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Using Freidlin-Wentzell sample path large deviations theory, we characterise the small-time behaviour of probabilities of a process following an uncorrelated local-stochastic volatility model.
As a corollary, we determine the small-maturity behaviour of the implied volatility under this class of processes.
We characterise the asymptotic smile and term structure of implied volatility in the Heston model at small maturities and all strikes. Using saddlepoint methods we derive a small-maturity expansion formula for call option prices, which we then transform into a closed-form expansion (including the leading-order and correction terms) for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and all Heston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order result of~\cite{FJ09I} which found the relationship between the zero-expiry smile and the diffusion coefficients. We solve for in/out-of-the-money and at-the-money cases; in the latter case our proof involves subleading-order saddlepoint approximation along a suitable path of integration.