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 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 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 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 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 An Iterative Method for Pricing American Options Under Jump Diffusion Models written by Santtu Salmi and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou's and Merton's jump-diffusion models show that the resulting iteration converges rapidly.

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 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 Pricing Derivatives Under L vy Models written by Andrey Itkin and published by . This book was released on 2017 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 A Laplace Space Approach to American Options written by Jingtang Ma and published by . This book was released on 2016 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we extend the lower-upper bound approximation (LUBA) idea of Broadie and Detemple [Broadie, M., Detemple, J., (1996) American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies. 9(4): 1211-1250] to the Laplace space. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, for which there exist closed-form expressions for the Laplace transforms of the corresponding “capped (barrier) option prices”. The method is applied to a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures similar as Broadie and Detemple in the Laplace space. Finally we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option, and further utilize the early exercise premium (EEP) representation in the Laplace space to obtain the upper bound. We obtain explicit expressions in the case of the constant elasticity of variance (CEV) model (Wong and Zhao) and the double-exponential jump diffusion (DEJD) model (Leippold and Vasiljevic). Numerical examples show that our lower and upper bounds are accurate and efficient compared to results in the literature. To the best of authors' knowledge, it is the first time that the LUBA idea of Broadie and Detemple is applied to a model with jumps, and this solves an open question stated on page 1181 of Kou and Wang.

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 Option Pricing Using Fourier Space Time stepping Framework written by Vladimir Surkov and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MNd log2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.

Download or read book Valuation of American Options written by David Animante and published by . This book was released on 2016 with total page 55 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of American style equity options as hedging instrument has gained currency in recent times. This phenomenon devolves from the ever-expanding need by individuals, corporations and governments to hedge away their financial risks and the clarion call for derivative securities that give the holder increased flexibility in exercise. Nevertheless, pricing American options is complex and there exists no analytic solution to the problem except a profusion of approximation and finite difference techniques. Indeed, many researchers have shown that these methods cannot handle multifactor situations where the underlying asset follows a jump-diffusion process and where the derivative security depends on multiple sources of uncertainty such as stochastic volatility, stochastic interest rate among others. Monte-Carlo simulation techniques therefore developed out of the search for a pricing formula that has the capacity to accommodate all forms of uncertainty and at the same time able to produce speedy and accurate results. Some scholars at first rejected these techniques as yielding inaccurate results but in recent times, many researchers have demonstrated the efficacy of Monte-Carlo simulation in option pricing. The aim of this study is to assess the effectiveness of Monte-Carlo simulation methods in comparison with other option pricing techniques. To achieve this objective, the research builds an algorithm to compute Call and Put prices based on a wide range of input parameters. It also develops a model where volatility or interest rate is stochastic and a deterministic function of time. The results indicate that Monte-Carlo simulation techniques produce option values and exercise boundaries that are very similar to the Binomial, Barone-Adesi and Whaley as well as the Explicit Finite Difference methods. The results also show that the stochastic volatility and stochastic interest rate models yield slightly different but more accurate results. Consequently, the study recommends simulation techniques that incorporate multiple sources of uncertainty simultaneously for fast, efficient and more accurate option pricing.

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 The Pricing of Multiple Exercisable American style Real Options written by Yu Meng (Ph.D.) and published by . This book was released on 2012 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Real options embedded in a project provide management with the flexibility to alter initial investment decisions, thus making them a practical tool for project planning and budgeting. Additional values are contributed to the underlying project due to the flexibilities that are provided by real options. This dissertation presents two models for pricing multiple exercisable American real options, one that employs the binomial tree method and the other one that employs the finite difference method. Different examples of multiple exercisable real options are discussed to demonstrate the two pricing models. Interactions between options and reality constraints are also considered. These two methods are compared with each other at the end. This dissertation also addresses the problem of tracking early exercise boundaries in pricing American-style real options. It is shown that both models provide effective numerical solutions to the free boundary problem"--Abstract, leaf iii.

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.