Optimal Position Management for a Market Maker with Stochastic Price Impacts

TL;DR

This paper develops a stochastic control framework for a market maker managing positions under uncertain order flows and stochastic price impacts, providing solutions via BSDEs and BSRDEs with approximation schemes.

Contribution

It introduces a novel stochastic Hamilton-Jacobi-Bellman approach for multi-security position management with stochastic impacts and proposes an efficient perturbative approximation scheme.

Findings

Solution characterized by a system of BSDEs and BSRDEs.

Verification achieved using standard BSDE techniques.

Perturbative scheme converges with proven rate.

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.