  We consider call option prices in diffusion models close to expiry, in an
asymptotic regime ("moderately out of the money") that interpolates between the
well-studied cases of at-the-money options and out-of-the-money fixed-strike
options. First and higher order small-time moderate deviation estimates of call
prices and implied volatility are obtained. The expansions involve only simple
expressions of the model parameters, and we show in detail how to calculate
them for generic local and stochastic volatility models. Some numerical
examples for the Heston model illustrate the accuracy of our results.
