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A second-order positivity preserving numerical method for gamma equation. (English) Zbl 1329.91140

Summary: 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) \({\Gamma}=V_{SS}\) 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 \(L_1\) norm is shown and the existence of interface curves is investigated numerically. Results of numerical simulations are given and discussed.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
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