Mean-Variance Hedging of Options on Stocks with Markov Volatilities

TLDR

The paper studies hedging a European call option in a diffusion model where drift and volatility depend on a Markov jump process, rendering the market incomplete and perfect hedging impossible. The goal is to derive a hedging strategy for such options. The authors use a mean‑variance hedging approach proposed by Föllmer, Sondermann, and Schweizer, and compute the distribution of the integrated telegraph signal via order‑statistics arguments to obtain the option price. This yields a generalised Black–Scholes formula, with an explicit expression for the two‑state jump process involving the Kac process distribution.

Abstract

We consider the problem of hedging an European call option for a diffusion model where drift and volatility are functions of a Markov jump process. The market is thus incomplete implying that perfect hedging is not possible. To derive a hedging strategy, we follow the approach based on the idea of hedging under a mean-variance criterion as suggested by Föllmer, Sondermann, and Schweizer. This also leads to a generalization of the Black–Scholes formula for the corresponding option price which, for the simplest case when the jump process has only two states, is given by an explicit expression involving the distribution of the integrated telegraph signal (known also as the Kac process). In the Appendix we derive this distribution by simple considerations based on properties of the order statistics.