On splitting-based numerical methods for nonlinear models of European options

Abstract

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations.

Keywords:

• option pricing
• nonlinear Black–Scholes model
• operator splitting
• difference scheme
• positivity preserving
• monotonicity

2010 AMS Subject Classifications:

• 35B09
• 65M06
• 91G60

Acknowledgements

This research was supported by the European Union under Grant Agreement number 304617 (FP7 Marie Curie Action Project Multi-ITN STRIKE – Novel Methods in Computational Finance) and Bulgarian National Fund of Science under Project DID 02/37-2009.