Asian Journal of Pure and Applied Mathematics

This paper considered the notion of European call option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and Black-Scholes Partial Differential Equation (BSPDE) using Crank-Nicolson finite difference method. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula; and all results showed as follows: increase in volatility increases the values of option for both BS and CN, there are significant difference between BS and CN due to the changes of stock volatility, a little increase in the initial stock prices significantly increases the value of call option, when the strike price is greater than the initial stock price it devalues the call option, increasing interest rate at difference levels increases the value of call option, Sobolev energy estimates were used as asset value function to estimate asset prices at different interest rates. To this end, the graphical solutions and comparisons of other parameters were discussed for the purpose of decision making in time varying investment plans.