The Factor Structure in Equity Options

Abstract

Principal component analysis of equity options on Dow-Jones …rms reveals a strong factor structure. The …rst principal component explains 77% of the variation in the equity volatility level, 77% of the variation in the equity option skew, and 60% of the implied volatility term structure across equities. Furthermore, the …rst principal component has a 92% correlation with S&P500 index option volatility, a 64% correlation with the index option skew, and a 80% correlation with the index option term structure. We develop an equity option valuation model that captures this factor structure. The model allows for stochastic volatility in the market return and also in the idiosyncratic part of …rm returns. The model predicts that …rms with higher betas have higher implied volatilities, and steeper moneyness and term structure slopes. We provide a tractable approach for estimating the model on a large set of index and equity option data on which the model provides a good …t. The equity option data support the cross-sectional implications of the estimated model. JEL Classi…cation: G10; G12; G13.

Dynamic Factor Models for the Volatility Surface

“…Section 3.1 describes a general DFM, which nests all subsequent specifications. Section 3.2 discusses a restricted model recently proposed by Christoffersen et al (2013). This setup explicitly takes the structure of the implied volatility surface data into account.…”

“…Following this intuitively plausible idea, we adopt the setting of Christoffersen et al (2013), who extract the implied volatility level, moneyness slope, and maturity slope by regressing the implied volatility cross-section at a given time t on the moneyness and the maturity:…”

“…First, following Christoffersen et al (2013), we define moneyness as the ratio of the option's strike price K and the spot price S. Second, following the constructing of our volatility surface data, we take moneyness equal to the option Δ. As discussed in Section 2.1, the latter takes more properties of the data into account for defining the relative likelihood of the option ending up in-the-money.…”

“…While the idea that the latent factors in a DFM should capture the key features of the volatility surface along the maturity and moneyness dimensions is intuitively plausible, the approach of Christoffersen et al (2013) discussed above is quite restrictive in terms of the specification of the factor loadings. The third and final setup is a hybrid approach, aiming to combine the flexibility of the general DFM of Section 3.1 with the economically plausible factor interpretation of the restricted model in Section 3.2.…”

“…Second, we examine restricted model specifications, where the factors are forced to represent comovement in the implied volatilities along the moneyness and maturity dimensions. Such a setup is commonly used in the economics and finance literature, see, for example, Dumas, Fleming, and Whaley (1998), Christoffersen et al (2013), and Christoffersen, Goyenko, Jacobs, and Karoui (2012). Third, we propose spline-based models that offer a flexible approach to capture the shape of the implied volatility surface.…”