In this paper, we introduce a new time series model having the 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  Monte-Carlo simulation technique for a parameter calibration of the model for the S \& P 500 option prices. By the calibration, we can see that the stochastic exponential tail makes the model better to analyze the market option prices  .
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 .
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by the calibration