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.
|