Abdul-Qayyum M. Khaliq 

(Department of Mathematics, Western Illinois University) 

Numerical Simulation of Black-Scholes Model 
For American Option

 

Abstract

Development of modern option pricing began with the publication of the Black Scholes option pricing formula in 1973. Black & Scholes (1973) and Merton (1973) gave derivation of a model equation to compute the value of an option. This equation has  had such financial impact that Robert Merton and Myron Scholes shared the 1997 Nobel Prize for economics (Fischer Black having died in 1995). The Black Scholes formula computes the value of an option based on the strike price of the option, the risk free rate of interest, volatility of the stock, and the time until the option expires. The European Option can be exercised only at expiry date whereas an American Option has the additional feature that exercise is permitted at any time during the life of the option. This makes the valuation of an American option a free boundary problem.
We consider Black-Scholes formula with a penalty term. This term allows the applicability to be extended beyond the basic European option model. The inclusion of the forced case needs the development of stable and efficient numerical methods. Since option-pricing constraints are typically nonlinear, the reaction-diffusion-advection equation (extended Black-Scholes model) is solved by linearly implicit methods, which  avoid solving non-linear equations at each time step. The implementation of such  method by Linearly Implicit way is found efficient and stable when computing American pricing option. Numerical results for single and multi assist American options are presented.
Last updated by fass@amadeus.math.iit.edu  on 01/27/03