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 both 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). Since they seldom buy and sell simultaneously, and therefore hold long and/or short inventories, they also need to mitigate the risk associated with price changes and subsequently skew their quotes dynamically. In this paper, (i) we propose a general modelling framework which generalizes (and reconciles) the various modelling 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 modelling framework to the case of multi-asset market making and we obtain general closed-form approximations for the optimal quotes of a multi-asset market maker, and (v) we show how the model can be used in practice in the specific (and original) case of two credit indices.