Adaptive finite differences and IMEX time-stepping to price options under Bates model

Abstract

In this paper, we consider numerical pricing of European and American options under the Bates model, a model which gives rise to a partial-integro differential equation. This equation is discretized in space using adaptive finite differences while an IMEX scheme is employed in time. The sparse linear systems of equations in each time-step are solved using an LU-decomposition and an operator splitting technique is employed for the linear complementarity problems arising for American options. The integral part of the equation is treated explicitly in time which means that we have to perform matrix-vector multiplications each time-step with a matrix with dense blocks. These multiplications are accomplished through fast Fourier transforms. The great performance of the method is demonstrated through numerical experiments.

Keywords:

• option pricing
• numerical methods
• Bates model
• adaptive finite differences
• IMEXtime-stepping

2010 AMS Subject Classifications:

• 65M06
• 91G60

Acknowledgments

The work in this paper builds on the sequence of Master theses/project reports [31,32,37,38]. The authors are thankful for the early contributions of these students. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) through Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX) under Projects snic2014-3-24 and snic2015-6-77. We thank the referees for their suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was financed by the Swedish Research Council under contract number 621-2007-6388.