Option pricing and hedging for regime-switching geometric Brownian motion models

TL;DR

This paper derives a variance-optimal equivalent martingale measure for regime-switching geometric Brownian motion models, enabling improved option pricing and hedging strategies in incomplete markets with regime dynamics.

Contribution

It introduces a new measure where the regime process becomes non-homogeneous, extending option pricing theory to more realistic market models with regime switching.

Findings

The measure minimizes mean-variance hedging error.

Option prices and hedging strategies are computed via Monte Carlo simulations.

Applications demonstrate practical implementation in financial markets.

Abstract

We find the variance-optimal equivalent martingale measure when multivariate assets are modeled by a regime-switching geometric Brownian motion, and the regimes are represented by a homogeneous continuous time Markov chain. Under this new measure, the Markov chain driving the regimes is no longer homogeneous, which differs from the equivalent martingale measures usually proposed in the literature. We show the solution minimizes the mean-variance hedging error under the objective measure. As argued by \citet{Schweizer:1996}, the variance-optimal equivalent measure naturally extends canonical option pricing results to the case of an incomplete market and the expectation under the proposed measure may be interpreted as an option price. Solutions for the option value and the optimal hedging strategy are easily obtained from Monte Carlo simulations. Two applications are considered.