Local volatility, conditioned diffusions, and Varadhan’s formula

We study classes of stochastic volatility models and derive asymptotic formulae for the associated local volatility surface. This gives new insight into the geometry of a reasonable local volatility surface, especially in extreme moneyness regimes. Specifically, we show that in the Stein–Stein model the squared local volatility grows linearly in the moneyness variable, in surprising agreement with Lee’s celebrated moment formula for the growth of implied volatility. Mathematically, our key tool is a version of Varadhan’s formula.