Crank-Nicolson Finite Difference Method with Sobolev Space Energy Estimate Theorem for Capital Market Prices

In this study, we have Black-Scholes analytic formula and Crank- Nicolson (CN) finite difference method for valuation of European put option which has earned the interests of researchers for determining both analytic and approximate solutions to Partial Differential Equations (PDEs) with Sobolev space energy estimate theorem. The simulations of analytical and numerical were effectively carried out. The results showed as follows: increase in volatility increases the values of option for both BS and CN prices, 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 put option, when the strike price is greater than the initial stock price it increases put option values, Sobolev space energy estimates were used as asset value function to estimate asset prices at different maturity periods. Finally, the graphical solutions and comparisons of other parameters were discussed all in this paper which is informative to investors for the proper investment plans.