Daily Brief: Tail Risk & Lévy Flight
Sharks eat fish, securities eat information, and both have fat tails.
Last week I discussed how the price of assets behaves similar to heat diffusion - and how by placing indicators, one can anticipate price fluctuations before they happen.
This is essentially what market makers do: liquidity provided by high-frequency trading firms (HFTs) like HRT, Jane Street, CitSec, among others, by implementing these strategies algorithmically. For the most part, these firms do a pretty good job of making markets efficient, and driving down the bid/ask spread on securities.
However, finance is slightly more complicated — and the world is not simply described by thermal Gaussian noise. I’ll briefly describe one of the caveats to diffusive pricing: tail risk.
TLDR; Tail Risk refers to rare, extreme market events that can cause massive, unexpected losses, far beyond what the traditional diffusive pricing models (like Black Scholes and MPT) predict. Most daily fluctuations are irrelevant, but tail risk focuses on those rare, high-impact events that can drastically reshape portfolios.
“The market only moves 10 times a year; everything else is just noise." - Unnamed biotech executive.
Ignoring tail risk leaves investors exposed to the unpredictable, where the biggest damage can happen. Before seeing this in action, let’s remind ourselves precisely what we mean by the “just noise”.
Gaussian Noise
Previously we discussed how assets roughly behave like random-walkers that in aggregate exhibit diffusion — here’s a plot from that newsletter:
Mathematically this means that the mean-square distance, scales linearly with time:
However, sadly, the world is not as simple as quadratic diffusion (above). Indeed, there is a vast landscape of possible interactions between asset processes, and price fluctuations that are not mutually independent — leading to super-diffusive behavior.
We can see this explicitly — consider the daily ticker data for the S&P going back almost a century:
Each day, the S&P experiences fluctuations in the price of its portfolio bundle. Over this 100 year history the root-mean-square (sometimes called a standard deviation) of daily percentage shift is about 1 percent up or down on any given day, with a slight bias up:
If asset prices were truly diffusive and small fluctuations were uncorrelated, then one would expect these deviations to be Gaussian (Bell-Curve) distributed, with standard deviations of about 1%:
In other words, if the distribution of ticker price was Gaussian, and exhibited perfect diffusive qualities, then 99.73% of all trading days would lie within plus/minus 3 percent - that means witnessing a trading day greater than 3% is 1-in-1000 (or about once every 4 years).
Is this the case? NO!
Tail Risk
For every small fluctuation up or down, one will also find major drops like the 20% drop of Black Monday (shown below):
So, what drives these large deviations? While the bundle of assets that underly the tradable securities of the S&P can have a wide range of correlations — there is one thing that the price of assets are ALL strongly correlated to: the bullish/bearish sentiment on Wall Street.
This can have major consequences.
Let’s take a closer look at the daily rises and drops of the S&P over the last 100 years. We can plot these relative shifts in a histogram (remember 1 std. dev. is about 1% up or down):
For those familiar with Gaussian noise, here is a summary of the relative probability (remember 1sigma for S&P is about 1%)
2sigma fluctuation is about 1-in-a-20
3sigma fluctuation is about 1-in-a-370
5sigma fluctuation is about 1-in-a-1,000,000
7sigma fluctuation is about 1-in-a-400,000,000,000
There have only been ~24,000 trading days since 1928 — so if the random noise was truly Gaussian distributed then one would expect about 70 3sigma fluctuations.
Again, is this the case? No!
In fact, there have been over 400 such trading days - or about 1 in 60!
It’s even more dramatic for 5 sigma fluctuations, for which there have been over 100 such examples — in other words, 4 in 1000 — which is a far cry from 1 in a million!
So what’s going on? This is what we mean by tail risk. By plotting the daily shifts in a histogram we can clearly see that the sample is more closely modeled by a power law distribution with fat tails, rather than a Gaussian distribution with thin tails:
So we’ve learned something interesting about the stochastic behavior of securities — they are fat tailed — meaning they are incredibly sensitive to large deviations away from the mean value. We can get an even closer look at this by plotting these distributions on a log scale:
While both Gaussian noise and power law distributions both decrease in probability as one moves out to the extremes — the power laws hold much more space at the bottom, thereby permitting far tailed Black Swan events to wreak havoc on our financial systems.
Lévy Flight and Hunting Patterns
There is a term for random variables that have a tendency to jump in an uncontrolled (non-Gaussian) manner: physicists and mathematicians call this Lévy flight.
This type of random walk is ubiquitous in the world of hunting and foraging as it allows animals to seek out food, without getting stuck too long in food deserts. Sharks are a classic example of animals whose hunting patterns obey a path resembling this type of fat tailed randomness:
To get a better sense of how Levy flight differs from normal Brownian motion, below is a simulated example of Levy flight in 2 dimensions compared to some Gaussian random walks:
This happens to be a much better description of the price fluctuations for individual securities. So we should think of individual securities as Levy Flights in one dimension (shown below):
As we can see from these completely random simulations, the large jumps and drops of a 1-dimensional Levy flight much more closely resemble real-world security price data. In the real world, the jumps could be a beat and raise announced at quarterly earnings — whereas the hard drops might be a door blowing off the side of an airplane.
While sharks jump around looking for bigger fish, securities jump around seeking out better information about their market price.
Don’t believe me? Here’s a real world example below:
Now that’s Lévy flight!
In fact, this behavior doesn’t just occur for index funds like the S&P, but occurs across all strata of the capital stack. Indeed, the size of the company doesn’t seem to affect the non-Gaussian behavior of assets. I’ve plotted the daily jumps over the past 4 years for a Large (NVDA), a Mid (TMDX), and a Small Cap (SCPH) company:
The black and gray bands mark 2sigma and 4sigma fluctuations respectively. It’s worth noting that the Market Cap seems to be inversely proportional to the number of large fluctuations outside the typical range of volatility for each security. I'll explore the relationship between market cap and volatility in more detail in a future newsletter.
In closing…
Tail risk is the critical outlier that standard financial models miss. While traditional pricing models typically follow Gaussian-like diffusion, extreme market events occur far more frequently than predicted, driven by fat-tailed distributions.
Lévy flights better describe these jumps, with rare but massive fluctuations impacting portfolios. In short, most daily noise is irrelevant—it's the rare, high-impact events that matter. Ignore them at your peril.
This is Math Meets Money.