Short-time implied volatility of additive normal tempered stable processes

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are beta, related to the variance of jumps, and delta, related to the smile asymmetry. It has been observed, in option market data, that beta = 1 and delta = -1/2. In this paper, we prove that the implied volatility of these additive processes is consistent, in the short-time, with the equity market empirical characteristics if and only if beta = 1 and delta = -1/2.