Long-Expiry and Extreme-Strike Asymptotics — Lee's Moment Formula

The previous note established what happens to implied vol at short maturities. This note covers the complementary limits: what happens at very long maturities (the Fouque-Papanicolaou-Sircar regime, where volatility mean-reverts slowly), and what happens at extreme strikes (the Roger Lee moment formula, which links the slope of the smile's wings to the existence of moments of the return distribution). These are the two boundaries of the surface — the left-top corner (long-dated, far OTM) and the wings at any given maturity — and they impose structural constraints that no model can violate.

• state Roger Lee's moment formula and compute the maximum slope of the implied vol wings from a model's moment structure;
• apply the FPS slow mean-reversion expansion to compute the long-maturity ATM vol and skew;
• check whether a proposed surface satisfies Lee's moment bounds at the wings.

1. 3.1 Roger Lee's moment formula
2. 3.2 Checking Lee's bound in practice
3. 3.3 Long-maturity asymptotics (Fouque-Papanicolaou-Sircar)