Optimal market making strategies under inventory constraints

We study the problem of a market-maker acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset. We assume that the market maker has a contractual obligation to permanently quote bid and ask prices for the security and therefore to satisfy any sell or buy order from the asset’s investors. On the opposite side of the trades, there are investors who act as liquidity takers by submitting either sell or buy market orders. The arrival of buy (and sell) market orders submitted by the investors is assumed to follow a Cox process with regime-shifting Markov intensity. The role of the market maker is very important in the trading of illiquid assets as it acts as a facilitator of trades between different investors. The market maker may therefore benefit from the bid-ask spread but faces a number of constraints, in particular the liquidity and inventory constraints. The objective is to maximize the expected utility of the market maker’s terminal wealth. We characterize our objective functions as unique viscosity solutions to the associated HJB system. We further enrich our study with some numerical results.