Fat Tails in the Markets
April 19, 2012
One of the vexing issues in the derivatives market is the phenomenon of "volatility of volatility" or "vvol". Every trader’s volatility estimate has volatility and he knows that this is what gives rise to "fat tails" and the "gamma of the gamma", the fourth moment of a probability distribution. Many seasoned traders will tell you that "fat tails" – the probability of rare events, like stock markets falling by 10% in a day, happening far more frequently than what is predicted by the Normal probability distribution – don’t have to be necessarily caused by blown out variances. A fat tail can occur even when an asset has a low volatility but a very high volatility of volatility.
Option traders believe that fat tails are caused due to a variable volatility. In a Black-Scholes world, where volatility is constant, there would be no fat tails. However, the fact that volatility is variable (and stochastically so) gives rise to volatility of volatility. High volatility causes a market to move towards the tail of a probability distribution. According to many traders one of the main reasons for the higher price of in the money and out of the money options than the Black-Scholes value is because of the existence of volatility of volatility which causes the tails of the distribution to become fat.
Kurtosis – Statistical Interpretation of Fat Tails
Kurtosis is the fourth moment of a probability distribution. If is a random variable drawn from a probability distribution with the sample size and the first and the second moments are and then the kurtosis is given by the following expression:
This fourth moment measures the peakedness of the distribution and the heaviness of the tails of the distribution. In the financial markets and option trading parlance it is related to the notion of "fat tails". Heavy tails occur when the distribution exhibits positive kurtosis. In this case, most of the observations lie towards the extreme end of the probability distributions and there are only a few observations near the mean. The distribution therefore has distinct, sharp peak but it falls off very fast around the mean and make the tails heavy. Negative kurtosis is the opposite of this where most of the observations are clustered around the mean. Here, the mean is flatter.
Reference: The Book of Greeks, Rahul Bhattacharya, CFE School Publication (2012)
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