The Term Structure, Calendars and the Shape of Volatility

Term structure: one stock, many expiries, and the calendar

O8 treated implied volatility as a single number per stock, and that was a useful simplification and a real one — for one expiry. The moment a second expiry is quoted, the simplification breaks, because the two numbers differ. The pattern of those differences is the **term structure** of volatility, and it has an everyday interpretation: the front month carries the events that fall inside it and the back month carries everything further out, so a report, a court date or a central-bank meeting sits inside one expiry and raises its implied volatility relative to its neighbours. A $100 stock with earnings in six days showing 52% in the eight-day expiry and 38% in the thirty-five-day one is not saying the stock will move; it is saying that when this expiry dies, a specific known event will have happened inside it. Take the event out and the two curves are usually close to flat, tilted gently upward when the market is calm and downward when it is frightened. The **calendar spread** is the structure that trades this difference directly: sell the near expiry and buy the same strike in the far one, for a small net debit. What the position owns is the *gap* between the two expiries rather than any particular price. Because the two legs share a strike, they start with nearly opposite deltas that cancel, so the calendar is close to direction-neutral at the moment it is opened, and what remains is the difference between two thetas and two vegas. Near-dated options decay fastest — that is the shape of the curve O12 introduced — so the short leg bleeds faster than the long leg, and the position collects that difference as time passes. The reason it is a *structure* rather than a trade is the payoff: a calendar makes the most when the stock is near the shared strike as the near leg expires, and loses when the stock moves far enough that the near leg is in the money and the far one is not. So a calendar is a bet that the stock sits still through the first expiry, which is why selling the expensive event month and owning the quiet month is its classic use — and why it is the first lesson in this subject where being right about direction can lose and being wrong about it can win. A diagonal is the same construction with the strikes moved apart, so it adds a deliberate directional lean to the term-structure position: sell a higher-strike near-dated call and buy a lower-strike far-dated one and the position is paid for sitting still *and* keeps a bullish tilt. That extra tilt comes at a price, because the two legs no longer cancel deltas and the position now has a direction to be wrong about. The two structures together make the general point: once the chain has more than one row, the number of things a trader can hold a view about has grown — not just direction, not just the size of the move, but *when* the movement happens. That is the last dimension this subject adds, and it is the one that most often decides whether an otherwise correct view makes money. The same stock, two expiries — Front expiry, eight days, containing a report: Implied volatility 52% — the event is priced inside it ← · Back expiry, thirty-five days, clear of the report: Implied volatility 38% — no event, so no event premium · Calendar: sell the front, buy the back, same strike: Owns the gap; nearly delta-neutral at the open · What it needs: The stock to sit near the strike until the front leg dies · What it loses to: A large move, which makes the near leg expensive to buy back When adjacent expiries sit many points apart, look for the event before assuming an opportunity. The gap is usually a diary entry, not a mispricing.

Skew and the smile: the surface across strikes

Expiries are one direction of the surface. Strikes are the other, and the pattern there is the more surprising of the two: for most equity indexes, out-of-the-money **puts trade at a higher implied volatility than out-of-the-money calls**, at the same expiry. If implied volatility were a single forecast about the stock’s future movement, all the strikes would show the same number — a flat line across the chain — and the honest way to see what is happening is to ask what a flat line would mean: the market would be charging the same for a 5% fall and a 5% rise. It does not, and it has not for as long as the data exists. The pattern is called **skew** when it is asymmetric in this way, and a **smile** when both wings are bid above the at-the-money strike, which is what a single-name equity chain more often shows because a takeover or a biotech result can be just as violent upward as downward. The standard explanation is the **leverage effect**: when a stock falls, its debt-to-equity ratio rises and its equity becomes more volatile, so a decline is followed by more turbulence than a rally of the same size. Add to that the demand side — institutions and funds buy downside protection for reasons that have nothing to do with a forecast, and that steady bid is what makes the puts expensive. What follows for a learner is a discipline rather than a theory: **the skew is the price of insurance, not a prediction of a crash.** A chain showing 32% for the at-the-money strike and 44% for a 10%-out-of-the-money put is saying that a lot of people want that contract and few want to sell it, which is why a put can be expensive and still be the right purchase for a position that cannot afford the fall. The test is the same one O16 applied to the collar — what is the position paying for the floor, in dollars, and can it afford that every period it is renewed? Two practical consequences follow from the shape. The first is that the same protection can be bought at different prices along the wing: a put spread — long a nearer strike, short one further out — buys most of the protection for a fraction of the premium by giving away the extreme tail, which is precisely the part of the wing where implied volatility is highest. That is why put spreads and collars are the standard institutional hedge and an outright long put is not: the steepest part of the skew is the part the spread sells. The second is that skew has its own term structure, and it is usually steeper in the near months and flatter further out, because near-dated insurance is bought against specific events while long-dated insurance is bought against a general state of the world. A surface that stays steep all the way out is a market that expects trouble to persist; a surface that flattens out quickly is one where the fear is dated. Reading both directions together — expiry and strike — is what the practitioner calls reading the surface, and the useful habit is to look at it before choosing a structure rather than after. • Expiry direction: the front month carries the events inside it, so its implied volatility rises relative to the next expiry out. • Strike direction: equity puts usually quote above calls of the same expiry — skew — because of the leverage effect and the demand for protection. • Skew is the price of insurance, not a forecast; the test is the dollar cost of the floor, not whether it is “cheap”. • A put spread or a collar buys most of the floor for a fraction of the premium by selling the steepest part of the wing. • Skew has term structure too: steep near, flatter out means the fear is dated. The failure mode of the surface is treating either direction as a signal on its own. A high front-month implied volatility is not “expensive premium to sell” until the event inside it is identified, because the event is exactly what the seller is taking the other side of — and a steep put skew is not a warning that a crash is coming, it is a statement about who is buying and who is willing to sell.

Sources

• Term structure, calendars and diagonals as the same view at two expiriesNatenberg, “Option Volatility and Pricing”
• The volatility surface, the skew and the crash premiumGatheral, “The Volatility Surface: A Practitioner’s Guide”
• The leverage effect and why equity index skew is persistently negativeChristie, “The stochastic behavior of common stock variances” (Journal of Financial Economics)