [τ, ] {0,,..., k} R be a C 1 function with respect to time with q {0,,..., k}, θ, q lqq t [τ, ], θ t, 0 0 and hen 0 γ t θ t, q 1 2 γ2 σ 2 q 2 θ t, q θ t, q + H θ θ Proof: Let α > 0. 6

7 Let us consider a point t α, q α such that θ t α, q α θ t α, q α α t α = If t α and q α 0 then: sup θ t, q θ t, q α t t,q [τ, ] {0,...,k} t θ t α, q α t θ t α, q α α Now, by definition of the functions θ and θ 0 γ [ t θ t α, q α t θ t α, q α ] [ θ t + H α, qα θ t α, q α θ t H α, qα θ t α, q ] α Now, by definition of t α, q α, since q α 0: θ t α, q α θ t α, q α θ t α, q α θ t α, q α θ t α, q α θ t α, q α θ t α, q α θ t α, q α Since H is a decreasing function, we have θ t H α, qα θ t α, q α θ t H α, qα θ t α, q α his leads to 0 γα which is not possible. herefore t α = or q α = 0 so that: = max sup θ t, q θ t, q α t t,q [τ, ] {0,...,k} sup θ, q θ, q, sup θ t, 0 θ t, 0 α t q {0,...,k} t [τ, ] 0 hus t, q θ t, q θ t, q α and sending α to 0 proves our result. We are now ready to solve the equation HJ θ Proposition 2. here exists a unique function θ : [0, ] {0,, 2,..., q 0 } R, C 1 in time, solution of HJ θ. Proof: We proceed by induction on q. For q = 0, we have by definition θ t, 0 = 0. Now, for a given q {, 2,..., q 0 }, let us suppose that θ, q : [0, ] R is a C 1 function q q. hen, the ODE 0 = γ t θ t, q 1 2 γ2 σ 2 q 2 θ t, q θ t, q + H with the terminal condition θ, q = lqq 7

8 satisfies the assumptions of Cauchy-Lipschitz theorem. Consequently, there exists a solution t θ t, q on a maximal interval that is a sub-interval of [0, ] and we want to show that this sub-interval is [0, ] itself. o prove that, let suppose by contradiction that t, ] is the maximal interval with t > 0. Let us notice that, because H is positive, t θ t, q+ 1 2 γσ2 q 2 t is decreasing. Hence, the only possibility for t, ] to be a maximal interval in [0, ] is that lim t t + θ t, q = +. But, if we consider η > 0, k = q and τ = t + η, the two functions θ = θ and θ t, q = 1 γ H 0 t satisfy the assumptions of the above comparison principle. Hence, η > 0, t [ t + η, ], θ t, q 1 γ H 0 t, in contradiction with the fact that lim t t + θ t, q = +. Hence, t θ t, q is defined on [0, ] and this proves the result. 2.2 Verification theorem and optimal quotes Now, we can solve the initial optimal control problem and find the optimal quotes at which the trader should send his limit orders. heorem 1 Verification theorem and optimal quotes. Let us consider the solution θ of the system HJ θ. hen, u t, x, q, s = exp γx + qs + θ t, q defines a solution to HJB and is the value function of the optimal control problem, i.e.: u t, x, q, s = sup E δ At [ exp γ X t,x,δ ] + q t,q,δ S t,s lqt,q,δ where At is the set of predictable processes on [t, ], bounded from below and where: dsτ t,s = σdw τ, S t,s t = s dxτ t,x,δ = S τ + δ τ dn τ, X t,x,δ t = x dqτ t,q,δ = dn τ, q t,q,δ t = q where the point process has stochastic intensity λ τ τ with λ τ = Λ δ τ 1 qτ. Moreover the optimal ask quote St a = S t + δ t, for q t > 0, is characterized by: θ t, q t θ t, q t δ t = δ where δ p is uniquely characterized, as in Lemma 1, by: δ p 1 γ log 1 γ Λ δ p Λ δ p Proof: From the very definition of θ and u it is straightforward to see that u is a solution of HJB. 8 = p

9 We indeed have that the boundary condition and the terminal condition are satisfied. For q we have: t u t, x, q, s σ2 2 ssu t, x, q, s + sup Λ δ [u t, x + s + δ, q, s u t, x, q, s] δ = γu t, x, q, s t θ t, q σ2 γ 2 q 2 u t, x, q, s + sup Λ δu t, x, q, s exp γδ θ t, q θ t, q 1 δ [ = u t, x, q, s γ t θ t, q 1 2 σ2 γ 2 q 2 ] + sup Λ δ 1 exp γδ θ t, q θ t, q = 0 δ [ = u t, x, q, s γ t θ t, q 1 ] 2 σ2 γ 2 q 2 θ t, q θ t, q + H = 0 Now, we need to verify that u is indeed the value function associated to the problem and to prove that our candidate δ t is indeed the optimal control. o that purpose, let us consider a control δ At and let us consider the following processes for τ [t, ]: dsτ t,s = σdw τ, S t,s t = s dxτ t,x,δ = S τ + δ τ dn τ, X t,x,δ t = x dqτ t,q,δ = dn τ, q t,q,δ t = q where the point process has stochastic intensity λ τ τ with λ τ = Λ δ τ 1 qτ 7. Now, let us write Itô s formula for u 8 : + t u, X t,x,δ, qt,q,δ, St,s = u t, x, q, s τ u τ, X t,x,δ τ, qt,q,δ τ, St,s τ + u τ, X t,x,δ τ + St,s τ + δ τ, q t,q,δ τ t + t + σ2 2 2 ssu τ, X t,x,δ τ, St,s σ s u τ, X t,x,δ τ, qt,q,δ + u τ, X t,x,δ τ + St,s τ + δ τ, q t,q,δ τ t τ, St,s, qt,q,δ τ, St,s τ dτ u τ, X t,x,δ τ, qt,q,δ τ, St,s τ λ τ dτ τ, St,s τ τ dw τ u τ, X t,x,δ τ, qt,q,δ τ, St,s τ dm τ where M is the compensated process associated to N for the intensity process λ τ τ. Now, we have to ensure that the last two integrals consist of martingales so that their mean is 0. o that purpose, let us notice that s u = γqu and hence, since the process q t,q,δ takes values between 0 and q, we just have to prove that: 7 his intensity being bounded since δ is bounded from below. 8 he equality is still valid when q τ = 0 because of the boundary condition for u and because the intensity process is then assumed to be 0. 9

10 and Hence: We have: [ E t [ E t u τ, X t,x,δ τ, qt,q,δ τ, St,s τ u τ, X t,x,δ τ + St,s τ + δ τ, q t,q,δ τ [ E t u τ, X t,x,δ τ, qt,q,δ τ, St,s τ ] 2 dτ < +, St,s u τ, Xτ t,x,δ, qτ t,q,δ, Sτ t,s 2 exp 2γ θ exp exp 2γ θ exp 2γx q δ + 2q inf τ ] λ τ dτ < + 2γX t,x,δ τ τ [t, ] St,s τ exp 2γ θ exp 2γx q δ 1 + exp 2γq [ E t u τ, X t,x,δ τ, q t,q,δ τ ] [, Sτ t,s 2 dτ = E t exp 2γ θ exp 2γx q δ t because of the law of inf τ [t, ] S t,s τ. 1 + E ] λ τ dτ < + + qτ t,q,δ Sτ t,s 1 infτ [t, ] Sτ t,s <0 inf u τ, X t,x,δ τ, qt,q,δ [ exp 2γq τ [t, ] St,s τ τ, St,s τ inf ] 2 dτ τ [t, ] St,s τ ] < + Now, the same argument works for the second and third integrals, noticing that δ is bounded from below and that λ is bounded. Hence, since we have, by construction 9 τ u τ, X t,x,δ τ, qt,q,δ τ, St,s τ + σ2 + u τ, X t,x,δ τ + St,s τ + δ t, q t,q,δ τ, St,s τ we obtain that [ E u, X t,x,δ 2 2 ssu τ, X t,x,δ τ u τ, X t,x,δ, qt,q,δ τ, St,s τ τ, qt,q,δ τ, St,s τ λ τ 0 ] [ ], q t,q,δ, S t,s = E u, X t,x,δ, qt,q,δ, St,s u t, x, q, s and this is true for all δ At. Since for δ t = δ t we have an equality in the above inequality by construction of the function δ, we obtain that: [ sup E δ At u, X t,x,δ ] [, q t,q,δ, S t,s u t, x, q, s = E u, X t,x,δ ], q t,q,δ, S t,s i.e. [ sup E exp γ X t,x,δ δ At ] + q t,q,δ S t,s lqt,q,δ 9 his inequality is also true when the portfolio is empty because of the boundary conditions. 10

11 [ u t, x, q, s = E exp γ X t,x,δ ] + q t,q,δ S t,s lqt,q,δ his proves that u is the value function and that t δ t is optimal. heorem 1 provides a simple way to compute the optimal quotes. One has indeed to solve the triangular system of ODEs HJ θ to obtain the function θ. Numerically, this does not constitute any difficulty. hen, once θ has been computed, the optimal quotes are given by the simple expression δ characterized by θ t,q t θ t,q t δ p 1 γ log 1 γ Λ δ p Λ δ p = p and can then be easily computed using Newton s method for instance. where the function δ is implicitly o better understand the model, we are now going to relate our results with those of [25], provide asymptotic results about the quotes and discuss the role of the parameters. 3 Examples and properties In the above part, we generalized a model already used in [25], in which the intensity functions had exponential shape 10 : Λ δ = A e k δ. In the case of exponential intensity, we can write the results of [25] in a slightly more general case than in the original paper using the language of this paper. In fact, the reason why closed-form solutions can be obtained in the exponential case is because the equation HJ θ simplifies to a linear system of equations once we replace the unknown θ by exp k θ : Proposition 3. Let us consider θ : [0, ] {0,, 2,..., q 0 } R, C 1 in time, solution of HJ θ when Λ δ = A e k δ. Let us define w = exp k θ. hen, w solves: with t w t, q = 1 2 γk σ 2 q 2 w t, q A 1 + γ k 1 γ w t, q k w, q = e k lqq, w t, 0 = 1 and the optimal quote, for q t > 0, is given by: δ t = 1 w t, q t log + 1 k w t, q t γ log 1 + γ k We provide below numerical approximations of both the function θ t, q and the optimal control function δ t, q that will motivate general results. he first result presented in this section concerns the behavior of the quotes as the time horizon increases. It generalizes 10 Another example has been considered in the literature on optimal execution, namely power-law intensity functions in the risk-neutral case see [13]. hese functions are only defined for positive δ and do not enter the framework developed above, although this framework can be adapted to take these intensity functions into account. 11

12 an asymptotic result obtained for exponential intensity functions in [25]. he second one, presented in the next section, concerns the limit regime as the size of orders goes to 0. Figure 1: Solution θ t, q for Λ δ = Ae kδ, q 0 = 600, = 50, = 1200 s, σ = 0.3 ick.s 1 2, A = 0.1 s 1, k = 0.3 ick 1, γ = ick 1 and l = 3 ick. he index q {0, 50,..., 600} of each curve can be read from the terminal values. Figure 2: Optimal control δ t, q for Λ δ = Ae kδ, q 0 = 600, = 50, = 1200 s, σ = 0.3 ick.s 1 2, A = 0.1 s 1, k = 0.3 ick 1, γ = ick 1 and l = 3 ick. he lower the quote the larger the inventory q {50, 100,..., 600}. Figure 1 and Figure 2 represent respectively the solution θ and the optimal quotes 12

13 δ t, q as given by heorem 111. We see on Figure 1 that θ is not a monotonic function of q. his means that the certainty equivalent of holding a quantity q of shares, once the MtM value of these shares has been removed is not a monotonic function of q. At the time horizon, the function is decreasing but far from two effects are at stake. On one hand, when there are more shares in the portfolio, there will be more trades and hence more opportunities to make money through limit orders: this goes in the direction of an increasing function θ t,. On the other hand, the larger the inventory to liquidate, the more price risk, and this goes in the direction of a decreasing function θ t,. In spite of this absence of monotonicity, Figure 1 and Figure 2 suggest to study the asymptotic case and the result obtained in [25] indeed generalizes to nearly any intensity function: Proposition 4 Asymptotic behavior. Assume 12 that lim p + H p = 0 and σ > 0. he asymptotic behavior for θ is: lim θ 1 0, q = H γ2 σ 2 q 2 = θ q q {,2,...,q} he resulting asymptotic behavior for the optimal quote is: 1 lim + δ t=0 = δ H 1 2 γ2 σ 2 q0 2 = δ Proof: Let us define 13 for q {0,, 2,..., q 0 }: θ q = q {,2,...,q} 1 H 1 2 γ2 σ 2 q 2 Let us define for t [0, ] and q {0,,..., q 0 }, θ r t, q = θ t, q. hen, we want to prove that: q {0,,..., q 0 }, lim t + θr t, q = θ q We proceed by induction. he result is true for q = 0. Let us suppose that the result is true for q for some q {,..., q 0 }. hen: ϵ > 0, t q, t t q, θ r t, q θ q ϵ Now, since H is a strictly decreasing function we have for t t q : 1 θ 2 γ2 σ 2 q 2 r + H t, q θ q + ϵ γ t θt, r q 1 θ 2 γ2 σ 2 q 2 r + H t, q θ q ϵ 11 One may wonder why we choose a risk aversion parameter γ = his figure seems small but it has in fact an important impact since the shares are sold by groups of his is guaranteed if lim δ + δλ δ = his is well-defined because of the assumptions on H and σ, and because γ > 0. In the risk-neutral case, there is indeed no upper bound for the optimal quotes when. his constitutes an important difference between our model and the model of Bayraktar and Ludkovski [13]. 13

14 or equivalently: θ r H t, q θ q + ϵ θ H q θ q θ γ t θt, r r q H t, q θ q ϵ θ H q θ q and Hence, for t t q : As a consequence, if there exists t q t t q, θ r t, q θ q ϵ. θ r t, q > θ q + ϵ t θ r t, q < 0 θ r t, q < θ q ϵ t θ r t, q > 0 t q such that θ r t q, q θ q ϵ then, In particular, if θ r t q, q θ q ϵ then, t t q, θ r t, q θ q ϵ. Now, if θ r t q, q > θ q + ϵ, then there are two possibilities. he first one is that the function t t q θ r t, q is decreasing and in that case it is bounded from below by θ q ϵ and must converge. Since lim t + θ r t, q = θ q, the only possible limit for θ r t, q is θ q. he second possibility is that t t q θ r t, q is not a decreasing function and in that case there must exists t q t q such that θ r t q, q θ q + ϵ. Since θ r t q, q θ q ϵ, we now that t t q, θ r t, q θ q ϵ. Finally, if θ r t q, q < θ q ϵ, then there are two possibilities. he first one is that the function t t q θ r t, q is increasing and in that case it is bounded from above by θ q+ϵ and must converge. Since lim t + θ r t, q = θ q, the only possible limit for θ r t, q is θ q. he second possibility is that t t q θ r t, q is not an increasing function and in that case there must exists t q t q such that θ r t q, q θ q ϵ. Since θ r t q, q θ q + ϵ, we now that t t q, θ r t, q θ q ϵ. he conclusion is that lim sup t + θ r t, q θ q ϵ. Sending ϵ to 0, we get the result for θ. he result for the optimal quote is then straightforward. hese asymptotic formulae deserve some comments. Firstly, regarding the above discussion on monotonicity, we know that H is a decreasing function and we then see on the asymptotic limit that θ is either a decreasing function when 1 2 γ2 σ 2 2 > H 0 or a function that is first increasing and then decreasing otherwise. Secondly, coming to the optimal quotes and the role of the parameters, we can analyze the way δ depends on σ, γ and Λ. he best way to proceed is to use the expression for H found in Lemma 1 and to notice that an equivalent way to define δ is through the following implicit characterization: 1 2 γσ2 q0 2 = γλ δ Λ δ 2 Λ δ It is then straightforward to see that δ decreases as σ increases. An increase in σ corresponds to an increase in price risk and this provides the trader with an incentive to speed up the execution process. herefore, it is natural that the asymptotic quote is a decreasing 14

15 function of σ. Differentiating the above expression with respect to γ, we see that the asymptotic quote decreases as the risk aversion increases. An increase in risk aversion forces indeed the trader to reduce both non-execution risk and price risk and this leads to posting orders with lower prices. Now, if one replaces the intensity function Λ by λλ where λ > 1, then it results in an increase in δ. his is natural because when the rate of arrival of market orders increases, the trader is more likely to liquidate his shares faster and posting deeper into the book allows for larger profits. In addition to the asymptotic regime, we can consider different sizes of orders. We provide below a numerical approximation of the quotes for two different sizes, when the intensity function is scaled by the size of the orders: Figure 3: Optimal quotes for q {50, 100,..., 600}, for q 0 = 600, = 1200 s, σ = 0.3 ick.s 1 2, A = 0.1 s 1, k = 0.3 ick 1, γ = ick 1 and l = 3 ick. Line: Λδ = Ae kδ and = 50. Dotted line: Λδ = 2Ae kδ and = 25. We see on Figure 3, that there is little difference between the two cases we considered. his is linked to the existence of a limit regime as 0 and the next section is dedicated to its analysis. he limiting equation for θ will turn out to be a classical equation in optimal liquidation theory see section 5. 4 Limit regime 0 In the preceding section the size of the order sent by the trader was fixed at, a size that is supposed to be small with respect to q 0. As a consequence, the question of the limiting behavior when tends to 0 is relevant 14. o that purpose we need to make an assumption 14 Although this is not recalled, it is assumed that is always chosen as a fraction of q 0. 15

16 on the behavior of the intensity function with respect to the order size. he right scaling already used above for the numerics underlying Figure 3 is to suppose that Λ δ = Λδ With this scaling assumption, along with additional technical hypotheses, our goal is to prove the following heorem that is rather technical and echoes the results obtained by [13], here with risk aversion i.e. γ > 0 whereas [13] deals with a risk-neutral case: heorem 2 Limit regime 0. Let us suppose that: ΛδΛ δ < 2Λ δ 2 lim δ + δλδ = 0 l is a continuous function For a given > 0, let us define θ c on [0, ] [0, q 0] by: { θt, c θ t, 0, if q = 0 q = θ t, k + 1, if q k, k + 1] hen θ c converges uniformly toward a continuous function θ : [0, ] [0, q 0] R that solves in the viscosity sense: γ t θt, q γ2 σ 2 q 2 H q θt, q = 0, on [0, 0, 1] θt, q = 0, on [0, ] {0} θt, q = lqq, on { } [0, q 0 ] where Hp = γ sup δ Λδδ p and where the terminal condition and the boundary condition are in fact satisfied is the classical sense. o prove this theorem, we first need to study H and the convergence of the Hamiltonian functions H towards H. We start with a counterpart of Lemma 1 that requires ΛδΛ δ < 2Λ δ 2. Lemma 2. Let us define Lp, δ = Λδ δ p. p R, δ Lp, δ reaches its maximum at δ p uniquely characterized by: Moreover, p δ p is a C 1 function. Subsequently H is a C 1 function with: δ p + Λ δ p Λ δ p = p H p = γ Λ δ p 2 Λ δ p Proof: he proof is similar to the proof of Lemma 1. Now we can state a result about convergence that also provides a uniform bound for the hamiltonian functions: Lemma 3. H converges locally uniformly towards H when 0 with p R, H p Hp. 16

17 Proof: For a fixed x 0, the function f : R + 1 e x f0 = x. is decreasing function with Hence, 0 < <, sup Λδ 1 e γδ p δ p sup Λδ 1 e γ δ p δ p γ sup Λδδ p δ p his gives: H p H p Hp Now, because H is continuous, using Dini s theorem, if we prove that convergence is pointwise, convergence will be locally uniform. We are then left with the need to prove pointwise convergence of H toward H. Using Lemma 1 and Lemma 2 we see that it is sufficient to prove that δ converges pointwise towards δ. But the sequence of functions f x = x 1 γ log 1 γ Λx Λ x is an increasing sequence of increasing functions and hence by the unique characterizations of δ p and δ p, we see that δ p increases as decreases to 0 and is bounded from above by δ p which is the only possible limit. Hence δ p δ p as 0 and this proves the result. Now, we will provide a uniform bound for the θ that will be important in the proof of heorem 2. Proposition 5 Bounds for θ. t [0, ], q {0,,..., q 0 }, Proof: lq 0 q γσ2 q0 2 t θ t, q 1 H0 t γ o prove these inequalities, we use the comparison principle enounced above. If θ t, q = 1 γ H 0 t then: θ, q = 0 lqq θ t, 0 = 1 γ H 0 t 0 and γ t θ t, q 1 2 γ2 σ 2 q 2 θ t, q θ t, q + H = γ 1 γ H γ2 σ 2 q 2 + H 0 = 1 2 γ2 σ 2 q 2 0 Hence θ t, q θ t, q = 1 γ H 0 t. he uniform upper bound is then obtained using Lemma 3. Now, if θ t, q = lq 0 q γσ2 q0 2 t then: θ, q = lq 0 q 0 lqq θ t, 0 = 1 γ lq 0q γ2 σ 2 q 2 0 t 0 17

18 and γ t θ t, q 1 2 γ2 σ 2 q 2 θ t, q θ + H t, q = 1 2 γ2 σ 2 q γ2 σ 2 q 2 + H γ2 σ 2 q 2 0 q 2 0 Hence θ t, q θ t, q = lq 0 q γσ2 q0 2 t. We are now ready to start the proof of heorem 2. Proof of heorem 2: We first introduce the following half-relaxed limit functions: θt, x = lim sup j + lim sup sup{θt c, q, t t + q q 1 0 j } θt, x = lim inf lim inf j + 0 inf{θc t, q, t t + q q 1 j } θ and θ are respectively upper semi-continuous and lower semi-continuous and the goal of the proof is to show that they are equal to one another and solution of the above partial differential equation. Step 1: θ and θ are respectively subsolution and supersolution of γ t θt, q γ2 σ 2 q 2 H q θt, q = 0, on [0, 0, 1] θt, q = 0, on [0, ] {0} θt, q = lqq, on { } [0, q 0 ] where the conditions are to be understood in the viscosity sense. o prove this point, let us consider a test function ϕ C 1 [0, ] [0, q 0 ] and t, q a local maximum of θ ϕ. Without loss of generality, we can assume that ϕt, q = θt, q and consider r > 0 such that: the maximum is global on the ball of radius r centered in t, q. outside of this ball ϕ 2 sup θ this value being finite because of the uniform bound obtained in the above Proposition. Following Barles-Souganidis methodology [12], we know that there exists a sequence n, t n, q n n such that: n 0, t n, q n t, q θ c n t n, q n θt, q t n, q n is a global maximum of θ c n ϕ Now, because of the definition of θ c n, if t, q [0, 0, q 0 ] we can always suppose that q n n and t n and then, by definition of θ n : γ t θ n t n, q n γ2 σ 2 qn 2 θn t n, q n θ n t n, q n n H n = 0 n Because H n is decreasing we have, by definition of t n, q n : 18

19 Hence: θn t n, q n θ n t n, q n n ϕtn, q n ϕt n, q n n H n H n n n Similarly, since t n < we have, for h sufficiently small: hese inequalities give: θ n t n + h, q n θ n t n, q n ϕt n + h, q n ϕt n, q n t θ n t n, q n t ϕt n, q n γ t ϕt n, q n γ2 σ 2 q 2 n H n ϕtn, q n ϕt n, q n n n 0 Using now the convergence of t n, q n towards t, q and the local uniform convergence of H n towards H we eventually obtain the desired inequality: γ t ϕt, q γ2 σ 2 q 2 H q ϕt, q 0 We see that the boundaries corresponding to q = q 0 and t = 0 play no role. However, we need to consider the cases t = and q = 0. If t = and q 0 then there are two cases. If there are infinitely many indices n such that t n < then the preceding proof still works. Otherwise, for all n sufficiently large, θ n t n, q n = lq n q n and hence, passing to the limit, θt, q = lq q. Eventually, we indeed have that: min γ t ϕ, q γ2 σ 2 q 2 H q ϕ, q, θ, q + lq q 0 If q = 0 then there are also two cases. If there are infinitely many indices n such that qn > 0 and t n < then the initial proof still works. Otherwise, for n sufficiently large θ n t n, q n = lq n q n or θ n t n, q n = 0 and hence, passing to the limit, we obtain θt, q = 0. Eventually, we indeed have that: min γ t ϕt, 0 H q ϕt, 0, θt, 0 0 We have proved that θ is a subsolution of the equation in the viscosity and one can similarly prove that θ is a supersolution. Step 2: q [0, q 0 ], θ, q = θ, q = lqq. We consider the test function ϕt, q = C ϵ t+ 1 ϵ q q ref 2 where q ref [0, q 0 ] is fixed, where ϵ > 0 is a constant and where C ϵ is a constant that depends on ϵ and will be fixed later. Let t ϵ, q ϵ be a maximum point of θ ϕ on [0, ] [0, q 0 ]. hen: θ, q ref θt ϵ, q ϵ C ϵ t ϵ 1 ϵ q ϵ q ref 2 his inequality gives q ϵ q ref as ϵ 0. 19

20 Now, γ t ϕt ϵ, q ϵ γ2 σ 2 q ϵ 2 H q ϕt ϵ, q ϵ = γc ϵ γ2 σ 2 q ϵ 2 H γc ϵ H 2q 0 ϵ 2 ϵ q ϵ q ref Hence, if we consider C ϵ = 1 γ H 2q 0 ϵ + 1, we see that we must have either t ϵ = and θt ϵ, q ϵ lq ϵ q ϵ or for sufficiently small ϵ, only in the case where q ref = 0, θt ϵ, q ϵ 0. If q ref 0, we then write, for ϵ sufficiently small: θ, q ref θt ϵ, q ϵ C ϵ t ϵ 1 ϵ q ϵ q ref 2 θt ϵ, q ϵ lq ϵ q ϵ Sending ϵ to 0 we obtain θ, q ref lq ref q ref If q ref = 0, then we know from the above two inequalities that: θ, q ref θt ϵ, q ϵ C ϵ t ϵ 1 ϵ q ϵ q ref 2 θt ϵ, q ϵ max lq ϵ q ϵ, 0 = 0 Hence q [0, q 0 ], θ, q lqq and the same proof works for the supersolution 15 to get q [0, q 0 ], θ, q lqq. As a consequence q [0, q 0 ], θ, q lqq θ, q and eventually, because θ, q θ, q: q [0, q 0 ], θ, q = lqq = θ, q. Step 3: t [0, ], θt, 0 = 0 Concerning the boundary condition corresponding to q = 0 we can apply the same ideas but only to the supersolution θ. Let us consider indeed t ref [0, and the test function ϕt, q = C ϵ q 1 ϵ t t ref 2. hen, let t ϵ, q ϵ be a minimum point of θ ϕ on [0, ] [0, q 0 ]. We have: his inequality gives t ϵ t ref as ϵ 0. Now, θt ref, 0 θt ϵ, q ϵ + C ϵ q ϵ + 1 ϵ t ϵ t ref 2 γ t ϕt ϵ, q ϵ γ2 σ 2 q ϵ 2 H q ϕt ϵ, q ϵ = 2γ t t ref ϵ γ2 σ 2 q ϵ 2 H C ϵ 2γ ϵ γ2 σ 2 q 0 2 H C ϵ Since lim p Hp = +, we can always choose C ϵ so that the above expression is strictly negative. 15 he only difference is that we have to pass to the limit in the case q ref = 0 to obtain the conclusion. 20

21 As a consequence, for ϵ sufficiently small, we must have q ϵ = 0 and θt ϵ, 0 0. Consequently: θt ref, 0 θt ϵ, q ϵ + C ϵ q ϵ + 1 ϵ t ϵ t ref 2 0 his result being already true for t ref =, we have that θt, 0 0, t [0, ] and in fact θt, 0 = 0, t [0, ] because of the definition of θt, 0. Step 4: t [0, ], there exists a sequence t n, q n such that t n t, q n 0, t n, q n t, 0 and θt n, q n 0 o prove this claim, we prove that g t = θ t, converges uniformly in t toward 0 as 0. By definition g 0 = l and g t = 1 2 γσ γ H g t. he stationary state of the above ODE is g = H γ2 σ 2 2 and H γ2 σ 2 2 H γ2 σ 2 2 as soon as <. Hence g is positive for sufficiently small. As a consequence, because H is decreasing, for sufficiently small we know that g is increasing, bounded from below by l and bounded from above by g. Now g t 1 γ H g t. Hence, g t and this gives g t G 1 γ Consequently, 0 g t γ H y dy l, where Gx = x 0 γ H y dy t 1 Hy dy. l g t G 1 γ Gx and it is sufficient to have lim x + x = +. But this is true as soon as lim x + Hx = 0, which is guaranteed by lim δ + δλδ = 0. Step 5: Comparison principle: θ θ. Let us consider α > 0 and the maximum M = max t,q [0, ] [0,q0 ] θt, q θt, q α t. If m = max t [0, ] θt, 0 θt, 0 α t < M then we distinguish two cases. Case 1: m 0. We introduce Φ ϵ t, q, t, q = θt, q θt, q α t q q 2 ϵ t t 2 ϵ. Let us consider t ϵ, q ϵ, t ϵ, q ϵ a maximum point of Φ ϵ. We have M Φ ϵ t ϵ, q ϵ, t ϵ, q ϵ and we are going to prove that lim inf ϵ 0 Φ ϵ t ϵ, q ϵ, t ϵ, q ϵ 0. We have Φ ϵ t ϵ, q ϵ, t ϵ, q ϵ Φ ϵ t ϵ, q ϵ, t ϵ, q ϵ and hence q ϵ q ϵ 2 consequence, t ϵ t ϵ 0 and q ϵ q ϵ 0. ϵ + t ϵ t ϵ 2 ϵ is bounded. As a 21

22 Now, if for all ϵ sufficiently small we have t ϵ, q ϵ, t ϵ, q ϵ [0, 0, q 0 ] [0, 0, q 0 ] then we have: and 2γ t ϵ t ϵ ϵ + γα γ2 σ 2 qϵ 2 H2 q ϵ q ϵ 0 ϵ 2γ t ϵ t ϵ ϵ γ2 σ 2 q ϵ 2 q ϵ q H2 ϵ 0 ϵ Hence γα γ2 σ 2 q 2 ϵ q ϵ 2 0 and this is a contradiction as we send ϵ to 0. he consequence is that there exists a sequence ϵ n 0 such that t ϵn, q ϵn, t ϵ n, q ϵ n verifies t ϵn =, n or q ϵn = 0, n or t ϵ n, n or q ϵ n, n. hen lim sup n + Φ ϵn t ϵn, q ϵn, t ϵ n, q ϵ n lim sup θt ϵn, q ϵn θt ϵ n, q ϵ n α t ϵn n + max θt, q θt, q α t max0, m 0 t,q { } [0,q 0 ] [0, ] }q 0 } Hence in that case M 0. Case 2: m > 0. In that case, we replace θ by θ + m in the above case and we obtain, instead of M 0, the inequality M m which contradicts our hypothesis. It remains to consider the case m = M. We know then that the maximum M is attained at a point t max, 0 and we suppose that M > 0. From Step 4, we consider a sequence t n, q n such that t n t max, q n 0, t n, q n t max, 0 and θt n, q n 0. We define: Ψ n t, q, t, q = θt, q θt, q α t t t 2 t n t max q q 2 1 q n his function attains its maximum at a point t n, qn, t n, q n. inequality Ψ n t max, 0, t n, q max Ψ n t n, qn, t n, q n : We first consider the θt max, 0 θt n, q n α t max t n t max θt n, q n θt n, q n α t n t n t n 2 t n t max We then have t n t n 0 and q n q n 0. q n q n q n 1 Hence lim sup n θt n, q n θt n, q n α t n M. Now, the maximum M is also given by lim n θt max, 0 θt n, q n α t max and we obtain that the penalization term q n q n q n 1 2 converge to

23 his gives q n = q n + q n + oq n > 0. Now, if we have infinitely many n such that t n = then: M = lim θt max, 0 θt n, q n α t max sup θ, q θ, q = 0 n q [0,q 0 ] and this contradicts our hypothesis. Otherwise, for all n sufficiently large: 2γ t n t n t n t max γ2 σ 2 q 2 n H 2 q n q n 1 0 q n q n Now, going to the subsolution, if there are infinitely many n such that t n =, then: M = lim θt max, 0 θt n, q n α t max sup θ, q θ, q = 0 n q [0,q 0 ] in contradiction with our hypothesis. Else, if q n = 0 and θt n, q n 0 for infinitely many n then we obtain M = lim n θt max, 0 θt n, q n α t max t n t max 0 straightforwardly and this contradicts our hypothesis. Hence, the viscosity inequality must be satisfied and we get: 2γ t n t n t n t max + γα γ2 σ 2 qn 2 H 2 q n qn 1 0 q n q n Combining the two inequalities eventually leads to αγ 0 as n and this is a contradiction. We have obtained that M 0 and hence θ θ α and, sending α to 0 we get θ θ. his proves that θ = θ is in fact a continuous function that we call θ, solution of the PDE introduced above. We have: θt, q = θt, q lim inf 0 θc t, q lim sup θt, c q θt, q = θt, q 0 Hence θt, q = lim 0 θ c t, q and, by the same token, lim 0,t,q t,q θ c t, q = θt, q so that the convergence is locally uniform and then uniform on the compact set [0, ] [0, q 0 ]. 5 Link with Almgren-Chriss and interpretation of Λ for negative δ 5.1 Back to Almgren-Chriss In the above section we proved that θ converges to θ which is the unique continuous viscosity solution 16 of 16 Uniqueness of continuous solutions can be shown using similar comparison principle arguments as in the above proof. 23

24 γ t θt, q γ2 σ 2 q 2 H q θt, q = 0, on [0, 0, 1] θt, q = 0, on [0, ] {0} θt, q = lqq, on { } [0, q 0 ] with the limit condition and boundary condition satisfied in the classical sense. Now, we are going to relate this equation to a classical equation in the Almgren-Chriss model of optimal liquidation. he intuition behind this link between our framework in the limit regime 0 and the Almgren-Chriss framework is that non-execution risk vanishes as tends to 0. Hence, the only remaining risk is price risk, corresponding to the 1 2 γ2 σ 2 q 2 term in the above equation. o see more precisely the correspondence between the two approaches, let us write the hamiltonian function as: Hp = γ sup Λδδ p δ = γ sup vλ 1 v p v>0 = γ sup vλ 1 v pv v 0 where the last equality holds since lim δ + δλδ = 0. Hence, if we define for v > 0, fv = Λ 1 v, then we can define the hamiltonian function Hp = sup v 0 fvv pv and write the partial differential equation for θ as: t θt, q γσ2 q 2 H q θt, q = 0, on [0, 0, 1] θt, q = 0, on [0, ] {0} θt, q = lqq, on { } [0, q 0 ] his equation is the Hamilton-Jacobi equation associated to the Almgren-Chriss optimal liquidation problem with an instantaneous market impact function per share f and with a final discount lq q at the time horizon 17. However, the instantaneous market impact function sometimes also phrased execution cost function has here a rather unusual form since fv is negative for v < Λ0 and positive for v > Λ0, whereas it is usually a positive function. his must be interpreted in a very simple way: if one needs to obtain an instantaneous volume inferior to Λ0, then one will choose a positive δ, that is he will send a limit order 18 this makes sense since non-execution risk disappears in the limit regime 0; on the contrary, if one needs an instantaneous volume greater than Λ0, then he will rely on a market order δ < 0. he above discussion only makes sense at the limit, when non-execution risk does not exist anymore. However, it provides some intuition about the way to understand the model, especially when it comes to negative δ. In particular, we will see that the above correspondence between our model and a model à la Almgren-Chriss provides a possible way to solve one of the main problems of the model discussed in [25]: the interpretation of the exponential intensity functions for negative values of δ. 17 In the usual Almgren-Chriss framework, this discount is in fact 1 q >0 to force liquidation but the theory can easily be adapted to incorporate a finite discount. 18 since we assumed that the reference price is the first bid quote. 24

25 5.2 Application to the choice of Λ he model we discuss in this paper does not consider explicitly market order or limit orders but rather considers that there is, for each price s a = s + δ, an instantaneous probability to obtain a trade at that price. In practice, this interpretation is well suited for limit orders, that is for positive values of δ, but we need to provide an interpretation for negative values of δ and similarly provide an intuition for the signification of the intensity function Λ on the entire real line. In practice, since the model has been designed to liquidate a position with limit orders, it should not be used if the liquidation under scrutiny evidently requires liquidity-consuming orders. However, it may happen, because of a slow execution, that the optimal quote in the model turns out to be negative 19 after some time. his issue of negative δ was present in our initial model with exponential intensity functions see [25]. Although the exponential form was justified for some stocks when δ 0, the intensity function was also exponential for δ < 0 and this choice was rather dictated by mathematical needs than by empirical rationale. Now, since Λ or in practice Λ can be chosen, we can improve by far the initial model. First, using statistics on execution, we can estimate the probability to be executed at any positive distance of the first bid limit. In practice the profile of the empirical intensity for positive δ is decreasing and may not be convex, especially when the bid-ask spread is large. his is an important remark because in the case of a non-convex Λ the hamiltonian function H has no reason to be convex either and then, the equation HJ θ is not directly associated to a deterministic control problem. hen, once the function Λ has been calibrated for positive δ, several natural choices are possible for Λ δ when δ 0. Instead of extending the function for negative δ using a specific functional form as in [25], we can assign to Λ δ a constant value when δ 0, this constant being equal to the value of Λ his corresponds to a very conservative choice that basically prevents the use of market order since the optimal quote will always be positive. However, a better choice consists in using the parallel made between the usual literature and our framework when the limit regime 0 is considered. If we indeed omit the non-execution risk, we can consider, for δ 0, that Λ δ = 1 f 1 δ = 1 sup{v 0, fv δ} where v fv is an instantaneous market impact function average execution cost per share that is typically equal to nought for small values of v and increasing. his choice for Λ is however subject to several comments. First, the function Λ must satisfy the hypotheses of the model. In particular, it must be decreasing and it may happen that, although the specifications for positive δ and negative δ are decreasing, the function is not decreasing on the entire real line. Since the function δ 0 1 f 1 δ is rather a lower bound to Λ δ because we have to take account of the non-execution risk, we can always scale the function δ 0 Λ δ so that the resulting function Λ is decreasing and strictly decreasing if we smooth the function. Second, a question remains regarding the interpretation of the model when an optimal quote δ turns out to be negative. he answer we propose, in line with the parallel made with Almgren-Chriss-like models, is to send a market order of size Λδ = Λ δ. 19 Although quotes evolve continuously between execution times, the practical use of the model see [25] requires to stay in the limit order book for some time and the optimal quote at some point may in practice be a strictly negative figure. 20 Or in fact Λ tick size. 25

26 Conclusion In this paper, we analyze the optimal liquidation strategy of a trader with a single-stock portfolio using limit orders. he classical literature on optimal liquidation, following Almgren- Chriss, only answers the question of the optimal liquidation rhythm and a new literature has recently emerged that uses either dark pools or limit orders to tackle the issue of the actual optimal way to liquidate. Our paper provides a general model for optimal liquidation with limit orders and extends both [13] that only considers a risk-neutral framework and [25] that was restricted to exponential intensity functions. Our results can be extended easily with the adjunction of a drift term 21 and more importantly to the case of a multi-stock portfolio, taking into account correlation between stock prices. Another improvement would also consist in linking the Brownian motion which drives the price and the point process modeling execution. Research in this direction has recently been made by Cartea, Jaimungal and Ricci [18] to model market making and a simplified version of their approach could be introduced in our model. For practical purposes, the most important improvement would consist in introducing the bid-ask spread of the market in order to better account for changes in the probability to be executed at a given price. All these improvements are part of an ongoing research work. References [1] A. Alfonsi, A. Fruth, and A. Schied. Constrained portfolio liquidation in a limit order book model. Banach Center Publ, 83:9 25, [2] A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit order books with general shape functions. Quantitative Finance, 102: , [3] A. Alfonsi and A. Schied. Optimal trade execution and absence of price manipulations in limit order book models. SIAM J. Financial Mathematics, 1: , [4] A. Alfonsi, A. Schied, and A. Slynko. Order book resilience, price manipulation, and the positive portfolio problem [5] R. Almgren. Optimal trading in a dynamic market [6] R. Almgren. Optimal trading with stochastic liquidity and volatility [7] R. Almgren and N. Chriss. Value under liquidation. Risk, 1212:61 63, [8] R. Almgren and N. Chriss. Optimal execution of portfolio transactions. Journal of Risk, 3:5 40, [9] R. Almgren and J. Lorenz. Adaptive arrival price. Journal of rading, 20071:59 66, [10] R.F. Almgren. Optimal execution with nonlinear impact functions and tradingenhanced risk. Applied Mathematical Finance, 101:1 18, [11] Marco Avellaneda and Sasha Stoikov. High-frequency trading in a limit order book. Quantitative Finance, 83: , [12] G. Barles and P.E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. In Decision and Control, 1990., Proceedings of the 29th IEEE Conference on, pages IEEE, One exception concerns the asymptotic behavior when the drift goes above a certain threshold. 26