### 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 estimated nonparametrically and factor risk premia can vary nonlinearly with states. The model is estimated using regressions, with minimal assumptions on factor and option return dynamics. Using index options, we characterize the conditional risk premia for the market return, market variance, and tail and intermediary risk factors. Unconditionally, all risk premia have the expected sign and meaningful magnitudes. Market and variance risk premia display pronounced time-variation, spike during crises, and always have the expected sign. Combined, market return and variance explain more than 90% of option return variation.

A pdf version is available here.

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

**Abstract:** We evaluate the empirical validity of the compound option framework. In a model where corporate securities are options on a firm’s assets, option contracts on these can be viewed as options on options, or *compound options*. We estimate a model with priced asset variance risk and find that it jointly explains the level and time variation of both equity index (SPX) and credit index (CDX) option prices well out-of-sample. This suggests that the two options markets are priced consistently, contrary to recent findings. We show that variance risk is important for establishing pricing consistency between equity, credit, and related derivatives.

A pdf version is available here.

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

**Abstract:** We show that default risk is a primary predictor of the cross-sectional and time-series variation of the comovement between corporate bond and stock returns. Bonds of less creditworthy firms behave more like the issuing firms’ stock, thereby increasing stock-bond comovement. We rationalize this finding with a new credit-risk model featuring a factor structure and stochastic volatility. The model identifies key forces underlying stock-bond comovement and reproduces the positive predictability with default risk uncovered empirically. Finally, we show that a dynamic trading strategy investing in bonds and stocks of the most creditworthy firms generates high diversification benefits and Sharpe ratios out-of-sample.

A pdf version will be available soon.