The Experts below are selected from a list of 60909 Experts worldwide ranked by ideXlab platform
Jérôme Lelong - One of the best experts on this subject based on the ideXlab platform.
-
Pricing Parisian options using Laplace Transforms
Bankers Markets & Investors : an academic & professional review, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this work, we propose to price Parisian options using Laplace Transforms. Not only do we compute the Laplace Transforms of all the different Parisian options, but we also explain how to invert them numerically. We prove the accuracy of the numerical inversion.
-
Pricing double barrier Parisian options using Laplace Transforms
International Journal of Theoretical and Applied Finance, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace Transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class C infinity. This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times.
-
pricing double barrier parisian options using Laplace Transforms
International Journal of Theoretical and Applied Finance, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace Transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class $\mathcal{C}^\infty$ (see Theorem 5.1). This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times (see Theorem 5.3).
Céline Labart - One of the best experts on this subject based on the ideXlab platform.
-
Pricing Parisian options using Laplace Transforms
Bankers Markets & Investors : an academic & professional review, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this work, we propose to price Parisian options using Laplace Transforms. Not only do we compute the Laplace Transforms of all the different Parisian options, but we also explain how to invert them numerically. We prove the accuracy of the numerical inversion.
-
Pricing double barrier Parisian options using Laplace Transforms
International Journal of Theoretical and Applied Finance, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace Transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class C infinity. This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times.
-
pricing double barrier parisian options using Laplace Transforms
International Journal of Theoretical and Applied Finance, 2009Co-Authors: Céline Labart, Jérôme LelongAbstract:In this article, we study a double barrier version of the standard Parisian options. We give closed formulas for the Laplace Transforms of their prices with respect to the maturity time. We explain how to invert them numerically and prove a result on the accuracy of the numerical inversion when the function to be recovered is sufficiently smooth. Henceforth, we study the regularity of the Parisian option prices with respect to maturity time and prove that except for particular values of the barriers, the prices are of class $\mathcal{C}^\infty$ (see Theorem 5.1). This study heavily relies on the existence of a density for the Parisian times, so we have deeply investigated the existence and the regularity of the density for the Parisian times (see Theorem 5.3).
Ward Whitt - One of the best experts on this subject based on the ideXlab platform.
-
power algorithms for inverting Laplace Transforms
Informs Journal on Computing, 2007Co-Authors: Efstathios Avdis, Ward WhittAbstract:This paper investigates ways to create algorithms to invert Laplace Transforms numerically within a unified framework proposed by Abate and Whitt (2006). That framework approximates the desired function value by a finite linear combination of transform values, depending on parameters called weights and nodes, which are initially left unspecified. Alternative parameter sets, and thus algorithms, are generated and evaluated here by considering power test functions. Real weights for a real-variable power algorithm are found for specified real powers and positive real nodes by solving a system of linear equations involving a generalized Vandermonde matrix, using Mathematica. The resulting power algorithms are shown to be effective, with the parameter choice being tunable to the transform being inverted. The powers can be advantageously chosen from series expansions of the transform. Experiments show that the power algorithms are robust in the nodes; it suffices to use the first n positive integers. The power test functions also provide a useful way to evaluate the performance of other algorithms.
-
a unified framework for numerically inverting Laplace Transforms
Informs Journal on Computing, 2006Co-Authors: Joseph Abate, Ward WhittAbstract:We introduce and investigate a framework for constructing algorithms to invert Laplace Transforms numerically. Given a Laplace transform \hat{f} of a complex-valued function of a nonnegative real-variable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) \approx f_n (t) \equiv \frac{1}{t} \sum_{k = 0}^{n}\omega_{k}\hat{f}\biggl(\frac{\alpha_{k}}{t}\biggr),\quad 0 where the weights ωk and nodes αk are complex numbers, which depend on n, but do not depend on the transform \hat{f} or the time argument t. Many different algorithms can be put into this framework, because it remains to specify the weights and nodes. We examine three one-dimensional inversion routines in this framework: the Gaver-Stehfest algorithm, a version of the Fourier-series method with Euler summation, and a version of the Talbot algorithm, which is based on deforming the contour in the Bromwich inversion integral. We show that these three building blocks can be combined to produce different algorithms for numerically inverting two-dimensional Laplace Transforms, again all depending on the single parameter n. We show that it can be advantageous to use different one-dimensional algorithms in the inner and outer loops.
-
computing Laplace Transforms for numerical inversion via continued fractions
Informs Journal on Computing, 1999Co-Authors: Joseph Abate, Ward WhittAbstract:It is often possible to effectively calculate probability density functions (pdf's) and cumulative distribution functions (cdf's) by numerically inverting Laplace Transforms. However, to do so it is necessary to compute the Laplace transform values. Unfortunately, convenient explicit expressions for required Transforms are often unavailable for component pdf's in a probability model. In that event, we show that it is sometimes possible to find continued-fraction representations for required Laplace Transforms that can serve as a basis for computing the transform values needed in the inversion algorithm. This property is very likely to prevail for completely monotone pdf's, because their Laplace Transforms have special continued fractions called S fractions, which have desirable convergence properties. We illustrate the approach by considering applications to compute first-passage-time cdfs in birth-and-death processes and various cdf's with non-exponential tails, which can be used to model service-time cdf's in queueing models. Included among these cdf's is the Pareto cdf.
-
numerical inversion of multidimensional Laplace Transforms by the laguerre method
Performance Evaluation, 1998Co-Authors: Joseph Abate, Gagan L Choudhury, Ward WhittAbstract:Abstract Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace Transforms to multidimensional Laplace Transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ϵ-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.
-
on the laguerre method for numerically inverting Laplace Transforms
Informs Journal on Computing, 1996Co-Authors: Joseph Abate, Gagan L Choudhury, Ward WhittAbstract:The Laguerre method for numerically inverting Laplace Transforms is an old established method based on the 1935 Tricomi--Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (i) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (ii) developing systematic methods for scaling, and (iii) using Wynn's (epsilon)-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of Transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux's theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples.
Jasson Vindas - One of the best experts on this subject based on the ideXlab platform.
-
a multidimensional tauberian theorem for Laplace Transforms of ultradistributions
Integral Transforms and Special Functions, 2020Co-Authors: Lenny Neyt, Jasson VindasAbstract:We obtain a multidimensional Tauberian theorem for Laplace Transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces...
-
complex tauberian theorems for Laplace Transforms with local pseudofunction boundary behavior
Journal D Analyse Mathematique, 2019Co-Authors: Gregory Debruyne, Jasson VindasAbstract:We provide several Tauberian theorems for Laplace Transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace Transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.
-
a multidimensional tauberian theorem for Laplace Transforms of ultradistributions
arXiv: Functional Analysis, 2019Co-Authors: Lenny Neyt, Jasson VindasAbstract:We obtain a multidimensional Tauberian theorem for Laplace Transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a convex acute cone of $\mathbb{R}^{n}$, also established here.
-
optimal tauberian constant in ingham s theorem for Laplace Transforms
Israel Journal of Mathematics, 2018Co-Authors: Gregory Debruyne, Jasson VindasAbstract:It is well known that there is an absolute constant $$\mathfrak{C}$$ > 0 such that if the Laplace transform $$G(s) = \int_0^\infty {\rho (x)} {e^{ - sx}}dx$$ of a bounded function ρ has analytic continuation through every point of the segment (−iλ, iλ) of the imaginary axis, then $$G(s) = |\int_0^\infty {\rho (\mu )} du - G(0)| \leqslant \frac{{\text{C}}}{\lambda }\mathop {limsup}\limits_{x \to \infty } |\rho (x)|$$ The best known value of the constant $$\mathfrak{C}$$ was so far $$\mathfrak{C}$$ = 2. In this article we show that the inequality holds with $$\mathfrak{C}$$ = π/2 and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace Transforms.
Joseph Abate - One of the best experts on this subject based on the ideXlab platform.
-
a unified framework for numerically inverting Laplace Transforms
Informs Journal on Computing, 2006Co-Authors: Joseph Abate, Ward WhittAbstract:We introduce and investigate a framework for constructing algorithms to invert Laplace Transforms numerically. Given a Laplace transform \hat{f} of a complex-valued function of a nonnegative real-variable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) \approx f_n (t) \equiv \frac{1}{t} \sum_{k = 0}^{n}\omega_{k}\hat{f}\biggl(\frac{\alpha_{k}}{t}\biggr),\quad 0 where the weights ωk and nodes αk are complex numbers, which depend on n, but do not depend on the transform \hat{f} or the time argument t. Many different algorithms can be put into this framework, because it remains to specify the weights and nodes. We examine three one-dimensional inversion routines in this framework: the Gaver-Stehfest algorithm, a version of the Fourier-series method with Euler summation, and a version of the Talbot algorithm, which is based on deforming the contour in the Bromwich inversion integral. We show that these three building blocks can be combined to produce different algorithms for numerically inverting two-dimensional Laplace Transforms, again all depending on the single parameter n. We show that it can be advantageous to use different one-dimensional algorithms in the inner and outer loops.
-
computing Laplace Transforms for numerical inversion via continued fractions
Informs Journal on Computing, 1999Co-Authors: Joseph Abate, Ward WhittAbstract:It is often possible to effectively calculate probability density functions (pdf's) and cumulative distribution functions (cdf's) by numerically inverting Laplace Transforms. However, to do so it is necessary to compute the Laplace transform values. Unfortunately, convenient explicit expressions for required Transforms are often unavailable for component pdf's in a probability model. In that event, we show that it is sometimes possible to find continued-fraction representations for required Laplace Transforms that can serve as a basis for computing the transform values needed in the inversion algorithm. This property is very likely to prevail for completely monotone pdf's, because their Laplace Transforms have special continued fractions called S fractions, which have desirable convergence properties. We illustrate the approach by considering applications to compute first-passage-time cdfs in birth-and-death processes and various cdf's with non-exponential tails, which can be used to model service-time cdf's in queueing models. Included among these cdf's is the Pareto cdf.
-
numerical inversion of multidimensional Laplace Transforms by the laguerre method
Performance Evaluation, 1998Co-Authors: Joseph Abate, Gagan L Choudhury, Ward WhittAbstract:Abstract Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace Transforms to multidimensional Laplace Transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ϵ-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.
-
on the laguerre method for numerically inverting Laplace Transforms
Informs Journal on Computing, 1996Co-Authors: Joseph Abate, Gagan L Choudhury, Ward WhittAbstract:The Laguerre method for numerically inverting Laplace Transforms is an old established method based on the 1935 Tricomi--Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (i) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (ii) developing systematic methods for scaling, and (iii) using Wynn's (epsilon)-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of Transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux's theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples.
-
numerical inversion of Laplace Transforms of probability distributions
Informs Journal on Computing, 1995Co-Authors: Joseph Abate, Ward WhittAbstract:We present a simple algorithm for numerically inverting Laplace Transforms. The algorithm is designed especially for probability cumulative distribution functions, but it applies to other functions as well. Since it does not seem possible to provide effective methods with simple general error bounds, we simultaneously use two different methods to confirm the accuracy. Both methods are variants of the Fourier-series method. The first, building on Dubner and Abate (Dubner, H., J. Abate. 1968. Numerical inversion of Laplace Transforms by relating them to the finite Fourier cosine transform. JACM 15 115–123.) and Simon, Stroot, and Weiss (Simon, R. M., M. T. Stroot, G. H. Weiss. 1972. Numerical inversion of Laplace Transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6 596–607.), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. L. 1978. An inversion technique for the Laplace transform with applications....
