Inferring Volatility Dynamics and Risk Premia from the S&P 500 and VIX Markets

Abstract

This paper studies the information content of the S&P 500 and VIX markets on the volatility of the S&P 500 returns. We estimate a flexible affine model based on a joint time series of underlying indexes and option prices on both markets. An extensive model specification analysis reveals that jumps and a stochastic level of reversion for the variance help reproduce risk-neutral distributions as well as the term structure of volatility smiles and of variance risk premia. We find that the S&P 500 and VIX derivatives prices are consistent in times of market calm but contain conflicting information on the variance during market distress.

VIX term structure and VIX futures pricing with realized volatility

“…Such a pattern motivates the separation of the two types of premium with different parameters in the modeling, and also provides weak evidence of possible “market segmentation” between the short and long ends of the volatility term structures. Such market segmentation is mentioned in variance swap markets (Bardgett, Gourier, & Leippold, ) and motivates a nonlinear two‐component GARCH model for modeling the term structure of the variance risk premium proposed by Bormetti, Corsi, and Majewski (). We also provide results based on all of the available information by combining all likelihoods, and the main results remain unaffected.…”

Joint pricing of VIX and SPX options with stochastic volatility and jump models

“…However, we see that all the models generate a positive skew in VIX option implied volatilities when the Feller condition is relaxed. This contrasts the conventional Gatheral (2008), Baldeaux and Badran (2014) and Bardgett et al (2013). Also, notice how the implied volatility curves of the SV and SVJ models have the same shape, as the SVJ model only adds flexibility to the SPX return distribution and not to the terminal distribution of the VIX.…”

“…This paper aims to fill this vacuum. Bardgett et al (2013) study a similar subject, namely, the pricing performance of stochastic volatility and jump models based on time series of VIX and SPX options over a long period. Moreover, they estimate the variance risk premium of the model.…”

Instantaneous squared VIX and VIX derivatives

“…Further, we use the informative VIX term structure data to determine the mean‐reverting speed and sequentially calibrate the volatility and jump parameters in the ISVIX process to market prices of VIX options. As noted by Duan and Yeh (), Duan and Yeh (), and Bardgett et al (), the 30‐day VIX is not sufficient for estimating risk‐neutral stochastic volatility models. In fact, the two‐factor stochastic volatility model of ISVIX requires the use of VIX term structure data.…”

“…For example, Branger et al () compare consistent and log VIX models by focusing on both the first and the second moments of the VIX risk‐neutral distribution in addition to pricing errors. Bardgett et al () conduct a comprehensive analysis by combining time series of SPX, VIX, and their options data and employing an accurate approximation and a powerful filter method to get an estimation of a total of around 25 parameters and three latent variables at one time. They find that a stochastic central tendency of volatility and jumps in volatility are important in capturing volatility smiles in both the SPX and VIX markets and the tail of variance risk‐neutral distribution, respectively.…”