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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
Downloaded By: [2007-2008 Korea Advanced Institute of Science & Technology] At: 13:32 12 August 2008
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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