A finitedifference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives
Volume 20, Issue 4, pp 283291
http://dx.doi.org/10.22436/jmcs.020.04.02
Publication Date: February 21, 2020
Submission Date: August 14, 2019
Revision Date: January 01, 2020
Accteptance Date: January 16, 2020
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 BlackScholes equation for option price into a quasilinear parabolic equation for the second derivative of the option price. We develop unconditionally monotone finitedifference schemes of secondorder of local approximation on uniform grids for the initial boundary value problem for the Gamma equation. Twoside estimates of the solution of the scheme are established. By means of regularization principle, the previous results are generalized for construction of unconditionally monotone finitedifference scheme (the maximum principle is satisfied without constraints on relations between the coefficients and grid parameters) of the secondorder of approximation on uniform grids for this equation. With the help of difference maximum principle, the twoside estimates for difference solution are obtained at the arbitrary nonsignconstant input data of the problem. A priori estimate in the maximum norm \(C\) is proved. It is interesting to note that the proven twoside 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 finitedifference 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, 283291
AMA Style
Hieu Le Minh, Hanh Truong Thi Hieu, Thanh Dang Ngoc Hoang, A finitedifference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives. J Math Comput SCIJM. (2020); 20(4):283291
Chicago/Turabian Style
Hieu, Le Minh, Hanh, Truong Thi Hieu, Thanh, Dang Ngoc Hoang. "A finitedifference 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): 283291
Keywords
 Gamma equation
 maximum principle
 twoside estimates
 monotone finitedifference scheme
 quasilinear parabolic equation
 scientific computing
MSC
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