Market Making in High-Frequency Trading via Mathematical Modelling

Abstract

This project attempts to understand microstructure modelling of tick-by-tick asset price via a semi-Markov model. It has been observed in the literature that such models are capable of reproducing various stylized facts of market microstructure, such as mean reversion and volatility clustering. We perform mathematical analyses of certain functionals of the stock price dynamics. In particular, these functionals are expressed using the conditional expectation of stock price. As an application of the mathematical analyses of the functional, we investigate the market making problem of the agent. Typically an agent optimally submits limit orders at the best ask and best bid prices. It has been shown in the literature that this problem can be solved using a Hamilton-Jacobi-Bellman equation, and a viscosity solution to such HJB equations has been obtained. However, we have obtained a classical solution to a related linear PDE, and this indicates that one can obtain a classical solution to the HJB equation with further investigation.