Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

FX Smile in the Heston Model

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Heston model; vanilla option; stochastic volatility; Monte Carlo simulation; Feller condition; option pricing with FFT;

Error Estimates for Certain Cubature Formulae

We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule (G) over cap (2l+1) is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with G(l). The advantages of (G) over cap (2l+1) are that it exists also when H2l+1 does not, and that the numerical construction of (G) over cap (2l+1), based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1

Stieltjes-type polynomials on the unit circle

29 pages, no figures.-- MSC2000 codes: Primary 65D32, 42A10, 42C05; Secondary 30E20.MR#: MR2476567Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the Gauss-Kronrod rule, provided that the integrand is analytic in a neighborhood of T.The work of B. de la Calle received support from Dirección General de Investigación (DGI), Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and from UPM-CAM under grants CCG07-UPM/000-1652 and CCG07-UPM/ESP-1896. The work of G. López was supported by DGI under grant MTM2006-13000-C03-02 and by UC3M-CAM through CCG06-UC3M/ESP-0690. The work of L. Reichel was supported by an OBR Research Challenge Grant.Publicad

Non-Intrusive, High-Dimensional Uncertainty Quantification for the Robust Simulation of Fluid Flows

Uncertainty Quantification is the field of mathematics that focuses on the propagation and influence of uncertainties on models. Mostly complex numerical models are considered with uncertain parameters or uncertain model properties. Several methods exist to model the uncertain parameters of numerical models. Stochastic Collocation is a method that samples the random variables of the input parameters using a deterministic procedure and then interpolates or integrates the unknown quantity of interest using the samples. Because moments of the distribution of the unknown quantity are essentially integrals of the quantity, the main focus will be on calculating integrals. Calculating an integral using samples can be done efficiently using a quadrature or cubature rule. Both sample the space of integration in a deterministic way and several algorithms to determine the samples exist, each with its own advantages and disadvantages. In the one-dimensional case a method is proposed that has all relevant advantages (positive weights, nested points and dependency on the input distribution). The principle of the introduced quadrature rule can also be applied to a multi-dimensional setting. However, if negative weights are allowed in the multi-dimensional case a cubature rule can be generated that has a very small number of points compared to the methods described in literature. The new method uses the fact that the tensor product of several quadrature rules has many points with the same weight that can be considered as on

Pricing contingent claims on credit and carbon single and multiple underlying assets

This thesis proposes alternative ways to price contingent claims written on portfolios of credit instruments as well as on carbon underlying assets. On the first topic of this research we tackle the pricing of Collateralized Debt Obligations (CDOs) by introducing two different approaches through the application of respectively Johnson SB distributions and entropy optimization principles, in contrast to market standard pricing approaches based on variations of the Gaussian copula model. The relevance of this topic is in line with the events that unfolded during the “credit crunch” of mid-2007 to early 2009, when CDOs made headlines as being responsible for more than $542 billion in losses through writedowns by financial institutions. On the second topic we propose a pricing methodology for Emission Reduction Purchase Agreement (ERPA) contracts. These are instruments based on carbon as an asset class and created by the emergence of an international carbon market that followed the adoption of the Kyoto Protocol (KP) to the United Nations Framework Convention on Climate Change (UNFCCC) in December 1997. ERPAs are of vital importance to the function of KP’s market mechanisms and the carbon markets at large as they formalize transactions of emissions reduction offsets between sellers and buyers, more specifically transactions involving Certified Emission Reductions (CERs). We propose a pricing methodology based on stochastic modeling of CER volume delivery risk and carbon prices as the two main drivers underlying ERPAs, and apply it to a case study on a run-of-river hydro power CDM project activity in China