A Study of Discrete Time Pricing American Fuzzy Put Option Model on Fuzzy Future Contract in Seller's Perspective using a Special Class of Fuzzy Numbers

A fuzzy analogue of the binomial option pricing tree model introduced by Cox et.al [1] was proposed by Yoshida [2] to study fuzzy American put option model in an uncertain environment using discrete time fuzzy stochastic process. Later, Muzzioli [3] used non overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in American put option pricing model admitting impreciseness only in volatility while Xcaojian Yu [4] persumed impreciseness in both risk-free interest rate and volatility of the underlying stock involving non-overlapping trapezoidal fuzzy numbers. K. Meenakshi et al. [5] defined new fuzzy risk-neutral probability measures using general trapezoidal fuzzy numbers to study "Problem of Pricing American Fuzzy Put Option Buyer’s Model". The two up and down jump factors and the risk-free interest rate fluctuates oftenly and hence are uncertain in nature. When financial investors come across a high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping type of fuzzy numbers will not be sufficient to predict the underlying fuzzy stock prices as the fuzzy stock prices would go up only in the up state and would go down only in the down state of the fuzzy binomial tree. The uncertainty associated in the above stated fuzzy option pricing parameters could not be captured completely here. Such a scenario needs attention. To handle such a situation, we need to consider fuzzy numbers which are not only non - overlapping but partially or fully overlapping and/ or contained in too. Also not much consideration had been given to American fuzzy put option model involving fuzzy martingales. This paper will consider the study of the fuzzy analogue of martingale pricig theory in the context of fuzzy option pricing theory.

In this study, we discuss American Fuzzy Put Option Seller’s Model (AFPOSM) based on fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they could be either overlapping or non-overlapping partially or completely and/or contained in using the fuzzy risk-neutral probability measures introduced by us. We record a computational procedure to obtain the fuzzy profit and loss (PL) values of sellers using two-period fuzzy binomial tree model wherein the fuzzy stock price and the fuzzy future price following discrete time fuzzy stochastic process. The same is performed by deploying the real stock market data obtained from the website [6] that includes American style Microsoft Corporation (MSFT) shares of the year 2018 under the bullish (the constant initial fuzzy stock price goes above the constant fuzzy strike price) scenario. Utilizing the PL values, we estimate an optimal exercise time and an optimal exercise price of sellers. Also the discounted fuzzy intrinsic values of AFPOSM on fuzzy future contract are validated. The computations carried out in AFPOSM are performed using MATLAB 2016a software.