Volatility Library

Article by: Torben G. Andersen, Tim Bollerslev, and Francis X. Diebold
Published by: Wharton School, Univ. of Penn.
Date: Jul 2002

“Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.”

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Article by: Gurdip Bakshi and Nikunj Kapadia
Published by: The Journal of Derivatives
Date: Fall 2003

“The research indicates that index option prices incorporate a negative volatility risk premium, thus providing a possible explanation of why Black-Scholes implied volatilities of index options on average exceed realized volatilities. This examination of the empirical implication of a market volatility risk premium on 25 individual equity options provides some new insights.

“While the Black-Scholes implied volatilities from individual equity options are also greater on average than historical return volatilities, the difference between them is much smaller than for the market index. Like index options, individual equity option prices embed a negative market volatility risk premium, although much smaller than for the index option — and idiosyncratic volatility does not appear to be priced.

“These empirical results provide a potential explanation of why buyers of individual equity options leave less money on the table than buyers of index options.”

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Article by: Jian Du and Nikunj Kapadia
Published by: University of Massachusetts, Amherst
Date: May 2011

“Both volatility and the tail of the stock return distribution are impacted by discontinuities (‘large jumps’) in the stock price process. In this paper, we construct model-free volatility and tail indexes in a manner that clearly distinguishes one from the other. Our tail index measures time-variation in jump intensity, and is constructed non-parametrically from the identical set of 30-day index options used to construct a volatility index. We use the indexes to examine the relative economic importance of volatility and tail in predicting market returns over 1996-2009. Over this period, our predictability regressions indicate that the rst order impact of jumps is through stock return volatility rather than the tail of the distribution.”

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