Taleb’s critique of the bell curve
© 2006, Reprinted with permission of Active Trader magazine (www.activetradermag.com)
If a set of values is said to have a normal distribution, it means the values adhere to the form of the standard “bell curve” (Figure A). In a bell-curve distribution, a one-standard-deviation calculation above and below the mean will contain approximately 68 percent of all the values in the set. For example, if the average of several values is 1.21 and the one standard deviation calculation is 0.11, 68 percent of the values will be somewhere between 1.10 (1.21 - 0.11) and 1.32 (1.21 +0.11).
However, Taleb doesn’t believe this approach applies to financial markets because market prices are arbitrary. While the bell curve may be a valid way to analyze the height of humans — the majority will be average height and the extreme outliers are rare events — in the markets, these outliers (crashes and sharp spikes) actually occur more often than the bell-curve distribution shows. When Taleb is looking for “fat tails,” he’s betting on the possibility an extraordinary move might occur in that instrument.
Rejecting the bell curve has huge implications for finance because standard risk-management tools (Sharpe ratio, value at risk, etc.) assume volatility represents real risk. Taleb argues against this assumption and is also critical of the Black-Scholes option pricing model, which uses volatility as one of six parameters of an option’s price.
Instead of using the Black-Scholes formula to price OTM options, Taleb uses a relatively new distribution theory called Power Laws, developed by Yale Professor Benoit Mandelbrot, author of The (Mis)behavior of Markets. Mandelbrot explains in his book, “A Power Law applies to positive or negative price movements in many financial instruments. It leaves room for many more big price swings than would the bell curve.”
In a bell-curve distribution, the number of large deviations is drastically reduced as values hit their extremes, causing traders to ignore these rare events. For example, the probability of a three-standard-deviation event is one in 740, and the probability of a six-standard-deviation event is one in 1 billion. However, the ratio between a five-standard-deviation move and a 10-standard-deviation occurrence is much lower than in the first instance (one in 3,500,000 vs. one in 130*1021), which means it becomes much less equal as the curve increases.
In contrast, a power law distribution has a constant ratio, meaning the change in probability between a sixand 12-standard-deviation event is the same as the ratio between a two-and four-deviation move. For power laws, as the numbers get higher, the ratio stays the same.
