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Skew Lognormal Cascade DistributionStephen Lihn / research archive

Quantitative finance
2008 to 2026

Data atlas / volatility

VIX Daily Log Return

A positive skew and a strong fit. Volatility rises fast and relaxes slowly.

Kurtosis 2 to 51990 to 2008Positive skew
VIX volatility index daily log return with a skew lognormal cascade fit.
The VIX fit from 1990 to 2008.

This is a very nice fit. The VIX is skewed to the positive side. Whenever there is volatility, the VIX rises very fast. It relaxes slowly when volatility subsides.

Because the VIX is mean reverting, the price increases do not stick. That distinction separates volatility from gold, where higher prices can persist.

Data source: Cboe volatility index data.

Related fits

Compare the shape.

10 Year Treasury Daily Log ReturnBond / 1962 to 2008XAU Index Daily Log ReturnCommodity / 1983 to 2008

Source shelf

Data and further reading

12 entries
  1. Cboe VIXOfficial volatility index information from Cboe.
  2. Cboe GVZThe Cboe dashboard for the gold volatility index.
  3. Federal Reserve Economic DataEconomic time series from the Federal Reserve Bank of St. Louis.
  4. Yahoo FinanceMarket data and historical quotes used for several series.
  5. Lihn 2008 on SSRNThe 2008 working paper on the scale invariant Brownian motion model.
  6. Lihn 2009 on SSRNThe working paper on higher order lognormal cascade distributions.
  7. Clark 1973Subordinated stochastic process model for speculative prices.
  8. Fernholz 2002Stochastic portfolio theory and the diversity index.
  9. Ornstein Uhlenbeck processA reference overview of the mean reverting process.
  10. Stable distributionBackground on stable distributions and heavy tails.
  11. Lognormal distributionDefinition and properties of the lognormal distribution.
  12. Stochastic portfolio theoryA reference overview of stochastic portfolio theory.