Fat Tails and Options Selling, Part I

One of the buzzwords in the financial community in 2008 was “fat tail” – the outlier that breaks price returns from a normal distribution. Perhaps the most famous comment came from David Viniar, CFO of Goldman Sachs (ticker: GS), who said their models “were seeing things that were 25-standard deviation moves, several days in a row.” For perspective, it would theoretically take several repetitions of the entire lifetime of our universe to have a reasonable chance of a 25-sigma event occurring.

It’s obvious that markets are not governed by a normal distribution, but just how much of a variance occurs in individual stocks? What about the diversification provided by a sector ETF? And how can this make you a better options trader?

Because financials have been at the heart of the storm, consider the daily change in the stock price of Citigroup (C) since the end of 1998 – over 2,500 trading days. The graph below shows a distribution of the real one-day percentage changes in Citigroup’s stock price, compared to a set of theoretical data created from a model that uses the rolling (trailing) 30-day volatility combined with a random Gaussian number generator. Real returns actually compare quite well with the model.

What happens if we zoom in, and focus on the supposed “fat tail” areas where models are supposed to break down? Again, the model is surprisingly accurate in sum.

Wasn’t modeling ability supposed to break down when stress-tested against large moves? Two points: first, no attempt was made to build the model to match reality; a reasonable recent period to estimate volatility was simply picked. Second, this is an average, so the actual error on a day-to-day basis could be significant.

Part of the reason the above model matched reality fairly well is that it actually used less data than the version you are about to see – the key factor is that by being entirely reliant on recent market signals, the model can adjust to rapidly rising volatility and account for the larger moves.

For comparison, here is a chart of cumulative volatility over the life of this Citigroup data set. Every day, we can say with increasing statistical significance that we understand the underlying volatility of Citigroup’s stock price.

Regenerating the original graph using new expected stock price changes based off the additional data yields the following – note how most of the returns clustered around the middle still match to some degree.

The real surprise, of course, comes from a zoom-in that focuses on the fat tail moves:

Using more data makes the forecasting worse, not better – an important principle to keep in mind across many disciplines. What does this say? The markets are not efficient, but they do typically have a good idea of near-term price action, and signal that quite clearly. We will more fully explore this phenomenon with diversified ETFs and its implications for options trading next time.