Multiscale methods for the valuation of American options with stochastic volatility

Abstract

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments.

Keywords:

• American option pricing
• stochastic volatility
• Heston's model
• parabolic boundary value problem
• free boundary
• monotone multigrid method
• multigrid efficiency

2010 AMS Subject Classification :

• 65M55
• 35J86
• 65N30
• 65D07
• 91B70

Acknowledgements

We thank Christoph Schwab, who originally pointed out the problem of stochastic volatility after completion of the papers 27 28 dealing with MMG methods with higher order B-splines. We also thank Ernst Eberlein, Andrej Palczweski and two anonymous referees on their remarks on modelling issues concerning boundary and Feller's condition, see Remark 2.1. We express our gratitude to two anonymous referees for their useful remarks on the whole manuscript. This work was supported in part by the Deutsche Forschungsgemeinschaft (SFB 611, Universität Bonn) and the Institute for Mathematics and its Applications (IMA) at the University of Minnesota with funds provided by the National Science Foundation (NSF).