A semi-Markovian approach to model the tick-by-tick dynamics of stock price

TL;DR

This paper introduces a semi-Markov model for tick-by-tick stock price dynamics, deriving related PDEs and applying the framework to market making strategies with optimal control solutions.

Contribution

It develops a semi-Markov process model for high-frequency stock prices and connects it to PDEs for pricing and optimal control in market making.

Findings

Derived a semi-Markov process model for stock prices

Established existence and uniqueness of solutions to associated PDEs

Obtained explicit optimal controls for market making

Abstract

We model the stock price dynamics through a semi-Markov process obtained using a Poisson random measure. We establish the existence and uniqueness of the classical solution of a non-homogeneous terminal value problem and we show that the expected value of stock price at horizon can be obtained as a classical solution of a linear partial differential equation that is a special case of the terminal value problem studied in this paper. We further analyze the market making problem using the point of view of an agent who posts the limit orders at the best price available. We use the dynamic programming principle to obtain a HJB equation. In no-risk aversion case, we obtain the value function as a classical solution of a linear pde and derive the expressions for optimal controls by solving the HJB equation.