Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics

Abstract

We develop unconditionally monotone nite-difference schemes of second-order of local approxi-mation on uniform grids for the initial boundary problem value for the Gamma equation. Two-sideestimates of the solution of the scheme are established. We consider the initial boundary valueproblem for the so called Gamma equation, which can be derived by transforming the nonlinearBlack-Scholes equation for option price into a quasilinear parabolic equation for the second derivativeof the option price, and for its exact solution the two-side estimates are obtained. By means of regu-larization principle, the previous results are generalized for construction of unconditionally monotonenite-difference scheme (the maximum principle is satised without constraints on relations betweenthe coeffcients and grid parameters) of second order of approximation on uniform grids for this equa-tion. With the help of difference maximum principle, the two-side estimates for difference solutionare obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in themaximum norm C is proved. It is interesting to note that the proven two-side estimates for differ-ence solution are fully consistent with differential problem, and the maximal and minimal values ofthe difference solution do not depend on the diffusion and convection coeffcients. Computationalexperiments, conrming the theoretical conclusions, are given.