Parameter-uniform numerical methods for problems with layer phenomena: Application in mathematical finance

Abstract

Mathematical modeling in financial mathematics leads to the Cauchy problem for the parabolic Black-Scholes equation with the value of a European call option. By changing of variables, the problem is a singularly perturbed equation with the perturbation parameter epsilon in (0,1]; For finite values of the parameter, the solution of the Cauchy problem has different types of singularities: the unbounded domain; the piecewise smooth initial function and its unbounded growth at infinity; an interior layer generated by the piecewise smooth initial function for small values of the parameter epsilon, etc.Primarily, we are interested in approximations to both the solution and its first order derivative in a neighborhood of the interior layer generated by the piecewise smooth initial function. For this purpose, a new method, the singularity splitting method, is constructed. The numerical results verifies that using this method, we can approximate epsilon-uniformly both the solution of the boundary value problem and its first order derivative in x with convergence orders close to 1 and 0.5, respectively, whereas the classical finite difference method does not.Moreover, in order to construct adequate grid approximations for the singularity of the interior layer type, we consider the boundary value problem in bounded domain with appearing of interior and boundary layers. The singularity of the boundary layer is stronger than that of the interior layer, which makes it difficult to construct special numerical methods suitable for the adequate description of the singularity of the interior layer type. Using the method of piecewise uniform meshes that condense in a neighborhood of the boundary layer and the singularity splitting method, a special finite difference scheme is constructed that make it possible to approximate epsilon-uniformly the solution of the boundary value problem on the whole domain, its first order derivative in x on the whole domain except the discontinuity point outside a neighborhood of the boundary layer, and also the first order spatial derivative multiplied by the parameter epsilon in a finite neighborhood of the boundary layer.