Recent Developments in Fuzzy Sets Approach in Option Pricing

Affiliation(s)

Department of Supply Chain Management, University of Manitoba, Winnipeg, Canada.

Department of Statistics, University of Manitoba, Winnipeg, Canada.

Department of Supply Chain Management, University of Manitoba, Winnipeg, Canada.

Department of Statistics, University of Manitoba, Winnipeg, Canada.

ABSTRACT

Recently there has been growing interest in fuzzy option pricing. Carlsson and Fuller [1] were the first to study the fuzzy real options and Thavaneswaran *et al.* [2] demonstrated the superiority of the fuzzy forecasts and then derived the membership function for the European call price by fuzzifying the interest rate, volatility and the initial value of the stock price. In this paper, we discuss recent developments in fuzzy option pricing based on Black-Scholes models. Fuzzy coefficient Black-Scholes partial differential equations (PDE) are derived. Membership function of the call price is given. The asset-or-nothing option by fuzzifying the maturity value of the stock price using adaptive fuzzy numbers is also discussed in some detail.

Cite this paper

S. Appadoo and A. Thavaneswaran, "Recent Developments in Fuzzy Sets Approach in Option Pricing,"*Journal of Mathematical Finance*, Vol. 3 No. 2, 2013, pp. 312-322. doi: 10.4236/jmf.2013.32031.

S. Appadoo and A. Thavaneswaran, "Recent Developments in Fuzzy Sets Approach in Option Pricing,"

References

[1] C. Carlson and R. Fuller, “On Possibilistic Mean Value and Variance of Fuzzy Numbers,” Fuzzy Sets and Systems, Vol. 122, No. 2, 2001, pp. 315-326. doi:10.1016/S0165-0114(00)00043-9

[2] A. Thavaneswaran, S. S. Appadoo and A. Paseka, “Weighted Possibilistic Moments of Fuzzy Numbers with Applications to GARCH Modeling and Option Pricing,” Mathematical and Computer Modelling, Vol. 49, No. 1-2, 2009, pp. 352-368. doi:10.1016/j.mcm.2008.07.035

[3] S. S. Appadoo, “Pricing Financial Derivatives with Fuzzy Algebraic Models: A Theoretical and Computational Approach,” Ph.D. Thesis, University of Manitoba, Winnipeg, 2006.

[4] A. Thavaneswaran, K. Thiagarajah and S. S. Appadoo, “Fuzzy Coefficient Volatility (FCV) Models With Applications,” Mathematical and Computer Modelling, Vol. 45, No. 7-8, 2007, pp. 777-786. doi:10.1016/j.mcm.2006.07.019

[5] U. Cherubini, “Fuzzy Measures and Asset Prices: Accounting for Information Ambiguity,” Applied Mathematical Finance, Vol. 4, No. 3, 1997, pp. 135-149. doi:10.1080/135048697334773

[6] H. Ghaziri, S. Elfakhani and J. Assi, “Neural Networks Approach to Pricing Options,” Neural Network World, Vol. 10, 2000, pp. 271-277.

[7] N. Trenev, “A Refinement of the Black-Scholes Formula of Pricing Options” Cybernetics and Systems Analysis, Vol. 37, No. 6, 2001, pp. 911-917. doi:10.1023/A:1014542217599

[8] Z. Zmeskal, “Generalised Soft Binomial American Real Option Pricing Model (Fuzziestochastic Approach),” European Journal of Operational Research, Vol. 207 No. 2, 2010, pp. 1096-1103. doi:10.1016/j.ejor.2010.05.045

[9] C. Carlsson and R. Fuller, “A Fuzzy Approach to Real Option Valuation,” Fuzzy Sets and Systems, Vol. 139, No. 2, 2003, pp. 297-312. doi:10.1016/S0165-0114(02)00591-2

[10] K. Thiagarajah and A. Thavaneswaran, “Fuzzy Coefficient Volatility Models with Financial Applications,” Journal of Risk Finance, Vol. 7, No. 5, 2006, pp. 503-524. doi:10.1108/15265940610712669

[11] X. Weidong, W. Chongfeng, W. J. Xu and H. Y. Li, “A Jump-Diffusion Model for Option Pricing under Fuzzy Environments,” Insurance: Mathematics and Economics, Vol. 44, No. 3, 2004, pp. 337-344.

[12] S.-H. Hoa and S.-H. Liao, “A Fuzzy Real Option Approach for Investment Project Valuation,” Expert Systems with Applications, Vol. 38, No. 12, 2011, pp. 15296-15302.

[13] M. L. Guerra, L. Sorini and L. Stefanini, “Parametrized Fuzzy Numbers for Option Pricing,” IEEE International Conference on Fuzzy Systems, London, 2007, pp. 727-732.

[14] W. Xu, W. Xu, H. Li and W. Zhang, “A Study of Greek Letters of Currency Option under Uncertainty Environments,” Mathematical and Computer Modelling, Vol. 51 No. 5-6, 2010, pp. 670-681. doi:10.1016/j.mcm.2009.10.041

[15] P. Nowak and M. Romaniuk, “Computing Option Price for Levy Process with Fuzzy Parameters,” Vol. 201, No. 16, 2010, pp. 206-210.

[16] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[17] F. Black and M. Scholes, “The Valuation of Option Contracts and a Test of Market Efficiency,” Journal of Finance, Vol. 27, No. 2, 1972, pp. 399-417. doi:10.2307/2978484

[18] A. Thavaneswaran, S. S. Appadoo and J. Frank, “Binary Option Pricing using Fuzzy Numbers,” Applied Mathematics Letters, 2012.

[1] C. Carlson and R. Fuller, “On Possibilistic Mean Value and Variance of Fuzzy Numbers,” Fuzzy Sets and Systems, Vol. 122, No. 2, 2001, pp. 315-326. doi:10.1016/S0165-0114(00)00043-9

[2] A. Thavaneswaran, S. S. Appadoo and A. Paseka, “Weighted Possibilistic Moments of Fuzzy Numbers with Applications to GARCH Modeling and Option Pricing,” Mathematical and Computer Modelling, Vol. 49, No. 1-2, 2009, pp. 352-368. doi:10.1016/j.mcm.2008.07.035

[3] S. S. Appadoo, “Pricing Financial Derivatives with Fuzzy Algebraic Models: A Theoretical and Computational Approach,” Ph.D. Thesis, University of Manitoba, Winnipeg, 2006.

[4] A. Thavaneswaran, K. Thiagarajah and S. S. Appadoo, “Fuzzy Coefficient Volatility (FCV) Models With Applications,” Mathematical and Computer Modelling, Vol. 45, No. 7-8, 2007, pp. 777-786. doi:10.1016/j.mcm.2006.07.019

[5] U. Cherubini, “Fuzzy Measures and Asset Prices: Accounting for Information Ambiguity,” Applied Mathematical Finance, Vol. 4, No. 3, 1997, pp. 135-149. doi:10.1080/135048697334773

[6] H. Ghaziri, S. Elfakhani and J. Assi, “Neural Networks Approach to Pricing Options,” Neural Network World, Vol. 10, 2000, pp. 271-277.

[7] N. Trenev, “A Refinement of the Black-Scholes Formula of Pricing Options” Cybernetics and Systems Analysis, Vol. 37, No. 6, 2001, pp. 911-917. doi:10.1023/A:1014542217599

[8] Z. Zmeskal, “Generalised Soft Binomial American Real Option Pricing Model (Fuzziestochastic Approach),” European Journal of Operational Research, Vol. 207 No. 2, 2010, pp. 1096-1103. doi:10.1016/j.ejor.2010.05.045

[9] C. Carlsson and R. Fuller, “A Fuzzy Approach to Real Option Valuation,” Fuzzy Sets and Systems, Vol. 139, No. 2, 2003, pp. 297-312. doi:10.1016/S0165-0114(02)00591-2

[10] K. Thiagarajah and A. Thavaneswaran, “Fuzzy Coefficient Volatility Models with Financial Applications,” Journal of Risk Finance, Vol. 7, No. 5, 2006, pp. 503-524. doi:10.1108/15265940610712669

[11] X. Weidong, W. Chongfeng, W. J. Xu and H. Y. Li, “A Jump-Diffusion Model for Option Pricing under Fuzzy Environments,” Insurance: Mathematics and Economics, Vol. 44, No. 3, 2004, pp. 337-344.

[12] S.-H. Hoa and S.-H. Liao, “A Fuzzy Real Option Approach for Investment Project Valuation,” Expert Systems with Applications, Vol. 38, No. 12, 2011, pp. 15296-15302.

[13] M. L. Guerra, L. Sorini and L. Stefanini, “Parametrized Fuzzy Numbers for Option Pricing,” IEEE International Conference on Fuzzy Systems, London, 2007, pp. 727-732.

[14] W. Xu, W. Xu, H. Li and W. Zhang, “A Study of Greek Letters of Currency Option under Uncertainty Environments,” Mathematical and Computer Modelling, Vol. 51 No. 5-6, 2010, pp. 670-681. doi:10.1016/j.mcm.2009.10.041

[15] P. Nowak and M. Romaniuk, “Computing Option Price for Levy Process with Fuzzy Parameters,” Vol. 201, No. 16, 2010, pp. 206-210.

[16] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[17] F. Black and M. Scholes, “The Valuation of Option Contracts and a Test of Market Efficiency,” Journal of Finance, Vol. 27, No. 2, 1972, pp. 399-417. doi:10.2307/2978484

[18] A. Thavaneswaran, S. S. Appadoo and J. Frank, “Binary Option Pricing using Fuzzy Numbers,” Applied Mathematics Letters, 2012.