Download or read book On the Acceleration of Explicit Finite Difference Methods for Option Pricing written by Stephen O'Sullivan and published by . This book was released on 2016 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: Implicit finite difference methods are conventionally preferred over their explicit counterparts for the valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latters' efficiencies. Implicit methods, however, are difficult to implement for all but the most simple of pricing models whereas explicit techniques are easily adapted to complex problems. In this work we present an acceleration technique for explicit finite difference schemes called Super-Time-Stepping (STS) for the first time in a financial context. Furthermore, we introduce a novel method for describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal walltime W required to attain a solution with an error of magnitude E. For European and American put option test cases we demonstrate degrees of acceleration over standard explicit methods resulting in efficiencies comparable, or superior, to a set of implicit scheme benchmarks. We conclude that STS is a powerful tool for the numerical pricing of options and propose it as the method-of-choice for exotic financial intruments such as those requiring multi-dimensional descriptions on adaptive meshes.

Download or read book Numerical Solution Of The American Option Pricing Problem The Finite Difference And Transform Approaches written by Carl Chiarella and published by World Scientific. This book was released on 2014-10-14 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: The early exercise opportunity of an American option makes it challenging to price and an array of approaches have been proposed in the vast literature on this topic. In The Numerical Solution of the American Option Pricing Problem, Carl Chiarella, Boda Kang and Gunter Meyer focus on two numerical approaches that have proved useful for finding all prices, hedge ratios and early exercise boundaries of an American option. One is a finite difference approach which is based on the numerical solution of the partial differential equations with the free boundary problem arising in American option pricing, including the method of lines, the component wise splitting and the finite difference with PSOR. The other approach is the integral transform approach which includes Fourier or Fourier Cosine transforms. Written in a concise and systematic manner, Chiarella, Kang and Meyer explain and demonstrate the advantages and limitations of each of them based on their and their co-workers' experiences with these approaches over the years.

Download or read book Computational Methods for Option Pricing written by Yves Achdou and published by SIAM. This book was released on 2005-07-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book allows you to understand fully the modern tools of numerical analysis in finance.

Download or read book Pricing Options Under Heston s Stochastic Volatility Model Via Accelerated Explicit Finite Differencing Methods written by Conall O'Sullivan and published by . This book was released on 2010 with total page 41 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method. For European and American put options under Heston's stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiencies to a benchmark implicit scheme. We conclude that STS is a powerful tool for the numerical pricing of options and propose them as the method-of-choice for exotic financial instruments in two and multi-factor models.

Download or read book Pricing the European Call Option by the Explicit Finite difference Method written by Hyeonju Kim and published by . This book was released on 2007 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Time discrete Method Of Lines For Options And Bonds The A Pde Approach written by Gunter H Meyer and published by World Scientific. This book was released on 2014-11-27 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: Few financial mathematical books have discussed mathematically acceptable boundary conditions for the degenerate diffusion equations in finance. In The Time-Discrete Method of Lines for Options and Bonds, Gunter H Meyer examines PDE models for financial derivatives and shows where the Fichera theory requires the pricing equation at degenerate boundary points, and what modifications of it lead to acceptable tangential boundary conditions at non-degenerate points on computational boundaries when no financial data are available.Extensive numerical simulations are carried out with the method of lines to examine the influence of the finite computational domain and of the chosen boundary conditions on option and bond prices in one and two dimensions, reflecting multiple assets, stochastic volatility, jump diffusion and uncertain parameters. Special emphasis is given to early exercise boundaries, prices and their derivatives near expiration. Detailed graphs and tables are included which may serve as benchmark data for solutions found with competing numerical methods.

Download or read book Mathematical Modeling And Methods Of Option Pricing written by Lishang Jiang and published by World Scientific Publishing Company. This book was released on 2005-07-18 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the unique perspective of partial differential equations (PDE), this self-contained book presents a systematic, advanced introduction to the Black-Scholes-Merton's option pricing theory.A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs. In particular, the qualitative and quantitative analysis of American option pricing is treated based on free boundary problems, and the implied volatility as an inverse problem is solved in the optimal control framework of parabolic equations.

Download or read book Accurate Finite Difference Methods for Option Pricing written by Jonas Persson and published by . This book was released on 2006 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Methods for American Option Pricing with Nonlinear Volatility written by Wen Wang and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is organized as follows: Chapter 1 is an introduction to option pricing theory; Chapter 2 focuses on theoretical model of uncertain volatility; Chapter 3 introduces the numerical methods; Chapter 4 shows the experiment results; Chapter 5 summarizes the work and points out some future research directions.

Download or read book A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Levy Models written by Rama Cont and published by . This book was released on 2004 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed for the Merton jump-diffusion model and for the Variance Gamma model with smooth and non-smooth payoff functions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.

Download or read book High Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility Jump Models written by Bertram Düring and published by . This book was released on 2017 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 efficiency and hedging performance to standard finite difference 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 LU-factorisation of a sparse matrix to perform the option price valuation. It 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.

Download or read book Adaptive Partial Differential Equation Methods for Option Pricing written by Guanghuan Hou and published by . This book was released on 2008 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This project investigates the application of finite difference schemes to option pricing problems. In particular, an adaptive mesh method is introduced to deal with difficulties that arise in the numerical approximation of PDE's in presence of discontinuities, such as a barrier. Compared to an equidistant mesh, this adaptive mesh method substantially increases the numerical accuracy with the same number of grid points. Several finite difference schemes for pricing American options are studied and compared both in one dimensional case and two dimensional case. The behaviors of the price and hedge factors of various types of barrier options and American barrier options are further studied in details.

Download or read book Numerical Methods for Fractional Black Scholes Equations and Variational Inequalities Governing Option Pricing written by Wen Chen and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: [Truncated abstract] The aim of this thesis is to develop numerical methods for pricing options whose underlying follow a Levy process, establish convergence theories for these numerical methods and demonstrate the accuracy, effectiveness and practicality of the developed methods numerically. We first propose a second order finite difference method (FDM) for the one-dimensional fractional Black-Scholes (FBS) equation governing the valuation of European options on a single asset. We then show that both the continuous and discretized FBS equations are uniquely solvable and establish the convergence of the numerical solution generated by the FDM to the viscosity solution of the ontinuous FBS equation by proving that the FDM is consistent, stable and monotone. We then propose a power penalty method for a fractional-order dierential linear complementarity problem (LCP) arising in the valuation of American options on a single asset.

Download or read book Fourth Order Accurate Implicit Finite Difference Method for Evaluating American Options written by International Business Machines Corporation. Research Division and published by . This book was released on 1998 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We present a numerical method for valuing vanilla American options on a single asset that is fourth order accurate in the log of the asset price, and second order accurate in time. The method overcomes the standard difficulty encountered in developing high order accurate finite difference schemes for valuing American options, that is the lack of smoothness in the option price at the critical boundary. To do this we make special corrections to the right hand side of the difference equations near the boundary so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr. The method can also be used for evaluating American options depending on more than one asset whenever estimates of the location of the critical boundary are available. Furthermore, the provable error estimates we obtain also allow development of extrapolation techniques. Results of numerical experiments comparing our method with more standard finite difference methods are provided."

Download or read book Option Pricing Under Generalized Tempered Stable Processes written by and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Pure jump L é vy processes have been the focus of recent work in mathematical finance. They allow for more realistic representation of stock price dynamics and numerically tractable models for option pricing. In this dissertation we propose a new option pricing model using the generalized tempered stable process and investigate numerical methods in pricing vanilla and exotic instruments. The Markov property of the price allows us to express option prices as solutions of partial integro-differential equations which involve, in addition to a second order differential operator, a non-local integral term. We develop an efficient implicit-explicit finite difference method based on the second order backward differentiation formula for solving them. Numerical results show that the scheme is consistent, convergent and stable. Monte Carlo simulation is also investigated for the valuation of path dependent options. A compound Poisson approximation is applied in the simulation of stock sample paths. We show how small jumps can be further approximated by a Brownian motion. In order to reduce the standard errors in simulation, we make use of the control variate technique. Finally, we calibrate the GTSP model to the S & P500 call options and present some properties of the risk neutral processes implied from option prices. The implied volatility surface is reproduced to demonstrate that the model is able to capture important features of the option market.

Download or read book Numerical Methods for Option Pricing written by Éanna Black and published by . This book was released on 2008 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Issues in Logic Operations and Computational Mathematics and Geometry 2013 Edition written by and published by ScholarlyEditions. This book was released on 2013-05-01 with total page 1187 pages. Available in PDF, EPUB and Kindle. Book excerpt: Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Random Structures and Algorithms. The editors have built Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Random Structures and Algorithms in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.