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Short Communications

Mean-variance Hedging of options on stocks with Markov volatilities

G. B. Di Masi^a, Yu. M. Kabanov^b, W. J. Runggaldier<sup>a

a</sup> Universita di Padova, Dipartimento di Matematica Рurа ed Applicata, Padova, Italy
^b Central Economics and Mathematics Institute, RAS

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 Fцllmer, 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.

Keywords: Black–Scholes formula, call option, stochastic volatility, incomplete market, meanvariance hedging, Kac process.

Received: 05.07.1993

 English version: Theory of Probability and its Applications, 1994, 39:1, 172–182

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