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Author: Manisha Goswami
Publisher:
ISBN:
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Languages : en
Pages :
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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.
Author: Manisha Goswami
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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.
Author: Arun Chockalingam
Publisher:
ISBN:
Category :
Languages : en
Pages : 30
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Book Description
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.
Author: Suchandan Guha
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Languages : en
Pages :
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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.
Author: Jonathan Ziveyi
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Languages : en
Pages :
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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.
Author: Carl Chiarella
Publisher: World Scientific
ISBN: 9814452629
Category : Options (Finance)
Languages : en
Pages : 223
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Book Description
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"
Author: Jean-Pierre Fouque
Publisher: Cambridge University Press
ISBN: 9780521791632
Category : Business & Economics
Languages : en
Pages : 222
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Book Description
This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.
Author: Carl Chiarella
Publisher:
ISBN:
Category :
Languages : en
Pages : 19
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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.
Author: Dimitrios Gkamas
Publisher:
ISBN:
Category :
Languages : en
Pages : 388
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Author: Carl Chiarella
Publisher:
ISBN:
Category :
Languages : en
Pages : 43
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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.
Author: Hak Min Lim
Publisher:
ISBN:
Category :
Languages : en
Pages : 59
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