Document Type : Research Paper


1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, Al 35209, USA

2 Department of Mathematics, Shahid Bahonar University of Kerman, Kerman and Institute for Studies in Theoretical Physics and Mathematics(IPM), Tehran, Iran


We use the basic binomial option pricing method but allow some
or all the parameters in the model to be uncertain and model this uncertainty
using fuzzy numbers. We show that with the fuzzy model we can, with a
reasonably small number of steps, consider almost all possible future stock
prices; whereas the crisp model can consider only n + 1 prices after n steps.


[1] S. S. Appadoo, R. K. Thulasiram, C. R. Bector and A. Thavaneswaran, Fuzzy algebraic
option pricing technique- a fundamental investigation, Proceedings ASAC Conference 2004,
Quebec City, Quebec.
[2] J. J. Buckley and E. Eslami, Introduction to fuzzy logic sand fuzzy sets, Springer, Heidelberg,
Germany, 2002.
[3] J. J. Buckley and Y. Qu, On using -cuts to evaluate fuzzy equations, Fuzzy Sets and Systems,
38(1990), 309-312.
[4] J. J. Buckley, T. Feuring and E. Eslami, Applications of fuzzy sets and fuzzy logic to economics
and engineering, Springer, Heidelberg, Germany, 2002.
[5] J. C. Cox and M. Rubinstein, Options markets, Prentice-Hall, Englewood Cliffs, NJ, 1985.
[6] D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications, Academic Press,
N.Y., 1980.
[7] M. Durbin, All About derivatives, McGraw-Hill, NY, NY, 2006.
[8] Frontline Systems (
[9] G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic, Prentice Hall, Upper Saddle River, N.J.,
[10] S. Muzzioli and C. Torricelli, A model for pricing an option with a fuzzy payoff, Fuzzy
Economic Review, 6(2001), 40-62.
[11] S. Muzzioli and C. Torricelli, A multiperiod binomial model for pricing options in an uncertain
world, Proceedings Second Int. Symposium Imprecise Probabilities and Their Applications,
Ithaca, NY, 2001, 255-264.
[12] H. T. Nguyen and E. A. Walker, A first course in fuzzy logic, Second Edition, CRC Press,
Boca Raton, FL., 2000.
[13] H. Reynaerts and M. Vanmaele, A sensitivity analysis for the pricing of european call options
in a binary tree model, Proceedings Fourth Int. Symposium Imprecise Probabilities and Their
Applications, Univ. Lugano, Switzerland, 2003, 467-481.
[14] H. A. Taha, Operations research, Fifth Edition, Macmillan, N.Y., 1992.
[15] R. G. Tompkins, Options analysis, Revised Edition, Irwin Professional Publishing, Chicago,
USA, 1994.
[16] M. A. Wong, Trading and investing in bond options, John Wiley and Sons, NY, NY, 1991.
[17] H. -C. Wu, Pricing European options based on the fuzzy pattern of black-scholes formula,
Computers and Operations Research, 31(2004),1069-1081.