Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Abstract

In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable, Carr–Geman–Madan–Yor and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.

Keywords:

• barrier options pricing
• Lévy process
• fractional diffusion equations
• Krylov subspace methods
• circulant preconditioner

2010 AMS Subject Classifications:

• 26A33
• 65F10
• 65L12
• 91G20
• 91G60

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Some papers consider $v=C\mathrm{\Gamma }\left(Y\right)\left[\right(M-1{)}^Y-M^Y+(G+1{)}^Y-G^Y]$, while others deem it as $v=C\mathrm{\Gamma }(-Y)\left[\right(M-1{)}^Y-M^Y+(G+1{)}^Y-G^Y]$. From [6], the latter one will be used in this paper.

Additional information

Funding

This work was supported by the research grants MYRG068(Y2-L2)-FST13-DD and MYRG071(Y3-L2)-FST13-LSL from University of Macau and 115/2013/A3 from FDCT of Macao.