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
Date Written: March 17, 2016
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
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