×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alefeld, G.; Herzberger, J., Introduction to interval computation, (1983), Academic Press New York
[2] Andrés, J.; Terceňno, G., Estimating a fuzzy term structure of interest rates using fuzzy regression techniques, European journal of operational research, 154, 808-818, (2004)
[3] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economics, 81, 637-659, (1973) · Zbl 1092.91524
[4] Carlsson, C.; Fullér, R., On possibilistic Mean value and variance of fuzzy numbers, Fuzzy sets and systems, 122, 315-326, (2001) · Zbl 1016.94047
[5] Carr, P.; Wu, L., Time-changed Lévy processes and option pricing, Journal of financial economics, 71, 113-141, (2004)
[6] Chrysafis, K.A.; Papadopoulos, B.K., On theoretical pricing of options with fuzzy estimators, Journal of computational and applied mathematics, (2007)
[7] Cox, E., The fuzzy systems handbook, (1994), Academic press New York
[8] Dubois, D.; Prade, H., The Mean value of a fuzzy number, Fuzzy sets and systems, 24, 279-300, (1987) · Zbl 0634.94026
[9] Dubois, D.; Prade, H., Possibility theory, (1988), Plenum Press New York · Zbl 0645.68108
[10] Duffie, D.; Singleton, J.P., Transform analysis and option pricing for affine jump-diffusions, Econometrica, 68, 1343-1376, (2000) · Zbl 1055.91524
[11] Fullér, R., On product-sum of triangular fuzzy numbers, Fuzzy sets and systems, 41, 83-87, (1991) · Zbl 0725.04002
[12] Fullér, R.; Majlender, P., On weighted possibilistic Mean and variance of fuzzy numbers, Fuzzy sets and systems, 136, 363-374, (2003) · Zbl 1022.94032
[13] Harrison, J.M.; Kreps, D.M., Martingales and arbitrage in multiperiod securities markets, Journal of economic theory, 20, 381-408, (1979) · Zbl 0431.90019
[14] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of financial studies, 6, 327-344, (1993) · Zbl 1384.35131
[15] Hull, J., Options, futures, and other derivative securities, (2000), Prentice Hall New Jersey
[16] Kaufmann, A.; Gupta, M.M., Introduction to fuzzy arithmetic theory and applicatios, (1985), Von nostrand Reinhold Company New York
[17] Kou, S.G., A jump-diffusion model for option pricing, Management science, 48, 1086-1101, (2002) · Zbl 1216.91039
[18] Kou, S.G.; Wang, H., Double exponential jump-diffusion model, Management science, 50, 1178-1192, (2004)
[19] Kwakernaak, H., Fuzzy random variables I: definitions and theorems, Information science, 15, 1-29, (1978) · Zbl 0438.60004
[20] Lee, C.F.; Tzeng, G.H.; Wang, S.Y., A new application of fuzzy set theory to the black – scholes option pricing model, Expert systems with applications, 29, 330-342, (2005)
[21] Merton, R.C., Option pricing when underlying stock returns are discontinuous, Journal of financial economics, 3, 125-144, (1976) · Zbl 1131.91344
[22] Moore, R.E., Interval analysis, (1966), Prentice-Hall, Englewood Cliffs New Jersey · Zbl 0176.13301
[23] Moore, R.E., Methods and applications of interval analysis, (1979), SIAM Philadelphia · Zbl 0417.65022
[24] Muzzioli, S.; Torricelli, C., A multiperiod binorimial model for pricing options in a vague world, Journal of economic dynamics and control, 28, 861-887, (2004) · Zbl 1179.91245
[25] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422, (1986) · Zbl 0592.60004
[26] Ribeiro, R.A.; Yager, R.R.; Zimmermann, H.J.; Kacprzyk, J., Soft computing in financial engineering, (1999), Physica-Verlag Heidelberg · Zbl 0924.90025
[27] Rogers, L.C.G., Arbitrage from fractional Brownian motion, Mathematical finance, 7, 95-105, (1997) · Zbl 0884.90045
[28] Samorodnitsky, G.; Taqqu, M.S., Stable non-Gaussian random processes: stochastic models with infinite variance, (1994), Chapman and Hall New York · Zbl 0925.60027
[29] Shapiro, A.F., Fuzzy random variables, Insurance: mathematics and economics, (2008)
[30] Simonelli, M.R., Fuzziness in valuing financial instruments by certainty equivalents, European journal of operational research, 135, 296-302, (2001) · Zbl 1051.91018
[31] Wang, G.; Zhang, Y., The theory of fuzzy stochastic processes, Fuzzy sets and systems, 51, 161-178, (1992) · Zbl 0782.60039
[32] Wu, Hsien-Chung, Pricing European options based on the fuzzy pattern of black – scholes formula, Computers & operations research, 31, 1069-1081, (2004) · Zbl 1062.91041
[33] Wu, Hsien-Chung, European option pricing under fuzzy environments, International journal of intelligent systems, 20, 89-102, (2005) · Zbl 1079.91045
[34] Wu, Hsien-Chung, Using fuzzy sets theory and black – scholes formula to generate pricing boundaries of European options, Applied mathematics and computation, 185, 136-146, (2007) · Zbl 1283.91184
[35] Yoshida, Y., The valuation of European options in uncertain environment, European journal of operational research, 145, 221-229, (2003) · Zbl 1011.91045
[36] Yoshida, Y.; Yasuda, M.; Nakagami, J.; Kurano, M., A new evaluation of Mean value for fuzzy numbers and its application to American put option under uncertainty, Fuzzy sets and systems, 157, 2614-2626, (2006) · Zbl 1171.91347
[37] Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606
[38] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, Part 1. information sciences, 8, 199-249, (1975), Part 2. Information Sciences 8, 301-353; Part 3, Information Sciences 9, 43-80 · Zbl 0397.68071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.