Classical (It�� diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential L��vy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see \cite{Tank} for an overview), and more recently rough volatility models \cite{AlosLeon, Fukasawa}. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential L��vy models and fractional stochastic volatility models.

25 pages. Typos corrected, introduction updated, and title shortened