Parametric integration by magic point empirical interpolation

Gaß, Maximilian; Glau, Kathrin

doi:10.1093/imanum/drx072

Cited by 3 publications

(5 citation statements)

References 20 publications

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“…Now, since u := f − Π X f vanishes on X, the sampling inequality of Theorem 10 and the norm equivalence (11) give…”

Section: Convergence For Uniform Points and Translational Invariant K...mentioning

confidence: 99%

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Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces

Santin,

Karvonen,

Haasdonk

2020

Preprint

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In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals, including integration functionals.For a large class of functionals, we prove convergence results for the approximation by means of uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed.Beyond the theoretical investigations, we demonstrate numerically that our algorithm is effective in treating various integration densities, and that it is even very competitive when compared to existing methods for Uncertainty Quantification.

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“…Now, since u := f − Π X f vanishes on X, the sampling inequality of Theorem 10 and the norm equivalence (11) give…”

Section: Convergence For Uniform Points and Translational Invariant K...mentioning

confidence: 99%

“…We mention also that other data-based algorithms to approximate linear functionals exists in different settings, and are actively investigated e.g. in the setting of empirical interpolation and reduced order modelling [?, 43,11,1].…”

Section: Introductionmentioning

confidence: 99%

Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces

Santin,

Karvonen,

Haasdonk

2020

Preprint

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“…The proof is provided in [18]. In view of a self-contained presentation we present the proof in detail in Appendix B.…”

Section: Theorem 42mentioning

confidence: 99%

“…The proof is provided in [18]. Compared to the proof of Theorem 4.2 in Appendix B, the only difference is that now the analyticity properties of H 1 with respect to the parameters p are exploited to derive an estimate for the best n-term approximation of U.…”

Section: Theorem 43mentioning

confidence: 99%

“…(iii) In view of the representation of the option prices Price K,T,q as parametric Fourier integrals in (8), we use the magic point integration algorithm to approximate parametric Fourier transforms that we call MagicFT, as introduced in [18]. From perspective (i), magic point integration for parametric option pricing is an alternative to standard quadrature rules.…”

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confidence: 99%

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Magic Points in Finance: Empirical Integration for Parametric Option Pricing

Gaß¹,

Glau²,

Mair³

2017

SIAM J. Finan. Math.

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Abstract. We propose an offline-online procedure for Fourier transform based option pricing. The method supports the acceleration of such essential tasks of mathematical finance as model calibration, realtime pricing, and, more generally, risk assessment and parameter risk estimation. We adapt the empirical magic point interpolation method of Barrault et al. [C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667-672] to parametric Fourier pricing. In the offline phase, a quadrature rule is tailored to the family of integrands of the parametric pricing problem. In the online phase, the quadrature rule then yields fast and accurate approximations of the option prices. Under analyticity assumptions, the pricing error decays exponentially. Numerical experiments in one dimension confirm our theoretical findings and show a significant gain in efficiency, even for examples beyond the scope of the theoretical results.Key words. parametric integration, Fourier pricing, magic point interpolation, empirical interpolation, offlineonline decomposition, calibration, affine processes, Fourier transform, sparse integration AMS subject classifications. 68Q25, 68R10, 65T99, 91G60DOI. 10.1137/16M11013011. Introduction. Most of the option pricing methods based on Fourier transforms are aimed at the evaluation of individual option prices. For this scenario case, existing pricing tools have achieved impressive performance. For real-time applications and those involving repeated evaluations, particularly fast run-times are crucial. Therefore, Fast Fourier transforms (FFTs) have become highly popular in order to reduce computational complexity when prices are required simultaneously for a large set of different strikes, following the seminal works of [8] and [36]. See also the monograph [5]. In this paper we shift the focus from the pricing problem for one strike or several strikes to the full parametric option pricing problem, considering all parameters such as strike, maturity, and model parameters.For a given model and option type, various applications require the evaluation of Fourier pricing routines repeatedly for different parameter constellations. We mention three of those applications: First, during calibration of financial models with Fourier methods, the optimization relies on multiple evaluations of a Fourier integral for varying model and option parameters. In particular, this is the case in modern approaches to model calibration that take into account parameter uncertainty such as are proposed in [25] and [22], as well as in [30] where, additionally, a consistent variation of parameters over time is considered. Second, each (intra-)day recalibration leads to new model parameters for which several option prices and

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Sampling based approximation of linear functionals in reproducing kernel Hilbert spaces

2021

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Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces.In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective, and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation.In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the P -greedy target-data-independent selection rule, and can additionally be proven to be optimal when they fully exploit adaptivity (f -greedy).Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in Reproducing Kernel Hilbert Spaces, as they allow us to compare adaptive interpolation with non-adaptive best nonlinear approximation.

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Parametric integration by magic point empirical interpolation

Cited by 3 publications

References 20 publications

Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces

Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces

Magic Points in Finance: Empirical Integration for Parametric Option Pricing

Sampling based approximation of linear functionals in reproducing kernel Hilbert spaces

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