In this paper we develop approximating formulas for European options prices based on short term asymptotics, i.e. when time-to-maturity tends to zero. The analysis is performed in a general setting where stochastic volatility and jumps drive the dynamics of stock returns. In a numerical study we show that the closed form approximation is accurate for a broad range of option parameters typically encountered in practice. An empirical application illustrates its use in calibrating observed smiles of S&P 500 index options, and in getting new insight into the dependence of the volatility of volatility and jump size distribution on the spot volatility. We also demonstrate that the volatility of volatility function inferred from option prices agrees with its estimate from the time series of spot volatilities inferred from the same observed option prices.