Efficient and stable numerical solution of the Heston–Cox–Ingersoll–Ross partial differential equation by alternating direction implicit finite difference schemes

Abstract

This paper concerns the numerical solution of the three-dimensional Heston–Cox–Ingersoll–Ross partial differential equation for the fair values of European-style financial options. We follow the method-of-lines approach and first semidiscretize on suitable nonuniform Cartesian spatial grids by applying finite difference schemes. The main aim of this paper is to investigate various prominent alternating direction implicit schemes for the effective time discretization of the obtained, very large, semidiscrete systems. For this purpose an extensive numerical study is performed, among others, for arbitrary correlation factors, for time-dependent mean-reversion levels, for short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. Also, the approximation of the hedging quantities Delta and Gamma is discussed and double barrier knock-out call options are considered.

Keywords:

• alternating direction implicit schemes
• financial option pricing
• finite difference methods
• hedging quantities
• Heston–Cox–Ingersoll–Ross partial differential equation

2010 AMS Subject Classifications:

• 65L04
• 65M06
• 65M20
• 91G60

Acknowledgements

The author is grateful to Karel in ’t Hout for sharing his knowledge and for his valuable remarks. This work has been supported financially by the Research Foundation – Flanders, FWO contract no. G.0125.08.