We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

http://www.cmap.polytechnique.fr/~rama/pide.pdf

A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump-diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling. Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump-diffusions. An additional 160 pages of online appendices are available on a Web page that supplements the book. Audience This book is written for graduate students in science and engineering who seek to construct models for scientific applications subject to uncertain environments. Mathematical modelers and researchers in applied mathematics, computational science, and engineering will also find it useful, as will practitioners of financial engineering who need fast and efficient solutions to stochastic problems. Contents List of Figures; List of Tables; Preface; Chapter 1. Stochastic Jump and Diffusion Processes: Introduction; Chapter 2. Stochastic Integration for Diffusions; Chapter 3. Stochastic Integration for Jumps; Chapter 4. Stochastic Calculus for Jump-Diffusions: Elementary SDEs; Chapter 5. Stochastic Calculus for General Markov SDEs: Space-Time Poisson, State-Dependent Noise, and Multidimensions; Chapter 6. Stochastic Optimal Control: Stochastic Dynamic Programming; Chapter 7. Kolmogorov Forward and Backward Equations and Their Applications; Chapter 8. Computational Stochastic Control Methods; Chapter 9. Stochastic Simulations; Chapter 10. Applications in Financial Engineering; Chapter 11. Applications in Mathematical Biology and Medicine; Chapter 12. Applied Guide to Abstract Theory of Stochastic Processes; Bibliography; Index; A. Online Appendix: Deterministic Optimal Control; B. Online Appendix: Preliminaries in Probability and Analysis; C. Online Appendix: MATLAB Programs

Applied Stochastic Processes and Control for Jump-Diffusions

An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed-point iteration scheme are presented. For typical model parameters, it is shown the error is reduced by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform method. Numerical tests of convergence for a variety of options are presented.

/pdf/robust-numerical-methods-for-contingent-claims-under-jump-kxzvkhx9d5.pdf

Robust numerical methods for contingent claims under jump diffusion processes

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, i...

An exact and explicit solution for the valuation of American put options

This paper deals with the numerical solution of the Heston partial differential equation (PDE) that plays an important role in financial option pricing theory, Heston (1993). A feature of this time-dependent, twodimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on non-uniform grids, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored timediscretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig–Sneyd scheme, the Modified Craig–Sneyd scheme, and the Hundsdorfer–Verwer scheme, each of which contains a free parameter. ADI schemes were not originally developed to deal with mixed spatialderivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston PDE. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In ’t Hout & Welfert (2007, 2009) we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term. It is expected that these schemes will be useful also for general two-dimensional convection-diffusion-reaction equations with mixed derivative terms.

/pdf/adi-finite-difference-schemes-for-option-pricing-in-the-1j68aleuc6.pdf

ADI finite difference schemes for option pricing in the Heston model with correlation

The Pricing Equations. Analysis of Finite Difference Methods. Special Issues. Coordinate Transformations. Numerical Examples. Index.

Pricing Financial Instruments: The Finite Difference Method

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Pricing Financial Instruments: The Finite Difference Method