The distribution of realized stock return volatility

Abstract

We examine “realized” daily equity return volatilities and correlations obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average. We find that the unconditional distributions of realized variances and covariances are highly right-skewed, while the realized logarithmic standard deviations and correlations are approximately Gaussian, as are the distributions of the returns scaled by realized standard deviations. Realized volatilities and correlations show strong temporal dependence and appear to be well described by long-memory processes. Finally, there is strong evidence that realized volatilities and correlations move together in a manner broadly consistent with latent factor structure.

Introduction

Financial market volatility is central to the theory and practice of asset pricing, asset allocation, and risk management. Although most textbook models assume volatilities and correlations to be constant, it is widely recognized among both finance academics and practitioners that they vary importantly over time. This recognition has spurred an extensive and vibrant research program into the distributional and dynamic properties of stock market volatility.⁴ Most of what we have learned from this burgeoning literature is based on the estimation of parametric ARCH or stochastic volatility models for the underlying returns, or on the analysis of implied volatilities from options or other derivatives prices. However, the validity of such volatility measures generally depends upon specific distributional assumptions, and in the case of implied volatilities, further assumptions concerning the market price of volatility risk. As such, the existence of multiple competing models immediately calls into question the robustness of previous findings. An alternative approach, based for example on squared returns over the relevant return horizon, provides model-free unbiased estimates of the ex post realized volatility. Unfortunately, however, squared returns are also a very noisy volatility indicator and hence do not allow for reliable inference regarding the true underlying latent volatility.

The limitations of the traditional procedures motivate our different approach for measuring and analyzing the properties of stock market volatility. Using continuously recorded transactions prices, we construct estimates of ex post realized daily volatilities by summing squares and cross-products of intraday high-frequency returns. Volatility estimates so constructed are model-free, and as the sampling frequency of the returns approaches infinity, they are also, in theory, free from measurement error (Andersen, Bollerslev, Diebold and Labys, henceforth ABDL, 2001a).⁵ The need for reliable high-frequency return observations suggests, however, that our approach will work most effectively for actively traded stocks. We focus on the 30 stocks in the Dow Jones Industrial Average (DJIA), both for computational tractability and because of our intrinsic interest in the Dow, but the empirical findings carry over to a random sample of 30 other liquid stocks. In spite of restricting the analysis to actively traded stocks, market microstructure frictions, including price discreteness, infrequent trading, and bid–ask bounce effects, are still operative. In order to mitigate these effects, we use a five-minute return horizon as the effective “continuous time record”. Treating the resulting daily time series of realized variances and covariances constructed from a five-year sample of five-minute returns for the 30 DJIA stocks as being directly observable allows us to characterize the distributional features of the volatilities without attempting to fit multivariate ARCH or stochastic volatility models.

Our approach is directly in line with earlier work by French et al. (1987), Schwert (1989), Schwert (1990a), Schwert (1990b), and Schwert and Seguin (1991), who rely primarily on daily return observations for the construction of monthly realized stock volatilities.⁶ The earlier studies, however, do not provide a formal justification for such measures, and the diffusion-theoretic underpinnings provided here explicitly hinge on the length of the return horizon approaching zero. Intuitively, following the work of Merton (1980) and Nelson (1992), for a continuous time diffusion process, the diffusion coefficient can be estimated arbitrarily well with sufficiently finely sampled observations, and by the theory of quadratic variation, this same idea carries over to estimates of the integrated volatility over fixed horizons. As such, the use of high-frequency returns plays a critical role in justifying our measurements. Moreover, our focus centers on daily, as opposed to monthly, volatility measures. This mirrors the focus of most of the extant academic and industry volatility literatures and more clearly highlights the important intertemporal volatility fluctuations.⁷ Finally, because our methods are trivial to implement, even in the high-dimensional situations relevant in practice, we are able to study the distributional and dynamic properties of correlations in much greater depth than is possible with traditional multivariate ARCH or stochastic volatility models, which rapidly become intractable as the number of assets grows.

Turning to the results, we find it useful to segment them into unconditional and conditional aspects of the distributions of volatilities and correlations. As regards the unconditional distributions, we find that the distributions of the realized daily variances are highly non-normal and skewed to the right, but that the logarithms of the realized variances are approximately normal. Similarly, although the unconditional distributions of the covariances are all skewed to the right, the realized daily correlations appear approximately normal. Finally, although the unconditional daily return distributions are leptokurtic, the daily returns normalized by the realized standard deviations are also close to normal. Rather remarkably, these results hold for the vast majority of the 30 volatilities and 435 covariances/correlations associated with the 30 Dow Jones stocks, as well as the 30 actively traded stocks in our randomly selected control sample.

Moving to conditional aspects of the distributions, all of the volatility measures fluctuate substantially over time, and all display strong dynamic dependence. Moreover, this dependence is well-characterized by slowly mean-reverting fractionally integrated processes with a degree of integration, d, around 0.35, as further underscored by the existence of very precise scaling laws under temporal aggregation. Although statistically significant, we find that the much debated leverage effect, or asymmetry in the relation between past negative and positive returns and future volatilities, is relatively unimportant from an economic perspective. Interestingly, the same type of asymmetry is also present in the realized correlations. Finally, there is a systematic tendency for the variances to move together, and for the correlations among the different stocks to be high/low when the variances for the underlying stocks are high/low, and when the correlations among the other stocks are also high/low.

Although several of these features have been documented previously for U.S. equity returns, the existing evidence relies almost exclusively on the estimation of specific parametric volatility models. In contrast, the stylized facts for the 30 DJIA stocks documented here are explicitly model-free. Moreover, the facts extend the existing results in important directions and both solidify and expand on the more limited set of results for the two exchange rates in Andersen, Bollerslev (2001a), ABDL (2001a (2001b) and the DJIA stock index in Ebens (1999a). As such, our findings set the stage for the development of improved volatility models—possibly involving a simple factor structure, which appears consistent with many of our empirical findings—and corresponding out-of-sample volatility forecasts, consistent with the distributional characteristics of the returns.⁸ Of course, the practical use of such models in turn should allow for better risk management, portfolio allocation, and asset pricing decisions.

The remainder of the paper is organized as follows. In Section 2 we provide a brief account of the diffusion-theoretic underpinnings of our realized volatility measures, along with a discussion of the actual data and volatility calculations. In Section 3 we discuss the unconditional univariate return volatility and correlation distributions, and we move to dynamic aspects, including long-memory effects and scaling laws, in Section 4. In Section 5 we assess the symmetry of responses of realized volatilities and correlations to unexpected shocks. We report on multivariate aspects of the volatility and correlation distributions in Section 6, and in Section 7 we illustrate the consistency of several of our empirical results with a simple model of factor structure in volatility. We conclude in Section 8 with a brief summary of our main findings and some suggestions for future research.