Analysis of optimal liquidation in limit order books

Blair, James W. and Johnson, Paul V. and Duck, Peter W. (2015) Analysis of optimal liquidation in limit order books. Preprint. (Submitted)

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Abstract

In this paper we study the optimal trading strategy of a passive trader who is trading in the limit order book. Using a combined approach of accurate numerical methods and asymptotical analysis we examine the problem using different stochastic processes to model the asset price, as well as introducing a proportional resilience for the limit order book. This results in more complex equations to solve than when examined under the case of standard Brownian motion, allowing us to perform interesting analytical (asymptotic) analysis which adds insight into the solution space. Under Geometric Brownian Motion, we reduce the resulting four-dimensional Hamilton-Jacobi-Bellman partial differential equation (PDE) to a novel three-dimensional non-linear PDE, as well as rescaling the variables to reduce the number of input parameters by two. We use numerical methods to solve the PDE before asymptotically examining it in several limits, with each approach informing and confirming the other. We find the transition from a time-varying solution to a perpetual-type solution results in the development of singular behaviour, and this transition is examined in some detail. Finally we emphasise the adaptability of our proposed methodologies by implementing the same methods on a mean-reverting process for the asset price. Throughout the paper we also analyse the resulting trading strategies from a financial perspective. The trading strategies we develop are asset-price dependent, which to our knowledge is a unique concept in the passive optimal trading literature, and is arguably more realistic.

Item Type:	Article
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 49 Calculus of variations and optimal control; optimization MSC 2010, the AMS's Mathematics Subject Classification > 91 Game theory, economics, social and behavioral sciences
Depositing User:	Mr James Blair
Date Deposited:	25 May 2015
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2299

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