hard · Quantitative Finance derivatives
An implied-volatility surface is quoted in total implied variance w(k,T)=σ_imp^2(k,T),T as a function of log-moneyness k=ln(K/F_T). To be free of static (calendar and butterfly) arbitrage, the surface must satisfy specific conditions.
Which pair correctly captures the necessary no-arbitrage requirements?
- w must be increasing in T for every fixed k (calendar), and the Durrleman / Gatheral 'g-function' g(k)≥ 0 at every slice (butterfly)
- σ_imp, not w, must be increasing in T, and w must be globally convex in k instead, for every fixed maturity slice.
- w must be increasing in T at fixed strike K, not at fixed k, and w must instead be monotone in k so as to rule out negative call spreads.
- w must instead be concave in T to reflect mean reversion, and partial^2 w/partial k^2≤ 0 is what's needed to keep the density nonnegative everywhere.
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