The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model

We characterize the asymptotic smile and term structure of implied volatility in the Heston model at small maturities. 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 Forde and Jacquier [Int. J. Theor. Appl. Finance, 12 (2009), pp. 861--876], which found the relationship between the zero-expiry smile and the diffusion coefficients.