3. The volatility smile: where the model stops adding up

Derivatives 3: Volatility

The previous module talked about the implied as though it were one number per asset and expiry. It is not, and that fact is probably the most interesting crack in the whole theory of options: the moment when the market tells an elegant model that the world is not like that.

What the model promised

The classic model assumes that the underlying's price moves continuously, without jumps, and that its returns follow a normal distribution with constant volatility. If that were true, there would be a single true volatility per asset and horizon, and every option of the same expiry should throw out the same implied, whatever its strike price.

Checking it is trivial: you take a whole expiry's option chain and back out each option's implied. The same figure does not come out. It never has.

What actually comes out

With a stock index at 100 and expiry a month away, a typical profile would be this: the put option with a strike of 90 trades with an implied of around 25 points, the one at the money hovers around 17, and the call option with a strike of 110 stays at around 15. The figures are illustrative, but the shape can be seen on any given day.

That curve is not a symmetrical smile: it is a lopsided grimace sloping down from left to right, steep on the low-strike side and almost flat on the high-strike one. In market jargon it is called the skew, and the word travels unchanged into every language. The smile proper, with both ends lifted equally, does show up in currencies, where an exchange rate can jump in either direction. And in some commodities it is inverted: in natural gas what frightens people is scarcity, so it is the call options that are dear.

Three reasons and none of them is a whim

• Falls are nothing like rises. An index rises in small steps over months and falls in a matter of days, and when it falls, everything falls at once: the correlation between shares shoots up precisely when diversification is needed. There is also a leverage effect, because when the value of a company's shares collapses its debt weighs relatively more and the stock becomes more volatile. That asymmetry does not fit in a constant-volatility model.
• There is structural demand for cover. Pension funds, insurers and managers buy put options systematically because their mandates or their regulators require it, and there is no comparable demand for far-off call options. Where there are permanent buyers and scarce sellers the price rises, and since it is expressed in implied terms, the implied rises.
• Real tails are fatter. The model assumes a normal distribution, in which extreme events are astronomically improbable. Markets produce them with an uncomfortable frequency, something Nassim Taleb has been pointing out for decades. Since you cannot change the formula everybody uses, you change the number you feed into it.

1987: the date the grimace appeared

Before October 1987, the curve of implieds by strike was reasonably flat: far-off protective options did not trade with a stable mark-up.

On October 19, 1987 the S&P 500 fell around 20% in a single session and the Dow Jones 22.6%. Under the classic model's assumptions, with the volatility of those months, a move like that is more than twenty standard deviations away: an event whose probability is so minuscule it would not fit inside the age of the universe. It happened on a Monday.

Since then the skew has appeared in equity indices and has never gone away. Mark Rubinstein christened it with a word that explains it well, crashophobia. Emanuel Derman and others spent years building models that would reproduce that shape rather than contradict it.

A point of precision is worth making, because it often gets lost: the skew is not a flaw waiting to be fixed. It is the practical way in which the trade uses a model it knows to be incomplete. The formula is kept because it serves as a common language and a different volatility is fed into each strike so that the price matches the market's. The implied stopped estimating future volatility and became a coordinate of the price.

The surface: add the expiry

Each expiry has its own curve. Stack them all up and you get a volatility surface, with two axes, strike and horizon, and an implied at every point. Three regularities:

• The skew is steepest in the short expiries and flattens out in the long ones. At one month, the difference between a far out-of-the-money put and the at-the-money one can be several points; at two years, far smaller.
• The term structure is usually upward-sloping in calm periods and inverts in frights, with the short end shooting above the long. It is the same phenomenon as the inversion of a yield curve and it means the same thing: the problem is perceived as immediate.
• A dated event deforms the surface only in the expiries that contain it, leaving a step between the one before and the one after.

A real case: the legacy of October 1987

What made that Monday instructive was not the magnitude, but the mechanism: part of the market was using portfolio insurance strategies that replicated a put option by selling futures automatically as the market fell. As it fell, they sold; as they sold, it fell further. A model that assumed continuous movement came up against a market where the hedging itself generated the jump.

Almost forty years later the imprint is still there in the prices. On any quiet session, the S&P 500 put option with a strike 10% below the index and a month of life trades with an implied several points above that of the at-the-money option. Nobody is predicting a crash that day. Insurance is being charged for against something that has already happened once.