A positivity-preserving numerical method for nonlinear Black-Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of the numerical solution. The numerical results for European put option and European butterfly spread are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.
Shengwu Zhou. Wei Li. Yu Wei. Cui Wen. "A Positivity-Preserving Numerical Scheme for Nonlinear Option Pricing Models." J. Appl. Math. 2012 1 - 20, 2012. https://doi.org/10.1155/2012/205686