Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics
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Abstract
This study develops unconditionally monotone finite-difference scheme of second-order of local approximation on uniform grids for the initial boundary problem value for the Gamma equation through the establishment of two-side estimates for the scheme’s solution. The study considers 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. By means of regularization principle, the previous study results were generalized for construction of unconditionally monotone finite-difference scheme (the maximum principle was satisfied without constraints on relations between the coefficients and grid parameters) of second order of approximation on uniform grids for this equation. With the help of difference maximum principle, the two-side estimates for difference solution were obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in the maximum norm C was proved. Interestingly, the proven two-side estimates for difference solution were fully consistent with differential problem, and the maximal and minimal values of the difference solution did not depend on the diffusion and convection coefficients. Finally, relevant computational experiments were given to confirm the above-named theoretical findings.
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