High frequency market making problem under a jump diffusion process

Abstract

In this thesis, we develop a high frequency market making strategy under a jump diffusion process. In the past literatures, high frequency market making strategies are all developed under the assumption that the underlying asset follows a diffusion process. However, in reality when a large size market buy (sell) order arrives, it is of high probability that the mid price of the stock immediately jumps to a higher (lower) value. To incorporate such impact of large size market orders, we choose to use a jump diffusion process, which is a combination of a diffusion process and a compound poisson process, to simulate the path of a stock's mid price. Based on the jump diffusion process that the underlying asset should follow, we propose the Hamilton-Jacobi-Bellman equation by the knowledge of stochastic control and dynamic programming principle. The HJB equation is an partial integro-differential equation that the optimal bid and ask quotes δ^a and δ^b should satisfy. Finally, we find the approximate expression of δ^a and δ^b by using taylor expansion and polynomial approximation of the utility function. The expression for δ^a and δ^b is the market making strategy under a jump diffusion process in a finite time horizon, and we call it 'jump' strategy. Lastly, we evaluate the performance of our 'jump' strategy by implementing it into simulated path of stock's price and real tick data of a stock Exxon Mobil Corporation. In addition, we also implement the traditional 'continuous' strategy, which is based on a diffusion process, into both simulated data and real data. Comparing the sharp ratio value of two strategies, the performance of our 'jump' strategy is generally better than that of the 'continuous' strategy.