Optimal Execution Strategies in Limit Order Books with General Shape Functions

Quantitative Finance (Impact Factor: 0.75). 08/2007; 10(2). DOI: 10.2139/ssrn.1510104
Source: arXiv

ABSTRACT We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: (accessed 16 February 2009) [CrossRef]View all references) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: (accessed 16 February 2009) [CrossRef]View all references). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

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    • "The traditional point of view to this problem (e.g. (Alfonsi and Schied, 2010), and (Alfonsi et al., 2007) ), optimal liquidation depends on exiting the sufficiently large limit order. As consequence, price impact is function of the shape and depth of the Limit Order Book (LOB). "
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    ABSTRACT: In an illiquid market as a result of a lack of counterparties and uncertainty about asset values, trading of assets is not being secured by the actual value. In this research, we develop an algorithmic trading strategy to deal with the discrete optimal liquidation problem of large order trading with different market microstructures in an illiquid market. In this market, order flow can be viewed as a Point process with stochastic arrival intensity. Interaction between price impact and price dynamics can be modeled as a dynamic optimization problem with price impact as a linear function of the self-exciting dynamic process. We formulate the liquidation problem as a discrete-time Markov Decision Processes where the state process is a Piecewise Deterministic Markov Process (PDMP), which is a member of right continuous Markov Process family. We study the dynamics of a limit order book and its influence on the price dynamics and develop a stochastic model to retain the main statistical characteristics of limit order books in illiquid markets.
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    • "Since our main goal of this paper is to demonstrate the influence of FTT in determining a trader's optimal trading, we only focus on a specific situation from the general setting. Alfonsi et al. (2010) provide a more general model setting with respect to the limit order book dynamics. Let P n be the average execution price for x t n . "
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    ABSTRACT: This paper provides the optimal position management strategy for a market maker who has to face uncertain customer orders in an "illiquid" market, where the market maker's continuous trading through a traditional exchange incurs stochastic linear price impacts. In addition, it is supposed that the market participants can partially infer the position size held by the market maker and their aggregate reactions affect the security prices. Although the market maker can ask its OTC counterparties to transact a block trade without causing a direct price impact in the exchange, its timing is assumed to be uncertain. Another important way for the market maker to reduce its position is to match an incoming customer order to the outstanding position being warehoused in its balance sheet. The solution of the problem is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we use an interesting connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties have been studied by Bismut (1976).
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