This small study demonstrates a substantial variation in patient's preferences with regard to postoperative chemoradiation for rectal cancer. Further studies in the preoperative setting are warranted.

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE. Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.

This paper will demonstrate how European and American option prices can be computed under the jump-diffusion model using the radial basis function (RBF)interpolation scheme. The RBF interpolation scheme is demonstrated by solving an option pricing formula, a onedimensional partial integro-differential equation (PIDE). We select the cubic spline radial basis function and propose a simple numerical algorithm to establish a finite computational range for the improper integral of the PIDE. This algorithm can improve the approximation accuracy of the integral with the application of any quadrature. Moreover, we offer a numerical technique termed cubic spline factorisation to solve the inversion of an ill-conditioned RBF interpolant, which is a well-known research problem in the RBF field. Finally, we numerically prove that in the European case, our RBF-interpolation solution is second-order accurate for spatial variables, while in the American case, it is second-order accurate for spatial variables and first-order accurate for time variables.

The aim of this paper is to show that option prices in jumpdiffusion models can be computed using meshless methods based on Radial Basis Function (RBF) interpolation instead of traditional meshbased methods like Finite Differences (FDM) or Finite Elements (FEM). The RBF technique is demonstrated by solving the partial integro-differential equation for American and European options on non-dividend-paying stocks in the Merton jump-diffusion model, using the Inverse Multiquadric Radial Basis Function (IMQ). The method can in principle be extended to Lévy-models. Moreover, an adaptive method is proposed to tackle the accuracy problem caused by a singularity in the initial condition so that the accuracy in option pricing in particular for small time to maturity can be improved.

We apply a new numerical method, the singular Fourier-Padé (SFP) method invented by Driscoll and Fornberg (2001, 2011), to price European-type options in Lévy and affine processes. The motivation behind this application is to reduce the inefficiency of current Fourier techniques when they are used to approximate piecewise continuous (non-smooth) probability density functions. When techniques such as fast Fourier transforms and Fourier series are applied to price and hedge options with non-smooth probability density functions, they cause the Gibbs phenomenon; accordingly, the techniques converge slowly for density functions with jumps in value or derivatives. This seriously adversely affects the efficiency and accuracy of these techniques. In this paper, we derive pricing formulae and their option Greeks using the SFP method to resolve the Gibbs phenomenon and restore the global spectral convergence rate. Moreover, we show that our method requires a small number of terms to yield fast error convergence, and it is able to accurately price any European-type option deep in/out of the money and with very long/short maturities. Furthermore, we conduct an error-bound analysis of the SFP method in option pricing. This new method performs favourably in numerical experiments compared with existing techniques.