A second-order positivity preserving numerical method for Gamma equation

In this work we consider Cauchy problem for the so called Gamma equation, which can be derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasi-linear parabolic equation for the second derivative (Greek) Γ=VSS of the option price V. We develop an efficient positivity preserving explicit numerical method for solving the model problem concerning different volatility terms. We prove that the obtained semi-discretization is positive and the corresponding full approximation also preserves this property, if the time step is restricted. The stability of the difference scheme in L1 norm is shown and the existence of interface curves is investigated numerically. Results of numerical simulations are given and discussed.