Estimates of Upper Bounds for Quadrature Errors

Donaldson, J. D.

doi:10.1137/0710003

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…Several approaches exist for estimating the error E Q N (f ) when f is holomorphic: (i) methods of contour integration [50,18], (ii) methods based on Hilbert space norm estimates [15,17] which consider E Q N as a linear functional on f , and (iii) methods based on the approximation theory [2,52]. Independent of the approach, the results are often comparable because the error bounds involve the supremum norm of f .…”

Section: Characterization Of the Optimal Damping Rulementioning

confidence: 99%

“…In the finite case, the contour C is closed and (3.3) represents an analytic function in the connected domain C \ [a, b] while we may take C to lie on the upper and lower edges of the real axis in the infinite case for large |x|. Discussions on choosing adequate contours are found in [22,18,17]. Moreover, precise estimates of H n (z) were derived in [18,23].…”

Section: Characterization Of the Optimal Damping Rulementioning

confidence: 99%

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Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

Bayer¹,

Hammouda²,

Papapantoleon³

et al. 2022

Preprint

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Efficient pricing of multi-asset options is a challenging problem in quantitative finance. When the Fourier transform of the density function is available, Fourier-based pricing methods become very competitive compared to alternative techniques because the integrand in the frequency space has often higher regularity than in the physical space. However, when designing a numerical quadrature method for most of these Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of the damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of the high dimensionality of the integration problem. To address these challenges, based on the extension of the one-dimensional Fourier valuation formula to the multivariate case, we propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of damping parameters based on a proposed heuristic optimization rule. Second, we use the adaptive sparse grid quadrature based on sparsification and dimension-adaptivity techniques to accelerate the convergence of the numerical quadrature in high dimensions. Through an extensive numerical study on the basket and rainbow options under the multivariate geometric Brownian motion and some multivariate Lévy models, we demonstrate the advantages of adaptivity and our damping parameter rule on the numerical complexity of the quadrature methods. Moreover, we reveal that our approach achieves substantial computational gains compared to the Monte Carlo method for different dimensions and parameter constellations.

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Section: Characterization Of the Optimal Damping Rulementioning

confidence: 99%

Section: Characterization Of the Optimal Damping Rulementioning

confidence: 99%

Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

Bayer¹,

Hammouda²,

Papapantoleon³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The asymptotic forms of the remainder for Gaussian quadratures has been found to be extremely useful in obtaining estimates and upper bounds for the remainder term, see Donaldson and Elliott [7], Donaldson [8] and Chawla and Jain [9]. For this purpose the form obtained by Barrett [3] for the Newton-Cotes formula has to be improved.…”

Section: Introductionmentioning

confidence: 99%