2. Calendars: selling the short expiry and buying the long one

Derivatives 8: Combined positions

We come back to the same index, now past the event of the previous module. The implied of the 30-day expiry has returned to 20% and that of the 90-day expiry is at 22%: the curve has its calm shape again, with the long maturities slightly above the short ones. Level 7 explained why that positive slope is the normal situation and what breaks it; here it is taken as read and traded on.

Selling the short expiry and buying the long one

A long calendar is exactly what the module's title says: you sell a near-dated option and buy another of the same type, with the same strike, at a more distant expiry. If the strikes are different as well it is called a diagonal, and everything that follows applies to it with a directional layer on top.

The logic is simple to state and deceptive to grasp: an option's time value is consumed faster the closer expiry is. You sell the slice of time that melts fast and buy the one that melts slowly.

The sum, with the same index at 100

• 100 call at 30 days, implied of 20%: 2.3 points, that is 230 euros. This is the one you sell.
• 100 call at 90 days, implied of 22%: 4.4 points, that is 440 euros. This is the one you buy.
• Net debit: 2.1 points, in round numbers 210 euros per structure.

That debit is also your maximum loss. The long leg can never be worth less than the short one, because it carries the same right for longer, so the position is never worth less than zero.

The best outcome comes if in thirty days' time the index is pinned at 100. The short call dies worthless and you are left with a 60-day call with the implied at 22%, worth some 3.6 points, 360 euros. Against 210 paid that is 150 euros of profit, 71%.

Now the extremes:

• Index at 115 at the short expiry: the short is worth 15 points and the long 15.2. The spread has narrowed to 0.2 points, 20 euros. You have lost 190 of the 210, 90%.
• Index at 85: both calls are worth almost nothing and the spread is left at 0.1 points, 10 euros. You lose 95%.

Why this is not a bet on direction

Look at the two extremes again: you lose almost the same if the index rises 15% as if it falls 15%. The position has no opinion about the direction of the move, it has an opinion about its size and about where the strike is. In that it resembles a short strangle, and that is where the first serious confusion comes in.

It is not a short strangle. A short strangle has an open-ended loss and the calendar has its loss capped at 210 euros. And above all: the short strangle gains from the implied falling in all its legs, whereas the calendar needs the long expiry's implied to hold up.

What you have really bought is the slope

Here is the heart of the module. The long leg has almost twice the vega of the short one, because sensitivity to volatility grows with the square root of the time remaining. In our position, net vega is around 8 euros per percentage point of implied.

But the relevant quantity is not the level of volatility, it is the difference between the short and the long implied. You bought that difference: you sold an implied of 20% and bought one of 22%. Anything that narrows that gap works against you, and anything that widens it works in your favor.

That is why it is worth saying plainly: a calendar is a position on the term structure of volatility, not on the price of the underlying. The price only decides whether you are still near the strike; the curve decides how much you get paid for being there.

If the curve flattens or inverts

Three cases, all with the index sitting still at 100, which is the outcome you were after:

• Curve intact (long at 22%): the position is worth 360 euros. You make 150.
• Curve flattened (the long falls to 20% and the short stays at 20%): the position is worth some 324 euros. You make 114. You got the place right and collect 24% less.
• Curve collapsed (the long falls to 15%): the position is worth some 243 euros. You make 33 euros on 210 risked, and that after thirty days of waiting pinned in the best possible scenario.

Inversion is worse still, and you do not even have to wait. Suppose that the day after opening a scare arrives and the short implied jumps from 20% to 30% while the long one stays at 22%. The short call you sold for 230 euros is now worth some 344 and the long one is still at 437. The spread is worth 93 euros against the 210 you paid: an unrealized loss of 117 euros, 56% of the position, without the index having moved a single point.

That is the calendar's specific risk and it appears on no expiry payoff diagram, because it is not an expiry risk: it is a risk of interim valuation.

Real case: March 2020, the curve the other way round

In March 2020, with the pandemic hitting markets, the VIX closed in the region of 82 points, close to its all-time highs, and the volatility futures curve turned inverted for weeks: the near expiries traded above the distant ones, exactly the reverse of the usual. Level 7 described that regime.

Translated into our position: anyone holding long calendars watched the leg they had sold become more expensive faster than the one they had bought, and on top of that the underlying was moving vertically, which is the structure's second executioner. Both things at once, and neither of them visible in the payoff profile drawn at the outset.

Hull explains the theoretical framework and Natenberg brings it down to the desk, but the operational summary fits in one sentence: the calendar gets paid for time passing and for the curve keeping its slope, and in March 2020 neither of those two things happened.