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A Simple Approach To Pricing American Options Under The Heston Stochastic Volatility Model
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Book Synopsis A Simple Approach to Pricing American Options Under the Heston Stochastic Volatility Model by : Natalia Beliaeva
Download or read book A Simple Approach to Pricing American Options Under the Heston Stochastic Volatility Model written by Natalia Beliaeva and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In a recent paper, NBZ [2010] present a multidimensional transform for generating path-independent trees for pricing American options under low dimensional stochastic volatility models. For this class of models, this approach has higher accuracy than the GARCH tree method of Ritchken and Trevor [1999], and is computationally more efficient than the Monte Carlo regression method of Longstaff and Schwartz [2001] as well as the lattice method of Leisen [2000]. In this paper, we give an explicit demonstration of the NBZ transform using the specific example of the Heston [1993] stochastic volatility model. This approach obtains highly accurate American option prices within a fraction of a second using the control variate method.
Book Synopsis Pricing American Options in the Heston Model by : Peter Ruckdeschel
Download or read book Pricing American Options in the Heston Model written by Peter Ruckdeschel and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce a refined tree method to compute option prices using the stochastic volatility model of Heston. In a first step, we model the stock and variance process as two separate trees and with transition probabilities obtained by matching marginal tree moments up to order two against the Heston model ones. The correlation between the driving Brownian motions is then incorporated by a node-wise adjustment of the probabilities. This adjustment, leaving the marginals fixed, optimizes the match between tree and model correlation. In some nodes, we are even able to further match moments of higher order. Numerically this gives convergence orders faster than 1/N, where N is the number of discretization steps. Accuracy of our method is checked for European option prices against a semi closed-form, and our prices for both European and American options are compared to alternative approaches.
Book Synopsis American Option Pricing Under Two Stochastic Volatility Processes by : Jonathan Ziveyi
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.
Book Synopsis The Heston Model and its Extensions in Matlab and C# by : Fabrice D. Rouah
Download or read book The Heston Model and its Extensions in Matlab and C# written by Fabrice D. Rouah and published by John Wiley & Sons. This book was released on 2013-08-01 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tap into the power of the most popular stochastic volatility model for pricing equity derivatives Since its introduction in 1993, the Heston model has become a popular model for pricing equity derivatives, and the most popular stochastic volatility model in financial engineering. This vital resource provides a thorough derivation of the original model, and includes the most important extensions and refinements that have allowed the model to produce option prices that are more accurate and volatility surfaces that better reflect market conditions. The book's material is drawn from research papers and many of the models covered and the computer codes are unavailable from other sources. The book is light on theory and instead highlights the implementation of the models. All of the models found here have been coded in Matlab and C#. This reliable resource offers an understanding of how the original model was derived from Ricatti equations, and shows how to implement implied and local volatility, Fourier methods applied to the model, numerical integration schemes, parameter estimation, simulation schemes, American options, the Heston model with time-dependent parameters, finite difference methods for the Heston PDE, the Greeks, and the double Heston model. A groundbreaking book dedicated to the exploration of the Heston model—a popular model for pricing equity derivatives Includes a companion website, which explores the Heston model and its extensions all coded in Matlab and C# Written by Fabrice Douglas Rouah a quantitative analyst who specializes in financial modeling for derivatives for pricing and risk management Engaging and informative, this is the first book to deal exclusively with the Heston Model and includes code in Matlab and C# for pricing under the model, as well as code for parameter estimation, simulation, finite difference methods, American options, and more.
Book Synopsis The Heston Model and Its Extensions in VBA by : Fabrice D. Rouah
Download or read book The Heston Model and Its Extensions in VBA written by Fabrice D. Rouah and published by John Wiley & Sons. This book was released on 2015-03-20 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: Practical options pricing for better-informed investment decisions. The Heston Model and Its Extensions in VBA is the definitive guide to options pricing using two of the derivatives industry's most powerful modeling tools—the Heston model, and VBA. Light on theory, this extremely useful reference focuses on implementation, and can help investors more efficiently—and accurately—exploit market information to better inform investment decisions. Coverage includes a description of the Heston model, with specific emphasis on equity options pricing and variance modeling, The book focuses not only on the original Heston model, but also on the many enhancements and refinements that have been applied to the model, including methods that use the Fourier transform, numerical integration schemes, simulation, methods for pricing American options, and much more. The companion website offers pricing code in VBA that resides in an extensive set of Excel spreadsheets. The Heston model is the derivatives industry's most popular stochastic volatility model for pricing equity derivatives. This book provides complete guidance toward the successful implementation of this valuable model using the industry's ubiquitous financial modeling software, giving users the understanding—and VBA code—they need to produce option prices that are more accurate, and volatility surfaces that more closely reflect market conditions. Derivatives pricing is often the hinge on which profit is made or lost in financial institutions, making accuracy of utmost importance. This book will help risk managers, traders, portfolio managers, quants, academics and other professionals better understand the Heston model and its extensions, in a writing style that is clear, concise, transparent and easy to understand. For better pricing accuracy, The Heston Model and Its Extensions in VBA is a crucial resource for producing more accurate model outputs such as prices, hedge ratios, volatilities, and graphs.
Book Synopsis American Option Pricing Under Stochastic Volatility by : Manisha Goswami
Download or read book American Option Pricing Under Stochastic Volatility written by Manisha Goswami and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The approximate method to price American options makes use of the fact that accurate pricing of these options does not require exact determination of the early exercise boundary. Thus, the procedure mixes the two models of constant and stochastic volatility. The idea is to obtain early exercise boundary through constant volatility model using the approximation methods of AitSahlia and Lai or Ju and then utilize this boundary to price the options under stochastic volatility models. The data on S & P 100 Index American options is used to analyze the pricing performance of the mixing of the two models. The performance is studied with respect to percentage pricing error and absolute pricing errors for each money-ness maturity group.
Book Synopsis A Simple New Formula for Options with Stochastic Volatility by : Steven L. Heston
Download or read book A Simple New Formula for Options with Stochastic Volatility written by Steven L. Heston and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper shows a relationship between bond pricing models and option pricing models with stochastic volatility. It exploits this relationship to find a new stochastic volatility model with a closed-form solution for European option prices. The model allows nonzero correlation between volatility and spot asset returns. When the correlation is unity the model contains the Black-Scholes [1973] model and Cox's [1975] constant elasticity of variance model as special cases. The option formula preserves the Black-Scholes property that changes in volatility are equivalent to changes in option expiration.
Book Synopsis The Evaluation of American Compound Option Prices under Stochastic Volatility Using the Sparse Grid Approach by : Carl Chiarella
Download or read book The Evaluation of American Compound Option Prices under Stochastic Volatility Using the Sparse Grid Approach written by Carl Chiarella and published by . This book was released on 2009 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we demonstrate a partial differential equation (PDE) approach to pricing American-type compound options where the underlying dynamics follow Heston's stochastic volatility model. This price is formulated as the solution to a two-pass free boundary PDE problem. A modified sparse grid approach is implemented to solve the PDEs, which is shown to be accurate and efficient compared with the results from Monte Carlo simulation combined with the Method of Lines.
Book Synopsis The Impact of Jumps on American Option Pricing by : Boda Kang
Download or read book The Impact of Jumps on American Option Pricing written by Boda Kang and published by . This book was released on 2019 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper analyzes the importance of asset and volatility jumps in American option pricing models. Using the Heston (1993) stochastic volatility model with asset and volatility jumps and the Hull and White (1987) short rate model, American options are numerically evaluated by the Method of Lines. The calibration of these models to S&P 100 American options data reveals that jumps, especially asset jumps, play an important role in improving the models' ability to fit market data. Further, asset and volatility jumps tend to lift the free boundary, an effect that augments during volatile market conditions, while the additional volatility jumps marginally drift down the free boundary. As markets turn more volatile and exhibit jumps, American option holders become more prudent with their exercise decisions, especially as maturity of the options approaches.
Book Synopsis An Analytical Approach to Pricing American Options Under Stochastic Volatility by : Zhe Zhang
Download or read book An Analytical Approach to Pricing American Options Under Stochastic Volatility written by Zhe Zhang and published by . This book was released on 2000 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis American Option Pricing Under Stochastic Volatility by : Suchandan Guha
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.
Book Synopsis Essays on American Options Pricing Under Levy Models with Stochastic Volatility and Jumps by : Ye Chen
Download or read book Essays on American Options Pricing Under Levy Models with Stochastic Volatility and Jumps written by Ye Chen and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In ``A Multi-demensional Transform for Pricing American Options Under Stochastic Volatility Models", we present a new transform-based approach for pricing American options under low-dimensional stochastic volatility models which can be used to construct multi-dimensional path-independent lattices for all low-dimensional stochastic volatility models given in the literature, including SV, SV2, SVJ, SV2J, and SVJ2 models. We demonstrate that the prices of European options obtained using the path-independent lattices converge rapidly to their true prices obtained using quasi-analytical solutions. Our transform-based approach is computationally more efficient than all other methods given in the literature for a large class of low-dimensional stochastic volatility models. In ``A Multi-demensional Transform for Pricing American Options Under Levy Models", We extend the multi-dimensional transform to Levy models with stochastic volatility and jumps in the underlying stock price process. Efficient path-independent tree can be constructed for both European and American options. Our path-independent lattice method can be applied to almost all Levy models in the literature, such as Merton (1976), Bates (1996, 2000, 2006), Pan (2002), the NIG model, the VG model and the CGMY model. The numerical results show that our method is extemly accurate and fast. In ``Empirical performance of Levy models for American Options", we investigate in-sample fitting and out-of-sample pricing performance on American call options under Levy models. The drawback of the BS model has been well documented in the literatures, such as negative skewness with excess kurtosis, fat tail, and non-normality. Therefore, many models have been proposed to resolve known issues associated the BS model. For example, to resolve volatility smile, local volatility, stochastic volatility, and diffusion with jumps have been considered in the literatures; to resolve non-normality, non-Markov processes have been considered, e.g., Poisson process, variance gamma process, and other type of Levy processes. One would ask: what is the gain from each of the generalized models? Or, which model is the best for option pricing? We address these problems by examining which model results in the lowest pricing error for American style contracts. For in-sample analysis, the rank (from best to worst) is Pan, CGMYsv, VGsv, Heston, CGMY, VG and BS. And for out-of-sample pricing performance, the rank (from best to worst) is CGMYsv, VGsv, Pan, Heston, BS, VG, and CGMY. Adding stochastic volatility and jump into a model improves American options pricing performance, but pure jump models are worse than the BS model in American options pricing. Our empirical results show that pure jump model are over-fitting, but not improve American options pricing when they are applied to out-of-sample data.
Book Synopsis American Options Under Stochastic Volatility by : Arun Chockalingam
Download or read book American Options Under Stochastic Volatility written by Arun Chockalingam and published by . This book was released on 2012 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of pricing an American option written on an underlying asset with constant price volatility has been studied extensively in literature. Real-world data, however, demonstrates that volatility is not constant and stochastic volatility models are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature for pricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American options under stochastic volatility which has relatively had much less attention from literature. First, we develop an exercise-policy improvement procedure to compute the optimal exercise policy and option price. We show that the scheme monotonically converges for various popular stochastic volatility models in literature. Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of option prices, exercise policies and implied volatilities on the market price of volatility risk and correlation between the asset and stochastic volatility.
Book Synopsis American Spread Option Models and Valuation by : Yu Hu
Download or read book American Spread Option Models and Valuation written by Yu Hu and published by . This book was released on 2013 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we investigate the American spread option pricing problem. Several approaches for the geometric Brownian motion model and the stochastic volatility model are developed. We also implement the above models and the numerical results are compared among different approaches.
Book Synopsis Numerical Solution Of The American Option Pricing Problem, The: Finite Difference And Transform Approaches by : Carl Chiarella
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.
Book Synopsis Pricing American Options Under Stochastic Volatility by : Elias Tzavalis
Download or read book Pricing American Options Under Stochastic Volatility written by Elias Tzavalis and published by . This book was released on 2003 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Quantitative Finance by : Maria Cristina Mariani
Download or read book Quantitative Finance written by Maria Cristina Mariani and published by John Wiley & Sons. This book was released on 2019-12-12 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a multitude of topics relevant to the quantitative finance community by combining the best of the theory with the usefulness of applications Written by accomplished teachers and researchers in the field, this book presents quantitative finance theory through applications to specific practical problems and comes with accompanying coding techniques in R and MATLAB, and some generic pseudo-algorithms to modern finance. It also offers over 300 examples and exercises that are appropriate for the beginning student as well as the practitioner in the field. The Quantitative Finance book is divided into four parts. Part One begins by providing readers with the theoretical backdrop needed from probability and stochastic processes. We also present some useful finance concepts used throughout the book. In part two of the book we present the classical Black-Scholes-Merton model in a uniquely accessible and understandable way. Implied volatility as well as local volatility surfaces are also discussed. Next, solutions to Partial Differential Equations (PDE), wavelets and Fourier transforms are presented. Several methodologies for pricing options namely, tree methods, finite difference method and Monte Carlo simulation methods are also discussed. We conclude this part with a discussion on stochastic differential equations (SDE’s). In the third part of this book, several new and advanced models from current literature such as general Lvy processes, nonlinear PDE's for stochastic volatility models in a transaction fee market, PDE's in a jump-diffusion with stochastic volatility models and factor and copulas models are discussed. In part four of the book, we conclude with a solid presentation of the typical topics in fixed income securities and derivatives. We discuss models for pricing bonds market, marketable securities, credit default swaps (CDS) and securitizations. Classroom-tested over a three-year period with the input of students and experienced practitioners Emphasizes the volatility of financial analyses and interpretations Weaves theory with application throughout the book Utilizes R and MATLAB software programs Presents pseudo-algorithms for readers who do not have access to any particular programming system Supplemented with extensive author-maintained web site that includes helpful teaching hints, data sets, software programs, and additional content Quantitative Finance is an ideal textbook for upper-undergraduate and beginning graduate students in statistics, financial engineering, quantitative finance, and mathematical finance programs. It will also appeal to practitioners in the same fields.
