Risk Latte - What is “Skew”?

What is "Skew"?

Rahul Bhattacharya
Mar 23, 2006

Once, at the turn of the millennium, there was a fierce argument between a trader and a derivatives sales guy of a commodity desk in a major bank. The trader had taken the volatility skew into pricing of a certain derivative product (a digital option) on the commodity and the price had come out very different from the salesmanˇ¦s initial estimate. Obviously, there was some explaining to do to the client and the salesman was desperate to sell the product to the client. The trader did his best to explain that there was a distinct curve around the ATM volatility and that the digital option embedded in the structure that the sales guy was selling needs to account for this volatility curve. Therefore, the price of the structure was coming out to be higher than expected. This was 2000 and ˇ§local volatilityˇ¨ was still not in fashion in many commodities market and traders would simply adjust the Black-Scholes pricing with the skew around ATM volatility to calculate the price of a digital (well, sort of back of the envelope).

Out of the money volatilities were higher - in fact, much higher in this case - than the ATM volatilities and the trader could clearly see that the out of the money volatilities were being constantly bid up. (A consequence of higher volatilities is higher options price, everything else being unchanged).

The salesman, let's call him Meek, said that there was nothing in the history of the price of the commodity which suggests there should be any skew-or asymmetry-in the distribution of the return of that asset. The sales guy was a statistician and his understanding of the skew was that a normal (Gaussian) distribution gets distorted on one side if there is a skew. Generally, a skew, or more appropriately a negative skew, means that the left side of a normal distribution- which accounted for negative return, or losses- is distorted, or is abnormal. And therefore, the left side is not a mirror image of the right side-the side that represents profits from a trade.

Meek was vehement in his argument:'if we can't find any distortions in the price/return data of the asset for the last couple of years then there was no rationale to use the skew in pricing of the ATM bet options". This was a sort of sacrilege to the trader and for him the salesman could as well be talking anthropology.

The trader, let's call him Vegard, replied:"if I look into your lifestyle and biological data for the last 35 years and conclude that you have not yet died, does it mean that I can say with very high confidence that you'll not die tomorrow*?" Ouch! That was a bit too much. But Vegard had a point.

In fact, Meek, the salesman, also had a point. He was totally right when he said that since for the last 5 years no asymmetries existed in the return (normal) distribution it didn't make sense to assume a distorted distribution for future. Skew is statistically defined as the third moment of a normal (Gaussian) distribution and measures the asymmetry in the distribution. If the negative (left) side of the return distribution has a distortion (the curve falls off more steeply than on the positive side) then it is indicative of the fact that there is a higher possibility of negative returns happening (relative to the positive returns). Translated into market terms this implies that there is a larger probability of negative returns (as compared to positive returns).

Vegard, the trader, thought that statistics as applied to markets was pure rubbish and what he was talking about, when he mentioned about the "curve around the ATM vols" was actually the volatility smile. To him the skew-or more accurately the volatility skew-was the higher value of the implied volatility the out of the money (OTM) options as compared to the at the money (ATM) options (something that is not predicted by the theoretical models of option pricing). This could mean- and it is only one of the many possibilities- that the investors may be expecting a crash and hence buying out of the money options as a hedge. This process was bidding up the out of the money volatilities. Since the market is placing a higher probability on a large negative return, i.e. a crash, this probability should be factored in the ATM options as well so that the trader doesn't end up selling the option cheap.

Both the trader and the salesman were correct and both were wrong.

*Quote: this is apparently very similar to a quote from Nassim Taleb's interview with a certain magazine. It is unclear if trader above paraphrased Taleb's quote. We are just reporting the verbal exchange as was reported to us by a third party.

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