Abstract
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.
Acknowledgments
The authors are grateful for the helpful comments of anonymous referees and editors.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 The standard NTS distribution is defined by the NTS distribution with and
under the condition that
and denoted to
, in many literature including Kim and Kim (Citation2018), Anand et al. (Citation2016), Anand et al. (Citation2017), and Kim et al. (Citation2015). In this paper, we change the parameterization for the convenience.
2 The second Wednesdays are free from the effect of Monday and the Triple Witching day (Friday). This selection is motivated by Carr et al. (Citation2003). The authors used the second Wednesday option prices on each month of 2000 for parameter estimation.
3 See Coleman and Li (Citation1994, Citation1996) and https://www.mathworks.com/help/optim/ug/lsqcurvefit.html for the details.
4 The implied volatility allows Black–Scholes price to equal the model price.
5 See Byrd et al. (Citation1999, Citation2000), Waltz et al. (Citation2006) and https://www.mathworks.com/help/optim/ug/fmincon.html.
6 In this simulation method, we use Piece-wise Cubic Hermite Interpolating Polynomial implemented in Matlab.