In our last post, we asked:

Are we able to find any advantages from Lévy flights in order to improve activities related with the stock markets?

As we always deliver the goods, we’ve looked for an example of a particular kind of Lévy Flights’ usefulness. In particular, we’re interested in ‘Truncated Lévy flights’ (TLF). Before showing the results obtained with this new model, let’s quickly review the “problems” associated with making assumptions concerning normally distributed returns.

“Returns are normally distributed”

It’s well known that the normal distribution model fails to describe the fat tails of markets. If we assume normality, this implies that the probability of a three-sigma event (a return three standard deviations below its arithmetic mean) has a probability of only 0.13%, once every 1000 times. However, we have checked this with daily returns of the S&P 500 Index (from January 1964 to nowadays) and the empirical probability is 0.6%. Then, the probability of a three-sigma event is almost five times greater than one would expect under a normal distribution.

Unfortunately, the Lévy stable distribution model is not the perfect solution to this problem, as it has fat tails, but leads to an infinite variance, thus complicating risk estimation, so…

What about trying a particular case of Lévy flight, one which matches with the properties we want?

This post seeks to introduce TLF’s in order to estimate the downside risk of an asset class. The selected downside risk metric is the monthly value-at-risk (VaR) with a confidence level of 99%. Selecting TLF for this task is suitable because it’s a model with a distribution somewhere between the normal and stable distribution:

1. an appropriately fat…
2. …but finite tail (= finite variance).

As you know, it’s not our style to fill the post with theory. With that in mind, here’s a biography to learn what a TLF consists of: definition, good article, and another good article.

We want to estimate the VaR from the TLF Cumulative Distribution Function (CDF). It also needs some kind of estimate from the real returns of S&P 500 index. Unfortunately, we don’t have a closed formula to do this, but we’re not alone in front of this limitation. In this article, a Spanish guy solves this using a discrete approximation mixing empirical distributions, percentiles and integrals. We followed his proposed method (go to pages 37 and 38) varying the way of calculating the integrals. We chose the numerical method Lobato Quadrature to approximate them.

So how do they compare?

If we compare the obtained VaR by this method with the VaR we would have obtained assuming normality, we can check that normality is very “lazy” forecasting because it tends to return too dramatic values of VaR. In almost all periods, the Normal approximation is too far from the real value of the monthly return. Because of this, this VaR is always true; real return never falls below this estimate.

Real immediate returns of the estimate date are shown in the graph below:

Unfortunately, we can say that our proposal isn’t appropriate either since it violates the compliance. In this sense, TLF VaR would be a worse estimate than the Normal VaR. The TLF estimated VaR is not as “lazy” as the normal approximation, but it’s actually greater than the real return more than 1% of the time, which doesn’t fit the definition of VaR.

So at this stage, the search is still on for better approximations… See you next time!