Journal of Applied Mathematics
Volume 2013 (2013), Article ID 623945, 9 pages
http://dx.doi.org/10.1155/2013/623945
Research Article

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

Jixiang Xu,1,2 Yanhua Tan,3 Jinggui Gao,4 and Enmin Feng1

1School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China
2School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China
3School of Sciences, Hebei University of Technology, Tianjin 300130, China
4College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266510, China

Received 24 November 2012; Accepted 20 January 2013

Academic Editor: Reinaldo Martinez Palhares

Copyright © 2013 Jixiang Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.