6. Concluding remarks

К оглавлению 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19

Our analysis of the intraday volatility patterns in the DM-$ foreign exchange

and S&P 500 equity markets documents how traditional time series methods

applied to raw high frequency returns may give rise to erroneous inference about

the return volatility dynamics. Explicit allowance for the influence of the strong

periodicity, as exemplified by our flexible Fourier form, is a necessary requirement

for discovery of the salient intraday volatility features. Moreover, adjusting

for the pronounced periodic structure appears critical in uncovering the complex

link between the short- and long-run return components, which may help to

explain the apparent conflict between the long-memory volatility characteristics

observed in interday data and the rapid short-run decay associated with news

arrivals in intraday data. More directly, however, our findings have immediate and

important implications for a large range of issues in the rapidly growing literature

using very high frequency financial data. Examples include investigations into the

lead-lag relationship among returns and volatility both within and across different

markets, the effect of cross listings of securities, the fundamental determinants

behind the volatility clustering phenomenon, the development of real time trading

and investment strategies and the evaluation of continuous option valuation and

hedging decisions. Only future research will reveal the extent of the biases induced

into these studies by the neglect of intraday periodic components.

Acknowledgements

We would like to thank Richard T. Bailie, the editor, an anonymous referee,

Dominique Guillaume, Robert J. Hodrick, Charles Jones, Stephen J. Taylor,

Kenneth F. Wallis, along with seminar participants at the Olsen and Associates

Research Institute for Applied Economics, the workshop on 'Market Micro

Structure' at the Aarhus School of Business, the Fall 1994 NBER Asset Pricing

Meeting at the Wharton School, the HFDF-I Conference in Ziirich, the 7th World

Congress of the Econometric Society in Tokyo, Duke University and the University

of California at Santa Barbara for helpful comments.

Appendix A. Data description

A.1. The Deutschemark-U.S. dollar exchange rate data

The DM-$ exchange rate data consist of all the quotes that appeared on the

interbank Reuters network during the October 1, 1992 through September 29,

1993 sample period. The data were collected and provided by Olsen and Associates.

Each quote contains a bid and an ask price along with the time to the nearest

even second. Approximately 0.36% of the 1,472,241 raw quotes were filtered out

using the algorithm described in Dacorogna et al. (1993). During the most active

trading hours, an average of five or more valid quotes arrive per minute; see

Bollerslev and Domowitz (1993). The exchange rate figure for each 5-minute

interval is determined as the interpolated average between the preceding and

immediately following quotes weighted linearly by their inverse relative distance

to the desired point in time. For instance, suppose that the bid-ask pair at 14.14.56

was 1.6055-1.6065, while the next quote at 14.15.02 was 1.6050-1.6055. The

interpolated price at 14.15.00 would then be exp{1/3. [ln(1.6055)+

!n(1.6065)]/2 + 2/3 • [ln(1.6050) + ln(1.6055)]/2} = 1.6055. The nth 5-minute

return for day t, Rt, ., is then simply defined as the difference between the

midpoint of the logarithmic bid and ask at these appropriately spaced time

intervals. This definition of the returns has the advantage, that it is symmetric with

respect to the denomination of the exchange rate. However, as noted by MiJller et

al. (1990), the numerical difference from the logarithm of the middle price is

negligible. All 288 intervals during the 24-hour daily trading cycle are used.

However, in order to avoid confounding the evidence in the correlation analysis

conducted below by the decidedly slower trading patterns over weekends, all the

returns from Friday 21.00 Greenwich mean time (GMT) through Sunday 21.00

GMT were excluded (see Bollerslev and Domowitz (1993) for a detailed analysis

of the quote activity in the DM-$ interbank market and a justification for this

' weekend' definition). Similarly, to preserve the number of returns associated with

one week we make no corrections for any worldwide or country specific holidays

that occurred during the sample period. All in all, this leaves us with a sample of

260 days, for a total of 74,880 5-minute intraday return observations i.e. R,,,,,

n = 1, 2 . . . . . 288, t = 1, 2 . . . . . 260.

A.2. The standard and poor's 500 stock index futures data

The intraday S&P 500 futures data are based on 'quote capture' information

provided by the Chicago Mercantile Exchange (CME) from January 2, 1986

152 T.G. Andersen, T. Bollerslet: / Journal of Empirical Finance 4 (1997) 115-158

through December 31, 1989. The data specify the time, to the nearest 10 seconds

and the exact price of the S&P 500 futures transaction whenever the price differs

from the previously recorded price 4~,. The calculation of the returns is based on

the last recorded logarithmic prices for the nearby futures contract over consecutive

five minute intervals. The price record covers the full trading day in the

futures market from 8.30 a.m. (central standard time) to 3.15 p.m. Although, the

New York Stock Exchange closes at 3.00 p.m., we retain the last three 5-minute

returns from the futures market in the analysis reported on below. The first return

for the trading day, i.e. from 8:30 to 8:35 a.m., constitutes another unusual time

interval. This period incorporates adjustments to the information accumulated

overnight, and consequently displays a much higher average return variability than

any other 5-minute interval. In effect, this is not a 5-minute return, and we

therefore delete it in the subsequent analysis. Alternatively, it would be possible to

account for this special return interval using dummy variables. However, any such

procedure is invariably ad hoc in nature. Furthermore, informal investigations

reveal little sensitivity to the exact treatment of the overnight returns. We thus

elect to work exclusively with the 5-minute returns. Following Chan et al. (1991),

we also exclude the October 15 through November 13, 1987 time period around

the stock market crash due to the frequent trading suspensions. Outside these four

weeks trading suspensions were rare, but did occur. In these instances the missing

prices were determined by linear interpolation, leading to identical returns over

each of the intermediate intervals. This obviously smoothes the series over the

missing data points which will mitigate the effect of sharp price changes subsequent

to a trading suspension. Experimentation with exclusion of trading days with

missing observations indicate that the findings pertaining to the degree of volatility

persistence reported on here are virtually unaffected by this interpolation. All in

all, these corrections result in a sample of 991 days, each consisting of 80 intraday

5-minute returns, for a total of 79,280 observations i.e. R,,,,, n -- 1, 2 . . . . . 80,

t=l,2 . . . . . 991.

Appendix B. Flexible Fourier form modeling of intraday periodic volatility

components

From Eq. (7), define,

xt,,, =- 21og[ IR, . , , - E(R,,,,)I] - log or,2 + log U = log s t2,n + log Z t2,n •

(A.I)

Our modeling approach is then based on a non-linear regression in the intraday

time interval, n, and the daily volatility factor, o-,,

x,,,, =f(0;o" t, n) + u, .... (A.2)

where the error, ut.,,-= log Z~,,- E(log Z~,,), is i.i.d, mean zero. In the actual

implementation the non-linear regression function is approximated by the following

parametric expression,

J [ n //2 D

f( O;cr,, n) =E°'tJi-o [ ~oj +/x,j~ +/-'.2j-~, + Ea,jI,,=,:i

i=1

+i Yl, J cos----~ + 6pj sin N '

where Nj-N-I~i_j,Ni=(N+ I)/2 and Nz=-N-I~i t,xi2=(N+ l)(N+

2)/6 are normalizing constants. For J = 0 and D = 0, Eq. (A.3) reduces to the

standard flexible Fourier functional form proposed by Gallant (1981, 1982).

Allowing for J > 1 and thus a possible interaction effect between o-/ and the

shape of the periodic pattern might be important in some markets, however. Each

of the corresponding J flexible Fourier forms are parameterized by a quadratic

component (terms with ix-coefficients) and a number of sinusoids (the 7- and

6-coefficients). Moreover, it may be advantageous to also include time specific

dummies for applications in which some intraday intervals do not fit well within

the overall regular periodic pattern (the A-coefficients).

Practical estimation is most easily accomplished using a two-step procedure.

Firstly, a generated x,, . series, 2t,,,, is obtained by replacing E(R,.,,) with the

sample mean of the returns, ~',., and cr t with the estimates from a daily volatility

model, say 6- t. Substituting ~t for o-, and treating ~-,.,, as the dependent variable

in the regression defined by Eqs. (A.2) and (A.3) allow the parameters to be

estimated by ordinary least squares (OLS). Note that from Eq. (3), 6, 2 represents

an estimate of M(sZ)o-t 2, so that after substitution for o- t in Eq. (A.2), the term

-log M(s 2) is implicitly included in the constant term in Eq. (A.3), /Zoo. Let

f , , , - f ( 0 ; 6 " , , n) denote the resulting estimate for the right hand side of Eq.

(A.3) 4v. The normalization T-I~,,_ I,N~,t=I,[T/N]St,n ~ 1, where [T/N] denotes

the number of trading days in the sample, then suggests the following estimator of

the intraday periodic component for interval n on day t,

T" exp(£,,,/2) ^ ~

st,,, y.~T/ff]EN= , e x p ( f , . / 2 ) (A.4)

Note that although the periodic modeling procedure is designed for fitting the average volatility pattern across the N intraday intervals, the second-stage estimation

of Eq. (A.3) is based on a time series regression that include all T intraday

returns. Utilizing this additional information in the data rather than simply fitting

the average intraday pattern, enhances the efficiency of the estimation.

The first step of our procedure involves the determination of the daily volatility

factor estimates i.e. ~t. Given the relative success of the daily MA(1)-GARCH(1,

1) models in explaining the aggregation results for the interdaily frequencies in

both markets, this appears to be a natural choice. Next, the number of interaction

terms, J and the truncation lag for the Fourier expansion, P, must be determined.

This is done primarily on the basis of parsimony i.e. for each of the return series

we choose the model that best matches the basic shape of the periodic pattern with

the fewest number of parameters. The resulting estimates for the DM-$ returns

with J=0and P=6are,

L,, =

n n 2

0.72 - 8.39 -- +5.59--

(1.06) (4.14) NI (4.14) N2

2"n'n 27rn 2~-2n 21r2n

- 2.51 cos-- - 0.40 sin-- - 0.38 cos-- +0.06sin

(--6.15) N (- 1044) N ( 3.71) N (2.70) N

27r3n 2rr3n 27r4n 27T4n

+ 0.42COS-- -- 0.09 sin-- -- 0.02 cos + 0.35 sin

(8.79) N (4.89) N (-0.53) N (20.48) N

27r5n 0.22 27r5n 27r6n 2rr6n

-- 0.12 cos-- +- sin 0.23 cos +0.01sin

(-5.38) N (13.35) N ( 12.67) N (0.45) N

where the numbers in parentheses indicate heteroskedastic robust t-statistics. It is

evident from the corresponding fit in Fig. 6a, that this representation provides an

excellent overall characterization of the average intradaily periodicity in the

DM-$ market. Consistent with Fig. 2a, the basic shape of the periodic pattern

appears invariant to the daily volatility level i.e. J = 0.

In contrast, our preferred model, the S & P 500, returns sets J = 1 and P = 2,

inferior overall fit. As seen in Fig. 2b, the volatility profile for the last fifteen

minutes of trading (intervals 78, 79 and 80) shows an abrupt change from the

overall smooth intraday pattern. Three dummy variables are included to minimize

the distortions that may otherwise arise from this distinct period i.e. d I = 78,

d 2 = 79 and d 3 = 80. The resulting fit depicted in Fig. 6b again testifies to the

success of this relatively simple procedure for modeling the periodicity in intraday

financial market volatility.

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