1,077 research outputs found
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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
Review of modern numerical methods for a simple vanilla option pricing problem
Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better
Discretization of fractional fully nonlinear equations by powers of discrete Laplacians
We study discretizations of fractional fully nonlinear equations by powers of
discrete Laplacians. Our problems are parabolic and of order
since they involve fractional Laplace operators . They
arise e.g.~in control and game theory as dynamic programming equations, and
solutions are non-smooth in general and should be interpreted as viscosity
solutions. Our approximations are realized as finite-difference quadrature
approximations and are 2nd order accurate for all values of . The
accuracy of previous approximations depend on and are worse when
is close to . We show that the schemes are monotone, consistent,
-stable, and convergent using a priori estimates, viscosity solutions
theory, and the method of half-relaxed limits. We present several numerical
examples.Comment: 19 pages, 5 figures, 1 tabl
On the local transformed based method for partial integro-differential equations of fractional order
High-order compact finite difference scheme for option pricing in stochastic volatility jump models
We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial -factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code
Fast and Efficient Numerical Methods for an Extended Black-Scholes Model
An efficient linear solver plays an important role while solving partial
differential equations (PDEs) and partial integro-differential equations
(PIDEs) type mathematical models. In most cases, the efficiency depends on the
stability and accuracy of the numerical scheme considered. In this article we
consider a PIDE that arises in option pricing theory (financial problems) as
well as in various scientific modeling and deal with two different topics. In
the first part of the article, we study several iterative techniques
(preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis
have been used to design various preconditioners to improve the convergence
criteria of iterative solvers. We implement a multigrid (MG) iterative method.
In fact, we approximate the problem using a finite difference scheme, then
implement a few preconditioned Krylov subspace methods as well as a MG method
to speed up the computation. Then, in the second part in this study, we analyze
the stability and the accuracy of two different one step schemes to approximate
the model.Comment: 29 pages; 10 figure
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
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