A Fuzzy Jump-Diffusion Option Pricing Model Based on the Merton Formula

Abstract

This paper proposes a fuzzy jump-diffusion (FJD) option pricing model based on Merton (J Financ Econ 3:125–144, 1976) normal jump-diffusion price dynamics. The logarithm of the stock price is assumed to be a Gaussian fuzzy number and the risk-free interest rate, diffusion, and jump parameters of the Merton model are assumed to be triangular fuzzy numbers to model the impreciseness which occurs due to the variation in financial markets. Using these assumptions, a fuzzy formula for the European call option price has been proposed. Given any value of the option price, its belief degree is obtained by using the bisection search algorithm. Our FJD model is an extension of Xu et al. (Insur Math Econ 44:337–344, 2009) fuzzy normal jump-diffusion model and has been tested on NIFTY 50 and Nikkei 225 indices options. The fuzzy call option prices are defuzzified and it has been found that our FJD model outperforms Wu et al. (Comput Oper Res 31:069–1081, 2004) fuzzy Black-Scholes model for in-the-money (ITM) and near-the-money (NTM) options as well as outperforms Xu et al. (Insur Math Econ 44:337– 344, 2009) model for both ITM and out-of-the-money (OTM) options.