Exploring the Theoretical Distributions that Best Fit the VIX

The VIX, also known as the CBOE Volatility Index, is a widely used measure of market volatility. It represents the market’s expectation of volatility over the next 30-day period. Traders and investors often rely on the VIX to gauge market sentiment and make informed decisions.

While the VIX itself is derived from the prices of options on the S&P 500 index, many wonder about the underlying theoretical distribution that best fits the VIX. In this article, we will explore some of the commonly used distributions and their suitability for modeling the VIX.

1. Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a popular choice for modeling various phenomena due to its simplicity. However, when it comes to the VIX, the normal distribution falls short in capturing its unique characteristics.

The VIX tends to exhibit skewness, meaning it has a non-symmetrical distribution. This is due to the fact that market volatility often spikes during periods of market stress, resulting in a right-skewed distribution.

2. Lognormal Distribution

The lognormal distribution is commonly used to model asset prices, as it allows for positive values only. However, when applied to the VIX, the lognormal distribution fails to capture the extreme values and fat tails observed in the index.

The VIX is known to experience occasional large spikes during market downturns, leading to a distribution that has more probability mass in the tails than what the lognormal distribution can accommodate.

3. Generalized Hyperbolic Distribution

The generalized hyperbolic distribution is a flexible distribution that can capture both skewness and kurtosis. It has been shown to provide a better fit to the VIX compared to the normal and lognormal distributions.

The generalized hyperbolic distribution allows for various shapes, including heavy tails and asymmetry. This makes it suitable for modeling the VIX, which often exhibits these characteristics.

4. Student’s t-Distribution

The Student’s t-distribution is another commonly used distribution for modeling financial data. It is known for its ability to capture fat tails, making it a potential candidate for fitting the VIX.

However, the Student’s t-distribution assumes that the data it models has finite variance. This assumption may not hold for the VIX, as market volatility can be unbounded during extreme events.


While no single theoretical distribution can perfectly capture the complex dynamics of the VIX, the generalized hyperbolic distribution has shown promise in providing a better fit compared to the normal and lognormal distributions.

It is important to note that the choice of distribution ultimately depends on the specific use case and the assumptions made about the underlying data. Traders and researchers should carefully consider the limitations and strengths of each distribution when attempting to model the VIX.

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