The Skewness Implied in the Heston Model and Its Application

Abstract

In this paper, we provide an exact formula for the skewness of stock returns implied in the Heston (1993) model by using a moment-computing approach. We compute the moments of Itô integrals by using Itô's Lemma skillfully. The model's affine property allows us to obtain analytical formulas for cumulants. The formulas for the variance and the third cumulant are written as time-weighted sums of expected instantaneous variance, which are neater and more intuitive than those obtained with the characteristic function approach. Our skewness formula is then applied in calibrating Heston's model by using the market data of the CBOE VIX and SKEW.

Inferring information from the S&P 500, CBOE VIX, and CBOE SKEW indices

“…Adding the VIX term structure information to a model can improve the option pricing performance only for some models. This paper also extends the Heston (1993) model for the CBOE SKEW in Zhang et al (2017) and further confirms the observation in Liu and van der Heijden (2016), that is, the estimation errors of true skewness by using the CBOE SKEW method are very large.…”

“…Zhang et al (2017) use a moment‐computing approach to drive a pricing formula for the CBOE SKEW based on the Heston (1993) model, while this paper broadly considers three typical affine models and uses an MGF method to calculate the CBOE SKEW. Our MGF method allows us to obtain a closed‐form pricing formula for the CBOE SKEW based on any affine models, while Zhang et al (2017) moment‐computing approach will generate a lot of integral terms in the CBOE SKEW pricing formula if the model becomes very sophisticated.…”

“…Similar to the VIX, the SKEW measures the perceived tail risk of the distribution of SPX returns over a 30‐day horizon by using the model‐free method in Bakshi, Kapadia, and Madan (2003). Zhang, Zhen, Sun, and Zhao (2017) provide an exact formula for the skewness of stock returns implied in the Heston (1993) model and separately use the SPX return, the CBOE VIX, and SKEW term structures to calibrate the model. Unfortunately, they do not study the option pricing performance after adding SKEW information.…”

Modeling VXX under jump diffusion with stochastic long‐term mean

“…We need to estimate the , and structural parameters as well as the daily and parameters. To do this we modify the efficient two‐step iterative estimation of Christoffersen et al (2009), also used by Luo and Zhang (2012) and Zhang et al (2017), among others.…”

“…Our model differs from the standard SVVJ model as the long‐term mean level of the instantaneous variance is stochastic and mean‐reverting. Implementing a stochastic allows for more realistic transient changes in the VIX and VIX futures term structures (Luo & Zhang, 2012; Zhang & Huang, 2010; Zhang et al, 2010, 2017). Further, there is a vast literature showing that at least two stochastic factors are needed to accurately model volatility dynamics (Andersen, Benzoni, & Lund, 2002; Bardgett et al, 2019; Bates, 2000, 2012; Branger, Kraftschik, & Völkert, 2016; Christoffersen, Heston, & Jacobs, 2009; Duarte & Kapadia, 2017; Egloff, Leippold, & Wu, 2010; Huang, Schlag, Shaliastovich, & Thimme, 2019; Kaeck & Alexander, 2012; Mencía & Sentana, 2013; Todorov, 2010).…”