## Friday, August 5, 2011

### Volatility: "VXX Options are Different" (VIX; VXX)

The VIX is up .23 at 31.89, the ETN is up 86 cents at \$29.75
A smart piece of writing from TheStreet:

The original Black Scholes option valuation formula assumed a log normal distribution for the underlying probability distribution for stock price changes. Some later option models assume a binomial distribution. In both cases the underlying model suffers from a fat tails problem which seems to raise its head just about every time there is a serious correction or some company specific news. Such models are good enough to trade, but admittedly imperfect.

One option market does not fit this category though and that is the iPath VIX Short Term options (VXX_). To understand why we need to look a little bit at how the VIX, the underlying index, is calculated. Essentially the exchange looks at a series of options and constructs an elaborate weighted average volatility of those options. The exact details need not concern us. It is sufficient simply to realize that it is an average of the implied volatilities of the options.

In the original Black Scholes model the trade would calculate or estimate the volatility of the underlying stock and then enter that volatility in the formula to arrive at a fair value for the options. No estimate of future direction was needed, just the volatility. In arriving at the implied volatility the process is reversed. We start with the current price of the option and solve the model for the market's estimate of the future volatility of the stock. It is these implied volatility numbers that are averaged to arrive at the current VIX index.

The important point is that the implied volatility is really an estimate of the standard deviation of future expected stock price moves. As anyone who still remembers their introductory statistics course knows that the standard deviation is calculated by summing the squares of the price then taking the square root. But the problem is that summing the squares of a random variable gives you a Chi Squared distribution and not a normal one. So the fundamental assumption in the Black Scholes model is destroyed. Essentially the distribution is the square root of a Chi Squared distribution.

The implications for traders are profound. First of all, any fair value calculation that an option calculator gives you is highly suspect. You cannot trust it and especially not with your money. The problem does not end there. The log normal assumption extends into all of the other metrics that an option trader typically uses. For example the delta used to estimate how the option will track the underlying will be wrong. The implied volatility calculations will also be wrong and should not be trusted. The same holds true for the theta, gamma, vega and rho computations. They are all suspect in the VXX options....MORE