Fractal Press

The problem in the world of finance is not just limited to using the Black-Scholes Model in options pricing. The “volatility smile” may or may not make up for the deficiencies in the model, but there are other far more insidious implications of assuming the world of the gaussian is relevant in the real world.

Consider the implications of infinite variance. Any distribution where the integral over the second moment does not converge (Pareto distribution with tail exponent < 2 will do it) will have an infinite or undefined variance. This has implications outside of the variance. It also invalidates the correlation measures, R-squared and other statistics.

A simple way to see this is to consider two random variables X and Y that are driven by stable infinite variance distributions. Let us see whether a correlation measure makes any sense. Remember that variance goes as X^2 and Y^2 and the correlation measures go as X*Y.

We have:

• X^2 ~= Variance of variable X
• Y^2 ~= Variance of variable Y
• (X+Y)^2 ~= Variance of combined stable distributed variables
• X*Y ~= Correlation measure

X*Y = 0.5*[(X+Y)^2 - X^2 - Y^2]

The sum of two random variables under stable distributions with infinite variance will result in a random variable driven by a stable distribution with infinite variance as well. So, all the terms on the right-hand-side of the equation above are infinite (undefined). This means the X*Y measure is undefined as well.

So correlation measures fall apart in domains with infinite variance. Distributions with the tail-exponent < 2 will have infinite variance, but given that estimating the tail exponent is notoriously difficult [1], any over-reliance on correlation measures is highly fragile to estimation errors.

In real-world samples of fractal domains (infinite variance domains), the correlation measure will show very high sensitivity to sample sizes and windows - it will jump all over the place with small changes in sample size.

Nassim Taleb’s arguments against the gaussian go far beyond those involving the volatility smile in options pricing. Once you identify a domain as possibly heavy-tailed be suspicious of any statistical measures: the “Sharpe Ratio,” R-squared, etc. There is little evidence that they work well in these domains. Over-reliance on these in decision-making (such as judging the performance of a hedge fund) can be disastrous.

[1] Levy-Stable Distributions Revisited: Tail Index > 2 Does Not Exclude The Levy Stable Regime (Weron 2001)

Note: I saw this framing of the connection between correlation measures and variance in one of Mandelbrot’s papers, I forgot which one.