Tempered Stable Processes with Time Varying Exponential Tails

Abstract: 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.