Assessing the Convergence of Three Up-and-Down Moves in Binomial Option Pricing Modelling to Black-Scholes-Merton in a Fuzzy Setting

Abstract

The fuzzy binomial option pricing (FBOP) method parameterizes the up and down multipliers via volatility, which is assumed to be a fuzzy number, and employs the Cox, Ross, and Rubinstein (CRR) structure for these multipliers. This paper examines alternative parameterizations for the up and down factors within the FBOP framework, specifically the Rendleman and Bartter (RB) and Trigeorgis (TRIG) binomial approximations. We have observed that although all three methodologies produce prices that converge to those of the Black-Scholes-Merton (BSM) model in a fuzzy setting, the RB approach typically results in smaller errors than TRIG and CRR. This superiority is more pronounced for at-the-money options than for other money degrees. While the RB model consistently shows the greatest convergence for both call and put options, its advantage is particularly evident in call options. The superiority of RB is also observed across all volatility scenarios (low, medium, and high). Additionally, Trigeorgis’ parameterization often provides a better approximation than CRR does, especially for out-of-the-money options and puts. Therefore, we believe it is worthwhile to explore alternative parameterizations for the up- and downfactors in future FBOP studies as alternatives to CRR.