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**Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs.**
*(English)*
Zbl 1175.91071

Summary: This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

### MSC:

91B25 | Asset pricing models (MSC2010) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

35K55 | Nonlinear parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

39A10 | Additive difference equations |

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\textit{R. Company} et al., ESAIM, Math. Model. Numer. Anal. 43, No. 6, 1045--1061 (2009; Zbl 1175.91071)

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