Optimal Position Management for a Market Maker with Stochastic Price Impacts

41 Pages Posted: 24 Mar 2015 Last revised: 5 Sep 2015

See all articles by Masaaki Fujii

Masaaki Fujii

University of Tokyo - Faculty of Economics

Date Written: September 5, 2015

Abstract

This paper deals with an optimal position management problem for a market maker who has to face uncertain customer order flows in an illiquid market, where the market maker’s continuous trading incurs a stochastic linear price impact. Although the execution timing is uncertain, the market maker can also ask its OTC counterparties to transact a block trade without causing a direct price impact. We adopt quite generic stochastic processes of the securities, order flows, price impacts, quadratic penalties as well as security borrowing/lending rates. The solution of the market maker’s optimal position-management strategy is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we make use of the connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties were studied by Bismut (1976). We also propose a perturbative approximation scheme for the resultant BSRDE, which only requires a system of linear ODEs to be solved at each expansion order. Its justification and the convergence rate are also given.

Keywords: stochastic control, liquidity, stochastic HJB, BSDE, inventory

JEL Classification: C61, G11, G21

Suggested Citation

Fujii, Masaaki, Optimal Position Management for a Market Maker with Stochastic Price Impacts (September 5, 2015). Available at SSRN: https://ssrn.com/abstract=2584430 or http://dx.doi.org/10.2139/ssrn.2584430

Masaaki Fujii (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

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