Options: A formula that is wrong, and indispensable

For centuries, options were traded without really knowing how to value them. Everyone priced them by guesswork.

In nineteen seventy-three, Fischer Black and Myron Scholes published a formula. The same year, Robert Merton generalised it.

That year, in Chicago, the first organised options exchange also opened. The market and its formula were born almost together.

The stroke of genius: an option can be copied by combining the share and a loan. If you can copy it, its price is the price of the copy.

To see it, there is a version you can draw: the binomial tree, published in nineteen seventy-nine.

A share is worth one hundred euros. Tomorrow, it will be worth one hundred and twenty, or eighty. Nothing else, for the example.

An option gives the right to buy it at one hundred. If the share rises, the option is worth twenty euros. If it falls, zero.

Now, buy half a share, so fifty euros, by borrowing forty. Say a zero rate, to keep it simple.

If the share rises: the half share is worth sixty, you repay forty, twenty is left. Exactly like the option.

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If it falls: the half share is worth forty, you repay forty, zero is left. Again like the option.

This copy cost you fifty minus forty: ten euros. So the option is worth ten euros. Otherwise, someone would gain without risk.

The most troubling part: we never needed to know whether the share was more likely to rise or fall.

Repeat the tree over thousands of tiny steps, and you get the Black and Scholes formula.

The formula needs five ingredients: the share price, the strike price, the time, the interest rate, and volatility.

The first four can be read on a screen. Future volatility, nobody knows.

So you do the reverse: start from the option's market price, and find the volatility that gives it back. That is implied volatility.

It tells you the agitation the market expects. The famous VIX index is one measure of it, for American shares.

If the model were right, all options on the same share would give the same implied volatility.

That is not the case. Options far from the current price often give a higher volatility. The curve smiles: that is the smile.

On stock indices, this smile tilts towards falls, especially since the crash of nineteen eighty-seven.

The smile is daily proof that the model is wrong. Big falls happen more often than it assumes.

In nineteen ninety-seven, Scholes and Merton received the Nobel prize. Fischer Black had died two years earlier.

Wrong, and yet used everywhere. Because it gives a common language: in this market, people no longer quote prices, they quote volatilities.

A model does not need to be true to be useful. You just need to know where it goes wrong.