Abstract
An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.
Highlights
An option is the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date at a fixed strike price
For American option, exercise is permitted at any time t ≤ T. It was shown in [1] that option pricing can be modeled using a partial differential equation of second order
We propose a non-standard finite difference method, where we use a non-local approximation of the reaction term of the generalized Black–Scholes equation, combined with an implicit time step technique
Summary
An option is the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date at a fixed strike price. European option can be exercised only on expiration date T. For American option, exercise is permitted at any time t ≤ T. It was shown in [1] that option pricing can be modeled using a partial differential equation of second order. In the financial market, the coefficients can depend on the time and the asset price. In such situation the analytical solution of the generalized Black–Scholes model is not available. Numerical simulations are essential to obtain information about the behavior of the solutions. It is important to develop numerical methods that reproduce the qualitative properties of the solution
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