Measuring Historical Volatility - Semantic Scholar · 2016-01-15 · 1 Measuring Historical Volatility In pricing options, anticipated volatility over the life of the option is the

Measuring Historical Volatility

Louis H. EderingtonUniversity of Oklahoma

Wei GuanUniversity of South Florida St. Petersburg

August 2004

Contact Info: Louis Ederington: Finance Division, Michael F. Price College of Business,University of Oklahoma, 205A Adams Hall, Norman, OK 73019, USA. Phone: (405) 325-5591Wei Guan: College of Business, University of South Florida St. Petersburg, 140 Seventh Avenue,St. Petersburg, FL 33701, USA. Phone: (727)-553-4945

Comments may be sent to the authors at [email protected] or [email protected].

mailto:[email protected]

mailto:[email protected].


Abstract

The adjusted mean absolute deviation is proposed as a simple-to-calculate alternative to

the historical standard deviation as a measure of historical volatility and input to option pricing

models. We show that when returns are approximately log-normally distributed, this measure

forecasts future volatility consistently better than the historical standard deviation. In the markets

we examine, it also forecasts as well or better than the GARCH model.

1


In pricing options, anticipated volatility over the life of the option is the crucial unknown

parameter. While sophisticated volatility estimation procedures, such as GARCH, are popular

among finance researchers, these require econometrics software which is difficult for the average

undergraduate student or casual options trader to obtain or master so have not found their way

into most derivatives texts. Instead, almost all derivatives textbooks instruct students to use

historical volatility, specifically the historical standard deviation, over some recent period as the

volatility input.1 Of course implied volatility provides an alternative (and theoretically better)

estimate of future volatility but for option pricing purposes, this measure suffers from an obvious

chicken and egg problem in that to calculate implied volatility requires the option price and to

calculate the appropriate price requires a volatility estimate. Hence, the historical standard

deviation of log returns is the volatility estimator touted in most textbooks and most commonly

reported on options websites.

One well-known short-coming of the historical standard deviation as an estimator of

future volatility is that only the information in past returns is considered, ignoring other possible

information sets, like knowledge of future scheduled events that might move the markets such as

a quarterly earnings report or an upcoming meeting of the Fed’s Open Market Committee.2

Another well-known potential problem is that all past squared return deviations back to an

arbitrary date are weighted equally in calculating the standard deviation and all observations

before that date are ignored (Engle, 2004, and Poon and Granger, 2003). Evidence on volatility

clustering and persistence indicates that more recent observations should contain more

information regarding volatility in the immediate future than older observations. Accordingly,

more sophisticated models, such as GARCH and the exponentially weighted moving average

model utilized by Riskmetrics, employ weighting schemes in which the most recent squared

2

return deviations receive the most weight and the weights gradually decline as the observations

recede in time.3

While the GARCH weighting scheme in which the most recent squared return deviation

receives the greatest weight and the weights decline exponentially is theoretically more appealing

than that of the historical standard deviation in which all observations are weighted equally back

to an arbitrary cutoff point, evidence is mixed on whether GARCH actually forecasts better. In a

comprehensive review of 39 studies comparing the historical standard deviation (or variance) and

GARCH, Poon and Granger (2003) report that 17 find that GARCH forecasts better while 22

find that the historical standard deviation (or variance) forecasts better.

We see another potential drawback to the historical standard deviation: that since the

historical standard deviation and variance are functions of squared return deviations, they could

be dominated by outliers. For example, in the S&P 500 market over the period from 7/5/67 to

7/11/02, the 1% largest daily return deviations account for 24.5% of the total squared return

deviations. Suppose daily returns from one month (specifically 20 trading days) are used to

forecast volatility over the next month. 81.8% of the months will contain no observations from

this 1% tail while 18.2% will contain 1 or more. The 24.5% figure implies that, if the underlying

distribution does not change, the standard deviation for months that contain one observation in

the 1% tail will tend to be 60% higher than that for months with none. Consequently, if a month

with an observation in the 1% tail precedes a month with none, the forecast standard deviation

will tend to be much too high.

One way to avoid giving extreme observations undue weight is to use a longer period to

measure the historical standard deviation. For instance, one might use returns over the last year

instead of over the last month. The problem with this approach is that volatility tends to cluster.

Specifically high (low) volatility one week, or month tends to be followed by high (low)

3

volatility the succeeding week or month. Measuring historical volatility over a long period, such

as a year, smooths out the clusters and loses the information in recent volatility.

An alternative proposed by Andersen and Bollerslev (1998) and others which gives less

weight to extreme observations without lengthening the data period is to switch from daily to

intraday data. For instance, if the trading day is 6 hours in length, switching from one month of

daily return observations to one month of hourly return observations increases the number of

observations 6 times and cuts the weight on any one observation to one sixth the weight on daily

observations. For GARCH type models, Andersen and Bollerslev (1998), Andersen et al (1999),

and Andersen et al (2003) report considerably improved volatility estimates using such intraday

data. However, intraday data is not readily available to students and traders, is costly, and is

more difficult to handle. Authors of derivative texts apparently feel volatility estimates based on

intraday data are not usually a viable or cost effective alternative since they do not tout this

approach and their example calculations of historical volatility invariably use daily data.

In our view, a possible alternative based on daily data which is less sensitive to extreme

observations while keeping the sample period short enough to reap the benefits of volatility

clustering is to measure the historical standard deviation in terms of absolute, instead of squared

return observations. Specifically, we propose using the historical mean absolute return deviation,

instead of the historical standard deviation to measure historical volatility and forecast future

volatility. Relatively, extreme observations are simply less extreme when measured in absolute

rather than squared terms. Moreover, the mean absolute return deviation, is easily calculated

using rudimentary statistical software such as EXCEL.

The major drawback of the mean absolute deviation is that it is distribution specific. That

is, unlike the sample standard deviation, the relation between the mean absolute deviation and the

population standard deviation differs between normal, beta, gamma, and other distributions.

However, since the primary use of historical volatility is as an input into the Black-Scholes

4

formula, which assumes log returns are normally distributed, basing the historical volatility

estimate on a log-normal distribution is a natural choice and imposes no additional assumption in

this application. Nonetheless, whether the historical standard deviation or the historical mean

absolute return deviation provides a better volatility forecast will likely depend on how closely

the actual distribution of returns approximates a normal distribution. This question is explored

below.

In this paper, we compare the ability of the historical mean absolute return deviation and

the historical standard deviation to forecast future volatility (measured as the standard deviation

of ex-post log returns) across a wide variety of markets, including the stock market, long and

short-term interest rates, foreign exchange rates and individual equities. Except for the

Eurodollar market at long horizons (and occasionally the Yen/Dollar exchange rate), the mean

absolute deviation forecasts future volatility better than the historical standard deviation in all

markets at almost all horizons. Not surprisingly, the major exception, Eurodollars, is the market

which is furthest from normality.

We also compare the mean absolute deviation volatility forecasts with those generated by

a GARCH(1,1) model. Their relative forecast ability seems to partially depend on how one

measures forecast accuracy. By one of our two measures, the mean absolute deviation clearly

dominates, by the other measure neither dominates. Since the GARCH model requires

specialized software, while the mean absolute deviation can be calculated using simple

spreadsheet software, such as EXCEL, and the mean absolute deviation forecasts as well or better

in most cases, there seems to be little payoff to the added expense and effort of GARCH type

models.

The remainder of the paper is organized as follows. In the next section, we explain the

adjusted mean absolute deviation measure of volatility and our measures of forecast accuracy.

5

Our data sets are described in section 2. Results are presented in 3 and section 4 concludes the

paper.

6

(1)

1. Measuring and Forecasting Volatility

Let Rt = ln(Pt/Pt-1) represent daily returns on a financial asset.4 The historical

standard deviation over the last n days is measured as:

where , the return deviation, and : is the expected return. Often : is replaced by the

sample mean and accordingly the r2 are divided by n-1, rather than n. The latter procedure

implicitly assumes that the expected return over the coming period equals the mean return in the

n-day period used to estimate STD(n). Given the low auto-correlation in returns there is no

justification for such an assumption and Figlewski (1997) shows that better forecasts are

normally obtained by setting :=0. In our calculations, we set : equal to the average daily return

over the entire data period. Since volatilities are normally quoted in annualized terms, the sum of

the squared daily return deviations is multiplied by 252, the approximate number of trading days

in a year.

As seen in equation 1, this volatility estimator assigns each squared past return deviation,

, after time t-n a weight of 1/n while observations before t-n receive a weight of zero. An

issue in applying this procedure is choosing the cutoff date n. While setting the length of the

period used to calculate historical volatility, n, equal to the length of the forecast period, s, is a

common convention, Figlewski (1997) finds that forecast errors are generally lower if the

historical variance is calculated over a longer period. Accordingly, we consider a variety of

sample period lengths and designate the length of the sample on which STD is based with (n).

For example, STD(20) indicates the annualized standard deviation of log returns over the last

twenty trading days.

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(2)

(3)

Our alternative measure of historical volatility is the adjusted mean absolute return

deviation over the same n days. The (unadjusted) mean absolute return deviation is:

However, in this unadjusted form, MAD(n) is a biased estimator of the population standard

deviation. While STD converges to F as sample size increases,5 MAD converges to F/k where k

depends on the distribution. If the distribution is normal, (Stuart and Ord, 1998). If the

data follow another distribution, k takes another value. Consequently, we use the adjusted mean

absolute deviation defined as:

This yields an unbiased estimator of F if the rt are normally distributed. Like STD(n), AMAD(n)

is annualized by multiplying by the square root of 252.

In the early part of the 20th century there was debate among statisticians about the

relative merits of STD and MAD as measures of volatility but use of the sample standard

deviation dominated since it is a consistent estimator of F regardless of the data distribution and

Fisher showed that it is more efficient for normal distributions. Nonetheless, it has been argued

by Staudte and Sheather (1990), and Huber (1996) among others that MAD is more robust, i.e.,

that it yields better estimates if the data are contaminated.

In this paper, STD(n) and AMAD(n) are compared in terms of their ability to forecast

actual volatility over various future horizons, s, corresponding to likely option times to

expiration. We (like all other studies in this area) measure this realized volatility as the standard

deviation of daily log returns over the period s. Specifically,

8

(4)

(5)

(6)

(7)

Note that STD(n) and AMAD(n) are both evaluated using the same measure of ex post volatility,

RLZ(s), the measure used in virtually all previous studies of forecasting ability.

We use two measures of the ability of STD(n) and AMAD(n) to forecast RLZ(s). The

first is the root mean squared forecast error:

where F(n)t designates STD(n)t or AMAD(n)t. The second is the mean absolute forecast error:

In this paper our main interest is which of these two simple volatility estimators, STD(n)

and AMAD(n), which are easily explained to and calculated by undergraduates, forecasts realized

volatility better. However, we are also interested in how AMAD(n) and STD(n) compare with

the more sophisticated ARCH-GARCH models proposed by academic researchers. By far the

most popular of the ARCH-GARCH models is Bollerslev’s original, the so-called GARCH(1,1)

model:

where vt represents the (unobserved) conditional variance at time t. Consequently, we also

measure how well this model forecasts actual volatility using RMSFE and MAFE.6

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2. Data and Procedures

We compare the volatility forecasting ability of these three models for four financial

assets with highly active options markets: the S&P 500 Index, the 10-year Treasury Bond rate,

the 3-month Eurodollar rate, and the Yen/Dollar exchange rate. We also collect data for five

equities chosen from those in the Dow-Jones Index: Boeing, GM, International Paper,

McDonald’s, and Merck. Daily return data for the five equities for the period 7/2/62 to 12/31/02

were obtained from CRSP tapes as were prices for the S&P 500 index. Daily interest rate and

exchange rate data were obtained from Federal Reserve Board files for the periods: 1/2/62-

12/31/03 for the 10 year bond rate, and 1/1/71-12/31/03 for the Eurodollar rate and the

Yen/Dollar exchange rate.

Using the three procedures, STD(n), AMAD(n), and GARCH, we forecast volatility over

horizons, s, of 10, 20, 40, and 80 market days or approximately two weeks, one month, two

months, and four months. Similarly, STD(n) and AMAD(n) are measured over historical periods,

n, of 10, 20, 40, and 80 market days. The GARCH procedure requires estimation of the

parameters " and $. This is done using 1260 daily return observations or approximately five

years of daily data. Consequently, our estimation periods for RLZ for the three models begin

approximately five years after the beginning dates for our data sets reported above, e.g., 7/5/67

for the S&P 500 index and the five equities. They end 120 trading days before the end of the data

sets, i.e., 7/11/02 for the S&P 500 index and the equities and 7/16/03 for the interest and

exchange rate series. To limit the computational burden, the GARCH model is re-estimated every

40 days.

Descriptive statistics for daily log returns are reported in Table 1 where the mean and

standard deviation are standardized for ease in interpretation. Skewness varies and is slight for

all except return on the S&P500 index. All except Merck exhibit excess kurtosis which is most

extreme for the S&P 500 index. Since the AMAD measure only provides a consistent estimate

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of the population standard deviation if the underlying distribution is normal, the Kolmogorov-

Smirnov D statistic test for normality shown in the last column is of particular interest. In all

nine markets, the normality null is rejected at the .01 level. However, the K-S D is considerably

higher for Eurodollar returns than any of the other log return series indicating a more serious

deviation from normality in that market. This is somewhat surprising since both skewness and

kurtosis are more serious for the S&P 500 index emphasizing that skewness and kurtosis do not

completely account for deviations from normality.

3. Results

Root mean squared forecast error (RMSFE) (equation 5) and mean absolute forecast error

(MAFE) (equation 6), measures of how accurately the three models forecast actual future

volatility (RLZ), are reported in Tables 2, 3, 4, and 5 for forecast horizons, s, of 10, 20, 40, and 80

trading days respectively. RMSFE statistics are reported in panel A and MAFE in panel B.

STD(n) and AMAD(n) are calculated using sample period lengths n of 10, 20, 40, and 80 trading

days. For each n, the measure, STD(n) or AMAD(n), with the lowest RMSFE and MAFE is

shown in bold. Also, in each market, the cell of the model with the lowest RMSFE or MAFE is

shaded. When this is the GARCH model, the STD(n) or AMAD(n) model with the lowest

RMSFE or MAFE is more lightly shaded.

Consider first the results in Table 2 when the models are used to forecast volatility over a

relatively short horizon of 10 trading days or two weeks. As shown by the cells in bold, in all nine

markets for all four n (a total of 36 pairwise comparisons) , the historical adjusted mean absolute

deviation anticipates actual future volatility better than the historical standard deviation. This

results holds whether forecast accuracy is measured in terms of the RMSFE in panel A or the

MAFE in panel B. In 31of the 36 pairwise comparisons, AMAD’s volatility forecasting ability is

significantly greater than STD’s at the 10% level according to Diebold and Mariano’s (1995) S1

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statistic. In 16 of 36 it is significantly better at the 5% level. Clearly AMAD dominates STD at

this forecast horizon.

As shown by the darkly and lightly shaded cells, among the AMAD models, forecast

accuracy is best when the mean absolute deviation is measured over 40 (4 markets) or 80 (5

markets) trading days. This parallels Figlewski’s (1997) finding that the historical standard

deviation is best measured over a period longer than the forecast horizon.

As shown by the darkly shaded cells, the AMAD model beats the GARCH model in all

nine markets according to the MAFE criterion. According to the RMSFE criterion however, the

evidence is mixed with AMAD forecasting better in four markets and GARCH in five.

Considering that the adjusted mean absolute deviation is easily calculated with standard

spreadsheet software while the GARCH model requires time, study, and sophisticated software,

the AMAD model seems the most time and cost effective for all but professional traders and

finance researchers. Of course, since it is dominated by AMAD, STD fares worse against

GARCH. In a majority of markets, GARCH forecasts better than the historical standard deviation

by both measures.

Turning to Table 3 in which the models are used to forecast volatility over a 20 trading day

horizon, the same results generally hold. However at this horizon, STD(10) has a lower MAFE

than AMAD(10) for GM and STD(80) has a lower RMSFE than AMAD(80) for Eurodollars. The

former is of little consequence since our results indicate that in all markets one should never use a

sample period as short as 10 trading days to forecast future volatility regardless of the horizon.

The latter reversal is more serious since in the Eurodollar market, STD(80) turns out to have the

lowest RMSFE of all the models. Recalling from above and Table 1 that the deviation from log

normality is most serious in this market, it is not surprising that AMAD should perform least well

in this market. Nonetheless, AMAD(80)’s MAFE is considerably lower than STD(80)’s and

AMAD beats STD for all n in all other markets by both criteria. Moreover in 12 of the 36

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pairwise comparisons, AMAD’s forecasting ability is significantly better than STD’s at the .05

level.

As we move to horizons of 40 and 80 days in Tables 4 and 5 respectively, STD beats

AMAD by one or both criteria in the Eurodollar market for several n and STD(80) beats

AMAD(80) by both criteria in the Yen/dollar exchange rate market. However, in all of these

cases, the null that the forecasting ability of AMAD and STD is equal cannot be rejected at the .10

level. In all the other markets for all measures based on 20 on more trading days, AMAD beats

STD by both criteria. Moreover, in nine comparisons at the 40 day horizon and six at the 80 day

horizon, AMAD’s forecasting ability is significantly better at the .05 level.

As we move to progressively longer forecast horizons, another trend emerges in that

estimates of historical volatility based on longer sample periods tend to forecast volatility better

than estimates based on shorter periods. Specifically, as the forecast horizon is lengthened the

instances in which AMAD(40) has a lower RMSFE or MAFE than AMAD(80) declines. As

shown in Table 5, at a forecast horizon of 80 trading days (about 4 months), AMAD(80) beats

AMAD(40) in all markets.

At all forecast horizons, AMAD (or STD) forecasts future volatility better than the

GARCH(1,1) model in all markets except Merck according to the MAFE criterion while results

are mixed according to the RMSFE criterion.

4. Conclusions

We have sought to determine whether there is a simple alternative estimator, that is one

simple enough to calculate using standard spreadsheet software and easily explained to the average

undergraduate student, which provides a better volatility estimate to use in option pricing models

than the historical standard deviation. We have found that when log-returns are approximately

normally distributed there is - specifically, the adjusted mean absolute return deviation, which is

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the absolute mean return deviation multiplied by the square root of B/2. In all except the short-

term interest rate market where the deviation from log-normality is most serious, the adjusted mean

absolute deviation forecasts actual volatility consistently better than the historical standard

deviation. We also find that this simple estimator compares very favorably with GARCH model

estimates. By one measure of forecast accuracy, the adjusted mean absolute return deviation beats

GARCH consistently, while by the other measure, they approximately break even.

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REFERENCES

Andersen, Torben, and Tim Bollerslev, 1998, Answering the skeptics: Yes standard volatility

models do provide accurate forecasts, International Economic Review 39 (#4), 885-905.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys, 2003, Modeling and forecasting

realized volatility, Econometrica 7 (#2), 579-625.

Andersen, T.G., T. Bollerslev, and S. Lange, 1999, Forecasting financial market volatility: sample

frequency vis-a-vis forecast horizon, Journal of Empirical Finance 6, 457-477.

Diebold, Francis X., and Roberto S. Mariano, 1995, Comparing predictive accuracy, Journal of

Business and Economic Statistics 13, 253-263.

Ederington, Louis H. and Jae Ha Lee, 1993, How Markets Process Information: News Releases

and Volatility, Journal of Finance 48 (September), 1161-1191.

Ederington, Louis H. and Jae Ha Lee, 2001, Intraday Volatility in Interest Rate and Foreign

Exchange Markets: ARCH, Announcement and Seasonality Effects, Journal of Futures

Markets 21 (#6), 517-52.

Ederington, Louis H. and Guan, Wei, 2005, Forecasting Volatility, forthcoming Journal of Futures

Markets, and currently available on SSRN.com.

Engle, Robert, 2004, Risk and volatility: econometric models and financial practice, American

Economic Review 94(3), 405-420.

Figlewski, Stephen, 1997, Forecasting volatility, Financial Markets, Institutions, and Instruments

vol. 6 #1, Stern School of Business, (Blackwell Publishers: Boston).

Huber, Peter, 1996, Robust Statistical Procedures, 2nd ed., (Society for Industrial and Applied

Mathematics: Philadelphia).

Hull, John C., 2003, Options, Futures, and Other Derivatives, 5th edition, Englewood Cliffs:

Prentice Hall.

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Poon, Ser-Huang and Clive Granger, 2003, Forecasting Volatility in Financial Markets: A Review,

Journal of Economic Literature 41 (#2), 478-539.

Staudte, Robert, and Simon Sheather, 1990, Robust Estimation and Testing, (John Wiley & Sons:

New York).

Stuart, Alan and Keith Ord, 1998, Kendall’s Advanced Theory of Statistics: Volume 1 Distribution

Theory, 6th edition, New York: Oxford University Press.

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Table 1 - Descriptive Statistics

Statistics are reported for annualized daily log returns for the following periods: S&P 500 andthe five individual equities: 7/5/67-7/11/02, T-Bonds: 11/02/66-7/16/03, Eurodollars and theYen/Dollar exchange rate: 11/04/75-7/16/03.

Mean

Standard

Deviation Skewness Kurtosis

Kolmogorov-

Smirnov D

S&P 500 0.066402 0.1537 -1.6081 39.893 .0583

T-Bonds -0.00602 0.1392 .0922 4.986 .0786

Eurodollars -0.06424 0.2430 -.3859 8.311 .2395

Yen/Dollar -0.03267 0.1043 -.5116 4.617 .0755

Boeing 0.098581 0.3369 .0212 5.242 .0556

GM 0.06834 0.2680 -.1853 7.351 .0470

Int. Paper 0.085882 0.2874 -.4092 13.128 .0459

McDonalds 0.12594 0.2898 -.3210 7.948 .0501

Merck 0.12818 0.2482 -.0516 3.375 .0444

Table 2 - Volatility Forecast Accuracy – 10 Trading Day Horizon

Root mean squared forecast errors (RMSFE) and mean absolute forecast errors (MAFE) are reported when the procedures listed in column 1 areused to forecast the standard deviation of returns over the next 10 trading days. STD(n) denotes the standard deviation calculated over the last ntrading days. AMAD(n) denotes the adjusted mean absolute deviation over the last n days. Data periods are reported in Table 1. For each n, the STD(n) or AMAD(n), with the lowest RMSFE and MAFE is shown in bold. In each market, the cell of the model with the lowest RMSFE orMAFE is shaded. When this is the GARCH model, the STD(n) or AMAD(n) with the lowest RMSFE or MAFE is lightly shaded.

Forecasting

Model

Markets

S&P 50010-year

T-Bond

90-day

EurodollarYen/Dollar Boeing GM Int’l Paper McDonalds Merck

Panel A - Root Mean Squared Forecast Errors (RMSFE)

STD(10) 0.07312 0.05717 0.12313 0.04790 0.15634 0.11596 0.12563 0.12338 0.10326

AMAD(10) 0.07023 0.05665 0.11976 0.04662 0.15101 0.11481 0.12003 0.12208 0.10197

STD(20) 0.07076 0.05325 0.11697 0.04327 0.14163 0.10928 0.11856 0.11314 0.09399

AMAD(20) 0.06616 0.05199 0.11317 0.04185 0.13465 0.10566 0.11244 0.11024 0.09162

STD(40) 0.06964 0.05216 0.11163 0.04215 0.13270 0.10321 0.11309 0.10750 0.08948

AMAD(40) 0.06477 0.05095 0.10834 0.04097 0.12638 0.09948 0.10798 0.10485 0.08729

STD(80) 0.07073 0.05296 0.11059 0.04148 0.12970 0.10113 0.10979 0.10957 0.08884

AMAD(80) 0.06661 0.05187 0.10885 0.04068 0.12504 0.09821 0.10473 0.10831 0.08712

GARCH 0.06461 0.05038 0.11192 0.04240 0.12716 0.09738 0.10741 0.10427 0.08586

Panel B - Mean Absolute Forecast Errors (MAFE)

STD(10) 0.04060 0.04142 0.08306 0.03469 0.10941 0.07839 0.08724 0.08448 0.07604

AMAD(10) 0.04021 0.04101 0.08253 0.03375 0.10750 0.07905 0.08528 0.08463 0.07576

STD(20) 0.03865 0.03823 0.08002 0.03143 0.09912 0.07336 0.08111 0.07608 0.06951

AMAD(20) 0.03719 0.03724 0.07747 0.02992 0.09570 0.07248 0.07861 0.07460 0.06817

STD(40) 0.03882 0.03789 0.07886 0.03045 0.09403 0.06979 0.07686 0.07211 0.06642

AMAD(40) 0.03679 0.03638 0.07342 0.02884 0.08943 0.06774 0.07440 0.06980 0.06412

STD(80) 0.04059 0.03896 0.07921 0.03027 0.09393 0.06856 0.07372 0.07241 0.06572

AMAD(80) 0.03797 0.03710 0.07378 0.02871 0.08877 0.06572 0.07050 0.07038 0.06366

GARCH 0.03721 0.03688 0.08217 0.03188 0.09482 0.06608 0.07428 0.07134 0.06452



Forecasting

Model

Markets

S&P 50010-year

T-Bond

90-day



STD(10) 0.07178 0.05346 0.11811 0.04358 0.14199 0.11011 0.11926 0.11448 0.09447

AMAD(10) 0.06830 0.05264 0.11521 0.04213 0.13579 0.10862 0.11292 0.11227 0.09296

STD(20) 0.06824 0.04855 0.10860 0.03910 0.12424 0.10092 0.10872 0.10127 0.08315

AMAD(20) 0.06291 0.04720 0.10520 0.03756 0.11644 0.09699 0.10217 0.09750 0.08097

STD(40) 0.06643 0.04694 0.10047 0.03714 0.11371 0.09306 0.10027 0.09631 0.07803

AMAD(40) 0.06109 0.04576 0.09876 0.03609 0.10730 0.08893 0.09456 0.09326 0.07611

STD(80) 0.06768 0.04722 0.09820 0.03572 0.10976 0.09008 0.09631 0.09805 0.07696

AMAD(80) 0.06302 0.04642 0.09864 0.03538 0.10580 0.08702 0.09059 0.09691 0.07547

GARCH 0.06141 0.04492 0.10012 0.03713 0.10638 0.08616 0.09506 0.09137 0.07286


STD(10) 0.03883 0.03888 0.08025 0.03161 0.09955 0.07424 0.08129 0.07676 0.07043

AMAD(10) 0.03835 0.03866 0.08198 0.03077 0.09787 0.07486 0.08012 0.07662 0.07027

STD(20) 0.03612 0.03522 0.07507 0.02832 0.08764 0.06676 0.07278 0.06555 0.06220

AMAD(20) 0.03456 0.03435 0.07455 0.02697 0.08422 0.06579 0.07027 0.06392 0.06105

STD(40) 0.03635 0.03425 0.07161 0.02655 0.08101 0.06135 0.06609 0.06210 0.05802

AMAD(40) 0.03434 0.03290 0.06830 0.02515 0.07701 0.05941 0.06332 0.06000 0.05641

STD(80) 0.03817 0.03495 0.07106 0.02617 0.07963 0.05903 0.06197 0.06241 0.05587

AMAD(80) 0.03545 0.03339 0.06798 0.02497 0.07583 0.05636 0.05852 0.06049 0.05442

GARCH 0.03468 0.03329 0.07588 0.02806 0.08102 0.05734 0.06346 0.06079 0.05442



Forecasting

Model

Markets

S&P 50010-year

T-Bond

90-day



STD(10) 0.07199 0.05278 0.11404 0.04291 0.13364 0.10540 0.11535 0.11053 0.09035

AMAD(10) 0.06808 0.05180 0.11119 0.04138 0.12666 0.10388 0.10864 0.10770 0.08904

STD(20) 0.06774 0.04739 0.10192 0.03758 0.11437 0.09430 0.10180 0.09792 0.07841

AMAD(20) 0.06210 0.04593 0.09948 0.03602 0.10606 0.09010 0.09462 0.09368 0.07629

STD(40) 0.06540 0.04480 0.09249 0.03440 0.10085 0.08574 0.09063 0.09223 0.07163

AMAD(40) 0.05993 0.04360 0.09266 0.03357 0.09525 0.08113 0.08407 0.08945 0.06996

STD(80) 0.06620 0.04422 0.08819 0.03206 0.09692 0.08228 0.08652 0.09377 0.07029

AMAD(80) 0.06095 0.04362 0.09085 0.03220 0.09360 0.07898 0.07990 0.09224 0.06870

GARCH 0.06009 0.04211 0.09092 0.03518 0.09216 0.07734 0.08623 0.08529 0.06436


STD(10) 0.04011 0.03855 0.07920 0.03112 0.09550 0.07098 0.07816 0.07492 0.06775

AMAD(10) 0.03954 0.03828 0.08204 0.03040 0.09322 0.07193 0.07691 0.07422 0.06792

STD(20) 0.03714 0.03464 0.07230 0.02691 0.08106 0.06228 0.06732 0.06422 0.05879

AMAD(20) 0.03589 0.03387 0.07335 0.02589 0.07741 0.06168 0.06498 0.06215 0.05804

STD(40) 0.03701 0.03285 0.06781 0.02469 0.07208 0.05664 0.05849 0.05926 0.05291

AMAD(40) 0.03509 0.03146 0.06770 0.02401 0.06919 0.05486 0.05560 0.05754 0.05176

STD(80) 0.03794 0.03326 0.06579 0.02368 0.07112 0.05392 0.05490 0.05997 0.05063

AMAD(80) 0.03517 0.03172 0.06488 0.02331 0.06831 0.05142 0.05092 0.05819 0.04905

GARCH 0.03531 0.03181 0.07232 0.02650 0.07231 0.05160 0.05736 0.05835 0.04788



Forecasting

Model

Markets

S&P 50010-year

T-Bond

90-day



STD(10) 0.07456 0.05419 0.11356 0.04231 0.13081 0.10505 0.11202 0.11459 0.09034

AMAD(10) 0.07063 0.05307 0.11158 0.04081 0.12424 0.10305 0.10467 0.11217 0.08913

STD(20) 0.07050 0.04827 0.10026 0.03638 0.11064 0.09348 0.09835 0.10184 0.07792

AMAD(20) 0.06468 0.04688 0.09923 0.03508 0.10325 0.08880 0.09000 0.09818 0.07574

STD(40) 0.06760 0.04482 0.08899 0.03232 0.09727 0.08470 0.08754 0.09601 0.07075

AMAD(40) 0.06141 0.04378 0.09073 0.03199 0.09207 0.07986 0.07931 0.09282 0.06873

STD(80) 0.06535 0.04307 0.08238 0.02940 0.08993 0.07881 0.08169 0.09272 0.06800

AMAD(80) 0.05926 0.04275 0.08678 0.03013 0.08680 0.07515 0.07367 0.09048 0.06607

GARCH 0.05994 0.04123 0.08351 0.03373 0.08315 0.07204 0.08343 0.08503 0.05890


STD(10) 0.04351 0.04004 0.08066 0.03119 0.09389 0.07250 0.07661 0.07836 0.06771

AMAD(10) 0.04288 0.03990 0.08443 0.03060 0.09199 0.07307 0.07488 0.07793 0.06752

STD(20) 0.04063 0.03569 0.07259 0.02683 0.07895 0.06348 0.06506 0.06724 0.05775

AMAD(20) 0.03934 0.03543 0.07576 0.02639 0.07604 0.06266 0.06253 0.06540 0.05725

STD(40) 0.03918 0.03346 0.06634 0.02413 0.07028 0.05695 0.05645 0.06175 0.05148

AMAD(40) 0.03727 0.03273 0.06946 0.02395 0.06828 0.05503 0.05307 0.06045 0.05045

STD(80) 0.03846 0.03314 0.06254 0.02215 0.06873 0.05270 0.05300 0.06047 0.04889

AMAD(80) 0.03567 0.03220 0.06516 0.02247 0.06586 0.05013 0.04884 0.05868 0.04767

GARCH 0.03638 0.03231 0.06763 0.02573 0.06653 0.05030 0.05684 0.06028 0.04366

1. Several textbooks do mention GARCH as a method for estimating volatility and referencesources but do not attempt to explain or describe it. The only text that seeks to teach studentshow to estimate volatility using GARCH that we have seen is Hull (2003).

2. See for instance Ederington and Lee (1993, and 2001). For an excellent review of the issuesin volatility forecasting see Poon and Granger (2003).

3. For a nice review of the highlights of this literature see Robert Engle’s Nobel prize acceptancespeech (Engle, 2004).

4. In the case of dividend paying stocks, the numerator is changed to Pt+Dt where Dt denotes anydividends paid over the period from t-1 to t.

5. While the sample variance is an unbiased estimator of F2, the sample standard deviation is abiased estimator of F in small samples. However, it is asymptotically unbiased and consistent.

6. The model in equation 7 yields a forecast variance for the next day, t+1. The forecastvariance over the next s days is obtained by successive forward substitution following theprocedure outlined in Ederington and Guan (2005).

ENDNOTES

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