Estimation of the Realized Volatility and the Average Realized Correlation of the DJIA

Andersen, Bollerslev, Diebold and Ebens (2001a) provide a detailed discussion on the theoretical justification for using the sum of intraday squared returns as a measure of realized stock volatility (see also Andersen, Bollerslev, Diebold and Labys (2001b)). This Section summarizes the above discussion and describes the m ethodology for estim ating the realized index volatility and the average realized correlation among the index’s components using high-frequency stock data. Assum e a N x l vector o f the logarithmic price process p t that follows a multivariate continuous time stochastic volatility diffusion:

dpt =Htd t +CltdWt (2.1)

where Wt is a standard A-dimensional Brownian motion, Qt is strictly stationary, and the time unit h is normalized to a trading day. The period return rt+h,t is then defined as the logarithm ic difference between the spot prices at t and at t+h:

r<+h.,=P,+h - P , (2.2)

Conditional on the sample path realization of p t and Q t, the distribution of the continuously com pounded returns rt+h,t is then:

W 1 - N ( ^ / t , „ d T , ^ a , „ d T ) (2.3)

where cr{//,+r,£2f+r}J=0 refers to the er-field generated by the sam ple paths of the drift p t+T and the diffusion matrix Ql+r for 0< i< h. Hence, the integrated diffusion matrix provides

‘. . . a natural m easure of the true latent h-period volatility’. Andersen et al (2001a) then suggest that, under weak regularity conditions,

Z W .a W ,A - f &,+rd * -> 0 (2.4)

as A —> 0, i.e. as the sam pling frequency of the returns increases. Thus, dependent on the sampling interval, the sum of sufficiently finely sampled intraday returns can provide a direct m easure o f ex post realized volatility that is asymptotically free of m easurem ent error.

The m ethodology in Andersen and Bollerslev (1997a) and in Andersen et al (2 0 0 1a) is adopted in order to extract a measure of realized volatilities and pairwise correlations of the D JIA ’s components using high frequency quotes. Similarly to the above papers, time-series of artificial intraday returns for each stock are constructed at five-m inute intervals. The five-minute sampling frequency is typically considered to be short enough so that the summ ation in (2.4) closely approximates the integrated volatility, and long enough to m inim ize the noise stemming from m arket-m icrostructure effects.

Given that trading hours in NYSE extend from 9:30 EST until 16:05 EST, one trading day can be decomposed in 79 five-minute intervals, such that A is equal to 1/79, or roughly 0.0127. Intraday spot levels are measured as the m idpoint of the best bid and the best ask prices recorded at or immediately before the 80 five-minute marks (note that best quotes are used instead of prices at which actual trades occurred). The corresponding 79 five-m inute returns in a trading day are then computed as the logarithmic differences between consecutive five-m inute marked spot prices.

Due to the use of a discrete sampling interval to approxim ate the continuous volatility process, Andersen et al (2001a) suggest that the presence o f negative serial correlation in the returns series as well as the inherent bid-ask spread are likely to bias the estim ation of the above volatility measure. In order to avoid or at least m inim ize the dependence in the mean of the observed intraday log-differences, dem eaned returns are estimated by fitting a sim ple MA(1) model for each of the five-minute series across the full ten-year sample. Also note that, similarly to previous em pirical studies, high- frequency returns refer only to intraday log-differences in spot prices, i.e. spot changes within the trading day, excluding any overnight price changes. Although this approach admittedly results in some loss of information with respect to the underlying returns

series, it is associated with a less noisy and well-behaved time-series. For notational simplicity, the resulting five-m inute demeaned return for stock i that is recorded at day t and at the five-m inute interval k is denoted by rt'+kA A. Hence, the realized daily covariance matrix cov, is given by:

c o v ,(i,y )= Y . (2-5)

where the elem ents in the diagonal of covt(i,j) refer to the intraday realized variances vft = {cov,(U )} of the thirty stocks in the DJIA. Similarly, the intraday realized covariances between stocks i and j at time t are given by the elements o f cov, outside the diagonal, with the intraday realized correlations denoted by corr,(i,j):

.N_ {cov,(1,7)}

c orrt ( i , j ) = --- --- ₍2.6)

Finally, the weighted average index correlation p ind,t at time t refers to the average correlation across all possible pairs of the DJIA components, scaled by each stock’s weight in the com position o f the index:

/, s S w,wjCOrrt ( i J ) (2.7)

where N gives the num ber o f stocks included in the index, i.e. thirty com ponent stocks in the case o f the Dow Jones Industrial Average, and w, refers to the weight o f stock i in the index’s com position. Since the DJIA is a value-weighted index, the weight w, o f stock i is essentially the ratio of the stock’s price divided by the sum of the prices o f all thirty components.

In addition to com puting realized volatilities for the DJIA constituent stocks and the average realized correlation of the parent index, the above time-series o f dem eaned

five-m inute returns of the individual stocks also allow the estimation of the index’s realized intraday volatility. M ore specifically, the index’s intraday return at day t and at the five-m inute interval k can be easily obtained as the weighted sum o f the constituents’ five-minute returns in (2.8).

N

A .A (2.8)

1=1

Sim ilarly to the m ethodology described above for the individual stocks, the DJIA realized variance v,^ , is proxied by the sum of squared intraday index returns, i.e.

vfnd,t = X [ ( ^ a . a ) 2] • Finally, the D JIA ’s realized volatility at time t is denoted by

<t=1.. [/i/A]

RV, and is m easured as the (annualized) squared root o f v ^ t .