  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.
