Fast Numerical Contour Integral Method for Fractional Diffusion Equations

Pang, Hong‐Kui; Sun, Hai‐Wei

doi:10.1007/s10915-015-0012-9

Cited by 35 publications

(20 citation statements)

References 42 publications

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“…, − 1, is given by (57). We find the parameters {ℓ }, = 1, 2 such that the condition (24) holds for all real values of > 0. A practicable way given by Zhou, Ma, and Sun [16] is to consider a sequence of 1 , 2 , .…”

Section: Theoremmentioning

confidence: 99%

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Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

Chen

2018

Discrete Dynamics in Nature and Society

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The aim of this paper is to study the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options pricing. Firstly, the governing free-boundary PDEs are modified as a unified form of PDEs on the fixed space region; then performing Laplace-Carson transform (LCT) leads to ordinary differential equations (ODEs) which involve the unknown inverse functions of free boundaries. Secondly, the inverse free-boundary functions are approximated and optimized by solving of the free-boundary values of the perpetual American volatility options. Finally, the ODEs are solved by the finite difference methods (FDMs), and the results are restored via the numerical Laplace inversion. Numerical results confirm that the NCIMs outperform the FDMs for solving free-boundary PDEs in regard to the accuracy and computational time.

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Section: Theoremmentioning

confidence: 99%

“…To analyze the spectrum of the coefficient matrix (61), we follow the idea in [24] and construct a Toeplitz matrix A to replace analyzing (61)…”

Section: Theoremmentioning

confidence: 99%

Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

Chen

2018

Discrete Dynamics in Nature and Society

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“…From the above generating function ( ) and the discussion of Pang and Sun [24], we know the spectrum Λ(B) lies the sectorial region Σ for any values of ∈ (0, /2). Since D is positive definite, the spectrum Λ(Ã) = Λ(DB) also lies in the same sectorial region Σ (see [24,Lemma 3.5]).…”

Section: Theorem 2 (I)mentioning

confidence: 99%

“…This is the vector version of Laplace inversion (41). Weideman et al [25] (also see Pang and Sun [24]) suggest to select the hyperbolic contour Γ, which can be parameterized by…”

Section: [K ( )]mentioning

confidence: 99%

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Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries

Zhou

2018

Discrete Dynamics in Nature and Society

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Laplace transform method (LTM) has a lot of applications in the evaluation of European-style options and exotic options without early exercise features. However the Laplace transform methods for pricing American options have unsatisfactory accuracy and suffer from the instability. The aim of this paper is to develop a Laplace transform method for pricing American Strangles options with the underlying asset price following the constant elasticity volatility (CEV) models. By approximating the free boundaries, the Laplace transform is taken on a fixed space region to replace the moving boundaries space. After solving the linear system in Laplace space, Gaver-Stehfest formula (GSF) and hyperbola contour integral method (HCIM) are applied to compute the Laplace inversion. Numerical results show that the LTM-HCIM outperform the LTM-GSF in regard to the accuracy and stability for the option values.

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Preconditioners for fractional diffusion equations based on the spectral symbol

Barakitis

Ekström

Vassalos

2022

Numerical Linear Algebra App

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It is well known that the discretization of fractional diffusion equations with fractional derivatives 𝛼 ∈ (1, 2), using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function  𝛼 , called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one-and two-dimensional cases. In this article we propose a new preconditioner denoted by   𝛼 which belongs to the 𝜏-algebra and it is based on the spectral symbol  𝛼 . Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one-dimensional case, the new preconditioner performs better in the more challenging multi-dimensional setting.

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Fast Numerical Contour Integral Method for Fractional Diffusion Equations

Cited by 35 publications

References 42 publications

Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries

Preconditioners for fractional diffusion equations based on the spectral symbol

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