Fast Fourier Transform for Discontinuous Functions

Fan, Guo‐Xin; Liu, Qing

doi:10.1109/tap.2004.823965

Cited by 27 publications

(22 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Though other values of α are allowed in principle, we need to know the payoff-transform itself in order to apply Cauchy's residue theorem, see [32,30,33]. This restriction on α will disappear when we switch to a discretised version of (16) in the next section. The Fourier transform of the damped continuation value can thus be calculated as the product of two functions, one of which, the extended characteristic function, is readily available in exponential Lévy models.…”

Section: (T S(t)) = S(t)mentioning

confidence: 99%

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

Lord¹,

Fang

Bervoets³

et al. 2007

SSRN Journal

View full text Add to dashboard Cite

Abstract. A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an M -times exercisable Bermudan option, the overall complexity is O(M N log 2 (N )) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.

show abstract

Section: (T S(t)) = S(t)mentioning

confidence: 99%

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

Lord¹,

Fang

Bervoets³

et al. 2007

SSRN Journal

View full text Add to dashboard Cite

show abstract

“…In practice, however, many functions to be transformed are discontinuous across the boundary of an irregular area. For example, in volume integral equation solvers in electromagnetics, some components of the unknown electric current density fields to be transformed are discontinuous across the material interfaces, which in general have arbitrary shapes; another example is the analysis of radiation patterns of reflector antennas and planar near-field to far-field transformation, where the Fourier transform integral of spatially limited functions with discontinuities at the boundary has to be evaluated [11,12]. For this kind of functions, however, there usually exist significant stair-casing errors due to the uniform Cartesian orthogonal grid required by the traditional 2D FFT algorithm, and the accuracy is limited since FFT is based on the trapezoidal quadrature scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Some works have been done to improve the accuracy for onedimension (1D) piecewise smooth functions [1,11,[13][14][15][16][17]. Direct extension of these algorithms to high dimensions still requires that the area is meshed into a Cartesian orthogonal grid [11], which is not flexible for an arbitrary boundary shape.…”

Section: Introductionmentioning

confidence: 99%

“…Direct extension of these algorithms to high dimensions still requires that the area is meshed into a Cartesian orthogonal grid [11], which is not flexible for an arbitrary boundary shape.…”

Section: Introductionmentioning

confidence: 99%

“…A fast algorithm for evaluating the Fourier transform of 2D discontinuous functions has been proposed in [12] as an extension of [11] to allow an arbitrary boundary shape. This method adopts a triangular mesh to conform with the arbitrary shape of the discontinuity boundary, the Gaussian quadrature to increase the accuracy, and 2D nonuniform fast Fourier transform (NUFFT) [18][19][20][21] to provide a computational complexity comparable with the traditional FFT algorithm.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Accurate Conformal Fourier Transform Method for 2d Discontinuous Functions

Zhu¹,

Liu

Shen³

et al. 2011

PIER

View full text Add to dashboard Cite

Abstract-Fourier transform of discontinuous functions are often encountered in computational electromagnetics. A highly accurate, fast conformal Fourier transform (CFT) algorithm is proposed to evaluate the finite Fourier transform of 2D discontinuous functions. A curved triangular mesh combined with curvilinear coordinate transformation is adopted to flexibly model an arbitrary shape of the discontinuity boundary. This enables us to take full advantages of high order interpolation and Gaussian quadrature methods to achieve highly accurate Fourier integration results with a low sampling density and small computation time. The complexity of the proposed algorithm is similar to the traditional 2D fast Fourier transform algorithm, but with orders of magnitude higher accuracy. Numerical examples illustrate the excellent performance of the proposed CFT method.

show abstract

Fast Fourier Transforms and Nufft

Liu

2005

Encyclopedia of RF and Microwave Engineering

View full text Add to dashboard Cite

The fast Fourier transform (FFT) algorithm is one of most fundamental and powerful computational tools in computational science and engineering. In this article, the development of the fast Fourier transform algorithm is reviewed, some representative applications in electromagnetics are illustrated, and some recent developments in extending the FFT algorithms to nonuniformly sampled data and to discontinuous functions are summarized.

show abstract

Fast Fourier Transform for Discontinuous Functions

Cited by 27 publications

References 11 publications

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

An Accurate Conformal Fourier Transform Method for 2d Discontinuous Functions

Fast Fourier Transforms and Nufft

Contact Info

Product

Resources

About