The Variance Gamma (V.G.) Model for Share Market Returns

Madan, Dilip B.; Seneta, E.

doi:10.1086/296519

Cited by 1,357 publications

(778 citation statements)

References 19 publications

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“…Black-Scholes (Black and Scholes, 1973), Merton (Merton, 1976), Heston (Heston, 1993), Bates (Bates, 1996), Normal inverse Gaussian (NIG) (see e.g. Barndorff-Nielsen, 1997 and the references therein ), variance Gamma (VG) (Madan and Seneta, 1990), BNS (Barndorff-Nielsen andShepard, 2001, 2002), CGMY (Carr et al, 2002), finite moment log-stable (FMLS) (Carr and Wu, 2003), NIG-CIR model (Carr et al, 2003b), a class of time-changed exponential Lévy models (Carr and Wu, 2004;Huang and Wu, 2004), etc. the characteristic function is available in closed form.…”

Section: Option Valuationmentioning

confidence: 99%

Sequential calibration of options

Lindström

Ströjby

Brodén

et al. 2008

Computational Statistics & Data Analysis

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Robust calibration of option valuation models to quoted option prices is non-trivial but crucial for good performance. A framework based on the state-space formulation of the option valuation model is introduced. Non-linear (Kalman) filters are needed to do inference since the models have latent variables (e.g. volatility). The statistical framework is made adaptive by introducing stochastic dynamics for the parameters. This allows the parameters to change over time, while treating the measurement noise in a statistically consistent way and using all data efficiently. The performance and computational efficiency of standard and iterated extended Kalman filters (EKF and IEKF) are investigated. These methods are compared to common calibration such as weighted least squares (WLS) and penalized weighted least squares (PWLS). A simulation study, using the Bates model, shows that the adaptive framework is capable of tracking time varying parameters and latent processes such as stochastic volatility processes. It is found that the filter estimates are the most accurate, followed by the PWLS estimates. The estimates from all of the advanced methods are significantly closer to the true parameters than the WLS estimates which overfits data. The filters are also faster than least squares methods. All calibration methods are also applied to daily European option data on the S&P 500 index, where the Heston, Bates and NIG-CIR models are considered. The results are similar to the simulation study and it can be seen that the overfitting is a real problem for the WLS estimator when applied complex models.

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Section: Option Valuationmentioning

confidence: 99%

Sequential calibration of options

Lindström

Ströjby

Brodén

et al. 2008

Computational Statistics & Data Analysis

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show abstract

“…With ν = 0 and σ = 0, we obtain the Variance Gamma Process (VGP), which was introduced to Finance by Madan and co-authors [53,52,51].…”

Section: Classes Of Lévy Processes Of Exponential Typementioning

confidence: 99%

Pricing of Discretely Sampled Asian Options Under Levy Processes

Xie

2012

SSRN Journal

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Abstract. We develop a new method for pricing options on discretely sampled arithmetic average in exponential Lévy models. The main idea is the reduction to a backward induction procedure for the difference W n between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, W n is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method.

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“…The VG model [17] arises by taking T t to be a gamma process with drift defined by the characteristic triple (b, 0, µ) with b ≥ 0 (usually, b is taken to be 0) and jump measure…”

Section: Examples Of Lsbmsmentioning

confidence: 99%

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Hurd

Kuznetsov

2009

J. Appl. Probab.

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In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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The Variance Gamma (V.G.) Model for Share Market Returns

Cited by 1,357 publications

References 19 publications

Sequential calibration of options

Sequential calibration of options

Pricing of Discretely Sampled Asian Options Under Levy Processes

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

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