Download or read book Adaptive Finite Difference Methods for Valuing American Options written by Duy M. Dang and published by . This book was released on 2007 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop space-time adaptive methods for valuing American options with strong emphasis on American put options. We examine the application of adaptive techniques to the Black-Scholes partial differential equation problem associated with an American put option in the context of non-uniform second-order finite differences. At certain timesteps, we obtain a redistribution of the spatial points based on a monitor function that attempts to equidistribute the error. The proposed finite difference discretization on non-uniform grids and redistribution of the spatial points lead to linear complementarity problems with M-matrices. The Projected Successive Over-relaxation and a penalty method are considered to handle the free boundaries. We study and compare the accuracy and efficiency of the considered methods. A complete proof of convergence and uniqueness of the projected SOR method under certain conditions is also presented.
Download or read book The Numerical Solution of the American Option Pricing Problem 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. Contents: Introduction; The Merton and Heston Model for a Call; American Call Options under Jump-Diffusion Processes; American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics OCo The Transform Approach; Representation and Numerical Approximation of American Option Prices under Heston; Fourier Cosine Expansion Approach; A Numerical Approach to Pricing American Call Options under SVJD; Conclusion; Bibliography; Index; About the Authors. Readership: Post-graduates/ Researchers in finance and applied mathematics with interest in numerical methods for American option pricing; mathematicians/physicists doing applied research in option pricing. Key Features: Complete discussion of different numerical methods for American options; Able to handle stochastic volatility and/or jump diffusion dynamics; Able to produce hedge ratios efficiently and accurately"
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 Finite Difference Methods for the Valuation Af American Options written by and published by . This book was released on 2012 with total page 87 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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 Finite Difference Methods for Pricing American Put Options written by Alexander Blaise Prideaux and published by . This book was released on 2005 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Comparison of Finite Difference Methods for Pricing American Options on Two Stocks written by Stéphane Villeneuve and published by . This book was released on 2001 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Modeling and Methods of Option Pricing written by Lishang Jiang and published by World Scientific. This book was released on 2005 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the perspective of partial differential equations (PDE), this book introduces 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.
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 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 Hybrid Laplace Transform and Finite Difference Methods for Pricing American Options Under Complex Models written by Jingtang Ma and published by . This book was released on 2017 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump-diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models.
Download or read book Valuing American Options by Simulation written by Francis A. Longstaff and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper presents a simple yet powerful new approach for valuing American options by simulation. The key to this approach is to use least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference and binomial techniques cannot be used. We illustrate this technique with a series of realistic examples ranging from the valuation of an American put in a single-factor setting to the valuation of a deferred American swaption in a twenty-factor string model of the term structure.
Download or read book A Comparison and Survey of Finite Difference Methods for Pricing American Options Under Finite Activity Jump Diffusion Models written by Santtu Salmi and published by . This book was released on 2014 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt: Partial-integro differential formulations are often used for pricing American options under jump-diffusion models. A survey on such formulations and numerical methods for them is presented. A detailed description of six efficient methods based on a linear complementarity formulation and finite difference discretizations is given. Numerical experiments compare the performance of these methods for pricing American put options under finite activity jump models.
Download or read book The Valuation of American Options written by Jie Ding and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Computational Methods for Option Pricing written by Yves Achdou and published by SIAM. This book was released on 2005-01-01 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors review some important aspects of finance modeling involving partial differential equations and focus on numerical algorithms for the fast and accurate pricing of financial derivatives and for the calibration of parameters. This book explores the best numerical algorithms and discusses them in depth, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries.
Download or read book Finite Difference Methods in Financial Engineering written by Daniel J. Duffy and published by John Wiley & Sons. This book was released on 2013-10-28 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method. In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature: Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options Early exercise features and approximation using front-fixing, penalty and variational methods Modelling stochastic volatility models using Splitting methods Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work Modelling jumps using Partial Integro Differential Equations (PIDE) Free and moving boundary value problems in QF Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.
Download or read book Front tracking Finite Difference Methods for the American Option Valuation Problem written by K. N. Pantazopoulos and published by . This book was released on 1996 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt:
