American Option Pricing Under Stochastic Volatility
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American Option Pricing Under Stochastic Volatility
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Author : Manisha Goswami
language : en
Publisher:
Release Date : 2008
American Option Pricing Under Stochastic Volatility written by Manisha Goswami and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.
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.
American Options Under Stochastic Volatility
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Author : Arun Chockalingam
language : en
Publisher:
Release Date : 2012
American Options Under Stochastic Volatility written by Arun Chockalingam and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
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.
American Option Pricing Under Stochastic Volatility
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Author : Suchandan Guha
language : en
Publisher:
Release Date : 2008
American Option Pricing Under Stochastic Volatility written by Suchandan Guha and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.
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.
American Option Pricing Under Two Stochastic Volatility Processes
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Author : Jonathan Ziveyi
language : en
Publisher:
Release Date : 2013
American Option Pricing Under Two Stochastic Volatility Processes written by Jonathan Ziveyi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
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.
The Numerical Solution Of The American Option Pricing Problem
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Author : Carl Chiarella
language : en
Publisher: World Scientific
Release Date : 2014-10-14
The Numerical Solution Of The American Option Pricing Problem written by Carl Chiarella and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-14 with Options (Finance) categories.
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"
The Evaluation Of American Option Prices Under Stochastic Volatility And Jump Diffusion Dynamics Using The Method Of Lines
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Author : Carl Chiarella
language : en
Publisher:
Release Date : 2008
The Evaluation Of American Option Prices Under Stochastic Volatility And Jump Diffusion Dynamics Using The Method Of Lines written by Carl Chiarella and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with categories.
This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen amp; Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.
Essays On American Options Pricing Under Levy Models With Stochastic Volatility And Jumps
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Author : Ye Chen
language : en
Publisher:
Release Date : 2019
Essays On American Options Pricing Under Levy Models With Stochastic Volatility And Jumps written by Ye Chen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with categories.
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.
The Evaluation Of American Compound Option Prices Under Stochastic Volatility Using The Sparse Grid Approach
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Author : Carl Chiarella
language : en
Publisher:
Release Date : 2009
The Evaluation Of American Compound Option Prices Under Stochastic Volatility Using The Sparse Grid Approach written by Carl Chiarella and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.
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.
An Analytical Approach To Pricing American Options Under Stochastic Volatility
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Author : Zhe Zhang
language : en
Publisher:
Release Date : 2000
An Analytical Approach To Pricing American Options Under Stochastic Volatility written by Zhe Zhang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with categories.
Binomial Option Pricing Under Stochastic Volatility And Correlated State Variables
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Author : Jimmy E. Hilliard
language : en
Publisher:
Release Date : 1998
Binomial Option Pricing Under Stochastic Volatility And Correlated State Variables written by Jimmy E. Hilliard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.
This article develops a method for valuing contingent payoffs for a non-constant volatility process via a simple recombining binomial tree. The direct application of the technology provides a way to price, for example, American calls or puts governed by a stock price process with stochastic volatility. The stock price and volatility diffusions may have non-zero correlations. This feature allows model prices consistent with the volatility smile. Numerical estimates of the hedge statistics (delta, gamma, and vega) are obtained directly from the tree.