Risk Premium, Variance Premium, and the Maturity Structure of Uncertainty

Abstract

Expected returns vary when investors face time-varying investment opportunities. Longrun risk models (Bansal and Yaron 2004) and no-arbitrage affine models (Duffie, Pan, and Singleton 2000) emphasize sources of risk that are not observable to the econometrician. We show that, for both classes of models, the term structure of risk implicit in option prices can reveal these risk factors ex-ante. Empirically, we construct the variance term structure implied in SP500 option prices. The variance term structure reveal two important drivers of the bond premium, the equity premium, and the variance premium, jointly. We also consider the term structure of higher-order risks as measured by skewness and kurtosis and still find that two factors are sufficient to summarize the information content from the term structure of risks. Overall, our results bode well for the ability of structural models to explain risk-returns trade-offs across different markets using only very few sources of risk. JEL classification: G12, G13 Bank classification: Financial markets; Asset pricing RésuméLes rendements espérés varient lorsque les occasions de placement qui s'offrent aux investisseurs ne sont pas constantes dans le temps. Les modèles avec risque de long terme (Bansal et Yaron, 2004) et les modèles affines basés sur l'absence d'arbitrage (Duffie, Pan et Singleton, 2000) mettent l'accent sur des sources de risque qui échappent à l'observation de l'économètre. Les auteurs montrent que, pour chacune de ces classes de modèles, la structure par terme des risques implicitement contenue dans les prix d'options peut révéler ces facteurs de risque a priori. De manière empirique, les auteurs construisent la structure par terme de la variance à partir des prix des options sur l'indice SP500. Cette structure par terme fait ressortir deux importants facteurs qui déterminent, conjointement, la prime obligataire, la prime sur actions et la prime liée à la variance. Les auteurs étudient par ailleurs la structure par terme des risques d'ordre supérieur quantifiés d'après les coefficients d'asymétrie et d'aplatissement. Là aussi, ils concluent que deux facteurs suffisent pour résumer le contenu informatif de la structure par terme des risques. Dans l'ensemble, les résultats obtenus augurent bien de la capacité des modèles structurels à rendre compte, à l'aide d'un nombre limité de facteurs, des arbitrages risquerendement qui s'exercent sur différents marchés.

Volatility term structures in commodity markets

“…The importance of considering the entire term structure has been widely documented for equity markets (e.g., Adrian & Rosenberg, ; Bakshi, Panayotov, & Skoulakis, ; Feunou, Fontaine, Taamouti, & Tédongap, ). In particular, these studies show that the volatility term structure is informative about, inter alia, risk premia, measures of real economic activity, business cycle risk, and the tightness of financial constraints.…”

The term structure of systematic and idiosyncratic risk

“…Feunou, Fontaine, Taamouti, and Tédongap () show that the principal components from the option‐implied variance term structure have predictive power for bond and equity returns. Our results indicate that factors capturing the slope of the term structure in the market level are related to expectations about the future variance and may help rationalize these findings.…”

The information content of the implied volatility term structure on future returns

“…Feunou et al () and Johnson () also use PCA to extract the information content of the VIX term structure.…”

“…2 When it was introduced in 1993, the VIX was compiled from the implied volatility of eight S&P 100 index options, comprising near at-the-money and nearby and second nearby calls and puts; however, ever since 2003, the VIX has been calculated from the prices of S&P 500 index options using a model-free formula with almost all of the available contracts (i.e., with a wide range of strike prices). 3 See, for example, Bakshi et al (2011), Feunou et al (2014), Luo and Zhang (2017), and Andreou et al (2017). the difference between the 30-day forward squared VIX and the 30-day squared current VIX level as the respective level and term structure variables. Second, we run a principal component analysis (PCA) of the squared VIX levels under various time horizons to generate the first and second components for our investigation; the first and second components represent the level and slope of the squared VIX term structure, respectively.…”