2. Forward volatility: between two future dates

Derivatives 7: Volatility as an asset

The figure that appears on screen next to a 60-day expiry is an average, and like every average it hides as much as it shows. It does not describe day 60: it describes the average rate priced in for the full sixty days, starting tomorrow. And the first thirty of those sixty are exactly the ones covered by the 30-day expiry. In other words, the long figure contains the short one whole. Separating one from the other is the most useful calculation in this level.

An implied refers to a stretch, never to an instant

Think of it this way: if you are told a car made a six-hour journey averaging 90 an hour, you do not know how fast it was going in the fifth hour. If you are also told the first three hours were done at 60, you can work out that the last three were done at 120. Implied volatility works the same way, with one difference that changes everything: what gets averaged is not volatility, but its square.

Volatilities do not add: variances add

Uncertainty accumulates in proportion to time when it is measured in variance, which is volatility squared. That is why level 3 divided by 16 to go from annual to daily rather than dividing by 256: time enters through the square root. With that rule, the calculation is primary-school arithmetic.

Suppose an implied of 16 at 30 days and 18 at 60 days, both at the money.

• The 30-day stretch accumulates 16 squared, which is 256, multiplied by 30 days: 7,680 units of variance.
• The 60-day stretch accumulates 18 squared, which is 324, multiplied by 60 days: 19,440 units.
• The difference, 11,760 units, is all the variance the market is pricing for days 31 to 60, and for those alone.
• Spread over those 30 days it works out at 392 a day, and the square root of 392 is 19.8.

That is forward volatility: almost 20 for the second month, a figure that appears on no screen. The 18 that does appear is the average of a known 16 and a hidden 19.8. Notice that the intuition of subtracting, 18 minus 16 equals 2, does not merely give a different number: it gives a number that means nothing.

The case that really matters: an event in the second stretch

Now suppose a company reports earnings in 40 days. The 30-day implied, which does not contain that date, is at 20. The 60-day one, which does contain it, is at 26.

• First stretch: 20 squared is 400, times 30 days, 12,000 units.
• Long stretch: 26 squared is 676, times 60 days, 40,560 units.
• The difference is 28,560 units over 30 days, which gives 952 a day, and its square root is almost 31.

The screen said 26 and the market is pricing almost 31 for the stretch where the earnings fall. Five points of difference on a number of 26 are not a nuance: they are the difference between believing something is expensive and believing it is cheap.

Isolating the day of the event

You can be more precise still. Of those 30 days in the second stretch, twenty-nine are ordinary days that ought to look like those of the first stretch, and one is the earnings day.

Twenty-nine days at a rate of 20 contribute 400 times 29, that is 11,600 units. Since the whole stretch accumulated 28,560, the day of the event is left with 16,960 units: more than the other twenty-nine combined. The square root of 16,960 is around 130, and applying the rule of dividing by 16 that amounts to a move of the order of 8% in that session.

An honest warning is due: that last step mixes calendar days with trading sessions, and options markets use calendars that weight weekends and holidays differently. The figure is an order of magnitude, not a reliable decimal. But as an order of magnitude it serves perfectly well for the only thing that matters: comparing it with what that stock has actually done at its last twelve earnings releases. If on average it moved 5%, the market is charging dearly. If it moved 11%, it is giving the protection away.

Five ways to ruin the calculation

• Subtracting implieds instead of variances. It is the universal mistake and it always gives too low a figure when the curve is rising.
• Mixing different strikes. You have to take the same relative point, at the money or the same delta, at both maturities. Otherwise what comes out measures skew, not maturity.
• Treating every day as equal. A long weekend, a Christmas Eve or a month with two public holidays do not contribute the same variance as a normal working week.
• Confusing price with forecast. Forward volatility carries the risk premium of level 3 inside it. It is what it costs, not what is going to happen, and on average it overstates the subsequent move.
• Forgetting that the short stretch can contain something too. If the first expiry includes a rate meeting, the figure being subtracted is already inflated and the result comes out artificially low.

Real case: France, April and May 2017

The French presidential elections of 2017 were held over two rounds, the first on April 23 and the second on May 7. The euro options market reflected that calendar with remarkable precision: expiries before April 23 traded at ordinary levels, those containing the first round were clearly more expensive and those spanning the second round as well embedded an additional variance that could be backed out exactly with the arithmetic of this module.

The interesting part came on Monday April 24. With the first-round result already known, the euro opened around 2% higher against the dollar and the implied of the short expiries collapsed within hours, even though the second round was still pending and was two weeks away. The reason is the one this module teaches: the variance the market was pricing for the stretch between the two rounds depended on the first round leaving several outcomes open. When it left only one as likely, that variance stopped making sense and its price went with it.

All of this is education about how prices form, not advice or a recommendation to trade.