Risk Latte - My Volatility has Volatility

My Volatility has Volatility

Rahul Bhattacharya
September 5, 2011

A month or two back, an equity derivatives salesman was completely crestfallen when the trader changed his price for an exotic product at the last moment. When the salesman protested to the trader in anguish, the trader replied, "I cannot help, my volatility has volatility."

One of the vexing issues in the derivatives market is the phenomenon of "volatility of volatility" or vvol, to use an options trader's lexicon. 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 an option 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.

Vvol, or volatility of volatility, is a statistical concept which means that volatility of a series of random variables is not constant, but varies. Historical volatility is measured by the standard deviation of stock price returns around a mean. If this standard deviation itself has a standard deviation then we say that the stock prices have volatility of volatility. Statisticians call it by an awful sounding name "heteroskedasticity" and it is related to the fourth moment, kurtosis, of a normal distribution. Heteroskedasticity in a Gaussian (Normal) distribution makes a Normal probability distribution display the fourth moment, when there should be none. One of the main reasons for heteroskedasticity or varying volatility is that large difference in the size of observations in a time series. As my dear friend, philosopher and guide, Justin P., who taught me option trading, used to say that the only beast that you cannot tame is Statistics.

If the above is too esoteric, then think of a simpler example. Say, you are watching a plane take off from the nearby airport and trying to measure the distance it travels every 5 seconds. Initially, say, in the first 10 to 20 seconds as you see the plane lift off from the runway, your measurements will be quite accurate, maybe even to the nearest meter. Then as the plane starts receding farther and farther away from you, the accuracy of your measurements will fall. This is because of the increased distance between you and the plane, the atmospheric distortions, and many other physical reasons. This data, filled with measurement errors, will be a heteroskedastic.

In the financial markets, the notion of variable volatility, the fact that standard deviation of stock returns would have a standard deviation, has given rise to what is known as the paradigm of "stochastic volatility" or "random volatility". In fact, the only way that volatility of an asset can have volatility is if it were random and behaved in a random fashion just like the underlying asset itself. Stochastic volatility is in the rationale for the higher order greeks (option sensitivities), such as vanna and volga, that traders use to capture risks of financial derivatives and is also intricately related to another powerful concept in derivatives trading, “vega convexity”.

Since the market crash of 1987, traders have known that the primary reason for the higher price of out of the money than what is implied by the theoretical Black-Scholes model is the existence of volatility of volatility. However, what many traders find absolutely infuriating is that volatility of volatility is never known in advance and nobody really knows how to this quantity correctly. What is done in most banks is that the quants try to calibrate vvol from option prices using a stochastic volatility model. However, whether this reflects the actual volatility of volatility of the underlying asset remains strongly debatable.

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