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Weighted possibilistic moments of fuzzy numbers with applications to GARCH modeling and option pricing. (English) Zbl 1165.91414

Summary: C. Carlson and R. Fullér [Fuzzy Sets Syst. 122, 315–326 (2001; Zbl 1016.94047)] have introduced possibilistic mean, variance and covariance of fuzzy numbers and R. Fullér and P. Majlender [Fuzzy Sets Syst. 136, 363–374 (2003; Zbl 1022.94032)] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. Recently, A. Thavaneswaran, K. Thiagarajah and S.S. Appadoo [Math. Comput. Modelling 45, No. 7–8, 777–786 (2007; Zbl 1165.91415)] have defined non-centered \(n\)th order possibilistic moments of fuzzy numbers. In this paper, we extend these results to centered moments and find the kurtosis for a class of FCA (Fuzzy Coefficient Autoregressive) and FCV (Fuzzy Coefficient Volatility) models. We also demonstrate the superiority of the fuzzy forecasts over the minimum square error forecast through a numerical example. Finally, we provide a description of option price specification errors using the fuzzy weighted possibilistic option valuation model.

MSC:

91G80 Financial applications of other theories
62P05 Applications of statistics to actuarial sciences and financial mathematics
03E72 Theory of fuzzy sets, etc.
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[1] Carlsson, C.; Fuller, R., On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122, 315-326 (2001) · Zbl 1016.94047
[2] Fuller, R.; Majlender, P., On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems, 136, 363-374 (2003) · Zbl 1022.94032
[3] Thavaneswaran, A.; Thiagarajah, K.; Appadoo, S. S., Fuzzy coefficient volatility (FCV) models with applications, Mathematical and Computer Modelling, 45, 777-786 (2007) · Zbl 1165.91415
[4] Cherubini, U., Fuzzy measures and asset prices, accounting for information ambiguity, Applied Mathematical Finance, 4, 135-149 (1997) · Zbl 1009.91006
[5] Medaglia, A. L.; Fang, S. C.; Nuttle, H. L.W.; Wilson, J. R., An efficient and flexible mechanism for constructing membership functions, European Journal of Operational Research, 139, 8495 (2002)
[6] Medasani, S.; Kim, J.; Krishnapuram, R., An overview of membership function generation techniques for pattern recognition, International Journal of Approximate Reasoning, 19, 391-417 (1998) · Zbl 0947.68555
[7] Appadoo, S. S.; Bhatt, S. K.; Bector, C. R., Application of possibility theory to investment decisions, Fuzzy Optimization and Decision Making, 7, 1 (2008) · Zbl 1136.91365
[8] Thiagarajah, K.; Thavaneswaran, A., Fuzzy coefficient volatility models with financial applications, Journal of Risk Finance, 7, 503-524 (2006)
[9] Nicholls, D. F.; Quinn, B. G., (Random Coefficient Autoregressive Models: An Introduction. Random Coefficient Autoregressive Models: An Introduction, Lecture Notes in Statistics, vol. 11 (1982), Springer: Springer New York) · Zbl 0497.62081
[10] Duan, J. C., The GARCH option pricing model, Mathematical Finance, 5, 13-32 (1995) · Zbl 0866.90031
[11] Heston, S.; Nandi, S., A closed-form GARCH option valuation model, Review of Financial Studies, 13, 585-625 (2000)
[12] Ghahramani, M.; Thavaneswaran, A., A note on GARCH model identification, Computers and Mathematics with Applications, 55, 2469-2475 (2008) · Zbl 1142.62394
[13] Jacquier, E.; Jarrow, R., Bayesian analysis of contingent claim model error, Journal of Econometrics, 94, 145-180 (2000) · Zbl 1009.62095
[14] Bakshi, G.; Cao, C.; Chen, Z., Empirical Performance of Alternative Option Pricing Models, Journal of Finance, 50, 2003-2049 (1997)
[15] Zimmermann, H. J., Fuzzy Set Theory and Its Applications (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Nowell
[16] Bodjanova, S., Median value and median interval of a fuzzy number, Information Sciences, 172, 73-89 (2005) · Zbl 1074.03018
[17] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[18] Grzegorzewski, P., Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems, 130, 321-330 (2002) · Zbl 1011.03504
[19] Appadoo, S. S.; Ghahramani, M.; Thavaneswaran, A., Moment properties of some time series models, The Mathematical Scientist, 30, 1, 50-63 (2005) · Zbl 1083.62081
[20] Thavaneswaran, A.; Appadoo, S. S.; Samanta, M., Random coefficient GARCH models, Mathematical and Computer Modelling, 41, 723-733 (2005) · Zbl 1079.62088
[21] Black, F.; Scholes, S. M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654 (1973) · Zbl 1092.91524
[22] Leland, H. E., Option pricing and replication with transactions costs, Journal of Finance, 40, 5, 1283-1301 (1985)
[23] Carlsson, C.; Fuller, R., A fuzzy approach to real option valuation, Fuzzy Sets and Systems, 139, 297-312 (2003) · Zbl 1055.91019
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