OPTIMAL LIQUIDATION TRAJECTORIES FOR THE ALMGREN–CHRISS MODEL

Arne Løkka and Junwei Xu
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Arne Løkka: Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK
Junwei Xu: Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK†Morgan Stanley Fixed Income Division, Institutional Securities 20 Bank Street Canary Wharf, London E14 4AD, UK

International Journal of Theoretical and Applied Finance (IJTAF), 2020, vol. 23, issue 07, 1-35

Abstract: We consider an optimal liquidation problem with infinite horizon in the Almgren–Chriss framework, where the unaffected asset price follows a Lévy process. The temporary price impact is described by a general function that satisfies some reasonable conditions. We consider a market agent with constant absolute risk aversion, who wants to maximize the expected utility of the cash received from the sale of the agent’s assets, and show that this problem can be reduced to a deterministic optimization problem that we are able to solve explicitly. In order to compare our results with exponential Lévy models, which provide a very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Lévy process approximation of such models. In particular we derive expressions for the Lévy process approximation of the exponential variance–gamma Lévy process, and study properties of the corresponding optimal liquidation strategy. We then provide a comparison of the liquidation trajectories for reasonable parameters between the Lévy process model and the classical Almgren–Chriss model. In particular, we obtain an explicit expression for the connection between the temporary impact function for the Lévy model and the temporary impact function for the Brownian motion model (the classical Almgren–Chriss model), for which the optimal liquidation trajectories for the two models coincide.

Keywords: Lévy processes; Almgren–Chriss model; algorithmic trading; optimal liquidation; optimal execution; constant absolute risk aversion; market impact; optimal control; Hamilton–Jacobi–Bellman equation (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1142/S0219024920500491

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