This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period 1962–1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. All the volatility measures move together countercyclically and help to predict GDP growth. Market volatility tends to lead the other volatility series. Factors that may be responsible for these findings are suggested.

Post-crash distributions inferred from S

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This paper shows that realised power variation and its extension we introduce here called realised bipower variation is somewhat robust to rare jumps. We show realised bipower variation estimates integrated variance in SV models --- thus providing a model free and consistent alternative to realised variance. Its robustness property means that if we have an SV plus infrequent jumps process then the di#erence between realised variance and realised bipower variation estimates the quadratic variation of the jump component. This seems to be the first method which can divide up quadratic variation into its continuous and jump components. Various extensions are given. Proofs of special cases of these results are given.

It is widely known that conditional covariances of asset returns change over time.

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This paper examines the effects of uncertainty about the stock return predictability on optimal dynamic portfolio choice in a continuous time setting for a long horizon investor. Uncertainty about the predictive relation affects the optimal portfolio choice through dynamic learning, and leads to a state-dependent relation between the optimal portfolio choice and the investment horizon. There is substantial market timing in the optimal hedge demands, which is caused by stochastic covariance between stock return and dynamic learning. The opportunity cost of ignoring predictability or learning is found to be quite substantial. How much should a “long horizon ” investor allocate to equity? The conventional wisdom says that a long horizon investor should invest more in equity because, over long horizons, aboveaverage returns tend to offset below-average returns. This is the notion of “time diversification.” Samuelson (1989, 1990), among others, has argued that the notion of “time diversification ” is spurious: when stock returns are i.i.d., for example, the optimal portfolio is independent of the horizon for an investor with an isoelastic utility function. When stock returns are predictable, however, the optimal stock allocation does depend on the investment horizon, even if the investor has an isoelastic utility.

INTRODUCTION The CBOE Market Volatility Index (VIX) is an average of S&P 100 option (OEX) implied volatilities. As such, it represents a market- consensus estimate of future stock market volatility. 1 The computation and dissemination of VIX on a real-time basis offers practitioners and academics an important new source of information. Practitioners, for This research was supported by the Futures and Options Research Center at the Fuqua School of Business, Duke University. We gratefully acknowledge the helpful comments and suggestions of Fischer Black, Mark Rubinstein, and two anonymous referees. We also thank participants at the University of Pennsylvania, the University of Texas at Dallas, and the University of Waterloo/KPMG Peat Marwick Thorne seminars, as well as attendees of the 1993 Conference on Financial Innovation: 20 Years of Black/Scholes and Merton (Duke University) and the 1994 Berkeley Program in Finance, Ojai Valley, California. Since OEX options are the mos

Abstract: This paper is an investigation into the determinants of asymmetries in stock returns. We develop a series of cross-sectional regression specifications which attempt to forecast skewness in the daily returns of individual stocks. Negative skewness is most pronounced in stocks that have experienced: 1) an increase in trading volume relative to trend over the prior six months; and 2) positive returns over the prior thirty-six months. The first finding is consistent with the model of Hong and Stein (1999), which predicts that negative asymmetries are more likely to occur when there are large differences of opinion among investors. The latter finding fits with a number of theories, most notably Blanchard and Watson’s (1982) rendition of stockprice bubbles. Analogous results also obtain when we attempt to forecast the skewness of the aggregate stock market, though our statistical power in this case is limited.

In this paper we review some recent work on limit results on realised power variation, that is sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.