Discrete Homotopy Analysis for Optimal Trading Execution with Nonlinear Transient Market Impact

Communications in Nonlinear Science and Numerical Simulation, 39:332-342, 2016.

21 Pages Posted: 19 Mar 2016 Last revised: 25 Apr 2016

See all articles by Gianbiagio Curato

Gianbiagio Curato

Scuola Normale Superiore

Jim Gatheral

CUNY Baruch College

Fabrizio Lillo

Università di Bologna

Date Written: March 17, 2016

Abstract

Optimal execution in financial markets is the problem of how to trade a large quantity of shares incrementally in time in order to minimise the expected cost. In this paper we study the problem of the optimal execution in the presence of nonlinear transient market impact. Mathematically such problem is equivalent to solve a strongly nonlinear integral equation, which in our model is a weakly singular Urysohn equation of the first kind. We propose an approach based on homotopy analysis method (HAM), whereby a well behaved initial trading strategy is continuously deformed to lower the expected execution cost. Specifically we propose a discrete version of the HAM, i.e. the DHAM approach, in order to use the method when the integrals to compute have no closed form solution. We find that the optimal solution is front loaded for concave instantaneous impact even when the investor is risk neutral. More important we find that the expected cost of the DHAM strategy is significantly smaller than the cost of conventional strategies.

Keywords: Optimal execution, Market impact, Homotopy analysis method, Nonlinear integral equations

Suggested Citation

Curato, Gianbiagio and Gatheral, Jim and Lillo, Fabrizio, Discrete Homotopy Analysis for Optimal Trading Execution with Nonlinear Transient Market Impact (March 17, 2016). Communications in Nonlinear Science and Numerical Simulation, 39:332-342, 2016.. Available at SSRN: https://ssrn.com/abstract=2749295

Gianbiagio Curato (Contact Author)

Scuola Normale Superiore ( email )

Piazza dei Cavalieri, 7
Pisa, 56126
Italy

Jim Gatheral

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Fabrizio Lillo

Università di Bologna ( email )

Via Zamboni, 33
Bologna, 40126
Italy

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