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Applied Economics Letters, 2008, 1–6, iFirst

The informational quality

of implied volatility and the

volatility risk premium

Stephen P. Ferris

a,

*, Woojin Kim

b

and Kwangwoo Park

c

a

Department of Finance, Trulaske College of Business, University of

Missouri, Columbia, MO 65203, USA

b

Korea University Business School, Seoul, 136-701, Republic of Korea

c

Graduate School of Finance, Korea Advanced Institute of Science and

Technology, Seoul, 130-722, Republic of Korea

This article examines the informational quality of implied volatility in

forecasting future realized volatility using daily S&P 500 and S&P 100

index option prices from 2000 to 2006. In contrast to many previous

studies, we find that implied volatility is an unbiased and efficient

estimator of future realized volatility. Unlike implied volatility estimates;

both historical and conditional volatility estimates using GARCH and

EGARCH models possess limited explanatory power. A delta-hedged

trading strategy with long positions in calls, however, generates

significantly negative profits that imply a misspecification of constant

volatility models. These results suggest that implied volatility estimates

from constant volatility models contain valuable information, even though

the model might be misspecified.

I. Introduction

By now it is fairly well established both in theory and

in practice that the volatility of the underlying stock

price process might not be constant. Some argue that

the minute-by-minute changes observed for implied

volatility are sufficient to conclude the implausibility

of such an assumption. Nevertheless, implied volati-

lity is typically estimated obtained by imputing it

from the Black–Scholes (1973) or Cox et al. (1979)

option pricing models which themselves are based on

the assumption of constant volatility.

Despite this apparent inconsistency, option traders

rely on implied volatilities derived from constant

volatility models when making trading decisions.

1

Traders apparently believe that important price

information is embedded in these implied volatility

estimates. Accordingly, researchers continue to test

and evaluate the information quality, or more

specifically, the forecasting ability of implied volati-

lity. Canina and Figlewski (1993) find that implied

volatility contains no incremental information

regarding future volatility, whereas Christensen and

Prabhala (1998), report that implied volatility is an

*Corresponding author. E-mail: Ferriss@missouri.edu

1

Since 1993, the Chicago Board option Exchange has been releasing information on market wide volatility through a

volatility index based on the S&P 500, namely the VIX. This index is updated minute by minute during each trading day.

These calculations are based on the constant volatility model of Black–Scholes up until September 2003. After that date, the

CBOE changed to a model-free procedure. They claim that ‘VIX has been considered by many to be the world’s premier

barometer of investor sentiment and market volatility’ (www.cboe.com).

Applied Economics Letters ISSN 1350–4851 print/ISSN 1466–4291 online ß2008 Taylor & Francis 1

http://www.tandf.co.uk/journals

DOI: 10.1080/13504850801935356

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efficient and unbiased estimator of future volatility

after controlling for certain econometric issues.

Other researchers such as Jorion (1995), Jackwerth

and Rubinstein (1996), Christensen and Prabhala

(1998), Fleming (1998), Ederington and Guan (2002)

and Bakshi and Kapadia (2003) conclude that

implied volatility is almost always upward biased

compared to historical and realized future volatility.

These researchers generally interpret their findings as

inconsistent with the constant volatility assumption.

In this study, we investigate the information

quality of implied volatility using more recent prices

for the S&P 500 (SPX) and S&P 100 (OEX) index

options. Our sample period extends from January

2000 to September 2006. In contrast to most previous

studies, we find strong support for the unbiasedness

and efficiency of implied volatility as an estimator

for realized future volatility.

Since we do not find an upward bias, we then

examine whether volatility is constant. We undertake

our investigation by adopting the methodology

of Bakshi and Kapadia (2003). They perform their

analysis by calculating the gains to a delta hedging

strategy which involves buying a call, shorting the

underlying index by the ratio of delta, and borrowing

or investing the remainder at the risk-free rate.

If the constant volatility assumption is correct,

this strategy should, on an average yield zero profits.

We find, however, that the gains to this strategy are

invariably negative, providing strong support for

a negative volatility risk premium and our consequent

rejection of the constant volatility assumption.

II. Data and Sample

Data source

We obtain daily SPX and OEX option data from

OptionMetrics for the sample period January 2000

through September 2006. OptionMetrics is a com-

prehensive dataset that contains daily price (bid and

ask quotes), volume and open interest information

for the entire US listed index and equity options

markets. OptionMetrics also contains pre-calculated

implied volatilities as well as other sensitivity values

for equity and index options. The implied volatilities

for the SPX and other European style options

are calculated by inverting the Black–Scholes

model. The implied volatilities for the OEX and

other American style options are calculated by

iterating the Cox et al. (1979) binomial model. Both

of these models rely on an assumption of constant

volatility. We obtain monthly SPX and OEX returns

from the CRSP monthly files while the monthly risk-

free rate is available from Professor Kenneth

French’s website. We use this data for our estimation

of the ARCH and GARCH models.

Sample construction

From this dataset we create three sub-samples

which we use in our subsequent empirical analysis.

The first sub-sample is a nonoverlapping monthly

series of implied volatilities, realized volatilities and

conditional volatility estimates from ARCH models

as described by Christensen and Prabhala (1998).

The second sub-sample is an overlapping daily series

of implied volatility, ex post realized volatility over

the remaining life of the option, and historical

volatility calculated using the past 60 calendar days.

These values are estimated following the procedures

of Canina and Figlewski (1993). Our final sub-sample

consists of a series of daily SPX option prices and

gains to a delta-hedged trading strategy constructed

similar to that of Bakshi and Kapadia (2003).

III. Relationship between Implied

Volatility and Realized Volatility:

Monthly Series

Conventional methodology

The conventional empirical analysis to assess the

forecasting performance of implied volatilities has

been largely focused on the following regression

specification,

t, realized ¼þt, implied þ"tð1Þ

where "

t

is an error term with mean zero. We estimate

the following additional specifications to compare the

relative performance of implied and historical

volatility in forecasting future volatility,

t, realized ¼þt,historical þ"tð2Þ

t, realized ¼þt, implied þt, historical þ"tð3Þ

where

t,historical

¼

t1,realized

for the nonoverlapping

monthly series sub-sample. This specification gener-

ates a first-order auto-regressive (i.e. AR(1)) process.

Further,

t,historical

¼

t,past60

for the overlapping daily

series sub-sample.

2

2

Fair and Shiller (1990) introduce the regression specification contained in Equation 3 which is generally referred as the

‘encompassing test’ in the forecasting literature.

2S. P. Ferris et al.

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OLS results for nonoverlapping monthly series

Table 1 presents the OLS regression results and the

Durbin–Watson statistics for the monthly series

contained in our first sub-sample for both the OEX

and SPX options. First, the adjusted R

2

values

and Durbin–Watson (DW) statistics are similar

to Christensen and Prabhala (1998). The Durbin–

Watson statistics are near to the value 2, implying

that serial correlation is not severe for the nonover-

lapping monthly series. The coefficient estimates, the

t-statistics for the coefficients, and the F-statistics for

tests of the joint hypotheses, however, demonstrate

stronger support for the unbiasedness of implied

volatility as an estimator of future volatility. The

results are slightly stronger for the SPX series.

Modelling volatility is an AR(1) process as in the

second regression specification has some explanatory

power, but the estimate is below one (¼1 is easily

rejected for both OEX and SPX series), the intercept

is significantly different from zero, while the adjusted

R

2

is less than half of those obtained from specifica-

tion (1). Comparing the results from regression

specifications (1) and (2), implied volatility appears

to be a better estimator than past volatility.

Even stronger support for unbiasedness can be

found in the third regression specification. Again, the

joint null hypothesis that ¼0, ¼1 and ¼0

cannot be rejected and the results are slightly stronger

for the SPX series. These findings are similar to those

reported by Christensen and Prabhala (1998) and

Christensen and Hansen (2002).

Implied volatility vs. conditional volatility estimates

from ARCH models

Day and Lewis (1992) compare the informational

content of implied volatility, past historical volatility

and conditional volatility estimates from GARCH

and EGARCH models using an OEX time-series

from March 1983 to December 1989. They report

that both implied volatility and GARCH forecasts

are significant in predicting future realized volatility

when used independently as regressors.

Consequently, we estimate the following model to

determine how our results might differ over the

sample period,

t,realized ¼þt,implied þt1,realized þt,estimated þ"t

We calculate three nested regressions to obtain

both GARCH and EGARCH estimates. The first

regression uses only the estimated conditional vola-

tility from the model as the regressor. Implied and

lagged historical volatility are added to the second

and third models.

In untabulated results we find that the conditional

volatility estimate from the GARCH model appears

to be a good estimator for future volatility when used

alone. The coefficient approximates to a value of 1

with a high p-value. But when implied volatility is

added to the regression, the GARCH estimate loses

its explanatory power. We conclude that implied

volatility remains a reliable predictor of future

realized volatility even when compared against more

sophisticated estimates of conditional volatility.

IV. Relationship between Implied

Volatility and Realized Volatility:

Daily Series

Canina and Figlewski (1993) provide strong evidence

against the informational quality of implied volatility

by using daily overlapping observation. Christensen

and Prabhala (1998) argue that this sampling method

is the main cause of the failure of implied volatility

Table 1. OLS regression results for nonoverlapping monthly series

Independent variables

Index option Intercept

t,implied

t1,realized

Adjusted R

2

DW H

0

F-statistics p-value

OEX 0.00 (0.02) 0.92 (6.72) 36% 1.86 ¼1¼0, ¼1 0.32 2.54 0.57 0.09

0.12 (5.33) 0.41 (3.96) 16% 2.01 ¼1¼0, ¼1 31.91 15.99 0.00 0.00

0.00 (0.00) 0.90 (4.74) 0.02 (0.20) 34% 1.89 ¼1¼0, ¼1, ¼0 0.26 1.68 0.61 0.18

SPX 0.01 (0.22) 0.97 (6.99) 38% 1.92 ¼1¼0, ¼1 0.06 1.85 0.81 0.16

0.11 (5.41) 0.40 (3.84) 15% 2.01 ¼1¼0, ¼1 32.74 16.41 0.00 0.00

0.01 (0.25) 0.99 (5.11) 0.02 (0.13) 36% 1.90 ¼1¼0, ¼1, ¼0 0.00 1.21 0.95 0.31

Notes: This table presents the OLS results for the 3 regression specifications described in section ‘OLS results for

nonoverlapping monthly series’. The sample consists of 80 nonoverlapping monthly series.

The t-statisatics for significance of each coefficient estimates are reported in parentheses.

The last two columns present F-statistics and p-values for the joint hypothesis presented in the column headed by H

0

.

Implied volatility and the volatility risk premium 3

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as an estimator for future volatility. They confirm the

results of Canina and Figlewski (1993) over the same

sample period using an identical sampling method

and conclude that implied volatility possesses no

incremental information.

Table 2 presents OLS regression results using the

overlapping daily series sub-sample across our sample

period. We observe that the coefficients, t-statistics

and adjusted R

2

are similar to the monthly regres-

sions, providing strong support for implied volatility.

In contrast to monthly regressions, the Durbin–

Watson (DW) statistics approach zero for all regres-

sions. These OLS results for the daily data sub-

sample provides strong support for the use of implied

volatility as an estimator of future volatility.

V. Gains to a Delta-Hedged Trading

Strategy

The empirical results presented in the previous two

sections provide support for implied volatility as an

unbiased and efficient estimator of future realized

volatility. These results lead us to ask whether in the

absence of any evidence of bias in the volatility

estimates, can we assume that volatility is constant?

To address this question, we calculate the gains to a

delta hedging strategy for SPX index options follow-

ing the procedure described by Bakshi and Kapadia

(2003). If volatility is constant, then options are

redundant securities and there is no risk premium for

volatility changes, so that the gains to a delta hedged

strategy should be zero. If volatility, however, is not

constant and this risk is priced in the market, then we

should expect to see negative gains to a delta hedged

strategy, since options provide hedging due to

the negative correlation between volatility and the

underlying price level.

Table 3 presents the results of this trading strategy.

The results are reported separately for options with

the nearest maturity (i.e. 2 weeks left until expiration)

and those with the second nearest maturity. Overall,

the gains to a delta hedged strategy are significantly

negative.

3

This result is consistent with Bakshi and

Kapadia (2003), where they find that the gains to

such a strategy yield negative returns for the SPX

during their sample period from 1988 to 1995.

Moreover, the magnitude of the negative gains

seems to have increased over our most recent

sample period of 2000 to 2006. In terms of dollar

gains, buying at-the-money call options and delta

hedging them yields an average loss nearly seven

times larger than that reported by Bakshi and

Kapadia (2003).

VI. Conclusions

A number of researchers test the forecasting ability of

the implied volatility estimates, contained in option

prices. Prior research, however, finds relatively

limited support for the informational quality

of implied volatility. Indeed, most empirical

analysis reports an upward biasedness in volatility

estimates.

In contrast, we find that implied volatility is an

unbiased estimator of the future realized volatility

when using most recent data from January 2000

through September 2006. Past historical volatility and

conditional volatility estimates from GARCH and

3

SEs are based on the sample SD divided by square root of number of observations, following the method used by Bakshi and

Kapadia (2003).

Table 2. OLS regression results for overlapping daily series

Independent variables

Index option Intercept

t,implied

t1,realized

Adjusted R

2

DW H

0

F-statistics p-value

OEX 0.01 0.45 0.94 8.34 39% 0.23 ¼1¼0, ¼1 0.31 9.14 0.58 0.00

0.09 4.55 0.52 4.73 17% 0.10 ¼1¼0, ¼1 18.82 10.35 0.00 0.00

0.01 0.41 0.97 7.55 0.04 0.36 39% 0.23 ¼1¼0, ¼1, ¼0 0.07 6.07 0.79 0.00

SPX 0.00 0.20 0.93 8.05 38% 0.25 ¼1¼0, ¼1 0.37 6.42 0.54 0.00

0.09 4.62 0.50 4.37 15% 0.11 ¼1¼0, ¼1 19.12 10.66 0.00 0.00

0.00 0.15 0.97 7.41 0.05 0.50 38% 0.26 ¼1¼0, ¼1, ¼0 0.04 4.26 0.85 0.01

Notes: This table presents the OLS results for the three regression specifications as described in Section IV. The sample

consists of 1644 overlapping daily series for OEX and 1645 for SPX. The t-statistics for significance of each coefficient

estimates are reported in parentheses. Last two columns present F-statistics and p-values for the joint hypothesis presented in

the column headed by H

0

.

4S. P. Ferris et al.

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Table 3. Gains to delta hedged trading strategy

(in $) /S(in %) /C(in %)

Moneyness (%) N1 2 All 1 2 All 1 2 All

10 to –7.5 4369 0.43 (9.59) 1.60 (13.40) 0.81 (16.23) 0.04 (9.90) 0.15 (13.50) 0.08 (16.50) 62.31 (12.59) 4.89 (0.44) 43.95 (9.00)

7.5 to –5 6326 0.77 (12.62) 2.39 (17.78) 1.25 (21.10) 0.08 (13.95) 0.22 (18.17) 0.12 (22.36) 51.33 (17.33) 6.34 (0.94) 37.93 (13.04)

5 to –2.5 8145 1.46 (19.39) 3.31 (23.29) 2.02 (29.47) 0.14 (22.19) 0.30 (23.90) 0.19 (32.07) 29.86 (19.78) 19.42 (14.09) 26.71 (23.54)

2.5 to –0 8921 2.19 (24.26) 4.27 (26.14) 2.85 (35.11) 0.20 (26.42) 0.38 (28.09) 0.26 (37.94) 13.39 (23.91) 15.17 (25.33) 13.96 (32.68)

0–2.5 8760 2.88 (29.16) 4.38 (25.47) 3.36 (38.56) 0.25 (30.32) 0.38 (27.36) 0.29 (40.60) 8.51 (29.21) 9.72 (26.30) 8.90 (38.55)

2.5–5 7482 2.29 (25.13) 3.60 (20.78) 2.72 (32.46) 0.19 (25.12) 0.30 (21.15) 0.23 (32.71) 4.11 (24.29) 5.41 (20.31) 4.53 (31.58)

5–7.5 6002 1.74 (20.25) 3.02 (17.83) 2.15 (26.84) 0.14 (18.62) 0.24 (16.47) 0.17 (24.76) 2.03 (17.66) 3.16 (15.58) 2.40 (23.50)

7.5–10 4986 1.20 (14.53) 2.27 (13.97) 1.55 (20.07) 0.10 (12.87) 0.18 (12.11) 0.12 (17.62) 1.07 (12.09) 1.82 (11.35) 1.32 (16.55)

Total 54 991 1.76 (50.11) 3.31 (55.59) 2.25 (73.75) 0.16 (45.16) 0.29 (53.80) 0.20 (68.27) 19.90 (17.72) 9.13 (10.99) 16.49 (20.51)

Notes: This table presents the gains to a delta hedged trading strategy as described in Section V.

The t-statistics for the null hypothesis that gains zero are reported in parenthesis.

The columns labelled with 1 represent options with nearest maturity with at least 2 weeks left until maturity and those with 2 represent options maturing in the next

expiration date.

is the delta hedged gain, Sis the underlying security price and Cis the call option price as described in Section V.

Implied volatility and the volatility risk premium 5

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EGARCH models possess only limited explanatory

power compared to the implied volatility estimates

during our sample period. Nevertheless, a delta

hedged trading strategy with long positions in calls

yields significantly negative profits, implying that

modelling volatility as a constant might be a

misspecification.

Our results suggest that implied volatility estimates

obtained from constant volatility models contain

valuable information, regardless of possible model

misspecification. If traders are convinced of the

validity of a model, then its implied volatility might

reflect useful information about future realized

volatility, even if it is misspecified. Considering that

the Black–Scholes (1973) and Cox et al. (1979)

binomial models are still widely used by traders and

industry practioners, such a process is likely. Our

results provide an insight concerning why implied

volatility inferred from a constant volatility model is

still widely used.

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