  In this paper, we introduce a new time series model having a stochastic
exponential tail. This model is constructed based on the Normal Tempered Stable
distribution with a time-varying parameter. The model 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. We empirically show the stochastic
skewness and stochastic kurtosis by applying the model to analyze S&P 500 index
return data. We present the Monte-Carlo simulation technique for the parameter
calibration of the model for the S&P 500 option prices. We can see that the
stochastic exponential tail makes the model better to analyze the market option
prices by the calibration.
