SSVI Surface Parameterization

The SVI parameterization fits one expiry's implied volatility smile with five parameters. A full volatility surface has many expiries, and fitting SVI separately to each one gives no guarantee that the resulting surface is free of calendar arbitrage, the requirement that total implied variance at a given strike never decreases as maturity increases, since otherwise you could construct a risk-free profit from a calendar spread.

Surface SVI (SSVI) solves this by linking every expiry's smile to a single shared "power-law" function of at-the-money total variance, so the whole surface is described by just a handful of global parameters instead of five per expiry. Because every slice is generated from the same underlying function evaluated at different maturities, SSVI can be shown, under a couple of simple parameter constraints (on the correlation-like skew parameter and the curvature parameter), to be free of both static arbitrage within a smile and calendar arbitrage across smiles, a guarantee that expiry-by-expiry SVI fitting doesn't come with automatically.

The tradeoff is flexibility: because all expiries share one functional form, SSVI can't perfectly reproduce every idiosyncratic wiggle a market's individual smiles might have, especially far from at-the-money. In practice it's used as a smoother, more robust global fit, sometimes as the base surface, with local expiry-specific adjustments layered on top only where the data demands it.