Optimal Position Management for a Market Maker with Stochastic Price Impacts

This paper provides the optimal position management strategy for a market maker who has to face uncertain customer orders in an iliquid market, where the market maker's continuous trading through a traditional exchange incurs stochastic linear price impacts. In addition, it is supposed that the market participants can partially infer the position size held by the market maker and their aggregate reactions affect the security prices. Although the market maker can ask its OTC counterparties to transact a block trade without causing a direct price impact in the exchange, its timing is assumed to be uncertain. Another important way for the market maker to reduce its position is to match an incoming customer order to the outstanding position being warehoused in its balance sheet. The solution of the problem 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 use an interesting connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties have been studied by Bismut (1976). --