# A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

Volume 20, Issue 4, pp 283--291
Publication Date: February 21, 2020 Submission Date: August 14, 2019 Revision Date: January 01, 2020 Accteptance Date: January 16, 2020
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### Authors

Le Minh Hieu - University of Economics, The University of Danang, Vietnam. Truong Thi Hieu Hanh - University of Economics, The University of Danang, Vietnam. Dang Ngoc Hoang Thanh - Department of Information Systems, School of Business Information Technology, University of Economics Ho Chi Minh city, Vietnam.

### Abstract

In the article, we consider the initial boundary value problem for the 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. We develop unconditionally monotone finite-difference schemes of second-order of local approximation on uniform grids for the initial boundary value problem for the Gamma equation. Two-side estimates of the solution of the scheme are established. By means of regularization principle, the previous results are generalized for construction of unconditionally monotone finite-difference scheme (the maximum principle is satisfied without constraints on relations between the coefficients and grid parameters) of the second-order of approximation on uniform grids for this equation. 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 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.

### Share and Cite

##### ISRP Style

Le Minh Hieu, Truong Thi Hieu Hanh, Dang Ngoc Hoang Thanh, A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives, Journal of Mathematics and Computer Science, 20 (2020), no. 4, 283--291

##### AMA Style

Hieu Le Minh, Hanh Truong Thi Hieu, Thanh Dang Ngoc Hoang, A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives. J Math Comput SCI-JM. (2020); 20(4):283--291

##### Chicago/Turabian Style

Hieu, Le Minh, Hanh, Truong Thi Hieu, Thanh, Dang Ngoc Hoang. "A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives." Journal of Mathematics and Computer Science, 20, no. 4 (2020): 283--291

### Keywords

• Gamma equation
• maximum principle
• two-side estimates
• monotone finite-difference scheme
• quasi-linear parabolic equation
• scientific computing

•  65N06

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