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The topic of volatility measurement and estimation is central to financial and more generally time-series econometrics. In this chapter, we begin by surveying models of volatility, both discrete and continuous, and then we summarize some selected empirical findings from the literature. In particular, in the first sections of this chapter, we discuss important developments in volatility models, with focus on time-varying and stochastic volatility as well as nonparametric volatility estimation. The models discussed share the common feature that volatilities are unobserved and belong to the class of missing variables. We then provide empirical evidence on “small” and “large” jumps from the perspective of their contribution to overall realized variation, using high-frequency price return data on 25 stocks in the DOW 30. Our “small” and “large” jump variations are constructed at three truncation levels, using extant methodology of Barndorff-Nielsen and Shephard (2006), Andersen, Bollerslev, and Diebold (2007), and Aït-Sahalia and Jacod (2009a, 2009b, 2009c). Evidence of jumps is found in around 22.8% of the days during the 1993–2000 period, much higher than the corresponding figure of 9.4% during the 2001–2008 period. Although the overall role of jumps is lessening, the role of large jumps has not decreased, and indeed, the relative role of large jumps, as a proportion of overall jumps, has actually increased in the 2000s.

This paper compares the out-of-sample forecasting performance of three long-memory volatility models (i.e., fractionally integrated (FI), break and regime switching) against three short-memory models (i.e., GARCH, GJR and volatility component). Using S&P 500 returns, we find that structural break models produced the best out-of-sample forecasts, if future volatility breaks are known. Without knowing the future breaks, GJR models produced the best short-horizon forecasts and FI models dominated for volatility forecasts of 10 days and beyond. The results suggest that S&P 500 volatility is non-stationary at least in some time periods. Controlling for extreme events (e.g., the 1987 crash) significantly improved forecasting performance.

This chapter proposes a nonparametric estimator of the risk neutral density (RND) based on cross-sectional European option prices. The authors recast the arbitrage-free equation for option pricing as a functional linear regression model where the regressor is a curve and the independent variable is a scalar corresponding to the option price. Then, the authors show that the RND can be viewed as the solution of an ill-posed integral equation. To estimate the RND, the authors use an iterative method called Landweber-Fridman (LF). Then, the authors establish the consistency and asymptotic normality of the estimated RND. These results can be used to construct a confidence interval around the curve. Finally, some Monte Carlo simulations and application to the S&P 500 options show that this method performs well compared to alternative methods.

This study examines several aspects of active portfolio management by equity hedge funds between 1996 and 2013. Consistent with the idea that cross-sectional return dispersion is a proxy for the market’s available alpha, our results show that equity hedge funds achieve their strongest performance during periods of elevated dispersion. The performance advantage is robust to numerous risk adjustments. Portfolio managers may use the current month’s dispersion to plan the extent to which the following month’s investment approach will be active or passive. We also estimate the active share for equity hedge funds and find an average of 53%. We further document the average annual expense ratio for managing hedge funds’ active share to be about 7%. This figure is remarkably close to active expense ratios reported previously for equity mutual funds, which may be interpreted as evidence of uniform pricing for active portfolio management services.