Optimal Market Dealing Under Constraints

We consider a market dealer acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset in a quote-driven market. The market dealer may benefit from the bid-ask spread but has the obligation to permanently quote both prices while satisfying some liquidity and inventory constraints. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to the above constraints. We characterize the value function as the unique viscosity solution to the associated HJB equation and further enrich our study with numerical results. The contributions of our study, as compared to previous studies [2], [12], [15], concern both the modelling aspects and the dynamic structure of the control strategies. Important features and constraints characterizing market making problems are no longer ignored. Indeed, along with the obligation to continuously quote bid and ask prices, we do not allow the market maker to stop quoting them when the stock inventory reaches its lower or higher bound. Furthermore, we no longer assume the existence of a reference price.