# Forecasting Stock Index Volatility: The Incremental Information in the Intraday High-Low Price Range

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**Forecasting Stock Index Volatility:**

**The Incremental Information in the**

**Intraday High-Low Price Range**

**Charles Corrado**

**Massey University - Albany**

**Auckland, New Zealand**

**Cameron Truong**

**University of Auckland**

**Auckland, New Zealand**

**November 2004**

**Abstract**

We compare the incremental information content of implied volatility and

intraday high-low range volatility in the context of conditional volatility

forecasts for three major market indexes: the S&P 100, the S&P 500, and

the Nasdaq 100. Evidence obtained from out-of-sample volatility forecasts

indicates that neither implied volatility nor intraday high-low range

volatility subsumes entirely the incremental information contained in the

other. Our findings suggest that intraday high-low range volatility can

usefully augment conditional volatility forecasts for these market indexes.

*JEL classification*: C13, C22, C53, G13, G14

*Keywords*: options; implied volatility; volatility forecasting

Please direct inquiries to: Charles Corrado, Department of Commerce, Massey

University - Albany, Private Bag 102 904 NSMC, Auckland, New Zealand.

[email protected]; or, Cameron Truong, Department of Accounting &

Finance, University of Auckland, Private Bag 92019, Auckland, New Zealand.

[email protected]

**Forecasting Stock Index Volatility:**

**The Incremental Information in the**

**Intraday Price Range**

**Abstract**

We compare the incremental information content of implied volatility and

intraday high-low range volatility in the context of conditional volatility

forecasts for three major market indexes: the S&P 100, the S&P 500, and

the Nasdaq 100. Evidence obtained from out-of-sample volatility forecasts

indicates that neither implied volatility nor intraday high-low range

volatility subsumes entirely the incremental information contained in the

other. Our findings suggest that intraday high-low range volatility can

usefully augment conditional volatility forecasts for these market indexes.

**I. Introduction**

Since the development of autoregressive conditional heteroscedasticity (ARCH)

models by Engle (1982) and their generalization (GARCH) by Bollerslev (1986, 1987),

ARCH modeling has become the bedrock for dynamic volatility models. While originally

formulated to forecast conditional variances as a function of past variances, the inherent

flexibility of ARCH modeling allows ready inclusion of other volatility measures as well.

Consequently, extensive research has focused on evaluating other volatility measures that

might improve conditional volatility forecasts. One popular volatility measure used to

augment ARCH forecasts is implied volatility from option prices. Lamoureux and

Lastrapes (1993) find that an ARCH model provides superior volatility forecasts than

implied volatility alone in a sample of 10 stock return series. However, Day and Lewis

(1992) report that a mixture of implied volatility and ARCH forecasts of future return

volatility for the S&P 100 stock index outperforms separate forecasts from implied

volatility or ARCH alone. More recently, Mayhew and Stivers (2003) find that implied

volatility improves GARCH volatility forecasts for individual stocks with high options

trading volume. They report that for stocks with the most actively traded options, implied

volatility reliably outperforms GARCH and subsumes all information in return shocks

beyond the first lag.

2

Another volatility measure that has become popular with the increasing

availability of intraday security price data is an intraday variance computed by summing

the squares of intraday returns sampled at short intraday intervals. Essentially, if the

security price path is continuous then increasing the sampling frequency yields an

arbitrarily precise estimate of return volatility (Merton, 1980). The efficacy of intraday

return variances has been demonstrated with foreign exchange data by Andersen et al.

(2001b), Andersen, Bollerslev, and Lange (1999), Andersen and Bollerslev (1998), and

Martens (2001) and with stock market data by Andersen et al. (2001a), Areal and Taylor

(2002), Fleming, Kirby, and Ostdiek (2003), and Martens (2002). Indeed as a competitor

to implied volatility, Taylor and Xu (1997), Pong, Shackleton, Taylor, and Xu (2003),

and Neely (2002) report that intraday return variances from the foreign exchange market

provide incremental information content beyond that provided by implied volatility

forecasts. By contrast, Blair, Poon, and Taylor (2001) find that the incremental

information content of intraday return variances for the S&P 100 stock index is scant and

that an implied volatility index published by the Chicago Board Options Exchange

(CBOE) provides the most accurate forecasts at all forecast horizons.

We extend the volatility forecasting literature cited above with the specific

objective of demonstrating the usefulness of the intraday high-low price range for

improving volatility forecasts for three major stock market indexes: the S&P 100, the

S&P 500, and the Nasdaq 100. This study represents the first attempt to compare the

effectiveness of the intraday high-low price range and implied volatility as forecasts of

future realized volatility for these market indexes.

We find that the intraday high-low range volatility estimator provides incremental

information content beyond that already contained in implied volatility indexes published

by the Chicago Board Options Exchange (CBOE). This is demonstrated by comparing

augmented volatility forecasts based around the asymmetric GARCH model developed

by Glosten et al. (1993) and Zakoian (1990), hereafter referred to as GJR-GARCH. Our

findings suggest that intraday high-low range volatility can usefully augment conditional

volatility forecasts for the three broad market indexes examined.

3

There are several reasons to consider the intraday high-low price range for

volatility measurement and forecasting. Firstly, high-low price range data has long been

available in the financial press and is often available when high-frequency intraday

returns data are not. Secondly, Andersen and Bollerslev (1998) point out that market

microstructure issues such as nonsynchronous trading effects, discrete price observations,

and bid-ask spreads, etc. may limit the effectiveness of intraday return variances as

volatility forecasts. For example, Andersen et al. (1999) report that sampling intraday

returns at one-hour intervals provided better results than sampling at 5-minute intervals in

their study of foreign exchange market volatility. The intraday high-low price range may

offer a useful alternative to an intraday return variance when market microstructure

effects are severe. Indeed, Alizadeh et al. (2002) suggest that, “Despite the fact that the

range is a less efficient volatility proxy than realized volatility under ideal conditions, it

may nevertheless prove superior in real-world situations in which market microstructure

biases contaminate high-frequency prices and returns.”

Thirdly, in addition to potential market microstructure biases Bai, Russell, and

Tiao (2001) point out that the estimation efficiency of an intraday return variance

estimator can be sensitive to non-normality in intraday returns data. As a basic

demonstration of potential sensitivity to non-normality, let

*r*d and

*r*h denote a one-day

return and an intraday return, respectively, such that the one-day return is the sum of

*n*

*n*intraday returns, i.e.,

*r*= ∑

*r*. Assuming that the

*n*intraday returns are identically,

*d*

*h*

*h*1

=

independently distributed (

*iid*), with an expected value of zero, i.e.,

*E*(

*r*) = 0 , then the

*h*

sum of the squared intraday returns is an unbiased estimator of the daily return variance.

*n*

*n*

2

*E*∑

*r*

=

*Var*∑

*r*

=

*Var r*

(1)

*h*

*h*

(

*d*)

*h*1=

*h*1=

Theoretically, the efficiency of the squared intraday returns volatility estimator specified

in equation (1) increases monotonically by dividing the trading day into finer increments.

A general statement of this proposition is provided by the following theorem:

4

**Theorem***n*

The variance of the squared intraday returns volatility estimator, i.e.,

2

*Var*∑

*r*

,

*h*

*h*1=

assuming

*iid*squared intraday returns with zero expected value is given by the expression

immediately below, in which

*Kurt*(

*r*d) and

*Kurt*(

*r*h) denote the kurtosis of daily

returns and intraday returns, respectively.

*n*

*n*

2

*Var*∑

*r*

= ∑

*Var r*

*h*

( 2

*h*)

*h*1=

*h*1

=

*n*

= ∑(

*E*(

*r*−

*Var r*

*h*)

( (

*h*))2

4

)

*h*1

=

=

*n*×(

*Var*(

*r*

*Kurt r*−

(2)

*h*))2 (

(

*h*) )1

= (

2

(

*Kurt r*−

*h*

)

*Var*(

*r*

×

*d*))

( ) 1

*n*

(

=

*Var*(

*r*

×

*Kurt r*− +

*d*))2

(

*d*)

2

3

*n*

The last equality on the right-hand side of equation (2) above is an immediate

consequence of the assumption of

*iid*intraday returns, for which the following

relationship holds as an adjunct to the Central Limit Theorem:1

*Kurt*(

*r*) − 3 =

*n*×(

*Kurt*(

*r*) − 3

(3)

*h*

*d*

)

Thus, with given values for the variance and kurtosis of daily returns, i.e.,

*Var*(

*r*d)

and

*Kurt*(

*r*d), the variance of the squared intraday returns volatility estimator declines

monotonically as

*n*increases.

However as shown in the last line of equation (2), the variance of the squared

intraday returns volatility estimator is bounded away from zero for non-normally

distributed returns with

*Kurt*(

*r*d) > 3. The theoretical relative efficiency of the squared

intraday returns volatility estimator to the squared daily return volatility estimator as a

function of return kurtosis is stated in equation (4) immediately below.

*Var*( 2

*r*

−

*d*)

*Kurt*(

*rd*) 1

=

(4)

*n*

2

2

*Var*∑

*r*

*Kurt*(

*r*− +

*d*)

3

*h*

*n*

*h*1=

1 An appendix provides a derivation.

5

With exactly normally distributed returns, i.e.,

*Kurt*(

*r*d) = 3, this relative efficiency

is bounded only by the number of intraday return intervals

*n*. However for plausible

kurtosis values, the relative efficiency in equation (4) can be severely bounded. For

example, a daily return kurtosis of

*Kurt*(

*r*d) = 4 with

*n*= 79 intraday return intervals

yields a theoretical relative efficiency of just 2.93.2

Parkinson (1980) shows that the intraday high-low price range volatility estimator

has a theoretical relative efficiency of 4.762 compared to a squared daily return.

However, this value assumes normally distributed returns. To assess relative efficiency

with non-normally distributed returns, we use Monte Carlo simulation experiments with

various return kurtosis values. We then simulate intraday returns over

*n*= 79 intraday

intervals for each of 100,000 trading days. Kurtotic intraday returns are generated by

random sampling from a mixture of normals, where with probability

*p*a random normal

variate is drawn with variance 2

σ and with probability 1-

*p*is drawn with variance 2

σ .

*p*

1−

*p*

The probability

*p*and the ratio of variances determine the kurtosis of the normals

mixture:

3(

4

4

*p*σ /σ

+1−

*p*

*p*

1−

*p*

)

*Kurtosis*= (

*p*σ /σ

+1−

*p*

*p*

−

*p*

)2

2

2

1

Following a convenient specification, we set

*p*= 1/

*Kurtosis*to solve for 2

σ as,

*Kurtosis*−1+ 3(

*Kurtosis*− 2)2 −1

2

)

σ

=

.

2

In each simulated trading day, we compute the sum of squared intraday returns, the

squared daily return, and the squared high-low range. Relative efficiencies computed

from these daily statistics averaged over 100,000 days are reported in the panel

immediately below.

2 Bai, Russell, and Tiao (2001) provide an extensive analysis of efficiency losses due to

kurtosis and other effects with non-

*iid*intraday returns.

6

Relative efficiencies of intraday variance estimators to

squared daily return estimator with varied kurtosis.

Daily

Squared intraday

return

Squared intraday

high-low range

Ratio

kurtosis returns estimator

estimator

3.5 4.786

2.960 1.617

4.0 2.944

2.632 1.118

4.5 2.289

2.462 0.930

5.0 1.997

2.385 0.837

Comparing relative efficiencies for the squared intraday returns estimator and the squared

intraday high-low range estimator as shown in the panel above, we see that for plausible

kurtosis values the squared intraday returns volatility estimator may not be greatly more

efficient than the squared high-low range estimator. Indeed, for daily kurtosis values

higher than about 4.3 the squared high-low range estimator is more efficient than the

squared intraday returns estimator. Further, Alizadeh et al. (2002) suggest that the

intraday high-low range is robust to microstructure noise, while the squared intraday

returns estimator can be quite sensitive to such noise.

**II. Data sources**

This study is based on returns for the S&P 100, S&P 500, and Nasdaq 100 stock

market indexes, along with daily implied volatilities for these indexes published by the

Chicago Board Options Exchange (CBOE). Ticker symbols for the implied volatility

indexes are VIX for the S&P 500, VXO for the S&P 100, and VXN for the Nasdaq 100.3

Our data set spans the period January 1990 through December 2003 for the S&P 100 and

S&P 500 stock indexes, and from January 1995 through December 2003 for the

Nasdaq 100 stock index.

3 The CBOE previously used the ticker VIX for S&P 100 implied volatility, but began

using VXO for S&P 100 implied volatility and VIX for S&P 500 implied volatility with

the introduction of the latter series.

7

**II.1. Daily index returns**

Daily index returns are calculated as the natural logarithm of the ratio of

consecutive daily closing index levels.

*r*

= ln

*c c*

(5)

*t*

(

*t t*1−)

In equation (5),

*r*t denotes the index return for day

*t*based on index levels at the close of

trading on days

*t*and day

*t*-1, i.e.,

*c*t and

*c*t-1, respectively.

**II.2. Daily high-low price range**

“..intuition tells us that high and low prices contain more information regarding to

volatility than do the opening and closing prices.” (Garman and Klass, 1980) For

example, by only looking at opening and closing prices we may wrongly conclude that

volatility on a given day is small if the closing price is near the opening price despite

large intraday price fluctuations. Intraday high and low values may bring more integrity

into an estimate of actual volatility.

In this study, we use the intraday high-low volatility measure specified in

equation (6), in which

*hi*t and

*lo*t denote the highest and lowest index levels observed

during trading on day

*t*.

(ln

*hi*−ln

*lo*

*t*

*t*)2

2

*RNG*=

(6)

*t*

4 ln 2

This intraday high-low price range was originally suggested by Parkinson (1980) as a

measure of security return volatility.4

**II.3 CBOE implied volatility indexes**

Implied volatilities have long been used by academics and practitioners alike to

provide forecasts of future return volatility. In addition to studies cited earlier,

Christensen and Prabhala (1998) overcome the methodological difficulties in Canina and

Figlewski (1993) and show that by using non-overlapping data and an instrumental

variables econometric methodology that implied volatility outperforms historical

4 Interesting extensions to Parkinson (1980) have been developed by Garman and Klass

(1980), Ball and Torous (1984), Rogers and Satchell (1991), Kumitomo (1992), and

Yang and Zhang (2000).

8

volatility as a forecast of future return volatility for the S&P 100 index. Corrado and

Miller (2004) update and extend the Christensen and Prabhala study and suggest that

implied volatility continued to provide a superior forecast of future return volatility

during the period 1995 through 2003.

In this study, we use data for three implied volatility indexes published by the

Chicago Board Options Exchange (CBOE). These implied volatility indexes are

computed from option prices for options traded on the S&P 100, the S&P 500, and the

Nasdaq 100 stock indexes.

The implied volatility indexes with ticker symbols VIX and VXN are based on

European-style options on the S&P 500 and Nasdaq 100 indexes, respectively. These

indexes are calculated using the formula stated immediately below, in which

*C*(

*K*,

*T*) and

*P*(

*K*,

*T*) denote prices for call and put options with strike price

*K*and time to maturity

*T*

stated in trading days. This formula assumes the option chain has strike prices ordered

such that

*K*

>

*K*. The two nearest maturities are chosen with the restriction that

*j*1

+

*j*

*T*≥ 22 ≥

*T*≥ 8 .

2

1

2

−

−

*VIX*

= ∑(− ) (

*T*22

*K*

*K*

*h*

)

*N*

*h*

*j*1

+

*j*1

1

−

∑

min

*C K*,

*T*,

*P K*,

*T*

(7)

2

( (

*j h*) (

*j h*))

−

*h*1

=

(

*T T*=

*K*

2

1 )

*j*1

*j*

Theoretical justification for this calculation method is provided by Britten-Jones and

Neuberger (2000).

The implied volatility index with ticker symbol VXO is based on American-style

options on the S&P 100 index.5 This index is calculated using the formula stated

immediately below in which

*IV*C(

*K*,

*T*) and

*IV*P(

*K*,

*T*) are implied volatilities for call and

put options, respectively, with strike

*K*and maturity

*T*. The at-the-money strike

*K*m

denotes the largest exercise price less or equal to the current cash index

*S*0. Hence, the

volatility index VXO is calculated using only option contracts with strike prices that

bracket the current cash index level.

1

2

∑∑(− )1

*j*+

*h*(

*T*−22

*S*−

*K*

*IV K*

*T*+

*IV K*

*T*

*h*

)(

,

,

0

*m*1

+ −

*j*) (

*C*(

*m*+

*j*

*h*)

*P*(

*m*+

*j*

*h*))

*j*=0

*h*1

*VXO*

=

=

(

(8)

*T*−

*T*

*K*

−

*K*

2

1 ) (

*m*1

+

*m*)

5 Authoritative descriptions of this implied volatility index are Whaley (1993) and

Fleming, Ostdiek, and Whaley (1995).

9

To be scaled consistently with the other daily volatility measures, the implied

volatility indexes VXO, VIX, and VXN are all squared and divided by 252, the assumed

number of trading days in a calendar year.

[TABLE 1 HERE]

**II.4 Descriptive statistics**

Table 1 provides a statistical summary of the volatility data used in this study.

Panel A reports the mean, maximum, minimum, standard deviation, and skewness and

kurtosis coefficients for squared daily returns, squared implied volatilities, and squared

high-low price ranges for the S&P 100 index. Panels B and C report descriptive statistics

for the S&P 500 and Nasdaq 100 indexes, respectively.

The period January 1990 through December 2003 yields 3,544 daily observations

for the S&P 100 and S&P 500 indexes and the period January 1995 through December

2003 yields 2,266 daily observations for the Nasdaq 100 index. Table 1 reveals noticeable

statistical differences among the three volatility measures. For example, in all panels of

Table 1 the average squared high-low range volatility is smaller than the average squared

daily return, which in turn is smaller than the average squared implied volatility.

Comparing volatility measures across S&P 100, S&P 500, and Nasdaq 100 indexes it is

evident that volatility for the Nasdaq 100 is highest among the three indexes. Indeed, the

average squared daily return for the Nasdaq 100 index is on average four to five times

larger in magnitude than average squared daily returns for the S&P 100 and S&P 500

indexes.

**III. Forecast methodology**

To model market volatility dynamics we draw on the GJR-GARCH model

specification developed by Glosten et al. (1993) and Zakoian (1990). This model attempts

to capture the asymmetric effects of good news and bad news on conditional volatility.

We augment the basic GJR-GARCH model with implied volatility and intraday high-low

price range volatility.

10