Monotone Finite-Difference Schemes with Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularizatio...

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Bibliographic Details
Published in:IEEE access 2020-01, Vol.8, p.1-1
Main Authors: Hieu, Le Minh, Hanh, Truong Thi Hieu, Thanh, Dang Ngoc Hoang
Format: Article
Language:English
Subjects:
Approximation
Black-Scholes equation
Boundary value problems
Convection
Dirichlet problem
Economics
Estimates
Europe
Exact solutions
financial engineering
Finite difference method
Finite difference methods
finite-difference scheme
Gamma equation
Mathematical analysis
Mathematical model
Maximum principle
numerical algorithm
numerical solution
Partial differential equations
Physics
Pricing
Regularization
regularization principle
stock price prediction
two-side estimates
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Summary:We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.
ISSN:2169-3536
2169-3536
DOI:10.1109/ACCESS.2020.2978594