A stochastic control approach to option market making

This paper presents a model for the market making of options on a liquid stock. 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 first consider that there is a risk-neutral agent who is willing to make markets in an option on the stock, with the aim of maximizing the expected terminal wealth at maturity. Using standard tools in optimal stochastic control, we provide analytical expressions for the optimal bid and ask quotes of the market maker. We then assume that the agent is risk-averse, and perturb the linear utility function by adding a variance term. In this setting, analytic approximations of the optimal bid and ask quotes are obtained. In the case where the stock price process follows a Heston model, Monte Carlo simulations are used to compare the optimal strategy to a ”zerointelligence” strategy, and to highlight the effects of some parameters misspecification on the performance of the strategy. JEL Classification: JEL: C51 Model construction and estimation, JEL: C52 Model evaluation and testing.