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Risk minimizing option pricing in a semi-Markov modulated market. (English) Zbl 1193.91155
The authors study risk minimizing option pricing in a semi-Markov modulated market. It is assumed that the stock price process follows a semi-Markov modulated geometric Brownian motion. The representation of the semi-Markov process as a stochastic integral with respect to a Poisson random measure is established. Using well-known results by H. Föllmer and M. Schweizer on risk minimizing hedging and on minimal martingale measures [Applied stochastic analysis, Pap. Workshop, London/UK 1989, Stochastic Monogr. 5, 389-414 (1990; Zbl 0738.90007)], it is shown that the risk minimizing price of a European call option satisfies a system of Black-Scholes equations. The locally risk minimizing prices of European barrier options and of second order compound options are presented and generalized for basket options. Some computational aspects are discussed.

91G20 Derivative securities (option pricing, hedging, etc.)
60K15 Markov renewal processes, semi-Markov processes
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