Abstract
We apply an exponential time integration scheme combined with a central difference scheme on a piecewise uniform mesh with respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time and spatial variables. It is proved that the scheme is unconditionally stable. Numerical results support the theoretical results.
Highlights
The pricing and hedging of derivative securities, known as contingent claims, is a subject of much practical importance
Options which can be exercised only on the expiry date are called European, whereas options which can be exercised any time up to and including the expiry date are classified as American
It was shown by Black and Scholes 1 that these option prices satisfy a second-order partial differential equation with respect to the time horizon t and the underlying asset price x
Summary
The pricing and hedging of derivative securities, known as contingent claims, is a subject of much practical importance. Options which can be exercised only on the expiry date are called European, whereas options which can be exercised any time up to and including the expiry date are classified as American It was shown by Black and Scholes 1 that these option prices satisfy a second-order partial differential equation with respect to the time horizon t and the underlying asset price x. It is well known that when using the standard finite difference method to solve those problems involving the convection-diffusion operator, such as the Black-Scholes differential operator, numerical difficulty can be caused. In 10, 11 numerical methods of option pricing models are studied by applying a standard finite volume method to obtain a difference scheme Their numerical schemes use central difference for a given mesh, but switch to upstream weighting for a small number of nodes, which are second-order spatial convergent.
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