A stochastic control approach for options market making

In this paper, we establish a model for market making in options whose underlying is perfectly liquid. In our model framework, the stock price follows a generic stochastic volatility model under the real-world probability measure P. Market participants price options on this stock under a risk-neutral pricing measure Q, and they may misspecify the parameters controlling the dynamics of the volatility process. We consider that there is an agent who is willing to make markets in an option on the stock with the aim of maximizing his expected utility from terminal wealth at the maturity of this option. Since market impact is an important feature in the microscopic time scale and should be taken into account in high frequency trading, we study di erent forms of this function argued in the recent literature. Through the use of optimal stochastic control, we provide exact expressions of optimal bid and ask quotes of the market making strategy in the case where the agent is risk-neutral. Afterward, we suppose that the agent is risk-averse and wants to reduce the variance of the nal wealth. In addition, this agent tries not to accumulate a large inventory in order not to have a signi cant exposure to market risk. For this purpose, we perturb the utility function by a penalty on the variance of nal wealth and also on accumulated inventory. Using singular perturbation with respect to the penalty parameter, we provide analytic approximations of the optimal bid and ask quotes. In order to con rm our theoretical results, we perform Monte Carlo simulations of the optimal market making strategy in the case where the stock price process follows a Heston model. We show that the opti- mal strategy is more pro table than a zero-intelligence strategy. Besides, we highlight the e ects of the misspeci cation of the parameters on the performance of the strategy.