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 Teor. Veroyatnost. i Primenen., 1994, Volume 39, Issue 1, Pages 211–222 (Mi tvp3771)

Short Communications

Mean-variance Hedging of options on stocks with Markov volatilities

G. B. Di Masia, Yu. M. Kabanovb, W. J. Runggaldiera

a Universita di Padova, Dipartimento di Matematica Ðurà 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 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.

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

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

Bibliographic databases:

Citation: G. B. Di Masi, Yu. M. Kabanov, W. J. Runggaldier, “Mean-variance Hedging of options on stocks with Markov volatilities”, Teor. Veroyatnost. i Primenen., 39:1 (1994), 211–222; Theory Probab. Appl., 39:1 (1994), 172–182

Citation in format AMSBIB
\Bibitem{Di KabRun94} \by G.~B.~Di Masi, Yu.~M.~Kabanov, W.~J.~Runggaldier \paper Mean-variance Hedging of options on stocks with Markov volatilities \jour Teor. Veroyatnost. i Primenen. \yr 1994 \vol 39 \issue 1 \pages 211--222 \mathnet{http://mi.mathnet.ru/tvp3771} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1348196} \zmath{https://zbmath.org/?q=an:0836.60075} \transl \jour Theory Probab. Appl. \yr 1994 \vol 39 \issue 1 \pages 172--182 \crossref{https://doi.org/10.1137/1139008} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RH52800008} 

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This publication is cited in the following articles:
1. Proc. Steklov Inst. Math., 237 (2002), 192–202
2. V. M. Radchenko, “Variance-minimizing hedging in the model with jumps at deterministic moments”, Theory Probab. Appl., 51:3 (2007), 536–545
3. Wang Yu., Yin G., “Quantile Hedging for Guaranteed Minimum Death Benefits with Regime Switching”, Stoch. Anal. Appl., 30:5 (2012), 799–826
4. Palmowski Z. Stettner L. Sulima A., “Optimal Portfolio Selection in An Ito-Markov Additive Market”, Risks, 7:1 (2019), 34
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