Partial Differential Equation Approach for Valuation of European Put Option Price using Two Stochastic Financial Models.

Abstract

An option is defined as a financial contract that provides the holder the right but not the obligation to buy or sell a specified quantity of an underlying asset in the future at a fixed price (called a strike price) at or before the expiration date of the option. This paper looked at two different models in finance which are the Constant Elasticity of Variance (CEV) model and the Black-Karasinski model. We obtained the various Partial Differential Equations (PDEs) option price valuation formulas for these two models using the riskless portfolio method through the elimination of the stochastic components from their respective Stochastic Differential Equations (SDEs). Also, numerical implementation of the derived Black-Scholes Parabolic PDEs options pricing equations for the SDEs models was carried out using the Explicit Finite Difference method which was implemented using MATLAB. Furthermore, results from the numerical examples using published data from the Nigeria Stock Exchange (NSE) show the various effects of increase in the underlying asset value (stock price) and increase in the risk free interest rate on the value of the European Put Option for these models. From the results, we see that an increase in the stock price and interest rate both yields a decrease in the value of the option price and hence guides the holder of the option with quality decision to not exercise his right on the option and may allow the option to expire.