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 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

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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.

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Keywords

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

MSC

•  65N06

References

• [1] G. Barles, H. M. Soner, Option pricing with transaction costs and a nonlinear Black--Scholes equation, Finance Stoch., 2 (1998), 369--397
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637--654
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] R. Company, L. Jódar, E. Ponsoda, C. Ballester, Numerical analysis and simulation of option pricing problems modeling illiquid markets, Comput. Math. Appl., 59 (2010), 2964--2975
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] E. Dremkova, M. Ehrhardt, A high-order compact method for nonlinear Black--Scholes option pricing equations of American options, Int. J. Comput. Math., 88 (2011), 2782--2797
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] I. Faragó, R. Horváth, Discrete maximum principle and adequate discretizations of linear parabolic problems, SIAM J. Sci. Comput., 28 (2006), 2313--2336
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] S. K. Godunov, V. S. Ryaben'kii, Difference Schemes (in Russian), Nauka, Moscow (1977)
  • MathSciNet
  • Google Scholar
  • MATH


• [8] O. Hyong-Chol, J. J. Jo, J. S. Kim, General properties of solutions to inhomogeneous Black--Scholes equations with discontinuous maturity payoffs, J. Differential Equations, 260 (2016), 3151--3172
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] L. M. Hieu, D. N. H. Thanh, S. Prasath, Second order monotone difference schemes with approximation on non-uniform grids for two-dimensional quasilinear parabolic convection-diffusion equations, Vestnik St. Petersburg University, Mathematics, 53 (2020), (In Press)


• [10] M. Jandačka, D. Ševčovič, On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. Appl. Math., 3 (2005), 235--258
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] M. N. Koleva, L. G. Vulkov, A second-order positivity preserving numerical method for Gamma equation, Appl. Math. Comput., 220 (2013), 722--734
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type (in Russian), Nauka, Moscow (1967)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] P. Matus, L. M. Hieu, L. G. Vulkov, Maximum principle for finite-difference schemes with non sign-constant input data (in Russian), Dokl. Nats. Akad. Nauk Belarusi, 59 (2015), 13--17
  • Google Scholar


• [14] P. Matus, L. M. Hieu, D. Pylak, Monotone finite-difference schemes of second-order accuracy for quasilinear parabolic equations with mixed derivatives, Diff. Equ., 55 (2019), 424--436
  • View Article


• [15] P. Matus, L. M. Hieu, L. G. Vulkov, Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations, J. Comput. Appl. Math., 310 (2017), 186--199
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] P. Matus, D. Poliakov, L. M. Hieu, On convergence of difference schemes for dirichlet IBVP for two-dimensional quasilinear parabolic equations with mixed derivatives and generalized solutions, Comput. Meth. Appl. Math., (2020), (In Press)
  • View Article


• [17] P. P. Matus, V. T. K. Tuyen, F. Z. Gaspar, Monotone difference schemes for linear parabolic equations with mixed boundary conditions (in Russian), Dokl. Nats. Akad. Nauk Belarusi, 58 (2014), 18--22
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York (2001)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH