# W-methods on the Heston problem (Soledad Perez Rodriguez, La Laguna)

27.05.2016 10:30

The Heston problem is an important partial differential equation in financial option pricing theory. It is a two-dimensional extension to the famous Black-Scholes equation that predicts the fair price of a call option when the asset price is considered to follow a stochastic process. In the derivation of the Heston model both the asset price and its variance are supposed to be stochastic. As a result the Heston equation involves a mixed derivative term due to the correlation between these two random processes. Analytical solutions of this problem are not available even for simple options so finding a good numerical approach is essential.

Several authors have applied finite differences for the spatial discretization of the Heston model. The success of the time integrator applied to the resulting semi-discretized initial value problem depends crucially on the strategy used to approximate the part of the Jacobian matrix corresponding to the mixed derivative term. Some methods of order two as the well-known Douglas method only reach order one on this problem when the Jacobian of the mixed derivatives

is neglected. K. J. in ‘t Hout et al. have proposed a splitting method called Modified Craig-Sneyd (MCS) scheme that achieves order two on the Heston PDE with unconditional stability and gives efficient results avoiding the computation of the Jacobian term corresponding to the mixed derivatives.

Other splitting methods that used inexact Jacobians are W-methods. We have found several W-methods of order two that combine the Approximate Matrix Factorization (AMF) technique to simplify the linear systems involved in the schemes and give better errors than the MCS method on the Heston model. In this talk we will present the implementation of these W-methods on this problem and the results obtained with fixed and variable time step sizes.

### Lieu

salle 624, Attention horaire inhabituel. Séminaire d'analyse numérique

### Organisé par

Section de mathématiques### Intervenant-e-s

Soledad Perez Rodriguez, Universidad de La Lagunaentrée libre