Some integrals involving Legendre polynomials providing combinatorial identities

Klemm, Anthony D.; Larsen, Sigurd Yves

doi:10.1017/s0334270000006883

Cited by 3 publications

(2 citation statements)

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“…We provide an analytical formula in proposition (7.1). Its computation is straightforward for small values of n. For n >> 1 several accuracy and stability issues arise because of rapid accumulation of round-off errors [34,35].…”

Section: Option Pricingmentioning

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: Option Pricingmentioning

confidence: 99%

“…M large does not have any implementation difficulty. However, for N >> 1, the computation of V n using (96) introduces instability and inaccuracy issues because of cancellations [35]. The objective in this section is to provide an alternative computational procedure for V n with machine accuracy.…”

Section: Alternate Computational Proceduresmentioning

confidence: 99%