Tempered stable processes with time-varying exponential tails

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.

Keywords:

• Option pricing
• Stochastic exponential tail
• Volatility of volatility
• Normal tempered stable distribution
• Lévy process

JEL Classification:

• C15
• C22
• G13
• G17

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 $\mu =0$ and $\gamma =\sqrt{1-{\text{β}}^2\left(\frac{2-\alpha }{2\theta }\right)}$ under the condition that $|\text{β}|<\sqrt{\frac{2\theta }{2-\alpha }}$ and denoted to $\text{stdNTS}(\alpha ,\theta ,\text{β})$, in many literature including Kim and Kim (2018), Anand et al. (2016), Anand et al. (2017), and Kim et al. (2015). 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. (2003). The authors used the second Wednesday option prices on each month of 2000 for parameter estimation.

3 See Coleman and Li (1994, 1996) 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. (1999, 2000), Waltz et al. (2006) 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${}^{TM}$.

Additional information

Funding

Kum-Hwan Roh gratefully acknowledges the support of Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korean government [grant number NRF-2020R1F1A1A01076001].