×

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.
PDFBibTeX XMLCite
Full Text: DOI

References:

[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) · Zbl 1010.03525
[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 · Zbl 0969.54002
[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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.