Unconditionally monotone finite difference scheme of the second-order approximation on uniform grids for the Gamma equation
In this article we consider the initial boundary-value problem for the so-called Gamma equation, which can be derived by transforming the nonlinear Black - Scholes equation for option price into a quasi-linear parabolic equation for the second de­rivative of option price, and for its exact solution the two-side estimates are obtained. By means of the regularization principle, the previous results are generalized to construct an unconditionally monotone finite-difference scheme (the maximum prin­ciple is satisfied without limitations on the relations between the coefficients and the grid parameters) of second-order ap­proximation on uniform grids for this equation. With the help of the difference maximum principle, the two-side estimates for a difference solution are obtained using the arbitrary non-sign-constant input data of the problem. The a priori estimate in the maximum norm C is proved. It is interesting to note that the proven two-side estimates for the difference solution are fully consistent with the differential problem, and the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients. Computational experiments confirming the theoretical conclusions are given.