## Journal of Applied Mathematics

### Pricing Currency Option Based on the Extension Principle and Defuzzification via Weighting Parameter Identification

#### Abstract

We present a fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option based on the extension principle. In order to keep consistent with the real market, we assume that the interest rate, the spot exchange rate, and the volatility are fuzzy numbers in the FG-K formula. The conditions of a basic proposition about the fuzzy-valued functions of fuzzy subsets are modified. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. To simplify the trade, the weighted possibilistic mean (WPM) value with a weighting function is adopted to defuzzify the fuzzy price to a crisp price. The numerical example shows our method makes the α-level set of fuzzy price smaller, which decreases the fuzziness. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the optimal parameter.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 623945, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807886

Digital Object Identifier
doi:10.1155/2013/623945

Zentralblatt MATH identifier
1266.91027

#### Citation

Xu, Jixiang; Tan, Yanhua; Gao, Jinggui; Feng, Enmin. Pricing Currency Option Based on the Extension Principle and Defuzzification via Weighting Parameter Identification. J. Appl. Math. 2013 (2013), Article ID 623945, 9 pages. doi:10.1155/2013/623945. https://projecteuclid.org/euclid.jam/1394807886

#### References

• M. B. Garman and S. W. Kohlhagen, “Foreign currency option values,” Journal of International Money and Finance, vol. 2, no. 3, pp. 231–237, 1983.
• F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economics, vol. 81, no. 3, pp. 637–659, 1973.
• M. Chesney and L. Scott, “Pricing european currency options: a comparision of the modified black-scholes model and a random variance model,” Journal of Financial and Quantitative Analysis, vol. 24, no. 3, pp. 267–284, 1989.
• K. Amin and R. Jarrow, “Pricing foreign currency options under stochastic interest rates,” Journal of International Money and Finance, vol. 10, no. 3, pp. 310–329, 1991.
• S. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, no. 2, pp. 327–344, 1993.
• D. Bates, “Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options,” Review of Financial Studies, vol. 9, no. 1, pp. 69–107, 1996.
• G. Sarwar and T. Krehbiel, “Empirical performance of alternative pricing models of currency options,” The Journal of Futures Markets, vol. 20, no. 3, pp. 265–291, 2000.
• P. Carr and L. R. Wu, “Stochastic skew in currency options,” Journal of Financial Economics, vol. 86, no. 1, pp. 213–247, 2007.
• L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
• Y. Yoshida, “The valuation of European options in uncertain environment,” European Journal of Operational Research, vol. 145, no. 1, pp. 221–229, 2003.
• Y. Yoshida, “A discrete-time model of American put option in an uncertain environment,” European Journal of Operational Research, vol. 151, no. 1, pp. 153–166, 2003.
• S. Muzzioli and C. Torricelli, “A multiperiod binomial model for pricing options in a vague world,” Journal of Economic Dynamics & Control, vol. 28, no. 5, pp. 861–887, 2004.
• H. C. Wu, “Using fuzzy sets theory and Black-Scholes formula to generate pricing boundaries of European options,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 136–146, 2007.
• W. D. Xu, C. F. Wu, W. J. Xu, and H. Y. Li, “A jump-diffusion model for option pricing under fuzzy environments,” Insurance: Mathematics & Economics, vol. 44, no. 3, pp. 337–344, 2009.
• L. H. Zhang, W. G. Zhang, W. J. Xu, and W. L. Xiao, “The double exponential jump diffusion model for pricing European options under fuzzy environments,” Economic Modelling, vol. 29, no. 3, pp. 780–786, 2012.
• F. Y. Liu, “Pricing currency options based on fuzzy techniques,” European Journal of Operational Research, vol. 193, no. 2, pp. 530–540, 2009.
• H. C. Wu, “European option pricing under fuzzy environments,” International Journal of Intelligent Systems, vol. 20, no. 1, pp. 89–102, 2005.
• W. J. Xu, W. D. Xu, H. Y. Li, and W. G. Zhang, “A study of Greek letters of currency option under uncertainty environments,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 670–681, 2010.
• M. L. Guerra, L. Sorini, and L. Stefanini, “Option price sensitivities through fuzzy numbers,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 515–526, 2011.
• R. Fullér and P. Majlender, “On weighted possibilistic mean and variance of fuzzy numbers,” Fuzzy Sets and Systems, vol. 136, no. 3, pp. 363–374, 2003.
• Y. Chalco-Cano, H. Román-Flores, M. Rojas-Medar, O. R. Saavedra, and M. D. Jiménez-Gamero, “The extension principle and a decomposition of fuzzy sets,” Information Sciences, vol. 177, no. 23, pp. 5394–5403, 2007.
• D. P. Bertsekas, A. Nedić, and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, Mass, USA, 2003.
• L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning–-I,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975.
• L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning–-II,” Information Sciences, vol. 8, no. 3, pp. 301–357, 1975.
• L. A. Zadeh, “The concept of a linguistic variablečommentComment on ref. [24c?]: We split this reference to [24a,24b,24c?]. Please check.and its application to approximate reasoning–-III,” Information Sciences, vol. 9, no. 1, pp. 43–80, 1975.
• T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1974.
• M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, vol. 36 of Stochastic Modelling and Applied Probability, Springer, Berlin, Germany, 2nd edition, 2005.