High-frequency market-making for multi-dimensional Markov processes

In this paper we complete and extend our previous work on stochastic control applied to high frequency market-making with inventory constraints and directional bets. Our new model admits several state variables (e.g. market spread, stochastic volatility and intensities of market orders) provided the full system is Markov. The solution of the corresponding HJB equation is exact in the case of zero inventory risk. The inventory risk enters into play in two ways: a path-dependent penalty based on the volatility and a penalty at expiry based on the market spread. We perform perturbation methods on the inventory risk parameter and obtain explicitly the solution and its controls up to first order. We also include transaction costs; we show that the spread of the market-maker is widened to compensate the transaction costs, but the expected gain per traded spread remains constant. We perform several numerical simulations to assess the effect of the parameters on the PNL, showing in particular how the directional bet and the inventory risk change the shape of the PNL density. Finally, we extend our results to the case of multi-aset market-making strategies; we show that the correct notion of inventory risk is the L2-norm of the (multi-dimensional) inventory with respect to the inventory penalties