A RBF based finite difference method for option pricing under regime-switching jump-diffusion model

Abstract

In this article, we presented a method for option pricing problem under regime-switching jump-diffusion models. We have proposed a numerical method for solving a partial integro-differential equation (PIDE) for pricing European option and for solving linear complementarity problem (LCP), to evaluate the price of American options. We use implicit explicit method for time semi discretization, followed by radial basis function based finite difference (RBF-FD) method for spatial discretization to solve PIDE. The proposed method is further extended to solve the LCP by coupling it with operator splitting method. Numerical simulation is done for European and American option to demonstrate efficiency and accuracy of the proposed method.     

A method for calculating meshless finite difference weights

An exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented. The method avoids the numerical ill conditioning associated with a small‐scale factor (ε) by delaying the limiting of ε → 0. The Gaussian Radial Basis function is approximated by a Taylor's series expansion in the variable ε, and the resultant matrix equation is solved using the Fraction Free LU decomposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

A nonstandard finite difference scheme for a strain-gradient theory

Strain-gradient theories have proved very useful in describing computational aspects of phase transforming materials such as shape memory alloys. A significant feature implicit to these theories is a relation between the driving force acting on a phase boundary and its velocity. Numerical calculation of the kinetic relation using standard finite difference methods shows significant quantitative and qualitative departures from the analytical kinetic relation. In this paper we derive a nonstandard finite difference scheme and show that the kinetic relation evaluated using this scheme displays the correct qualitative behaviour and matches the analytical solution significantly better quantitatively. In particular, the nonstandard finite difference scheme eliminates spurious lattice trapping in the kinetic relation.     

A generalized finite difference scheme for convection-dominated metal-forming problems

A simple but versatile numerical technique using generalized finite difference discretization has been developed for heat transfer problems involving high convective heat flow, irregular geometry and high local thermal gradients. Upwind differencing is utilized to stabilize the numerical oscillations often induced in convection-dominated heat transfer problems. An arbitrary irregular mesh scheme is adopted to treat the irregular geometry and to achieve high accuracy in zones having high thermal gradients. To demonstrate the validity of the formulation procedure, results predicted from the present scheme are compared with the analytical solution for a problem having a regular boundary. Application to a typical metal-forming process having curved boundaries is then included.     

A finite difference calculation procedure for three-dimensional turbulent separated flows

A general finite difference scheme has been proposed along with a three-dimensional co-ordinate transformation procedure for the prediction of three-dimensional fully elliptic flows. This numerical scheme has been successfully employed for the calculations of the three-dimensional turbulent separated flow in a rectangular diffuser. The complexity of the phenomena is seen to increase tremendously for the three-dimensional flows of this class.     

A finite difference perturbation procedure for non-linear elastic stability problems

A finite difference perturbation scheme is developed which allows a simple solution to a wide class of non-linear difurcation problems. The analysis shows that in order to determine the inital post bucking behaviour accurately, it is not necessary to solve more than the linear eigenvalue difference equation with similar accuracy.     

A hybrid finite difference/helmholtz integral scheme for ultrasonic scattering

Two techniques for analyzing the scattering of elastic waves by flaws are combined in a hybrid model. One is a finite difference scheme for handling the details of the interaction with the flaw, the other is a Helmholtz integral scheme for extending the results into the far field. Results are given for the diffraction coefficients for a semi-infinite thin crack, and for a 270° corner.     

A balanced expansion technique for constructing accurate finite difference advection schemes

A new procedure, called the Balanced Expansion Technique (BET), is employed to construct accurate finite difference advection schemes that, for the model equation considered, are neutrally stable. By applying BET systematically, the phase error can be made as small as one wishes. Test calculations with one-dimensional problems have confirmed the expected accuracy of these methods.     

Curvilinear finite difference method for biharmonic equation

The essence of the proposed curvilinear finite difference solution method is the direct solution of the transformed fourth order partial differentiall equation using the finite difference expressions based on the flexible structure so that the geometry generality of the finite element method can be approached. The examples of applications indicte that the curvilinear mesh can be used and the boundary condition on the curved boundary can be easily and accurately prescribed.     

A Fortran program to generate finite difference formulas

A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.     

A finite element model for Mindlin plates

In the general framework of Reissner-Mindlin theory, a plate model based on certain potential functions is discussed, together with its mechanical interpretation. A finite element implementation is also described and numerical results are reported.     

A finite element model for thin-walled members

Thin-walled members subject to generalized loading may be modelled by using a finite element approach with conventional stress and strain parameters. A solution method for a class of problems that would use these elements is described. A numberical example of an elastic beam is given.     

A third-order semi-implicit finite difference method for solving the one-dimensional convection-diffusion equation

A third-order semi-implicit five-point finite difference method is developed to solve the one-dimensional convection-diffusion equation, using the ‘weighted’ modified equation method. It is shown to have a large stability region, to be very accurate and computationally fast.     

A staggered mesh finite difference scheme for the computation of compressible flows

A simple high resolution finite difference technique is presented to approximate weak solutions to hyperbolic systems of conservation laws. The method does not rely on Riemann problem solvers and is therefore easy to extend to a wide variety of problems. The overall performance (resolution and CPU requirements) is competitive with other state-of-the-art techniques offering sharp non-oscillatory shocks and contacts. Theoretical results confirm the reliability of the approach for linear systems and non-linear scalar equations.     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

The finite difference technique for solving crack problems

A detailed examination of the Finite Difference method for solving crack problems is presented and discussed. The three classical mode I configurations (i.e. Centered Crack Plate, Double Edge Notch and Single Edge Notch) as well as an uncommon case (A Penny Shape Crack embedded in a circular plate in bending) are solved and discussed. The Stress Intensity Factors are computed by taking more than one (first) term in William's Series, using two or three points near the tip. This technique improves the accuracy and frees one from relying on the very first point near the tip as a measure base. In most cases, the accuracy was found to be between 1–3% for uniform mesh size in the order of 5% from the half crack length. No special imposed functions were used near the tip, which makes the technique competitive to the Finite Element method, especially for 3-D problems or cases where the degree of singularity is not known. The solution is found iteratively (a two step SOR method) and some techniques for quick convergence are discussed.     

Improved finite difference formulas for boundary value problems

This paper deals with the numerical solution of linear differential equations of fourth order by finite differences. It points out significant (but usually overlooked) errors which result from the conventional method of imposing the boundary conditions in such problems. Revised finite difference formulas are derived which apply near the boundaries and which eliminate the above errors. Three commonly encountered boundary conditions ar econsidered. These correspond, in the terminology of beam analysis, to a clamped end, to a simply supported end and to a free end. The improvement in accuracy that can be achieved with the revised formulas is illustrated by two representative examples. The revised formulas are shown to reduce the overall error of the numerical solution by a factor of about five in a typical case.     

High-order finite difference modal method for diffraction gratings

A new high-order finite difference modal method (FDMM) is developed for analyzing diffraction gratings in conical and classical mountings. The difference scheme is constructed by enforcing the internal interface conditions in each grating layer to high-order derivatives, and it gives a high order of accuracy for computing the eigenmodes of the grating layer. Between different layers, the interface conditions are implemented using a Fourier matching scheme and a point matching scheme. Compared with the standard Fourier modal method, the high-order FDMM is more efficient since the matrices in the discretized eigenvalue problems are sparse. Numerical examples are used to illustrate the performance of the method.     

A finite element model for liquid phase electroepitaxy

A finite element numerical simulation model for the liquid phase electroepitaxial growth process of gallium arsenide is presented. The basic equations obtained from the fundamental principles of electrodynamics of continua, the constitutive equations for the liquid and solid phases derived from a rational thermodynamic theory, and the associated interface and boundary conditions are presented for a two-dimensional axisymmetric growth cell configuration. The field equations are solved numerically by an adaptive finite element procedure. The effect of moving interfaces is taken into account. Numerical simulations are carried out for different convection levels by changing the value of the gravitational constant. Results show that convection has significant effect on the growth process under normal gravity conditions and results in thickness non-uniformity of the grown layers. The thickness non-uniformity leads to curved interfaces of growth and dissolution, which enhance convection.     

A finite element model for single-sided crack patchings

A finite element model is established for analyzing the behavior of cracked plates which are repaired with a single-sided patch. The formulation is based on the Reissner-Mindlin plate theory with an assumed variation of the transverse shear and normal stresses through the thickness of the cracked -plate and patch. The generalized stress-strain relations relating the transverse shear stress resultants and the adhesive stresses to the displacements of the plate and patch are established by using a variational principle. By means of the finite element model presented herein, single-sided crack patching problems can be solved with a reasonable estimate of the adhesive stresses and the stress intensity factor. Numerical examples are provided to illustrate the effects of the patch size on the stress intensity factor in the cracked plate and the stress distribution in the adhesive layer, and compared with results from the previous analysis.