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 study the ability of three-factor affine term-structure models to extract conditional volatility using interest rate swap yields for 1991–2005 and Treasury yields for 1970–2003. For the Treasury sample, the correlation between model-implied and EGARCH volatility is between 60% and 75%. For the swap sample, this correlation is rather low or negative. We find that these differences in model performance are primarily due to the timing of the swap sample, and not to institutional differences between swap and Treasury markets. We conclude that the ability of multifactor affine models to extract conditional volatility depends on the sample period, but that overall these models perform better than has been argued in the literature.

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.

We use intraday data to compute weekly realized moments for equity returns and study their time-series and cross-sectional properties. Buying stocks in the lowest realized skewness decile and selling stocks in the highest realized skewness decile generates an average return of 19 basis points the following week with a t-statistic of 3.70. This result is robust across a wide variety of implementations and is not captured by the Fama-French and Carhart factors. The relation between realized kurtosis and next week׳s stock returns is positive but not always significant. We do not find a strong relation between realized volatility and next week׳s stock returns.

We propose no-arbitrage term structure models with volatility factors that follow GARCH processes. The models’ tractability is similar to canonical affine term structure models, but they fit yield volatility much better, especially for long-maturity yields. This improvement does not come at the expense of a deterioration in yield fit. Because of the improved volatility fit, the model performs substantially better in pricing Treasury futures options. We conclude that the specification of the volatility factors is critical. Modeling volatility as a function of (lagged) squared innovations to factors improves on models where volatility is a linear function of the factors.

We introduce a top-down no-arbitrage model for pricing structured products. Losses are described by Cox processes whose intensities depend on economic variables. The model provides economic insight into the impact of structured products on financial institutions’ risk exposure and systemic risk. We estimate the model using CDO data and find that spreads decrease with higher interest rates and increase with volatility and leverage. Volatility is the primary determinant of variation in tranche spreads. Leverage and interest rates are more closely associated with rare credit events. Model-implied risk premiums and the probabilities of tranche losses increase substantially during the financial crisis.