### Publications and Peer Reviewed Papers

**The Factor Structure in Equity Options (2018), with P. Christoffersen and K. Jacobs**

*The Review of Financial Studies*, 31 (2), 595–637.

**Abstract:** Equity options display a strong factor structure. The first principal components of the equity volatility levels, skews, and term structures explain a substantial fraction of the cross-sectional variation. Furthermore, these principal components are highly correlated with the S&P500 index option volatility, skew, and term structure respectively. We develop an equity option valuation model that captures this factor structure. The model predicts that firms with higher market betas have higher implied volatilities, steeper moneyness slopes, and a term structure that co-varies more with the market. The model provides a good fit and the equity option data support the model’s cross-sectional implications.

A pdf version of the paper is available here.

**Beta Risk in the Cross-Section of Equities (2019), with A. Boloorforoosh, P. Christoffersen, and C. Gouriéroux**

*The Review of Financial Studies*, 33 (9), 4318–4366.

**Abstract: **We develop a conditional capital asset pricing model in continuous time that allows for stochastic beta exposure. When beta comoves with market variance and the stochastic discount factor (SDF), beta risk is priced, and the expected return on a stock deviates from the security market line. The model predicts that low-beta stocks earn high returns, because their beta positively comoves with market variance and the SDF. The opposite is true for high-beta stocks. Estimating the model on equity and option data, we find that beta risk explains expected returns on low- and high-beta stocks, resolving the “betting against beta” anomaly.

A pdf version of the paper is available here.

**A Tractable Framework for Option Pricing with Dynamic Market Maker Inventory and Wealth (2020), with K. Jacobs**

*The Journal of Financial and Quantitative Analysis*, 55(4), 1117-1162.

**Abstract:** We develop a tractable dynamic model of an index option market maker with limited capital. We solve for the variance risk premium and option prices as a function of the asset dynamics and market maker option holdings and wealth. The market maker absorbs end users’ positive demand and requires a more negative variance risk premium when she incurs losses. We estimate the model using returns, options, and inventory and find that it performs well, especially during the financial crisis. The restrictions imposed by nested existing reduced-form stochastic-volatility models are strongly rejected in favor of the model with a market maker.

A pdf version of the paper is available here.

**Option-Based Estimation of the Price of Co-Skewness and Co-Kurtosis Risk (2021), with P. Christoffersen, K. Jacobs, and M. Karoui**

*The Journal of Financial and Quantitative Analysis*, Forthcoming.

**Abstract:** We show that the prices of risk for factors that are nonlinear in the market return can be obtained using index option prices. The price of coskewness risk corresponds to the market variance risk premium, and the price of cokurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An analysis of factor models with coskewness risk indicates that the new estimates of the price of risk improve the models’ performance compared with regression-based estimates.

A pdf version of the paper is available here.

### Working Papers

**The Low-Minus-High Portfolio and the Factor Zoo (2019), with D. Andrei, and J. Cujean**

**Abstract:** Anomalies in the cross section of returns should not be regarded as evidence against the CAPM. Regardless whether the CAPM is rejected for valid reasons or by mistake, a single long-short portfolio will always explain, together with the market, 100% of the cross-sectional variation in returns. Yet, this portfolio need not proxy for fundamental risk. We show theoretically how factors based on valuation ratios (e.g, book-to-market), or on investment rates, can be proxies for this portfolio. More generally, the empiricist can uncover an infinity of proxies for this portfolio, thus unleashing the factor zoo.

A pdf version of the paper is available here.

**Modeling Conditional Factor Risk Premia Implied by Index Option Returns (2021), with K. Jacobs and P. Orlowski**

**Abstract: **We propose a novel factor model for option returns. Option exposures are modeled nonparametrically and factor risk premia may vary non-linearly with states. The model allows for estimation of factor risk premia and factor exposures using regressions, with minimal assumptions on the dynamics of factors and/or option returns. We implement the model using index option returns. The model explains expected option returns across moneyness and maturities, and its hedging performance is impressive. We obtain estimates of the average risk premia on the market index, the market variance, as well as factors associated with tail risk and intermediary risk, and we characterize the time variation in these risk premia. The signs of the average risk premia are consistent with economic intuition for all factors and risk premia spike during crises. The magnitudes of the risk premia on the market and variance factors are reasonable and they have the expected sign throughout the sample.

A pdf version will be available soon.

**Asset Variance Risk and Compound Option Prices (2021), with H. Doshi, J. Ericsson, and S. Seo**

**Abstract:** We examine the empirical validity of a compound option pricing framework. In a structural credit risk model where corporate securities are options on the issuing firm’s assets, option contracts on corporate securities can be viewed as options on options, or \textit{compound options}. We estimate a model with priced asset variance risk and find that the estimated model well captures the level and time variation of equity index (SPX) and credit index (CDX) option prices out-of-sample. Our analysis reveals that asset variance risk plays a key role in establishing the broad pricing consistency between equity, credit, and associated derivatives markets.

A pdf version will be available soon.

**Understanding the Comovement between Corporate Bonds and Stocks: The Role of Default Risk (2021), with A. Dickerson, A. Jeanneret, and P. Mueller**

**Abstract:** We study the comovement between corporate bonds and stocks. Using a sample of 2,123 firms over the period 1973-2019, we show that firm default risk is a critical driver of the covariance and correlation between bond and stock returns, both across firms and time. That is, corporate bonds issued by less creditworthy firms behave more like the issuing firms’ stock. The positive relationship between stock-bond comovement and default risk is not only economically meaningful, but also strongly non-linear and convex in the cross-section. We rationalize these findings with a new credit risk model of the firm featuring a factor structure in stochastic asset variance. This model sheds new light on the economic forces underlying the comovement between stock and corporate bond valuation. We also show that these results have fundamental implications for portfolio allocation decisions.

A pdf version will be available soon.