The Term Structure of Volatility

The smile describes how implied vol varies across strike; the term structure describes how it varies across maturity. Fix the moneyness (say at-the-money) and plot implied vol against expiry, and you get an upward or downward curve that carries just as much information as the smile, it tells you what the market expects volatility to do over time, and it is what calendar spreads and VIX-futures traders live on. Together, smile and term structure form the full implied-vol surface.

The two shapes

• Contango (upward-sloping): short-dated vol below long-dated. The normal, calm-market state, the market prices near-term quiet but assigns a higher long-run vol because something might happen over a longer horizon, and because vol mean-reverts up from a low spot level.
• Backwardation (downward-sloping / inverted): short-dated vol above long-dated. The stress state, a crisis or an imminent event (earnings, election, Fed) spikes near-term vol, while the market expects mean-reversion down to a calmer long-run level.

The shape is driven by mean reversion of volatility: if spot vol is low it is expected to rise (contango), if high to fall (backwardation), so the term structure is the market's forecast path of vol pulled toward its long-run mean, exactly the $\kappa, \theta$ mechanism of Heston.

Forward volatility and variance additivity

The clean way to reason about the term structure is through total variance, which is additive in time under independent increments. Define total implied variance to maturity $T$ as $w(T) = \sigma_{\text{imp}}^2(T)\,T$. The variance over a future window $[T_1, T_2]$, the forward variance, is the difference:

$\sigma_{\text{fwd}}^2(T_1, T_2) = \frac{\sigma_2^2\,T_2 - \sigma_1^2\,T_1}{T_2 - T_1} = \frac{w(T_2) - w(T_1)}{T_2 - T_1}.$

This is the volatility analogue of a forward interest rate: the vol the market implies for the period between two future dates, extracted from two spot vols. The forward volatility is its square root. A steep contango implies high forward vol; an inverted curve can imply forward variance that is low, or, if the curve is too inverted, negative, which is forbidden.

No-calendar-arbitrage: total variance must not decrease

The forward-variance formula reveals the fundamental constraint. Forward variance cannot be negative (you cannot have negative variance over a real period), so

$\boxed{\,\sigma_2^2\,T_2 \ge \sigma_1^2\,T_1 \quad\text{for } T_2 > T_1,\,}$

i.e. total implied variance $w(T) = \sigma^2 T$ must be non-decreasing in maturity. This is the calendar-spread no-arbitrage condition: a longer-dated option (same strike) must be worth at least as much as a shorter-dated one, because it contains all the shorter option's optionality plus more time. Note this constrains total variance, not vol itself, implied vol can and does fall with maturity (backwardation) as long as $\sigma^2 T$ still rises. A fitted surface that violates $\partial_T(\sigma^2 T) \ge 0$ admits a costless calendar arbitrage: sell the expensive short-dated, buy the cheap long-dated.

Trading the term structure: calendar spreads

A calendar spread, long a longer-dated option, short a shorter-dated one at the same strike, is the direct vehicle for a term-structure view. Its P&L profile:

• It is long vega on the back month, short vega on the front, so it profits when the term structure steepens (long vol relative to short) or when back-month vol rises.
• It is typically short front gamma / long theta in calm periods: the short near-dated leg decays fast, so a calendar collects theta if spot sits near the strike.
• Near a known event, the front-month vol is inflated (event premium); selling the front and buying the back is a bet the realized event move is smaller than the elevated front-month implied, an event-vol trade.

VIX futures are the macro version: the VIX futures curve is the term structure of 30-day forward variance, and its contango/backwardation drives the roll yield of vol ETPs.

Worked example

The 1-month ATM implied vol is 18% and the 3-month is 20% (contango). Total variances: $w_1 = 0.18^2\times\tfrac{1}{12} = 0.0027$ and $w_3 = 0.20^2\times\tfrac{3}{12} = 0.0100$. The forward vol over months 1–3 is

$\sigma_{\text{fwd}}(1\text{m},3\text{m}) = \sqrt{\frac{0.0100 - 0.0027}{3/12 - 1/12}} = \sqrt{\frac{0.0073}{0.1667}} = \sqrt{0.0438} = 20.9\%.$

The forward vol (20.9%) sits above both spot vols, the market implies the middle period will be more volatile than the near month, the signature of an upward term structure. Check the no-arbitrage bound: $w_3 = 0.0100 > w_1 = 0.0027$, so total variance rises, no calendar arbitrage.

What breaks in practice

• Event kinks. Earnings, FOMC, and elections put a spike in the term structure at a specific date, implied vol is not smooth in maturity; the maturities bracketing the event jump. Naive interpolation smears the event premium and mis-hedges it. Traders decompose implied variance into a diffusive part plus discrete event variance.
• The front end whips. Short-dated vol is far more volatile than long-dated (the curve pivots around a longer point), so front-month calendars carry big, fast-moving vega and gamma risk.
• Sampling and roll. Constant-maturity term structures (like the VIX's 30-day point) require interpolation that introduces roll and basis; the VIX-futures roll yield is the tradeable consequence.
• Curve shape mean-reverts unpredictably. Contango is normal but the transition to backwardation in a shock is abrupt and non-linear, short-vol carry trades that harvest contango are picking up nickels in front of that steamroller.

In interviews

Define the term structure (ATM implied vol vs maturity), name the two regimes, contango (calm, upward) and backwardation (stress, inverted), and tie the shape to mean reversion of vol. The technical core is variance additivity: forward variance $= (\sigma_2^2 T_2 - \sigma_1^2 T_1)/(T_2 - T_1)$, the vol analogue of a forward rate, and the no-calendar-arbitrage condition that total variance $\sigma^2 T$ must be non-decreasing (not vol itself, vol can fall). Be ready to compute a forward vol from two spot vols and to explain that a calendar spread trades the slope (long back, short front vega; short front gamma). Connect to VIX futures as the traded forward-variance curve.