Articlebronze2004

Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Bertram Düring · Michel Fournié · Ansgar Jüngel

52 citations0/0 files storedESAIM Mathematical Modelling and Numerical Analysis

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Stochastic processes and financial applications Advanced Numerical Methods in Computational Mathematics Differential Equations and Numerical Methods

Abstract

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

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References

1973The Pricing of Options and Corporate Liabilities 1973Theory of Rational Option Pricing 2005Theory of rational option pricing 1989Optimal replication of contingent claims under transaction costs 1990Market Liquidity, Hedging, and Crashes 1993European Option Pricing with Transaction Costs 1992Option Replication in Discrete Time with Transaction Costs 1992Market Manipulation, Bubbles, Corners, and Short Squeezes

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2009Numerical Methods for Non-Linear Black–Scholes Equations 2019A new higher order compact finite difference method for generalised Black–Scholes partial differential equation: European call option 2020A sixth order numerical method and its convergence for generalized Black–Scholes PDE 2013High-Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-Uniform Grids

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