Tempered stable processes with time-varying exponential tails

In this paper, we introduce a new time series model with a stochastic exponential tail. This model is constructed based on the Normal Tempered Stable distribution with a time-varying parameter. It captures the stochastic exponential tail, which generates the volatility smile effect and volatility term structure in option pricing. Moreover, the model describes the time-varying volatility of volatility and empirically indicates stochastic skewness and stochastic kurtosis in the S&P 500 index return data. We present a Monte-Carlo simulation technique for parameter calibration of the model for S&P 500 option prices and show that a stochastic exponential tail improves the calibration performance.