Taylor Series are Limits of Legendre Expansions

Fishback, Paul E.

doi:10.35834/mjms/1316092234

Cited by 2 publications

(3 citation statements)

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“…The Fourier transform for Chebyshev polynomials does not have a simple closed form and requires some numerical approximations (see discussion in [14]). Moreover the experiments show this formula is numerically stable for large n. Generally, the classical Legendre series offers the simplest method of representing a function using polynomial expansion means [15]. Also the recent analysis by Cohen and Tan [16] shows Legendre polynomial approximation yields an error at least an order of magnitude smaller than the analogous Taylor series approximation and the authors strongly suggest that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.…”

Section: Introductionmentioning

confidence: 94%

“…Legendre polynomial offers tractability property allowing to compute analytically many quantities of interests. For example, Legendre polynomial has an analytical formula for its Fourier transform as in (15), which is instrumental and used to recover the coefficients A n in the series expansion of the density function (26). The Fourier transform for Chebyshev polynomials does not have a simple closed form and requires some numerical approximations (see discussion in [14]).…”

Section: Introductionmentioning

confidence: 99%

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Option pricing with Legendre polynomials

Hok¹,

Chan

2017

Journal of Computational and Applied Mathematics

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Here we develop an option pricing method based on Legendre series expansion of the density function.The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in proposition 7.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Option pricing with Legendre polynomials

Hok¹,

Chan

2017

Journal of Computational and Applied Mathematics

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“…Then an open disk around x of radius less than the convergence radius of (3.20) is in the interior of the ellipse of convergence (3.21) for δ small enough. This was discussed for Legendre polynomials by Fishback [20].…”

Section: Generalized Taylor Seriesmentioning

confidence: 99%

Differentiation by integration using orthogonal polynomials, a survey

Diekema¹,

Koornwinder

2012

Journal of Approximation Theory

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This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added; accepted by J. Approx. Theor

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Taylor Series are Limits of Legendre Expansions

Cited by 2 publications

References 3 publications

Option pricing with Legendre polynomials

Option pricing with Legendre polynomials

Differentiation by integration using orthogonal polynomials, a survey

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