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.