Optimal market making

Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with static and dynamic components: they need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread), while mitigating the risk associated with price changes -- because they seldom buy and sell simultaneously, and therefore hold long or short inventories which expose them to market risk. In this paper, (i) we propose a general modeling framework which generalizes (and reconciles) the various modeling approaches proposed in the literature since the publication of the seminal paper ``High-frequency trading in a limit order book'' by Avellaneda and Stoikov, (ii) we prove new general results on the existence and the characterization of optimal market making strategies, (iii) we obtain new closed-form approximations for the optimal quotes, (iv) we extend the modeling framework to the case of multi-asset market making, and (v) we show how the model can be used in practice in the specific case of the corporate bond market and for two credit indices.