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Soft Computing

, Volume 23, Issue 24, pp 13351–13374 | Cite as

Option pricing and the Greeks under Gaussian fuzzy environments

  • Hong-Ming ChenEmail author
  • Cheng-Feng Hu
  • Wei-Chang Yeh
Methodologies and Application
  • 69 Downloads

Abstract

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.

Keywords

Gaussian fuzzy number Options Greeks SPX Optimization 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Andersen TG, Bollerslev T, Diebold FX, Ebens H (2001) The distribution of realized stock return volatility. J Financ Econ 61(1):43–76CrossRefGoogle Scholar
  2. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654MathSciNetCrossRefGoogle Scholar
  3. Buckley J (1987) The fuzzy mathematics of finance. Fuzzy Sets Syst 21(3):257–273MathSciNetCrossRefGoogle Scholar
  4. Carlsson C, Fullér R (2003) A fuzzy approach to real option valuation. Fuzzy Sets Syst 139(2):297–312MathSciNetCrossRefGoogle Scholar
  5. Chrysafis KA, Papadopoulos BK (2009) On theoretical pricing of options with fuzzy estimators. J Comput Appl Math 223(2):552–566MathSciNetCrossRefGoogle Scholar
  6. 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):1624–1643MathSciNetCrossRefGoogle Scholar
  7. Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626MathSciNetCrossRefGoogle Scholar
  8. Guerra ML, Sorini L, Stefanini L (2011) Option price sensitivities through fuzzy numbers. Comput Math Appl 61(3):515–526MathSciNetCrossRefGoogle Scholar
  9. Hull JC (2006) Options, futures, and other derivatives. Pearson Education India, DelhizbMATHGoogle Scholar
  10. Jarrow RA, Turnbull SM (2000) Derivative securities. South-Western Publishing, NashvilleGoogle Scholar
  11. 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 509–520Google Scholar
  12. Muzzioli S, De Baets B (2017) Fuzzy approaches to option price modeling. IEEE Trans Fuzzy Syst 25(2):392–401CrossRefGoogle Scholar
  13. 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 266:131–143MathSciNetCrossRefGoogle Scholar
  14. Passarelli D (2012) Trading options Greeks: how time, volatility, and other pricing factors drive profits, vol 159. Wiley, HobokenCrossRefGoogle Scholar
  15. Qin Z, Li X (2008) Option pricing formula for fuzzy financial market. J Uncertain Syst 2(1):17–21Google Scholar
  16. 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 503:1288–1296MathSciNetCrossRefGoogle Scholar
  17. Starczewski JT (2005) Extended triangular norms on Gaussian fuzzy sets. In: EUSFLAT conference, pp 872–877Google Scholar
  18. Sun Y, Yao K, Dong J (2018) Asian option pricing problems of uncertain mean-reverting stock model. Soft Comput 22(17):5583–5592CrossRefGoogle Scholar
  19. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wolkenhauer O (1997) A course in fuzzy systems and control. Int J Electr Eng Educ 34(3):282CrossRefGoogle Scholar
  21. Wu HC (2004) Pricing European options based on the fuzzy pattern of Black–Scholes formula. Comput Oper Res 31(7):1069–1081CrossRefGoogle Scholar
  22. Wu HC (2005) European option pricing under fuzzy environments. Int J Intell Syst 20(1):89–102CrossRefGoogle Scholar
  23. Xu W, Wu C, Xu W, Li H (2009) A jump-diffusion model for option pricing under fuzzy environments. Insur Math Econ 44(3):337–344MathSciNetCrossRefGoogle Scholar
  24. 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):670–681MathSciNetCrossRefGoogle Scholar
  25. 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):143–162MathSciNetCrossRefGoogle Scholar
  26. Yoshida Y (2003) The valuation of European options in uncertain environment. Eur J Oper Res 145(1):221–229MathSciNetCrossRefGoogle Scholar
  27. 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):2614–2626MathSciNetCrossRefGoogle Scholar
  28. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefGoogle Scholar
  29. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8(3):199–249MathSciNetCrossRefGoogle Scholar
  30. 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):780–786CrossRefGoogle Scholar
  31. Zhang WG, Xiao WL, Kong WT, Zhang Y (2015) Fuzzy pricing of geometric asian options and its algorithm. Appl Soft Comput 28:360–367CrossRefGoogle Scholar
  32. Zhou J, Yang F, Wang K (2015) Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming. J Intell Fuzzy Syst 30(1):71–87CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsTunghai UniversityTaichungTaiwan
  2. 2.Department of Applied MathematicsNational Chiayi UniversityChiayi CityTaiwan
  3. 3.Department of Industrial Engineering and Engineering ManagementNational Tsing Hua UniversityHsinchuTaiwan

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