Optimal Execution Comparison Across Risks and Dynamics, with Solutions for Displaced Diffusions

21 Pages Posted: 11 Apr 2013 Last revised: 10 May 2014

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Giuseppe Di Graziano

Deutsche Bank AG

Date Written: May 9, 2014

Abstract

We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion. Optimal strategies in the adapted class under various risk criteria, namely value-at-risk, expected shortfall and a new criterion called "squared asset expectation" (SAE), related to a version of the cost variance measure, are derived and compared. It is well known that displaced diffusions (DD) exhibit dynamics which are in-between arithmetic Brownian motions (ABM) and geometric Brownian motions (GBM) depending of the choice of the shift parameter. Furthermore, DD allows for changes in the support of the mid asset price distribution, allowing one to include a minimum permitted value for the mid price, either positive or negative. We study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although differences are not negligible for high levels of risk aversion and low market impact assets. This is illustrated with numerical examples.

Keywords: Optimal trade execution, Algorithmic trading, Displaced Diffusion, HJB equation, calculus of variations, risk measures, Value at Risk, Expected Shortfall, Squared-Asset Expectation, Market Impact

JEL Classification: C51, G12, G13

Suggested Citation

Brigo, Damiano and Di Graziano, Giuseppe, Optimal Execution Comparison Across Risks and Dynamics, with Solutions for Displaced Diffusions (May 9, 2014). Available at SSRN: https://ssrn.com/abstract=2247951 or http://dx.doi.org/10.2139/ssrn.2247951

Damiano Brigo (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www.imperial.ac.uk/people/damiano.brigo

Giuseppe Di Graziano

Deutsche Bank AG ( email )

Winchester House
1 Great Winchester Street
London, EC2N 2DB
United Kingdom

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