Fat Tail Options

The most common way to estimate the value of options is to use the Black-Scholes formula. If the price changes of a security are log-normally distributed, the Black-Scholes formula provides the theoretical price of so called European options on that security. Unfortunately, there is overwhelming evidence that price changes are not log-normally distributed. Instead, security price changes have what is often called fat tails. They also exhibit skewness. Fat tails can be modeled with so called Stable Distributions, also called Levy distributions and Levy-Pareto distributions. A Normal or Gaussian distribution is a special case of a stable distribution. The Cauchy distribution is another well known example of a stable distribution. In fact, the Gaussian and the Cauchy distributions are the only two stable distributions for which closed form mathematical formulas exist.

All stable distributions except the Gaussian distribution have unlimited variance. This means that large moves, "crash events", are much more common than would otherwise be expected. This is consistent with the behavior we observe in real capital markets.

There is great theoretical appeal with stable distributions. A stable distribution has a scale invariant feature - if the changes of a time series have a stable distribution, it does not matter what time scale you look at it. If daily changes have stable distributions, then weekly changes have stable distributions as well. One could almost argue that capital market models not using stable distributions are unrealistic (that is not completely true, though).

One of the implications of a capital market based on Stable Distributions is that it is not even theoretically possible to create perfect hedges using only the underlying security. In a crash scenario, the hedge will only partially protect the option. However, in most cases, the deviations from perfect hedge are small. There is still hope of being able to create perfect hedges for certain portfolios of options, although more research is needed on the subject.