The risk premia embedded in index options

Abstract

We study the dynamic relation between aggregate stock market risks and risk premia via an exploration of the time series of equity-index option surfaces. The analysis is based on estimating a general parametric asset pricing model for the risk-neutral equity market dynamics using a panel of options on the S&P 500 index, while remaining fully nonparametric about the actual evolution of market risks. We find that the risk-neutral jump intensity, which controls the pricing of left tail risk, cannot be spanned by the market volatility (and its components), so an additional factor is required to account for its dynamics. This tail factor has no incremental predictive power for future equity return volatility or jumps beyond what is captured by the current and past level of volatility. In contrast, the novel factor is critical in predicting the future market excess returns over horizons up to one year, and it explains a large fraction of the future variance risk premium. We contrast our findings with those implied by structural asset pricing models that seek to rationalize the predictive power of option data. Relative to those studies, our findings suggest a wider wedge between the dynamics of equity market risks and the corresponding risk premia with the latter typically displaying a far more persistent reaction following market crises.

Tail risk premia and return predictability

“…We also performed predictive regressions where VRP t and VIX t 2 were not averaged over the month with the results being very similar to the ones reported below for their monthly-averaged counterparts. 41 These differences between the VIX and LJV mirror the differences between the option-based estimates for the time-varying diffusive and jump intensity risks in the fully parametric three-factor stochastic volatility model recently estimated by Andersen, Fusari, and Todorov (2015). 42 Following a number of recent studies in the empirical asset pricing literature, we will refer to À VRPt as the variance risk premium, and correspondingly, VRPt À LJV t as the part of the premium due to "normal" sized price moves.…”

“…Hence, in our empirical investigations, we restrict our attention to the "special" compensation for jump tail risk. 14 Andersen, Fusari, and Todorov (2015) have recently estimated the separate components based on a standard two-factor stochastic volatility model augmented with a third latent time-varying jump intensity factor. 15 Intuitively, for τ↓0,…”

“…Santa-Clara and Yan (2010), in particular, find that an estimate of the equity risk premium due to jumps, as implied from options and a one-factor stochastic volatility jump-diffusion model, significantly predict subsequent market returns. Similarly, Andersen, Fusari, and Todorov (2015), relying on a richer multifactor specification, find that a factor directly related to the risk-neutral jump intensity helps forecast future market returns. Allowing for both volatility jumps and self-exciting jump intensities, Li and Zinna (2014) report that the predictive performance of the variance risk premium estimated within their model may be improved by separately considering the estimated jump component.…”

Herding and Stochastic Volatility

“…Andersen, Fusari and Todorov, 2015). By this procedure one finds a set of structural parameters for all short term options on a chosen day.…”

“…A large literature on affine stochastic volatility models has emerged focusing on improving the path-breaking model of Heston (1993): models that allow for jumps in the dynamics of the price of the financial asset, in order to account for large moves such in the case of crashes (Bates, 1996;Bakshi, Cao and Chen, 1997), models allowing that the long-run variance is itself a stochastic process, modeled as a diffusion process or as a discrete state Markov process (Bardgett, Gourier and Leippold, 2013;Kaeck and Alexander, 2012), models that allow for jumps in the dynamics of the variance (Eraker, 2004;Broadie, Chernov and Johannes, 2007), two-factor models that generates stochastic correlation between returns and volatility (Christoffersen, Heston and Jacobs, 2009), or three-factor models with jumps both in the dynamics of the underlying and of the volatility (Andersen, Fusari and Todorov, 2015).…”