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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.

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