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The VG process, from my inexpert point-of-view, seems to nearly perfectly model equity distributions.

For longer term options, there is little to no volatility, skewness, or kurtosis parameter skew.

For shorter term options, it is true as the authors claim that the VG model provides no more resolution than Black-Scholes-Merton since the skewness and kurtosis parameters tend to 1, and the volatility skew reappears.

What is the mathematical explanation for this phenomenon?

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up vote 3 down vote accepted

VG belongs in the family of variance-mean mixture models. Given a horizon $T$ the distribution of log-returns $f$ is a mixture of Gaussians $f_G$ with randomised mean and variance. The randomisation density is $g$ and its mean and variance increase with $T$. For the VG process this randomised factor is Gamma-distributed.

More concretely, denote with $f_G(x;\mu,\sigma^2)$ the Gaussian density, and with $g(s;\theta,T)$ the mixing density which has positive support. Then the log-return density is given by $$ f(x;\mu,\sigma,\theta,T) = \int_0^\infty f_G(x;\mu s,\sigma^2 s)\ g(s;\theta,T)\ ds $$ It follows that option prices can be written as a weighted average of Black-Scholes prices.

The higher moments (see p85) are of the form $$\text{skewness}=c_1/\sqrt{T}\text{, and kurtosis}=3+c_2/T$$ Hence as $T\rightarrow 0$ they both go to infinity, while as $T\rightarrow\infty$ the distribution becomes Gaussian.

The presence of higher moments for small $T$ manifests itself as a skew of short-term options. However, how pronounced this skew is will depend on what you have on the x-axis, namely strike price, moneyness standardised with time, or Delta.

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That is really an excellent answer! This is not my specialty, yet you zeroed in on the heart of the issue that even I could get it immediately. Thank you so much! –  user6500 Aug 3 at 15:23

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