This work considers pricing European call options and the study of Greek letters of options under a fuzzy environment. In the past work, stock prices are usually represented by symmetric triangular fuzzy numbers for the computational convenience while pricing options with uncertainty. It might not be enough to explain the stochastic nature of the underlining price in the option pricing formula. This work considers developing the fuzzy pattern of European call option under the assumption of the stock return being a Gaussian fuzzy number. The study of Greeks for the sensitivity analysis of the fuzzy call option price with respect to the change in the pricing variables is included. The empirical analysis and comparison on the fuzzy European option pricing based on the real market data of SPX options at CBOE are provided. Our results show that the fuzzy options are more close to the theoretical options derived from the Black–Scholes formula while employing Gaussian fuzzy stock returns for pricing European call options.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
The data source on the February 16, 2018, rate. http://www.optionistics.com/quotes/stock-option-chains/SPX.
The data source on the February 16, 2018, rate. https://www.federalreserve.gov/releases/h15/.
Andersen TG, Bollerslev T, Diebold FX, Ebens H (2001) The distribution of realized stock return volatility. J Financ Econ 61(1):43–76
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654
Buckley J (1987) The fuzzy mathematics of finance. Fuzzy Sets Syst 21(3):257–273
Carlsson C, Fullér R (2003) A fuzzy approach to real option valuation. Fuzzy Sets Syst 139(2):297–312
Chrysafis KA, Papadopoulos BK (2009) On theoretical pricing of options with fuzzy estimators. J Comput Appl Math 223(2):552–566
De Andrés-Sánchez J (2018) Pricing European options with triangular fuzzy parameters: assessing alternative triangular approximations in the Spanish stock option market. Int J Fuzzy Syst 20(5):1624–1643
Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626
Guerra ML, Sorini L, Stefanini L (2011) Option price sensitivities through fuzzy numbers. Comput Math Appl 61(3):515–526
Hull JC (2006) Options, futures, and other derivatives. Pearson Education India, Delhi
Jarrow RA, Turnbull SM (2000) Derivative securities. South-Western Publishing, Nashville
J Mezei, M Collan, P Luukka (2018) Real option analysis with interval-valued fuzzy numbers and the fuzzy pay-off method. In: Kacprzyk J, Szmidt E, Zadrożny S, Atanassov K, Krawczak M (eds) Advances in fuzzy logic and technology 2017. Springer International Publishing, Cham, pp 509–520
Muzzioli S, De Baets B (2017) Fuzzy approaches to option price modeling. IEEE Trans Fuzzy Syst 25(2):392–401
Muzzioli S, Ruggieri A, De Baets B (2015) A comparison of fuzzy regression methods for the estimation of the implied volatility smile function. Fuzzy Sets Syst 266:131–143
Passarelli D (2012) Trading options Greeks: how time, volatility, and other pricing factors drive profits, vol 159. Wiley, Hoboken
Qin Z, Li X (2008) Option pricing formula for fuzzy financial market. J Uncertain Syst 2(1):17–21
Shafi K, Latif N, Shad SA, Idrees Z, Gulzar S (2018) Estimating option greeks under the stochastic volatility using simulation. Phys A Stat Mech Appl 503:1288–1296
Starczewski JT (2005) Extended triangular norms on Gaussian fuzzy sets. In: EUSFLAT conference, pp 872–877
Sun Y, Yao K, Dong J (2018) Asian option pricing problems of uncertain mean-reverting stock model. Soft Comput 22(17):5583–5592
Wang X, He J, Li S (2014) Compound option pricing under fuzzy environment. J Appl Math 2014:875319. https://doi.org/10.1155/2014/875319
Wolkenhauer O (1997) A course in fuzzy systems and control. Int J Electr Eng Educ 34(3):282
Wu HC (2004) Pricing European options based on the fuzzy pattern of Black–Scholes formula. Comput Oper Res 31(7):1069–1081
Wu HC (2005) European option pricing under fuzzy environments. Int J Intell Syst 20(1):89–102
Xu W, Wu C, Xu W, Li H (2009) A jump-diffusion model for option pricing under fuzzy environments. Insur Math Econ 44(3):337–344
Xu W, Xu W, Li H, Zhang W (2010) A study of greek letters of currency option under uncertainty environments. Math Comput Model 51(5):670–681
Xu W, Liu G, Yu X (2018) A binomial tree approach to pricing vulnerable option in a vague world. Int J Uncertain Fuzziness Knowl Based Syst 26(01):143–162
Yoshida Y (2003) The valuation of European options in uncertain environment. Eur J Oper Res 145(1):221–229
Yoshida Y, Yasuda M, Ji Nakagami, Kurano M (2006) A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty. Fuzzy Sets Syst 157(19):2614–2626
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8(3):199–249
Zhang LH, Zhang WG, Xu WJ, Xiao WL (2012) The double exponential jump diffusion model for pricing European options under fuzzy environments. Econ Model 29(3):780–786
Zhang WG, Xiao WL, Kong WT, Zhang Y (2015) Fuzzy pricing of geometric asian options and its algorithm. Appl Soft Comput 28:360–367
Zhou J, Yang F, Wang K (2015) Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming. J Intell Fuzzy Syst 30(1):71–87
Conflict of interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by V. Loia.
About this article
Cite this article
Chen, H., Hu, C. & Yeh, W. Option pricing and the Greeks under Gaussian fuzzy environments. Soft Comput 23, 13351–13374 (2019). https://doi.org/10.1007/s00500-019-03876-w
- Gaussian fuzzy number