# 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

Yoshida [2] developed a fuzzy replica of Cox et al’s binomial option pricing tree model to explore the fuzzy American put option model in an uncertain environment utilising a discrete time fuzzy stochastic process. Later, Muzzioli [3] used non-overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in an American put option pricing model, admitting imprecision only in volatility, whereas Xcaojian Yu [4] used non-overlapping trapezoidal fuzzy numbers to assume imprecision in both the risk-free interest rate and the volatility of the underlying stock. To investigate the “Problem of Pricing American Fuzzy Put Option Buyer’s Model,” K. Meenakshi et al. [5] proposed new fuzzy risk-neutral probability measures utilising generic trapezoidal fuzzy numbers. The risk-free interest rate and the two up and down jump elements varies often, making them unpredictable in nature. When financial investors come across high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping fuzzy numbers will not be enough to predict the underlying fuzzy stock prices because the fuzzy stock prices will only go up in the up state of the fuzzy binomial tree and will only go down in the down state. The uncertainty associated with the fuzzy option pricing parameters mentioned above could not be fully conveyed here. A situation like this requires special treatment. To deal with such a circumstance, we must take into account fuzzy numbers that are not only non-overlapping but also partially or totally overlapping and/or included in. Also, not much thought had gone into it to a fuzzy martingales-based American fuzzy put option model The fuzzy counterpart of martingale pricing theory will be studied in the context of fuzzy option pricing theory in this work.

We discuss the American Fuzzy Put Option Seller’s Model (AFPOSM) in this paper, which is based on a fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they can be partially or completely overlapping or non-overlapping and/or contained in using the fuzzy risk-neutral probability measures introduced by us. The fuzzy profit and loss (PL) values of sellers are computed using a two-period fuzzy binomial tree model, with the fuzzy stock price and fuzzy future price following a discrete time fuzzy stochastic process. The same thing is done by Using genuine stock market data from the website [6] to simulate the bullish (constant initial fuzzy stock price exceeds constant fuzzy strike price) scenario for Microsoft Corporation (MSFT) shares in the year 2018. We estimate an ideal exercise time and an optimal exercise price for sellers using the PL values. On a fuzzy future contract, the discounted fuzzy intrinsic values of AFPOSM are also verified. The MATLAB 2016a programme is used to do the calculations in AFPOSM.

**Author(S) Details**

**K. Meenakshi
**Department of Mathematics, Stella Maris College (Autonomous), Affiliated to University of Madras, Chennai, India.

Felbin C. Kennedy

Felbin C. Kennedy

Department of Mathematics, Stella Maris College (Autonomous), Affiliated to University of Madras, Chennai, India.

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