Multigrid method for pricing European options under the CGMY process

Justin W. L. Wan; Justin W. L. Wan

doi:10.3934/math.2019.6.1745

AIMS Mathematics

2019, Volume 4, Issue 6: 1745-1767. doi: 10.3934/math.2019.6.1745

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Multigrid method for pricing European options under the CGMY process

Justin W. L. Wan ^,

David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada

Received: 30 September 2018 Accepted: 15 August 2019 Published: 27 December 2019
MSC : 65F10, 65M55

We propose a fast multigrid method for solving the discrete partial integro-differential equations (PIDEs) arising from pricing European options when the underlying asset is driven by an infinite activity Lévy process. We consider the CGMY model whose kernel singularity gets worse when the parameter Y approaches two. Due to the integral term, the discretization matrix is dense. In order to obtain an efficient multigrid method, we apply a fixed point iteration as a smoother for multigrid. In each smoothing step, we only need to solve a sparse matrix corresponding to the differential operator and compute a matrix-vector product involving the integral operator by a fast Fourier transform (FFT). We prove that the fixed point iteration smoother is effective reducing the high frequency components. Moreover, we also prove a two-grid convergence of the multigrid method by a local mode analysis. We demonstrate the effectiveness of the multigrid method by solving the option pricing equation under the CGMY model with finite and infinite variation processes.
- multigrid,
- two-grid analysis,
- Black-Scholes equation,
- CGMY model,
- PIDE
Citation: Justin W. L. Wan. Multigrid method for pricing European options under the CGMY process[J]. AIMS Mathematics, 2019, 4(6): 1745-1767. doi: 10.3934/math.2019.6.1745

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Abstract

We propose a fast multigrid method for solving the discrete partial integro-differential equations (PIDEs) arising from pricing European options when the underlying asset is driven by an infinite activity Lévy process. We consider the CGMY model whose kernel singularity gets worse when the parameter Y approaches two. Due to the integral term, the discretization matrix is dense. In order to obtain an efficient multigrid method, we apply a fixed point iteration as a smoother for multigrid. In each smoothing step, we only need to solve a sparse matrix corresponding to the differential operator and compute a matrix-vector product involving the integral operator by a fast Fourier transform (FFT). We prove that the fixed point iteration smoother is effective reducing the high frequency components. Moreover, we also prove a two-grid convergence of the multigrid method by a local mode analysis. We demonstrate the effectiveness of the multigrid method by solving the option pricing equation under the CGMY model with finite and infinite variation processes.

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