We study a an optimal high frequency trading problem within a marketmicrostructure model designed to be a good compromise between accuracy andtractability. The stock price is driven by a Markov Renewal Process (MRP),while market orders arrive in the limit order book via a point processcorrelated with the stock price itself. In this framework, we can reproduce theadverse selection risk, appearing in two different forms: the usual one due tobig market orders impacting the stock price and penalizing the agent, and theweak one due to small market orders and reducing the probability of aprofitable execution. We solve the market making problem by stochastic controltechniques in this semi-Markov model. In the no risk-aversion case, we provideexplicit formula for the optimal controls and characterize the value functionas a simple linear PDE. In the general case, we derive the optimal controls andthe value function in terms of the previous result, and illustrate how the riskaversion influences the trader strategy and her expected gain. Finally, byusing a perturbation method, approximate optimal controls for small riskaversions are explicitly computed in terms of two simple PDE's, reducingdrastically the computational cost and enlightening the financialinterpretation of the results.
展开▼
