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SVQ · Surface models

SVI parameterization

Also called: stochastic volatility inspired, raw SVI, SVI smile

SVI is a five-parameter formula for one expiry's implied total variance that fits equity smiles well and has known no-arbitrage conditions.

Definition

SVI, short for stochastic volatility inspired, is a parametric form for the implied volatility smile of a single expiry, introduced by Jim Gatheral in 2004. It models total implied variance as a function of log-moneyness with five parameters.

Its popularity comes from three properties: it fits listed equity smiles closely, its wings grow linearly in log-moneyness as theory requires, and conditions for avoiding arbitrage are well understood.

Intuition

Raw implied volatilities are noisy, sparse in the wings and sometimes inconsistent with each other. A good parametric form smooths the noise, extrapolates sensibly, and gives stable derivatives for Greeks and densities.

SVI's shape is a hyperbola: roughly linear in each wing with a smooth rounded minimum. Each parameter has a geometric meaning, which makes fits easier to inspect than a black-box spline.

How it's calculated

Raw SVI for total implied variance w = σ²T at log-moneyness k = ln(K/F):

w(k) = a + b · { ρ(k − m) + √[(k − m)² + σ²] }

  • a: overall level of variance.
  • b ≥ 0: angle between the wings (overall slope).
  • |ρ| < 1: rotation, controlling skew.
  • m: horizontal shift of the minimum.
  • σ > 0: smoothness of the vertex.

Variance must stay non-negative, which requires a + bσ√(1 − ρ²) ≥ 0. Wing slopes are b(1 + ρ) on the right and b(1 − ρ) on the left; Lee's moment formula bounds them, so b(1 + |ρ|) ≤ 2. Fits are usually done by weighted least squares on total variance, often with a reduced parameter set to stabilize the optimiser.

How to read it

Check the fitted parameters for plausibility: ρ near −1 means a very steep downside skew; σ near zero means a sharp, almost kinked minimum. Then check residuals by strike to see where the fit misses the market.

A good fit per expiry is not enough. Each slice must also avoid butterfly arbitrage, and adjacent slices must not cross in total variance, or the surface implies negative probabilities or calendar arbitrage.

Common mistakes

  • Assuming SVI is automatically arbitrage-free. Unconstrained fits can produce negative densities; the no-arbitrage conditions must be imposed or checked.
  • Fitting implied volatility instead of total variance with equal weights, which over-weights illiquid wings.
  • Letting each expiry fit independently, so slices cross and create calendar arbitrage.
  • Reading parameters across dates without noticing the optimiser jumped between local minima.

In Saqrid

SVQ can fit the surface with per-expiry arbitrage-free SVI with calendar clamping, or switch to a global SSVI, a smoothed fit or SABR.

Its Arb Scanner flags butterfly, calendar and Durrleman density violations as model-consistency checks and says whether each looks real or a fitting artifact.

Open SVQ.

  • 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.
  • Durrleman conditionThe Durrleman condition tests whether an implied variance smile implies a non-negative probability density, the smile form of no butterfly arbitrage.
  • Butterfly arbitrageButterfly arbitrage exists when option prices are not convex in strike, so a butterfly spread would cost less than zero to own.
  • Calendar arbitrageCalendar arbitrage arises when total implied variance falls as expiry lengthens at the same moneyness, implying a calendar spread with negative cost.
  • Implied volatility surfaceAn implied volatility surface maps the implied volatility of options across strike (or moneyness) and time to expiry for one underlying.

Go deeper

Saqrid Institute lesson: Arbitrage-Free Fits: SVI / Jump-Wing. Lesson text is open to read without an account.

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