Pricing And Hedging Options:A Numerical PDE Approach With Smoothing

There are many studies on options pricing since options are one of the most widely traded financial products.However,the study on exotic options is relatively rare.As the development of financial market,there will be much more complex option products.Thus,finding a general method to solve the option pricing and hedging problems is essential.This paper mainly focuses on the pricing and hedging problems of European call option,digital call option and a special case of barrier option via numerical solution of partial differential equations(PDEs).Firstly,this paper transfers the B-S PDE of option pricing into heat equation,which makes the numerical solving much easier than solving the B-S PDE directly.Besides,this paper discusses the most widely used numerical approach in solving PDEs,the implicit scheme and the Crank-Nicolson scheme and their implications in option pricing.The speed of convergence in implicit scheme is slower than that in Crank-Nicolson scheme,but the solution of implicit scheme is more stable.Thus,we recommend the implicit scheme in numerical option pricing.Also,the errors of numerical solution are huge and order of convergence in space dimension does not approximate theoretical value when the payoff function of option is not continuous or its first order derivative does not exist,especially when these discontinuous points are not on the grid of numerical solution.To handle this problem,this paper proposes a smoothing function approach.This approach changes the payoff of points near the discontinuous points via a smoothing function.Finally,this paper shows the relationship of Greeks and initial stock price.The Greeks change much more smoothly with initial stock price after smoothing payoff function than the case without smoothing payoff function.