First Principles: Volatility Skew - 1

Part 1: What is volatility skew and why does it exist?

The entire option chain, all the strikes, refer to the same underlying. Logically then, all strikes should have the same Implied Volatility.

Fig. 1 : Why do different strikes have different IVs?

But they obviously don’t!

This ‘skew’ in the IV vs Strikes graph is called Volatility Skew.

There are (/ could be) multiple reasons for this.

Let’s break it down, to /first principles/.

People invest in (the underlying stocks making up the) stock indices because of the benefits of diversification. Diversification helps reduce volatility of the portfolio while keeping most of the long term returns of the individual stocks making up the index.

But we have seen that this diversification breaks down in times of extreme stress and liquidity events. When such events occur, all stocks tend to go down together. No beta holds up, and correlations tend towards 1 — The index tends to behave like a single stock.

Moreover, the average magnitude of negative returns on down days, is larger than the average magnitude of positive returns on up days, on average.

Fig 2: Nifty Return Distribution since 2008-01-01

We have to remember that Implied Volatility is a BSM model concept. It exists only in the BSM world to normalize option prices and give us a scale to rank and compare the richness of option prices. BSM, however, assumes that the continuously compounded returns of the underlying stock (index) are normally distributed, which they obviously aren’t. It also assumes that the volatility of the underlying stock (index) stays constant throughout the life of the option, which it obviously also doesn’t.

This flaw of assumption is ‘fixed’ by option traders using a different IV at each strike. Since the underlying’s returns are actually not normally distributed in the real world, meaning ‘outliers’ are more common and higher in magnitude than a normal distribution would suggest, it would be wrong to price options assuming a normal distribution. Sellers and buyers agree about this, so they agree on higher prices than the model would otherwise deem ‘fair’. In much more interesting words:

Dirty Texas Hedge @HedgeDirty

Long gamma traders know in their bones tail events are not only more likely but will happen sooner. Bleeding theta doesn’t bother them because the asshole on the other side is a complacent laxbro who will get what’s coming. They have contrarian streaks and Taleb shrines

Dirty Texas Hedge @HedgeDirty

Long theta traders know in their bones neurotic pussies overpay for metaphysically impossible tail risks. Short gamma doesn’t bother them because the asshole on the other side is an over-Zerohedged paranoiac. They have colossal egos and insurance execs for fathers

NB: This is usually reversed for commodities, where the up-side vols are elevated due to the ‘panic effect’ arising from supply shortages driving the prices up.