A stochastic control approach to bid-ask price modelling

TL;DR

This paper introduces a stochastic control model for bid-ask prices of European assets, utilizing a Markov-modulated geometric Brownian motion and advanced change-of-measure techniques to derive price estimates.

Contribution

It presents a novel stochastic control framework incorporating Markov chain dynamics and Girsanov theorem extensions for bid-ask price modeling.

Findings

Derived a system of PDEs for price estimation

Established a dynamic programming principle for bid-ask prices

Provided a new approach for estimating prices beyond trader quotes

Abstract

This paper develops a model for the bid and ask prices of a European type asset by formulating a stochastic control problem. The state process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. A Girsanov theorem for Markov chains is implemented for the change of coefficients, including the diffusion coefficient which cannot be changed by the usual Girsanov theorem for Brownian motion. The price of a European type asset is then determined using an Esscher transform and a system of partial differential equations. A dynamic programming principle and a maximum/minimum principle associated with the stochastic control problem are then derived to model bid and ask prices. These prices are not quotes of traders or market makers but represent estimates in our model on which reasonable quantities could be traded.