Arbitrage-Free Constraints and the SVI Parameterisation

Interpolating and extrapolating an implied vol surface requires care: most natural functional forms introduce static arbitrage — portfolios of European options that have non-negative payoff in all scenarios and a negative price today. A surface with calendar spread arbitrage allows you to make risk-free money by selling the cheap short-dated call and buying the expensive long-dated one; a surface with butterfly arbitrage contains options whose prices violate the non-negativity of the risk-neutral density. Any model or parameterisation used in production must rule both out.

This note derives the three binding conditions (density positivity, no calendar arb, no butterfly arb) in terms of total variance $w(k, T)$, and then presents the SVI (Stochastic Volatility Inspired) parameterisation, a five-parameter per-slice functional form that the industry uses to interpolate the surface cleanly.

Learning objectives. By the end of this note you should be able to: