Hi:I've been trying to model stock returns with power-law tails on both sides with a probability distribution of two overlaid Pareto distributions. I define my quantile function thus:or with a location and scale parameter thus:x is the quantile (from 0 to 1) for which I want to find out the appropriate stock return, k determines the tail-heaviness, and a and b are location and scale.This gives quite interesting results, similar to the Gaussian but, as expected, with lots of extra kurtosis. Different from stable distributions, k can be larger than 2 and variance thus finite.Now this probability distribution is so simple that someone must have used it before. I wondered whether anyone of you know this to be a researched probability distribution, in particular whether the inverse of the quantile function above, i.e. the CDF, can be expressed, perhaps in terms of one of the common transcendental functions (intuitively, Gamma might be a candidate).

I did some very early-stage studies with a model similar to this back a number of years ago. The results where somewhat promising with the following observations:1. the k's were statistically different for upside and downside excursions2. Some stocks showed heavy or lighter tail deviations from the best-fit tail power-law model on one or both tailsI did not look for analytic solutions for the CDF, inverse, or moments because I was only interested in the properties of the tails. Ultimately, I found that the parameters for fitting the tails of the distribution were not persistent enough for my needs at the time (probably due to heteroskedasticity issues).

Yes, the k's appear to be different (alas, smaller, i.e. more heavy-tailed, for losses than for gains) for losses and gains. I'm playing around with this a little--one might either be satisfied that the simple three-parameter model I gave is obviously a better description of reality than the Gaussian, use two different k values, or use a k value "sweeping" from a small value for large losses to a large value for large gains.

QuoteOriginally posted by: LStabilityHi:I've been trying to model stock returns with power-law tails on both sides with a probability distribution of two overlaid Pareto distributions. The Student t distribution has power law tails and has often been used to model heavy tails in return distributions. There also exist skewed Student t distributions -- one can Google "skewed student t".

QuoteOriginally posted by: gelfandThe Student t distribution has power law tails and has often been used to model heavy tails in return distributions. There also exist skewed Student t distributions -- one can Google "skewed student t".True. In fact, it might turn out that the formula I gave actually can be reduced to the Student-t distribution. Still, the way I expressed it, it's quite easy to have two different k-values for the left and right tails, which appears to be useful to match the different tail fatnesses actually observed (partially caused by leverage, I'd presume), and my derivation (or rather heuristic justification) appears to be more defensible than using Student-t merely for it's mathematical properties--what it the interpretation of a degree of freedom in securities returns?