It is Winter break. Having recently finished reading Boyd and Vandenberghe’s Book on Convex Optimization I decided to take time out to read Hagen Kleinert’s paper on the Lévy distribution in finance. There’s a more complete description in Ch20. The paper shows that both the short- and long-time behavior of returns on the S&P500 index look like a Lévy distribution,

$$P(\Delta x|t) = \int_{-\infty}^\infty \frac{dp}{2\pi} e^{ip\Delta x - t H(p)}, \quad H(p) = -|c p|^\lambda $$ Here, \(\Delta x = x_t - x_0\) is the change in \(x\) over time \(t\).

This is the symmetric version of the general ‘Stable’ Lévy distribution implemented in scipy. Kleinert uses the alternate notation, \(\sigma^\lambda/2 = c^\lambda\).

It’s called stable because the characteristic function is obviously \( H(p) \), and adding random variables multiplies their characteristic functions together (the CF of the sum is the product of the CFs).

So, using \( dx(t) = \log(S(t+dt)/S(t)) \sim \) Lévy\((\lambda,c)\), \(\int_0^T dx \sim\) Lévy\((\lambda,T^{1/\lambda}c)\) is then the distribution of \(\log(S(t+T)/S(t))\)

It’s conceptually useful, and the graphs look nice, so what can we do with it?

Well, gains in the market from fancy prediction algorithms widely use Gaussian correlations (that is, Einstein’s Brownian motion on harmonic oscillators). Those correlations would allow you to ‘know’ the direction a stock is moving and then buy or sell to make a profit.

The Lévy distribution itself doesn't help with that directly. Instead, it is supposed to model the tails of the distribution better. Fits to real data show \(\lambda \sim 3/2\), and the tails fall off as \(\Delta x^{-5/2}\). That means the variance is infinite and Black Swan events are present in the model.

Using Bayesian methods to get correlations under the presence of Lévy distributions might help, but again the correlations are infinite, so we need to check other distributions (truncated Lévy, gamma, etc.).

Because it models fat tails, the Lévy distribution would be ideally suited for market risk analysis – for example to optimize a portfolio’s returns under a constraint that the probability of negative returns is less than one tenth of a percent or something. This brings us to Boyd and Vendenberghe’s book. Their ‘Example 4.8’ (p. 158) is a variant of the Markowitz portfolio optimization problem.

It might also help predict changes in a portfolio over time to help decide on a long-term allocation strategy. This latter problem is commonly thought of as retirement planning, and brings us to the next post – which will be on optimal control theory.