Using fuzzy sets theory and Black-Scholes formula to generate pricing boundaries of European options

Abstract The application by using extension principle in fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, the risk-free interest rate, volatility and stock price may occur imprecisely in the real world. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price in the financial market. Under these assumptions, the European call and put option price will turn into the fuzzy numbers, and the extension principle will be invoked to generate the pricing boundaries of the European call and put options. This will make the financial analyst who can pick any reasonable European option price with an acceptable belief degree for his/her later financial analysis. In order to obtain the belief degree, an optimization problem has to be solved.

[1]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[2]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[3]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[4]  Yuji Yoshida,et al.  The valuation of European options in uncertain environment , 2003, Eur. J. Oper. Res..

[5]  Hsien-Chung Wu,et al.  Pricing European options based on the fuzzy pattern of Black-Scholes formula , 2004, Comput. Oper. Res..

[6]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[7]  Hsien-Chung Wu,et al.  Mining interesting sequential patterns for intelligent systems , 2005 .

[8]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[9]  R. Ribeiro.  Soft computing in financial engineering , 1999 .

[10]  M. S. Bazaraa,et al.  Nonlinear Programming , 1979 .

[11]  Maria Rosaria Simonelli,et al.  Fuzziness in valuing financial instruments by certainty equivalents , 2001, Eur. J. Oper. Res..

[12]  Zdenek Zmeskal,et al.  Application of the Fuzzy - Stochastic Methodology to Appraising the Firm Value as a European Call Option , 2001, Eur. J. Oper. Res..

[13]  T. Apostol.  Mathematical Analysis , 1957 .

[14]  Volker Krätschmer,et al.  A unified approach to fuzzy random variables , 2001, Fuzzy Sets Syst..

[15]  R. Aumann.  INTEGRALS OF SET-VALUED FUNCTIONS , 1965 .

[16]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .