SVQ · Surface models
SSVI (surface SVI)
Also called: surface SVI, Gatheral–Jacquier SSVI
SSVI extends SVI from one expiry to a whole implied volatility surface, with explicit conditions that rule out butterfly and calendar arbitrage.
Definition
SSVI, or surface SVI, is a parameterization of the entire implied total variance surface introduced by Gatheral and Jacquier (2014). Instead of fitting each expiry separately, it ties all expiries together through the at-the-money total variance curve.
Its main appeal is that simple conditions on a few parameters guarantee the surface has no static arbitrage, either within each expiry or across expiries.
Intuition
Fitting SVI expiry by expiry often produces slices that cross each other, which implies calendar arbitrage. SSVI avoids this by describing the surface as one object: each slice is the same basic shape, scaled by how much variance the market prices up to that expiry.
The trade-off is flexibility. Because shape parameters are shared, SSVI can miss idiosyncratic features of individual expiries, so practitioners sometimes use it as a stable backbone and add per-slice corrections.
How it's calculated
With θt the at-the-money total implied variance at maturity t and k = ln(K/Ft):
w(k, θt) = (θt / 2) · { 1 + ρφ(θt)k + √[(φ(θt)k + ρ)² + 1 − ρ²] }
Here ρ controls skew and φ(θ) controls how curvature changes with maturity. A common choice is a power-law family such as φ(θ) = η / [θγ(1 + θ)1−γ].
Gatheral and Jacquier show the surface has no calendar arbitrage if θt is non-decreasing in t and θφ(θ) satisfies a bounded-growth condition, and no butterfly arbitrage if θφ(θ)(1 + |ρ|) < 4 and θφ(θ)²(1 + |ρ|) ≤ 4 for all θ.
How to read it
A good SSVI fit gives a smooth surface whose slices never cross and whose implied densities stay non-negative, which makes it a safe base for Greeks, local volatility and risk-neutral densities.
Inspect residuals by expiry. Systematic misses on short expiries or around events signal where the shared-parameter structure is too rigid and a slice-level adjustment may be needed.
Common mistakes
- Assuming SSVI matches every slice as closely as independent SVI fits. It trades some fit quality for consistency.
- Ignoring that the no-arbitrage guarantees only hold when the parameter conditions are enforced during fitting.
- Using an at-the-money variance curve θt that decreases with maturity, which breaks the calendar condition immediately.
- Extrapolating far beyond listed strikes and expiries as if the fitted wings were observed.
- Comparing fitted SSVI parameters across dates after changing the functional form of φ, which changes what the parameters mean.
In Saqrid
SVQ offers an arbitrage-free global SSVI fit as one of its surface options, alongside per-expiry SVI with calendar clamping, a smoothed fit and SABR.
Open SVQ.