2,181 research outputs found
A unified approach for the solution of the Fokker-Planck equation
This paper explores the use of a discrete singular convolution algorithm as a
unified approach for numerical integration of the Fokker-Planck equation. The
unified features of the discrete singular convolution algorithm are discussed.
It is demonstrated that different implementations of the present algorithm,
such as global, local, Galerkin, collocation, and finite difference, can be
deduced from a single starting point. Three benchmark stochastic systems, the
repulsive Wong process, the Black-Scholes equation and a genuine nonlinear
model, are employed to illustrate the robustness and to test accuracy of the
present approach for the solution of the Fokker-Planck equation via a
time-dependent method. An additional example, the incompressible Euler
equation, is used to further validate the present approach for more difficult
problems. Numerical results indicate that the present unified approach is
robust and accurate for solving the Fokker-Planck equation.Comment: 19 page
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts
In this paper, we study the benefits of using polyharmonic splines and node
layouts with smoothly varying density for developing robust and efficient
radial basis function generated finite difference (RBF-FD) methods for pricing
of financial derivatives. We present a significantly improved RBF-FD scheme and
successfully apply it to two types of multidimensional partial differential
equations in finance: a two-asset European call basket option under the
Black--Scholes--Merton model, and a European call option under the Heston
model. We also show that the performance of the improved method is equally high
when it comes to pricing American options. By studying convergence,
computational performance, and conditioning of the discrete systems, we show
the superiority of the introduced approaches over previously used versions of
the RBF-FD method in financial applications
Modelling FX smile : from stochastic volatility to skewness
Imperial Users onl
Tradable Schemes
In this article we present a new approach to the numerical valuation of
derivative securities. The method is based on our previous work where we
formulated the theory of pricing in terms of tradables. The basic idea is to
fit a finite difference scheme to exact solutions of the pricing PDE. This can
be done in a very elegant way, due to the fact that in our tradable based
formulation there appear no drift terms in the PDE. We construct a mixed scheme
based on this idea and apply it to price various types of arithmetic Asian
options, as well as plain vanilla options (both european and american style) on
stocks paying known cash dividends. We find prices which are accurate to in about 10ms on a Pentium 233MHz computer and to in a
second. The scheme can also be used for market conform pricing, by fitting it
to observed option prices.Comment: 13 pages, 5 tables, LaTeX 2
A numerical study of radial basis function based methods for option pricing under one dimension jump-diffusion model
The aim of this paper is to show how option prices in the Jump-diffusion model can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integro-differential equation (PIDE) in one-dimension for the Ameri-
can put and the European vanilla call/put options on dividend-paying stocks in the Merton and Kou Jump-diffusion models. The radial basis function we select is the Cubic Spline. We also propose a simple numerical algorithm for
finding a finite computational range of a global integral term in the PIDE so that the accuracy of approximation of the integral can be improved. Moreover, the solution functions of the PIDE are approximated explicitly by RBFs
which have exact forms so we can easily compute the global intergal by any kind of numerical quadrature. Finally, we will also show numerically that our scheme is second order accurate in spatial variables in both American and European cases
- …