Which loss function should be used when estimating and evaluating option valuation models? Many different functions have been suggested, but no standard has emerged. We emphasize that consistency in the choice of loss functions is crucial. First, for any given model, the loss function used in parameter estimation and model evaluation should be the same, otherwise suboptimal parameter estimates may be obtained. Second, when comparing models, the estimation loss function should be identical across models, otherwise inappropriate comparisons will be made. We illustrate the importance of these issues in an application of the so-called Practitioner Black-Scholes model to S&P 500 index options.

We nest multiple volatility components, fat tails, and a U-shaped pricing kernel in a single option model and compare their contribution in describing returns and option data. All three features lead to statistically significant model improvements. A U-shaped pricing kernel is economically most important and improves option fit by 17%, on average, and more so for two-factor models. A second volatility component improves the option fit by 9%, on average. Fat tails improve option fit by just over 4%, on average, but more so when a U-shaped pricing kernel is applied. Overall, these three model features are complements rather than substitutes: the importance of one feature increases in conjunction with the others.

Equity risk measured by beta is of great interest to both academics and practitioners. Existing estimates of beta use historical returns. Many studies have found option-implied volatility to be a strong predictor of future realized volatility. We find that option-implied volatility and skewness are also good predictors of future realized beta. Motivated by this finding, we establish a set of assumptions needed to construct a beta estimate from option-implied return moments using equity and index options. This beta can be computed using only option data on a single day. It is therefore potentially able to reflect sudden changes in the structure of the underlying company.

We uncover stylized facts of commodity futures’ price and volatility dynamics in the post-financialization period and find a factor structure in daily commodity volatility that is much stronger than the factor structure in returns. The common factor in commodity volatility relates to stock market volatility as well as to the business cycle. Model-free realized commodity betas with the stock market were high during 2008–2010 but have since returned to the pre-crisis level, close to 0. While commodity markets appear segmented from the equity market when considering only returns, commodity volatility indicates a nontrivial degree of market integration.

Standard option valuation models leave no room for option illiquidity premia. Yet we find the risk-adjusted return spread for illiquid over liquid equity options is 3.4% per day for at-the-money calls and 2.5% for at-the-money puts. These premia are computed using option illiquidity measures constructed from intraday effective spreads for a large panel of U.S. equities, and they are robust to different empirical implementations. Our findings are consistent with evidence that market makers in the equity options market hold large and risky net long positions, and positive illiquidity premia compensate them for the risks and costs of these positions. (JEL G12)

International equity markets are characterized by nonlinear dependence and asymmetries. We propose a new dynamic asymmetric copula model to capture long-run and short-run dependence, multivariate nonnormality, and asymmetries in large cross-sections. We find that correlations have increased markedly in both developed markets (DMs) and emerging markets (EMs), but they are much lower in EMs than in DMs. Tail dependence has also increased, but its level is still relatively low in EMs. We propose new measures of dynamic diversification benefits that take into account higher-order moments and nonlinear dependence. The benefits from international diversification have reduced over time, drastically so for DMs. EMs still offer significant diversification benefits, especially during large market downturns.

We build a new class of discrete-time models that are relatively easy to estimate using returns and/or options. The distribution of returns is driven by two factors: dynamic volatility and dynamic jump intensity. Each factor has its own risk premium. The models significantly outperform standard models without jumps when estimated on S&P500 returns. We find very strong support for time-varying jump intensities. Compared to the risk premium on dynamic volatility, the risk premium on the dynamic jump intensity has a much larger impact on option prices. We confirm these findings using joint estimation on returns and large option samples.

The cross-section of stock returns has substantial exposure to risk captured by higher moments in market returns. We estimate these moments from daily S&P 500 index option data. The resulting time series of factors are thus genuinely conditional and forward-looking. Stocks with high sensitivities to innovations in implied market volatility and skewness exhibit low returns on average, whereas those with high sensitivities to innovations in implied market kurtosis exhibit high returns on average. The results on market skewness risk are extremely robust to various permutations of the empirical setup. The estimated premium for bearing market skewness risk is between -6.00% and -8.40% annually. This market skewness risk premium is economically significant and cannot be explained by other common risk factors such as the market excess return or the size, book-to-market, momentum, and market volatility factors. Using ICAPM intuition, the negative price of market skewness risk indicates that it is a state variable that negatively affects the future investment opportunity set.

Under very general conditions, the total quadratic variation of a jump-diffusion process can be decomposed into diffusive volatility and squared jump variation. We use this result to develop a new option valuation model in which the underlying asset price exhibits volatility and jump intensity dynamics. The volatility and jump intensity dynamics in the model are directly driven by model-free empirical measures of diffusive volatility and jump variation. Because the empirical measures are observed in discrete intervals, our option valuation model is cast in discrete time, allowing for straightforward filtering and estimation of the model. Our model belongs to the affine class enabling us to derive the conditional characteristic function so that option values can be computed rapidly without simulation. When estimated on S&P500 index options and returns the new model performs well compared with standard benchmarks.

Many studies have documented that daily realized volatility estimates based on intraday returns provide volatility forecasts that are superior to forecasts constructed from daily returns only. We investigate whether these forecasting improvements translate into economic value added. To do so, we develop a new class of affine discrete-time option valuation models that use daily returns as well as realized volatility. We derive convenient closed-form option valuation formulas, and we assess the option valuation properties using Standard & Poor’s (S&P) 500 return and option data. We find that realized volatility reduces the pricing errors of the benchmark model significantly across moneyness, maturity, and volatility levels.