### basis functions and the generalized

"... Multi-dimensional option pricing using radial ..."

(Show Context)
### Evaluation Of Financial Options using Radial Basis Functions

"... A radial basis function �RBF � is a function Φ�x, xi � which depends only on the distance r between x ∈ � d and a fixed point xi ∈ � d. Φ�x, xi � ⩵ Φ � � x � xi �� Each function Φ�x, xi � is radially symmetric about the center xi. Since their discovery in the early 1970’�s ..."

Abstract
- Add to MetaCart

A radial basis function �RBF � is a function Φ�x, xi � which depends only on the distance r between x ∈ � d and a fixed point xi ∈ � d. Φ�x, xi � ⩵ Φ � � x � xi �� Each function Φ�x, xi � is radially symmetric about the center xi. Since their discovery in the early 1970’�s

### On Unsymmetric Collocation by Radial Basis Functions

- Appl. Math. Comput

"... . Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by E. Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The ..."

Abstract
- Add to MetaCart

. Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by E. Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system still was missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort. 1 Introduction A large variety of numerical techniques can be formulated as generalized interpolation problems on spaces of multivariate functions. An easy special case is provided by collocation methods. These use an N-dimensional space S of functions and N functionals 1 ; : : : ; ::: N . The space S is spanned by functions f 1 ; : : : ; f N , and then one looks for a func...

### Option

"... Abstract: In this paper, we have implemented a radial basis function (RBF) based method for solving the Black–Scholes partial differential equation. The application we have cho-sen is the valuation of European call options based on several underlying assets. We have shown that by appropriate choices ..."

Abstract
- Add to MetaCart

(Show Context)
Abstract: In this paper, we have implemented a radial basis function (RBF) based method for solving the Black–Scholes partial differential equation. The application we have cho-sen is the valuation of European call options based on several underlying assets. We have shown that by appropriate choices of the RBF shape parameter and the node point place-ment, the accuracy of the results can be improved by at least an order of magnitude. We have also looked at how and where to implement boundary conditions in more than one dimension.

### unknown title

"... parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs ..."

Abstract
- Add to MetaCart

(Show Context)
parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs

### the analysis of homogeneous Helmholtz, modified Helmholtz

"... Numerical investigation on convergence of boundary knot method in ..."

(Show Context)
### A Meshless, Integration-Free, and Boundary-Only RBF Technique

"... wencQifi.uio.no Abstract-Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation sy ..."

Abstract
- Add to MetaCart

(Show Context)
wencQifi.uio.no Abstract-Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete