Geodesic
Mean-variance Hedging of options on stocks with Markov volatilities
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 211-222

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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. 
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     author = {G. B. Di Masi and Yu. M. Kabanov and W. J. Runggaldier},
     title = {Mean-variance {Hedging} of options on stocks with {Markov} volatilities},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     publisher = {mathdoc},
     volume = {39},
     number = {1},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1994_39_1_a7/}
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G. B. Di Masi; Yu. M. Kabanov; W. J. Runggaldier. Mean-variance Hedging of options on stocks with Markov volatilities. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 211-222. http://geodesic.mathdoc.fr/item/TVP_1994_39_1_a7/