Fat Tail Option Calculator
The Fat Tail Option calculator makes use of Stable Distributions to estimate the theoretical value of European options. This provides a richer method with a better fit to real data and real capital market behavior than the common Black-Scholes formula. Especially, it can be used to take into account the existence of crash events.
The Fat Tail Option Calculator is available as a stand-alone application or as an Add-in to MS Excel. It is available as a free download.
See also the Fractal Financial Markets essay.
System Requirements
The Fat Tail Option Calculator requires Windows NT 4.0 or later or Windows 95 or later. If you want to use the Excel Add-in, Excel 97 or later is required.
As there are millions of complex numerical calculations involved in calculating the value of an option using stable distributions, we recommend that you use a modern computer with at least a Pentium 400 MHz processor.
Mathematical background
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.
A stable distribution is controlled by four parameters:
alpha - The alpha value ranges from 0 to 2 and measures the frequency of large moves. The lower the alpha value, the more large changes tend to occur. An alpha value of 2 is equivalent to the Gaussian distribution and a value of 1 represents the Cauchy distribution. For financial time series, alpha values between 1.8 and 2.0 seem to fit real data.
beta - The beta value ranges from -1 to 1 and measures the amount of skewness of the distribution. A distribution is skewed if large changes in one direction are more common than large changes in the other direction. A negative beta means that large negative changes are more common than large positive changes (and that small positive changes are more common than small negative changes). The Fat Tail Option calculator uses a beta value of -1. Other beta values give mathematical difficulties due to the unlimited variance, and are unrealistic.
mean - The mean value measures the average change. This is usually close to zero, and is calculated by the Fat Tail Option Calculator based on the arbitrage model used to value the options.
spread - The spread measures the size of changes. It is closely related to the volatility, but volatility can only be calculated for Gaussian distributions. For a Gaussian distribution, the spread is approximately equal to 0.7 times the volatility. An accurate estimate of the spread is the most important parameter in when valuing options. The Fat Tail Option Calculator expects a daily spread value. Typical values of this parameter for stock market indexes are between 1% and 2%.
The Fat Tail Option Calculator uses numerical integration rather than Monte Carlo simulations to calculate the value of options. This makes it faster and and makes the calculations more accurate, but it also makes it unable to take advantage of our advanced capital market models.
Estimating the Spread
In order to value options, you must first estimate the spread. Unfortunately, there is no way to directly observe the spread of a time series. Instead there are two major approaches. The first one is to calculate which spread value that is most consistent with actual market prices. This is also known as the implied spread. The other way is to use historical price changes. With historical price changes you can use a general purpose method, such as the maximum-likelihood method to estimate parameters. There is, however, a simpler method that provides pretty good results, based on the fractiles of stable distributions.
1. Calculate the 28% and 72% fractiles of the logarithmic changes (the value where 28% (and 72% respectively) of the changes are smaller than the value).
2. Calculate the difference of the fractiles.
3. Divide the difference by 1.654 to get the spread.
This method is almost completely independent of the values of the other parameters. A warning is in place when using historical values to estimate the spread. The spread varies from day to day. It is hard to make good predictions of future spread simply by looking at historical values.
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