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A jump-diffusion model for option pricing under fuzzy environments. (English) Zbl 1162.91388
Summary: Owing to fluctuations in the financial markets from time to time, the rate \(\lambda \) of Poisson process and jump sequence \(\{V_i\}\) in the Merton’s normal jump-diffusion model cannot be expected in a precise sense. Therefore, the fuzzy set theory proposed by L. A. Zadeh [Inf. Control 8, 338–353 (1965; Zbl 0139.24606)] and the fuzzy random variable introduced by H. Kwakernaak [Inf. Sci. 15, 1–29 (1978; Zbl 0438.60004)] and M. L. Puri and D. A. Ralescu [Math. Proc. Camb. Philos. Soc. 97, 151–158 (1985; Zbl 0559.60007) and J. Math. Anal. Appl. 114, 409–422 (1986; Zbl 0592.60004)] may be useful for modeling this kind of imprecise problem. In this paper, probability is applied to characterize the uncertainty as to whether jumps occur or not, and what the amplitudes are, while fuzziness is applied to characterize the uncertainty related to the exact number of jump times and the jump amplitudes, due to a lack of knowledge regarding financial markets. This paper presents a fuzzy normal jump-diffusion model for European option pricing, with uncertainty of both randomness and fuzziness in the jumps, which is a reasonable and a natural extension of the R. C. Merton [J. Financ. Econ. 3, No. 1-2, 125–144 (1976; Zbl 1131.91344)] normal jump-diffusion model. Based on the crisp weighted possibilistic mean values of the fuzzy variables in fuzzy normal jump-diffusion model, we also obtain the crisp weighted possibilistic mean normal jump-diffusion model. Numerical analysis shows that the fuzzy normal jump-diffusion model and the crisp weighted possibilistic mean normal jump-diffusion model proposed in this paper are reasonable, and can be taken as reference pricing tools for financial investors.

91G20 Derivative securities (option pricing, hedging, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
03E72 Theory of fuzzy sets, etc.
91G80 Financial applications of other theories
Full Text: DOI
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