Static-Arbitrage Constraints on the SVI Parameterization

The paper gives conditions under which the widely-used SVI parameterization of the implied volatility smile is free of butterfly and calendar-spread arbitrage, and exhibits a closed-form sub-class (SSVI) that is arbitrage-free by construction. We read it as the practical standard for representing a surface the desk can trust before any conditioning on it.

Raw SVI total variance w(k) = a + b{ ρ(k − m) + √((k − m)² + σ²) }; SSVI parameterizes w(k, θ) by the ATM total variance θ with constraints on the smoothing function φ(θ) that rule out butterfly and calendar arbitrage.

The contribution is less a model than a hygiene constraint: it separates surfaces that merely fit quotes from surfaces that are internally consistent as a risk-neutral object. A desk that conditions memos on skew or term structure must first know the surface it reads is arbitrage-free, or the conditioning inherits the arbitrage.

Any memo input derived from the surface — skew, term-structure inversion, wing behaviour — should be read off an arbitrage-free fit, not a raw interpolation of quotes.

Single-name surfaces below liquidity thresholds remain noisy; arbitrage-free fitting suppresses but does not eliminate the resulting parameter instability.