Smile Extensions: Displaced Diffusion, CEV, and Beyond

The standard LFM produces a flat implied vol surface: for a given caplet expiry, Black's formula gives the same vol for every strike. Real cap and floor markets exhibit a volatility smile or skew — out-of-the-money strikes command different implied vols than at-the-money options. A lognormal model that calibrates to ATM vols will systematically misprice OTM caps and floors, creating arbitrage relative to quoted prices. This lesson introduces three families of smile extensions for the LFM: (1) displaced diffusion (shifted lognormal), which adds a single parameter $\alpha$ generating a monotone skew and preserving analytical caplet formulae; (2) the constant-elasticity-of-variance (CEV) model, where vol scales as $F^\gamma$ for $0 < \gamma < 1$, producing a downward skew without shifting the support; and (3) a brief orientation to stochastic-vol extensions (SABR-LMM) for when a full smile (not just a skew) is needed. The lesson emphasises what each extension can and cannot match, and shows how each modifies the caplet pricing formula.