Download or read book Numerical Solutions for American Options on Assets with Stochastic Volatilities written by Jinliang Li and published by . This book was released on 2001 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: This dissertation discusses American options on assets with stochastic volatilities. First, it gives a proof of the solution uniqueness of the 2-D PDE to evaluate options for both general two-factor model and the model used in this dissertation. Second, it formulates the two factor American option as a 2-D PDE free boundary problem. Third, because the solution of this 2-D PDE free boundary problem is not a very smooth function and the free boundary changes rapidly near maturity, most of the numerical methods could fail to find a reasonable solution or the numerical solution has a large truncation error. Instead of solving this 2-D PDE directly, the difference between the solution of the original 2-D PDE free boundary problem and the solution of a 1-D parabolic equation with the same final condition is calculated. The difference function is very smooth in the entire region. We can solve this new 2-D PDE free boundary problem more accurately and more efficiently. Fourth, this paper uses a coordinate transformation to map the moving boundary to a fixed boundary and applies the Singularity Separating Method (SSM) technique to separate the free boundary and find the exact location of the free boundary (the optimal exercise price). This will be very useful for arbitrage activities. Fifth, it develops numerical methods to solve the new free boundary problem and focuses on the high order implicit finite difference method. It provides several methods to solve the nonlinear system. Sixth, It discovers the put-call symmetry relation between American options in the two factor stochastic volatility model. Seventh, it uses the extrapolation technique to improve the approximation accuracy of the numerical solution. Chapter 5 gives several numerical examples.
Download or read book Numerical Methods for Pricing American Put Options Under Stochastic Volatility written by Dominique Joubert and published by . This book was released on 2013 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Efficient Numerical Methods for Pricing American Options Under Stochastic Volatility written by Samuli Ikonen and published by . This book was released on 2005 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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 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 Numerical Methods for Pricing American Put Options Under Stochastic Volatility written by Dominique Joubert and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Early exercise boundary -- Free boundary value problem -- Linear complimentary problem -- Crank-Nicolson finite difference method -- Projected Over-Relaxation method (PSOR) -- Stochastic volatility -- Heston stochastic volatility model -- Vroeë uitoefengrens -- Vrye grenswaardeprobleem -- Linêere komplimentêre probleem -- Crank-Nicolson eindige differensiemetode -- Geprojekteerde oorverslappingsmetode (PSOR) -- Stogastiese volatiliteit -- Heston stogastiese volatiliteitsmodel.
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 American Option Pricing Under Stochastic Volatility written by Suchandan Guha and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: We developed two new numerical techniques to price American options when the underlying follows a bivariate process. The first technique exploits the semi-martingale representation of an American option price together with a coarse approximation of its early exercise surface that is based on an efficient implementation of the least-squares Monte Carlo method. The second technique exploits recent results in the efficient pricing of American options under constant volatility. Extensive numerical evaluations show these methods yield very accurate prices in a computationally efficient manner with the latter significantly faster than the former. However, the flexibility of the first method allows for its extension to a much larger class of optimal stopping problems than addressed in this paper.
Download or read book Nonlinear Option Pricing written by Julien Guyon and published by CRC Press. This book was released on 2013-12-19 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: New Tools to Solve Your Option Pricing Problems For nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research—including Risk magazine’s 2013 Quant of the Year—Nonlinear Option Pricing compares various numerical methods for solving high-dimensional nonlinear problems arising in option pricing. Designed for practitioners, it is the first authored book to discuss nonlinear Black-Scholes PDEs and compare the efficiency of many different methods. Real-World Solutions for Quantitative Analysts The book helps quants develop both their analytical and numerical expertise. It focuses on general mathematical tools rather than specific financial questions so that readers can easily use the tools to solve their own nonlinear problems. The authors build intuition through numerous real-world examples of numerical implementation. Although the focus is on ideas and numerical examples, the authors introduce relevant mathematical notions and important results and proofs. The book also covers several original approaches, including regression methods and dual methods for pricing chooser options, Monte Carlo approaches for pricing in the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection method and the particle method for calibrating local stochastic volatility models to market prices of vanilla options with/without stochastic interest rates, the a + bλ technique for building local correlation models that calibrate to market prices of vanilla options on a basket, and a new stochastic representation of nonlinear PDE solutions based on marked branching diffusions.
Download or read book Numerical Methods in Finance written by L. C. G. Rogers and published by Cambridge University Press. This book was released on 1997-06-26 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods in Finance describes a wide variety of numerical methods used in financial analysis.
Download or read book American Option Pricing Under Two Stochastic Volatility Processes written by Jonathan Ziveyi and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we consider the pricing of an American call option whose underlying asset dynamics evolve under the influence of two independent stochastic volatility processes as proposed in Christoffersen, Heston and Jacobs (2009). We consider the associated partial differential equation (PDE) for the option price and its solution. An integral expression for the general solution of the PDE is presented by using Duhamel's principle and this is expressed in terms of the joint transition density function for the driving stochastic processes. For the particular form of the underlying dynamics we are able to solve the Kolmogorov PDE for the joint transition density function by first transforming it to a corresponding system of characteristic PDEs using a combination of Fourier and Laplace transforms. The characteristic PDE system is solved by using the method of characteristics. With the full price representation in place, numerical results are presented by first approximating the early exercise surface with a bivariate log linear function. We perform numerical comparisons with results generated by the method of lines algorithm and note that our approach provides quite good accuracy.
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 Numerical Evaluation of American Options written by Liang Tan and published by LAP Lambert Academic Publishing. This book was released on 2009-11 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book we discuss various numerical evaluation problems for American options. Base on Black-Scholes framework, we establish partial differential complementarity problems (PDCP) for American options. Then we introduced various finite difference schemes to discretize the PDCP to obtain a system of Linear Complementarity Problems. The solution analysis and numerical algorithms are discussed. Next we study the pricing problem for American options whose payoff function are determined by two or more underlying assets. We formulate the two-asset American option pricing problem as two-dimensional PDCP. We first perform some state variable transformation and then introduce the ADI scheme and LOD scheme. After this, we discuss American option on an underlying asset with stochastic volatility. At last we consider the implied volatility problem for American options. We formulate a mathematical program with complementarity constraints (MPCC). Then we applied a penalty approach to solve the MPCC by utilizing the existing NLP tools. The parameter estimation problem for a mean-reverting stochastic volatility process is also considered.
Download or read book Numerical Analysis Of Stochastic Volatility Jump Diffusion Models written by Abdelilah Jraifi and published by LAP Lambert Academic Publishing. This book was released on 2014-06-30 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the modern economic world, the options contracts are used because they allow to hedge against the vagaries and risks refers to fluctuations in the prices of the underlying assets. The determination of the price of these contracts is of great importance for investors.We are interested in problems of options pricing, actually the European and Quanto options on a financial asset. The price of that asset is modeled by a multi-dimentional jump diffusion with stochastic volatility. Otherwise, the first model considers the volatility as a continuous process and the second model considers it as a jump process. Finally in the 3rd model, the underlying asset is without jump and volatility follows a model CEV without jump. This model allow better to take into account some phenomena observed in the markets. We develop numerical methods that determine the values of prices for these options. We first write the model as an integro-differential stochastic equations system "EIDS," of which we study existence and unicity of solutions. Then we relate the resolution of PIDE to the computation of the option value.
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 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 An Investigation of the Impact of Stochastic Interest Rates on the Pricing of Equity Options written by Peter Carayannopoulos and published by . This book was released on 1993 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:
