Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics - Le, Minh Hieu (H. : ĐHQGHN, 2020)
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-side estimates of the solution of the scheme are established. 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 quasilinear parabolic equation for the second derivative of 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 monotone nite-difference scheme (the maximum principle is satised without constraints on relations between the 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 solution are obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in the maximum 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 of the difference solution do not depend on the diffusion and convection coeffcients. Computational experiments, conrming the theoretical conclusions, are given.