%0 Journal Article %T A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives %A Hieu, Le Minh %A Hanh, Truong Thi Hieu %A Thanh, Dang Ngoc Hoang %J Journal of Mathematics and Computer Science %D 2020 %V 20 %N 4 %@ ISSN 2008-949X %F Hieu2020 %X 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. %9 journal article %R 10.22436/jmcs.020.04.02 %U http://dx.doi.org/10.22436/jmcs.020.04.02 %P 283--291 %0 Journal Article %T Option pricing with transaction costs and a nonlinear Black--Scholes equation %A G. Barles %A H. M. Soner %J Finance Stoch. %D 1998 %V 2 %F Barles1998 %0 Journal Article %T The pricing of options and corporate liabilities %A F. Black %A M. Scholes %J J. Polit. Econ. %D 1973 %V 81 %F Black1973 %0 Journal Article %T Numerical analysis and simulation of option pricing problems modeling illiquid markets %A R. Company %A L. Jódar %A E. Ponsoda %A C. Ballester %J Comput. Math. Appl. %D 2010 %V 59 %F Company2010 %0 Journal Article %T A high-order compact method for nonlinear Black--Scholes option pricing equations of American options %A E. Dremkova %A M. Ehrhardt %J Int. J. Comput. 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