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.

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.

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.

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, 56(1), 65-91.

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.

Modeling Conditional Factor Risk Premia Implied by Index Option Returns (2023), with K. Jacobs and P. Orlowski

The Journal of Finance, Accepted.

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. We estimate the model using index options to characterize the conditional risk premia for factors of interest such as the market return, market variance, tail and intermediary risk factors, higher moments, and the VIX term structure slope. Combined, market return and variance explain more than 90% of option return variation. Unconditionally, the magnitude of the variance risk premium is plausible. It displays pronounced time-variation, spikes during crises, and always has the expected sign.

A pdf version is available here.