SAQRID

SVQ · No-arbitrage

Calendar arbitrage

Also called: calendar spread arbitrage, term-structure arbitrage

Calendar arbitrage arises when total implied variance falls as expiry lengthens at the same moneyness, implying a calendar spread with negative cost.

Definition

Calendar arbitrage is a violation of monotonicity across expiries. At a fixed forward moneyness, an option with more time to expiry must be worth at least as much as an otherwise identical shorter one, which in volatility terms means total implied variance cannot decrease with maturity.

When a surface breaks this rule, a calendar spread (long the longer expiry, short the shorter) would appear to have negative cost.

Intuition

Extra time can only add uncertainty, never remove it. If the market priced less total variance for a longer horizon than a shorter one at the same relative strike, you could own the long option, sell the short one, and hold a position worth at least zero for a credit.

Note that implied volatility can fall with maturity, as it does in an inverted term structure. What cannot fall is volatility squared times time.

How it's calculated

With total implied variance w(k, T) = σimp(k, T)² × T at log-moneyness k = ln(K/FT):

∂w(k, T) / ∂T ≥ 0 for every k

In discrete form, slices must not cross: w(k, T2) ≥ w(k, T1) for T2 > T1. Checking at fixed forward moneyness rather than fixed strike matters whenever interest rates or dividends move the forward.

Fitting methods enforce this either globally (as SSVI does through a non-decreasing at-the-money variance curve) or by clamping adjacent per-expiry fits so they cannot cross.

Example: a 30-day option at 30% implied volatility has total variance 0.30² × 30/365 ≈ 0.0074. A 60-day option at the same moneyness priced at 20% has 0.20² × 60/365 ≈ 0.0066, lower than the shorter expiry, so the pair violates the calendar condition even though each volatility looks plausible on its own.

How to read it

Crossing slices most often appear around short expiries and scheduled events, and in illiquid wings where quotes are wide. A genuine-looking violation on liquid strikes is worth checking for stale data, dividend assumptions or a wrong forward before anything else.

Event days can create legitimately steep term structures, but they cannot make total variance decline with maturity.

Common mistakes

  • Checking at fixed strike instead of fixed forward moneyness, which creates false violations for assets with carry.
  • Confusing an inverted volatility term structure with calendar arbitrage.
  • Ignoring discrete dividends, which can shift forwards enough to fake a crossing.
  • Assuming per-expiry fits are consistent with each other without testing.

In Saqrid

SVQ's per-expiry arbitrage-free SVI fit uses calendar clamping, and the global SSVI option enforces consistency across expiries.

The Arb Scanner flags calendar violations as model-consistency checks and says whether each looks real or a fitting artifact.

Open SVQ.

  • Butterfly arbitrageButterfly arbitrage exists when option prices are not convex in strike, so a butterfly spread would cost less than zero to own.
  • SSVI (surface SVI)SSVI extends SVI from one expiry to a whole implied volatility surface, with explicit conditions that rule out butterfly and calendar arbitrage.
  • Implied volatility surfaceAn implied volatility surface maps the implied volatility of options across strike (or moneyness) and time to expiry for one underlying.
  • Durrleman conditionThe Durrleman condition tests whether an implied variance smile implies a non-negative probability density, the smile form of no butterfly arbitrage.

Go deeper

Saqrid Institute lesson: No-Arbitrage in Practice: The Arb Scanner. Lesson text is open to read without an account.

← All terms