Pricing American options under variance gamma

Hirsa, Ali; Madan, Dilip B.

doi:10.21314/jcf.2003.112

Cited by 103 publications

(73 citation statements)

References 12 publications

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“…The main insight is that matrix M in (30) is a sum of a Hankel and a Toeplitz matrix. (27) gives the following representation:…”

Section: Efficient Algorithmmentioning

confidence: 99%

“…They represent the state of the art for pricing options under the local volatility process. Generally speaking, however, the computational process with PIDE is rather expensive, especially for the infinite activity Lévy processes we are interested in, because they give rise to an integral in the PIDE with a weakly singular kernel [2,27,44].…”

Section: Introductionmentioning

confidence: 99%

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Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

2009

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We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional probability density functions. The computational complexity is O((M − 1)N log N ) with N a (small) number of terms from the series expansion, and M , the number of early-exercise/monitoring dates. This paper is the follow-up of [22] in which we presented the impressive performance of the Fourier-cosine series method for European options.

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“…The main insight is that matrix M in (30) is a sum of a Hankel and a Toeplitz matrix. (27) gives the following representation:…”

Section: Efficient Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

2009

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“…Recently, Lévy process models have become popular in the financial literature [1,2,3,4,5,6,7]. Option pricing, under exponential Lévy process with finite activity [5,8,9,10,11,12] and infinite activity [13,14,15,16,17] has been extensively studied. In these papers, various numerical methods were proposed for solving the option pricing Partial IntegroDifferential Equation (PIDE).…”

Section: Introductionmentioning

confidence: 99%

“…In the variance gamma case, a partially implicit method was suggested in [11], with the jump integral part split into a local and a non-local part. In [14], the jump integral part was also split into local and nonlocal parts with the local term computed using implicit timestepping and the nonlocal term computed using explicit timestepping.…”

Section: Introductionmentioning

confidence: 99%

Robust numerical valuation of European and American options under the CGMY process

Wang¹,

Wan²,

Forsyth³

2007

JCF

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We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting Partial Integro-Differential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

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“…Valuing American options in such models is however far from trivial, due to the weakly singular kernels of the integral terms appearing in the PIDE, as reported in, e.g., [2,6,10,11].…”

Section: Introductionmentioning

confidence: 99%

A Fast Method for Pricing Early-Exercise Options with the FFT

Lord¹,

Fang

Bervoets³

et al. 2007

Computational Science – ICCS 2007

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Abstract.A fast and accurate method for pricing early exercise options in computational finance is presented in this paper. The main idea is to reformulate the well-known risk-neutral valuation formula by recognizing that it is a convolution. This novel pricing method, which we name the 'CONV' method for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Lévy models, including the exponentially affine jump-diffusion models. For an M -times exercisable Bermudan option, the overall complexity is O (MN log(N )) with N grid points used to discretize the price of the underlying asset. It is also shown that American options can be very efficiently computed by combining Richardson extrapolation to the CONV method.

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Pricing American options under variance gamma

Cited by 103 publications

References 12 publications

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

Robust numerical valuation of European and American options under the CGMY process

A Fast Method for Pricing Early-Exercise Options with the FFT

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