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SVQ · No-arbitrage

Durrleman condition

Also called: Durrleman's condition, g(k) density condition

The Durrleman condition tests whether an implied variance smile implies a non-negative probability density, the smile form of no butterfly arbitrage.

Definition

The Durrleman condition is a test, stated directly on the implied total variance curve of one expiry, for whether that smile has no butterfly arbitrage. It defines a function g(k) of log-moneyness; the smile is arbitrage-free in strike exactly when g(k) is non-negative everywhere (together with mild limit conditions on call prices).

It is the standard way to check a fitted SVI or SSVI slice, because it works with the fitted curve and its first two derivatives rather than with option prices.

Intuition

Butterfly arbitrage is equivalent to the implied risk-neutral density turning negative. Computing that density from prices requires a second derivative in strike, which is noisy. The Durrleman function rewrites the same requirement in terms of the smile itself, so you can test a parametric fit analytically and see exactly which log-moneyness region causes the problem.

Violations usually appear where the smile bends too sharply or its wings are too steep relative to its level.

How it's calculated

With total variance w(k) = σimp(k)² T and primes denoting derivatives in k:

g(k) = (1 − k·w′ / (2w))² − (w′² / 4)·(1/w + 1/4) + w″ / 2

The implied risk-neutral density in log-moneyness is then

p(k) = g(k) / √(2π w(k)) · exp(−d−(k)² / 2), d−(k) = −k / √w − √w / 2

so the density is non-negative wherever g(k) ≥ 0. In practice g(k) is evaluated on a fine grid across, and somewhat beyond, the listed strikes.

How to read it

If g(k) dips below zero in a narrow region, the fit is implying negative probability there. Look at whether that region is supported by liquid quotes or is an artifact of extrapolation, and whether a constrained refit removes it without hurting the rest of the smile.

A clean g(k) does not mean the surface is fully arbitrage-free; calendar consistency across expiries is a separate check.

A quick sanity check is the flat smile: if w(k) is constant, then w′ = w″ = 0 and g(k) = 1, so the density is the ordinary lognormal one. Violations therefore come only from slope and curvature, which is why steep wings and sharp minima are the first places to look.

Common mistakes

  • Checking only at listed strikes, which can miss violations between them or in the extrapolated wings.
  • Applying the formula to implied volatility instead of total variance.
  • Treating a fit-induced violation as evidence of mispriced options.
  • Forgetting the separate calendar condition across maturities.

In Saqrid

SVQ's Arb Scanner flags Durrleman density violations, alongside butterfly, calendar and vertical checks, as model-consistency checks and says whether each looks real or like 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.
  • SVI parameterizationSVI is a five-parameter formula for one expiry's implied total variance that fits equity smiles well and has known no-arbitrage conditions.
  • Risk-neutral density (Breeden–Litzenberger)The risk-neutral density is the probability distribution of the future underlying price implied by option prices, recovered from their curvature in strike.
  • 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.

Go deeper

Saqrid Institute lesson: Wing Arbitrage & Negative Density. Lesson text is open to read without an account.

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