The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions. We transform the problem to an initial boundary value problem in dimensionless form. There are two parameters in the coefficients of the resulting linear parabolic partial differential equation. For a range of values of these parameters, the solution of the problem has a boundary or an initial layer. The initial function has a discontinuity in the first-order derivative, which leads to the appearance of an interior layer. We construct analytically the asymptotic solution of the equation in a finite domain. Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in $x$ and at the final time is not affected by the boundary. Also, we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values, while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied. We present numerical computations, which determine experimentally the parameter-uniform rates of convergence. We note that this rate is rather weak, due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.
|Number of pages||15|
|Journal||Numerical Mathematics: Theory, Methods and Applications|
|Publication status||Published - 2008|
- Finite difference methods
- Monotone robust method