This is not the document you are looking for? Use the search form below to find more!

Report home > World & Business

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

0.00 (0 votes)
Document Description
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.
File Details
  • Added: April, 16th 2010
  • Reads: 1502
  • Downloads: 203
  • File size: 433.85kb
  • Pages: 32
  • Tags: options, implied volatility, volatility forecasting
  • content preview
Submitter
  • Username: samanta
  • Name: samanta
  • Documents: 1258
Embed Code:

Add New Comment




Related Documents

Cyber Monday Online Offers _ The Way To Low-Price Shopping

by: appset867, 1 pages

of list areas and also special offers on the planet with millions of dollars price of merchandise offered

Exchanges of Cost Information in the Airline Industry

by: samanta, 34 pages

We empirically analyze exchanges of cost information in amulti-market oligopoly model for the airline industry with entry and incomplete informa- tiononmarginal costs. We develop an algorithm to ...

Empirical Analyses of Industry Stock Index Return Distributions for the Taiwan Stock Exchange

by: samanta, 19 pages

We study the daily return distributions for 22 industry stock indexes on the Ta i- wan Stock Exchange under the unconditionalhomoskedastic independent, identically distributed and the conditional ...

Double Bubble: The Implications of the Over-Valuation of the Stock Market and the Dollar

by: samanta, 19 pages

The stock market is over-valued by close to 50 percent, according to most economists who have examined stock prices and trends in corporate profits. The dollar may be over-valued by 30 percent, or ...

About a coincident index for the state of the economy

by: samanta, 32 pages

The construction of coincident indexes for the economic activity ofa country is a common practice since the fifties. The methodologies vary from heuristic methods to probabilistic or statistical ones ...

Is ThereaPositive Relationship between Stock Market Volatility and ...

by: wasil, 22 pages

This paper investigates whether evidence fora positive relationship between stock market volatility and the equity premium is more decisive when the volatility feedback effects of large and ...

Information in Accruals about the Quality of Earnings

by: shinta, 52 pages

We extend the analysis in Sloan (1996) to identify the source of information in accruals about earnings quality. Our results indicate that information in accruals about earnings quality is ...

S Fund, Small Capitalization Stock Index Investment Fund

by: henriette, 2 pages

S Fund Small Capitalization Stock Index Investment Fund Thrift Savings Plan ...

REAL SECTOR CONFIDENCE INDEX FROM THE BUSINESS TENDENCY SURVEY OF CBRT

by: samanta, 17 pages

The Business Tendency Survey (BTS) and the constructed real sector confidence index of the Central Bank of the Republic of Turkey (CBRT) are introduced in this survey. BTS conducted since December ...

The Dynamics of Price Elasticity of Demand in the Presence of Reference Price Effects

by: samanta, 13 pages

The authors derive an expression for the price elasticity of demand in the presence of reference price effects that includes a component resulting from the presence of gains and losses in consumer ...

Content Preview


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.
c.j.corrado@massey.ac.nz; or, Cameron Truong, Department of Accounting &
Finance, University of Auckland, Private Bag 92019, Auckland, New Zealand.
c.truong@auckland.ac.nz


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 rd and rh 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(rd) and Kurt(rh) denote the kurtosis of daily
returns and intraday returns, respectively.
n
n


2
Var r
= ∑Var r

h
( 2h)
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(rd)
and Kurt(rd), 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(rd) > 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(rd) = 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(rd) = 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
pσ /σ
+1− p
p
1− p
)
Kurtosis = (

pσ /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), rt 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., ct and ct-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 hit and lot denote the highest and lowest index levels observed
during trading on day t.
(ln hi −lnlo
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 IVC(K,T) and IVP(K,T) are implied volatilities for call and
put options, respectively, with strike K and maturity T. The at-the-money strike Km
denotes the largest exercise price less or equal to the current cash index S0. Hence, the
volatility index VXO is calculated using only option contracts with strike prices that
bracket the current cash index level.
1
2
∑∑(− )1j+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

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

 

 

Your download will begin in a moment.
If it doesn't, click here to try again.

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

Insert your wordpress URL:

example:

http://myblog.wordpress.com/
or
http://myblog.com/

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

From:

To:

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

Enter two words as shown below. If you cannot read the words, click the refresh icon.

loading

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

Copy html code above and paste to your web page.

loading