How Variance-Gamma Mixtures Shape Short-Term Option Skew
Summary
The note explains why a Variance-Gamma model can produce pronounced short-dated option skew even when its longer-horizon distribution appears closer to Gaussian. It describes the process as a variance-mean mixture: log returns are formed by averaging Gaussian distributions over a positive, Gamma-distributed random factor. Option prices can therefore be viewed as weighted averages of Black-Scholes prices.
The explanation points to the model’s higher moments: skewness scales inversely with the square root of time, while excess kurtosis scales inversely with time. These moments grow as the horizon shrinks and diminish at longer horizons, where the distribution approaches Gaussian. The resulting short-term skew is real, though how it appears depends on the horizontal coordinate used to compare options, such as strike, time-scaled moneyness, or delta. The note gives a conceptual account rather than calibration data or a numerical example, so it does not establish how large the effect will be for a particular market or parameter set.
Key ideas
- Variance-Gamma returns can be represented as a mixture of Gaussian distributions with randomized mean and variance.
- The mixing factor is Gamma-distributed and has positive support.
- Option values under the model can be expressed as weighted averages of Black-Scholes values.
- Skewness and kurtosis increase as the time horizon becomes very short and fade toward Gaussian behavior over longer horizons.
- The visible shape of implied volatility skew depends on whether options are compared by strike, scaled moneyness, or delta.
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# Why doesn't Variance-Gamma process flatten volatility skew for short term options?
# Why doesn't Variance-Gamma process flatten volatility skew for short term options?
The Variance-Gamma (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?
## Answer by Kiwiakos (score 7, accepted)
https://quant.stackexchange.com/a/14269
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.