Le Minh Hieu, Truong Thi Hieu Hanh, Dang Ngoc Hoang Thanh

## 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.

Keywords: Gamma equation, maximum principle, two-side estimates, monotone finite-difference scheme, quasi-linear parabolic equation, scientific computing.

## References

[1] O. Hyong-Chol, J.J. Jo, J.S. Kim, General properties of solutions to inhomogeneous Black-Scholes equations with discontinuous maturity payoffs, Journal of Differential Equations 260 (4) (2016) 3151–3172.
https://doi.org/10.1016/j.jde.2015.08.036.
[2] F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of political economy 81 (3) (1973) 637–654.
[3] G. Barles, H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Finance Stoch 2 (4) (1998) 369-397. https://doi.org/10.1007/s007800050046.
[4] R. Company, L. Jodar, E. Ponsoda, C. Ballester, Numerical analysis and simulation of option pricing problems modeling illiquid markets, Comput. Math. Appl. 59 (2010) 2964-2975.
https://doi.org/10.1016/j.camwa.2010.02.014.
[5] E. Demkova, M. Ehrhardt, A high-order compact merhod for nonlinear Black-Scholes option pricing of American options, Int. J. Comput. Math. 88 (13) (2011) 2782–2797. https://doi.org/10.1080/00207160.2011.558574.
[6] M. Jandacka, D. Sevcovic, On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. of Appl. Math. 3 (2005) 235–258. http://dx.doi.org/10.1155/JAM.2005.235.
[7] M.N. Koleva, L.G. Vulkov, A second-order positivity preserving numerical method for Gamma equation, Appl. Math. and Comput. 220 (2013) 722–734. https://doi.org/10.1016/j.amc.2013.06.082.
[8] A. Samarskii, The Theory of Difference Schemes, Marcel Dekker Inc., New York, Basel, 2001.
[9] S. Godunov, V. Ryabenkii, Difference Schemes, Nauka, Moscow, 1977 (in Russian).
[10] P. Matus, V.T.K. Tuyen, F. Gaspar, Monotone difference schemes for linear parabolic equations with mixed boundary conditions, Dokl. Natl. Acad. Sci. Belarus 58 (5) (2014), 18-22 (in Russian).
[11] I. Farago, R. Horvath, Discrete maximum principle and adequate discretizations of linear parabolic problems, SIAM J. Sci. Comput. 28 (6) (2006), 2313-2336. https://doi.org/10.1137/050627241.
[12] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Equations of Parabolic Type), Moscow: Nauka, 1967.
[13] P. Matus, L.M. Hieu, L.G. Vulkov, Maximum principle for finite-difference schemes with non sign-constant input data, Dokl. Nats. Akad. Nauk Belarusi. 59 (5) (2015) 13-17 (in Russian).
[14] P. Matus, L.M. Hieu, L.G. Vulkov, Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations, J. of Comput. Appl. Math. 310 (2017) 186-199.
https://doi.org/10.1016/j.cam.2016.04.006.
[15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964.