Discussion Papers In Economics
And Business

STOCK MARKET VOLATILITY AND THE FORECASTING ACCURACY
OF IMPLIED VOLATILITY INDICES

Kazuhiko NISHINA
Nabil MAGHREBI
Moo-Sung KIM

Discussion Paper 06-09

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSTOCK MARKET VOLATILITY AND THE FORECASTING ACCURACY
OF IMPLIED VOLATILITY INDICES

Kazuhiko NISHINA
Nabil MAGHREBI
Moo-Sung KIM

Discussion Paper 06-09

March 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSTOCK MARKET VOLATILITY AND THE FORECASTING ACCURACY
OF IMPLIED VOLATILITY INDICES

Kazuhiko NISHINA
Graduate School of Economics
Osaka University, Japan

Nabil MAGHREBI*
Faculty of Economics
Wakayama University, Japan

Moo-Sung KIM
College of Business Administration
Pusan National University, Korea

JEL Classification: C52, C53, G14
Keywords: Implied volatility index, Out-of-sample forecasting, GARCH modelling*Corresponding author:

Acknowledgement:

Sakaedani 930 Wakayama 640-8510, Japan
Phone: +81-73-457-7658
Fax: +81-73-457-7659
Email: nebilmg@eco.wakayama-u.ac.jp

The authors acknowledge with thanks financial support, Grant-in-aid
for scientific research, from the Japan Society for the Promotion of
Science.

STOCK MARKET VOLATILITY AND THE FORECASTING ACCURACY
OF IMPLIED VOLATILITY INDICES

ABSTRACTThis study develops a new model-free benchmark of implied volatility for
the Japanese stock market similar in construction to the new VIX based on
the S&P 500 index. It also examines the stochastic dynamics of the implied
volatility index and its relationship with realized volatility in both markets.
There is evidence that implied volatility is governed by a long-memory
process. Despite its upward bias, implied volatility is more reflective ofchanges in realized volatility than alternative GARCH models, which
account for volatility persistence and the asymmetric impact of news. The
implied volatility index is also found to be inclusive of some but not all
information on future volatility contained in historical returns. However, its
higher out-of sample performance provides further support to the rationale
behind drawing inference about future stock market volatility based on theincremental information contained in options prices.

INTRODUCTION
According to Heisenberg uncertainty principle in quantum theory, it is not possible tomeasure with perfect accuracy the position and momentum of a particle simultaneously.
This argument may apply with equal force to financial markets where the level and volatilityof asset prices cannot be measured with the same precision. The daunting difficulties
associated with this observation problem derive from the fact that these parameters arecomplementary and the measurement of one parameter with greater accuracy is likely to beaccompanied with some loss in measurement precision of the other. Whereas information
on stock price indices is being disseminated at increasingly higher data frequency, measures2

of future volatility with comparable accuracy and frequency are rather hardly available.
Assuming that the second moment of return distribution exists, it is possible to estimate
volatility as the variance or standard deviation from the history of returns. However, thevolatility implicit in option prices can provide an alternative measure of future return
variability. One of the merits of this implied volatility approach is that it results in atime-series of observations that reconciles differences in data frequency, albeit notnecessarily in accuracy, of price level and volatility.
The construction of a time-series of implied volatility is important to enhance ourunderstanding of how market volatility ebbs and flows in increasingly integrated financial
markets. Stock market volatility constitutes indeed an essential ingredient not only ininvestment and risk-hedging but in market regulation and monetary policy-making as well.
Market participants can view volatility quite differently and their anticipated reactions toeconomic events and the release of macroeconomic information may differ as well. The
development of an implied volatility index is important for policymaking purposes insofar
as it reveals the changing and perhaps stochastic beliefs of market participants. The
importance of such measure of consensus expectations about short-term volatility formonetary policies is evident for instance, in the explicit reference by the Bank of England to
the implied functions from options markets in its monetary policy meetings. There isevidence from Fornari (2004) that the implied volatility from swaption prices is responsive
to macroeconomic announcements such as the release of US economic indicators.
Furthermore, Neely (2005) shows that changes in the implied volatility from three-month
eurodollar interest rates are associated with major events about monetary policy, stock
markets and the real economy. Carr and Wu (2006) provide also evidence that the new VIX
implied volatility index from S&P 500 option prices is reflective of the increasing
uncertainty prior to monetary policy decisions about the Fed Fund target rate.
3

The new VIX index computed and disseminated by the Chicago Board of Options
Exchange is a model-free benchmark in the sense that it does not depend on a particular
option pricing model. This approach has the merit of avoiding measurement errors arisingfrom model misspecification. This index aggregates information in the term structure of
implied volatility and the nonlinear relationship with respect to exercise prices reflected byvolatility smiles, sneers and smirks. However, the absence of implied volatility benchmarks
for other international stock markets impedes the growth of international evidence on theusefulness of implied volatility for risk-hedging and policymaking purposes. It is the
purpose of the present study to develop a new VIX benchmark for the Japanese stock market.
The construction of this volatility benchmark is an acknowledgment of the fact that the
Nikkei 225 index is the underlying asset of several financial derivatives traded, albeit under
different contract specifications, on major Asia-Pacific derivatives markets.1
This paper examines the stochastic dynamics of the forward-looking implied
volatility index of Japanese and US equity markets. It assesses its relationship with realized
volatility by addressing in particular, the classical question of whether the implied volatility
index reveals incremental information beyond that contained in historical returns. It also
examines the empirical question which naturally arises as to whether drawing inference
about future stock market volatility based on implied volatility is justified by higher
out-of-sample forecasting ability. Based on a novel measure of forecasting performance
proposed by Blair, Poon and Granger (2001), this study assesses the forecasting ability of
implied volatility indices relative to alternative GARCH models. Thus, in addition to thedevelopment of the new VIX index for the Japanese equity market, this study differs also
1

Futures on Nikkei 225 stock average are traded on three derivatives markets, namely theSingapore Exchange Derivatives Trading Division, formerly the Singapore International
Monetary Exchange since September 3, 1986, on the Osaka Securities Exchange since September
3, 1988 and on the Chicago Mercantile Exchange since September 25, 1990.
4

from the existing literature as it uses a sample period covering fifteen years of daily options
prices, and spanning the Asian financial crisis in 1997, the Russian debt default and theLong-Term Capital Management (LTCM) crisis in 1998, the burst of the information
technology bubble in 2000, the Latin American debt crisis in 2002 and the 1990s
decade-long recession of the Japanese economy.
The remainder of the paper is structured as follows. The next section presents a
brief review of the literature on implied volatility. Section 3 examines the distributional
properties of implied volatility and the underlying benchmark returns. Section 4 assesses theinformational content of implied volatility by modelling its relationship with realized
volatility. Section 5 examines the forecasting performance of implied volatility relative to
GARCH models. Section 6 concludes the paper.
2. REVIEW OF LITERATURE
The literature on market volatility implicit in option premium seems to extend along several
strands of studies. The first class of research explores various issues related to the numericaldifficulties encountered in the determination of implied volatility. There are convergenceproblems which arise from the failure of the iterative process to equate the option’s market
price with its theoretical premium and the absence of closed-form solutions to option
pricing models. The literature examines the observed properties of implied volatility
function and their implications for option pricing and proposes some nonlinear numericalapproaches. There is evidence of ‘smile’ patterns in implied volatility functions based on
Black-Scholes (1973) option pricing model, where deep-in-the-money or out-of-the-money
options are associated with higher implied volatility than at-the-money options. Pena, Rubio
and Serno (1999) attribute such systematic volatility smiles to transaction costs and find asignificant inverse relationship between the degree of curvature and time remaining until
expiration. As noted by Hentschel (2003), the implied volatility functions derived from5

options away from the money are sensitive to various sources of measurement errors.
Arguably however, the observed functions are inconsistent with Black-Scholes
option pricing model, which assumes constant volatility that is independent of exercise
prices and time. In order to address these inherent inconsistencies and recover unbiased
estimates of implied volatility, approximation techniques have been advocated in several
studies such as Corrado and Miller (1996) and Chambers and Nawalkha (2001). A“model-free” approach is also proposed by Britten-Jones and Neuberger (2000) who adjust
arbitrary volatility processes to option prices, drawing upon the standard practice of fittinginterest rate processes to bond prices. The modelling of implied volatility as function of
exercise prices such as Ait-Sahalia and Lo (1998), can also be based on polynomial
smoothing, interpolation or splines smoothing of the pricing function. Furthermore, themodelling of option prices under GARCH process by Duan, Gauthier and Simonato (1999),
and Ritchken and Trevor (1999) is an attempt to capture the path-dependence of volatility
and its negative correlation with return.
It is also possible as in Hull and White (1987), and Heston (1993) inter alia, tosubstitute the constant volatility with the entire joint probability distribution of returns andchanges in volatility. Furthermore, the modelling of stochastic volatility is found by Balland Roma (1994), to be consistent with implied volatility smiles. However, this approachbased on the assumption of volatility being a deterministic function of asset price and timefaces difficulties of its own. Indeed, the dynamics of the market price of risk are hard to
estimate. Furthermore, Dumas, Fleming and Whaley (1998) provide evidence that thepredictive power and hedging performance of the deterministic volatility function are nothigher than Black-Scholes implied volatility. It is also shown that the estimateddeterministic function is itself unstable over time and Black-Scholes hedge-ratios are rather
more reliable.
6

The second class of studies in the literature on implied volatility focuses on its
relationship with realized volatility, its information content and significance for forecasting
purposes.2 Day and Lewis (1992) provide evidence that volatility implied by S&P 100 index
options contains useful information for forecasting market volatility. However, it is also
found to be inefficient in the sense that the conditional volatility based on GARCH
modeling contains incremental information beyond that reflected by implied volatility.
Further evidence of inefficiency is provided by Lamoureux and Lastrapes (1993) using
volatilities implicit in individual stock option prices. The empirical results from Canina andFiglewski (1993) are also suggestive of the absence of correlation between S&P 100 implied
volatility and future volatility. In contrast, the empirical evidence, from Jorion (1995) andAmin and Ng (1997), indicates that implied volatility provides efficient, albeit biased,
estimates of future volatility in foreign exchange markets. The implied volatility from S&P
100 option prices is also found by Fleming (1998) to constitute despite its upward bias, areliable estimate of future volatility. There is evidence that implied volatility is efficientwith respect to its past forecast errors and that such errors are orthogonal to parameters
usually embedded in ARCH models.
Arguably, the assessment of the relationship between implied volatility andrealized volatility is a joint test of the option pricing model and market efficiency. But, thedifficulties in reconciling this body of conflicting evidence may also stem from various
sources of measurement errors such as model misspecification and the failure to account for
dividend payments. As noted by Christensen and Prabhala (1998), there are also “maturity
mismatch” problems associated with early studies, which may arise from the assessment of
the predictive power (one-day or week-ahead horizons) of implied volatilities calculated
2

An interesting review of the literature on volatility forecasting including studies on the predictive

performance of implied volatility is provided by Poon and Granger (2003).
7

from options with much longer time remaining to expiration. Using monthly observations of
non-overlapping volatility, they explain away part of the bias in S&P 100 implied volatility
with regime shifts in options pricing around the October 1987 stock market crash. The
empirical results suggest also that implied volatility is associated with higher predictive
power than historical volatility. Using an index of implied volatility based on the S&P 100
options, Blair, Poon and Taylor (2001) find also that implied volatility is more accurate forout-of-sample forecasting than past realized volatility. There is also evidence from Whaley(2000) that this implied volatility index is indicative of the extent of investor fear and stress,
thereby reveals useful information about market sentiment. More recently, there is evidence
from Giot (2005) that implied volatility indices contain useful information for Value-at-Risk
modelling.
In light of the widely documented errors in implied volatility estimation, it isimportant to reduce measurement errors and to take into account the term structure andnonlinear relation between implied volatility and exercise prices. The new implied volatility
index, which provides an aggregate estimate of future volatility, is based on a model-freemethodology in order to avoid misspecification problems. The next section briefly describes
this model-free approach and examines its distributional properties in the Japanese and US
equity markets.
3. DISTRIBUTIONAL PROPERTIES OF IMPLIED VOLATILITY
The new VIX index represents a model-free approximation of a thirty-day return varianceswap rate.3 It is based on a hypothetical at-the-money option with fixed expiration and with
exercise price equal to the futures on the underlying index. The model-free methodology is

3

Unlike this model-free new VIX index, the original CBOE-VIX index based on the OEX bid-ask
quotes on S&P100 American options is calculated using Black-Scholes pricing model. Carr andWu (2006) provide an insightful analysis of the rationale underlying the calculation of these
indices and their major differences.
8

aimed at gathering information contained not only in the market prices of at-the-money ornear-the-money options but in the volatility structure across the prices of a portfolio ofin-the-money and out-of-the-money options as well. The implied volatility index isestimated by interpolating the implied volatilities from the nearest and next-term options,which are calculated as follows.
v =
2
i ,t

2

τi

∑

Ni
j

∆X ij ,t
X ij2,ter fiτ i⎞1⎛ F
G (X ij ,t ) − ⎜⎜ t − 1⎟⎟τ i ⎝ X i 0 ,t⎠2where τ i is the time expressed in minutes until the i th maturity of near or next-term options,r f is the risk-free interest rate until expiration, F is the forward equity index determinedfrom option prices and X 0 is the exercise price immediately below F . For each stripof N i out of-the-money options, the exercise price X j of the j th option corresponds to a call
if X j > F and a put otherwise. The spread between exercise prices ∆X j is equal to theaverage difference between the strike prices nearest to X j . For the highest (lowest) exercise
price, ∆X j is equal to the difference between that exercise price and the next lower (higher)
strike price. The closing price G (.) of each option is used to calculate its contribution
∆X ij
X ij2

e riτ i G (X ij ) to the implied volatility index. The contribution to implied volatility is anincreasing function of the exercise price for puts and a decreasing function of the strike priceof calls. The index uses the nearest and next-term expirations to span the fixed period of 30days to expiration. As expiration draws near, the rollover to the second and third contract
months takes place with eight days remaining to maturity in order to avoid pricing errors inoptions with imminent expiration. The interpolation process of the nearest and next-term
implied volatilities results in a single measure of implied volatility, which is annualized toobtain the new VIX index.
9

The sample period extends from January 1990 through December 2004, resulting
in daily observations and spanning 180 options expirations. There is evidence from Figure 1,
which describes the behavior of daily time-series of implied volatility for the S&P 500 and
Nikkei 225 indices that expectations of the future level of volatility vary over time andacross markets. There is a tendency for implied volatility in the US market to be lower thanin the Japanese market and to converge since the late 1990s. The peaks in implied volatility
for the Japanese market in the early 1990s coincide with the onset of the economic recession
and heightened uncertainty induced by lingering bad debt problems and the appreciation of
the yen, among other factors. The similarities in the behavior of these indices are also
reflective of major economic events including the Asian financial crisis in 1997 and theRussian debt and LTCM crises in 1998, among others. The burst of the information
technology bubble in 2000 is not associated with as significant an increase in implied
volatility as that associated with the Latin America debt crisis in 2002. There is a tendency
for implied volatilities to increase sharply but decrease rather on monotonous basis. This
volatility persistence suggests that shocks to the volatility process are likely to belong-lived.
The distributional moments reported in Table 1 indicate that the average sample
implied volatility in the US market is indeed significantly lower than in the Japanese market.
Expectations of higher Nikkei 225 volatility may be related to the stronger concerns over theJapanese economy during the post-bubble recession of the 1990s, which are reflected bynegative stock market returns. It is noted however that though the means of implied
volatilities are significantly different, their variances are not. The test of the null hypothesis
for the equality of means indicates also that the magnitude of daily log changes in implied
volatility are comparable across markets. In light of the distributional moments andnormality tests, there is little evidence that stock market returns and volatility are normally
10

distributed.
The autocorrelations for the levels of implied volatility reported in Table 2 remain
positive, suggesting that the autocorrelation function decays exponentially to zero. Thissuggests that the dynamics of implied volatility are stochastic and likely to be governed by along-memory process. In contrast, the evidence of negative autocorrelations for stock
market returns and changes in daily implied volatility suggests that the autocorrelation
functions are more likely to decay rather in oscillatory patterns. Also, the results of unit-root
tests indicate that all time-series of stock market returns and volatility are found to bestationary.
4. MODELLING THE RELATIONSHIP BETWEEN IMPLIED AND REALIZED VOLATILITIESThe standard deviations of returns reported in Table 1 can be converted into annualized
volatility estimates which are rather close to the sample averages of implied volatilities(19.40% against 19.88% for the S&P 500 index and 28.22% against 24.76% for the Nikkei
225 index). In order to assess the significance of the long-term relationship between theforward-looking implied volatility index and actual volatility, an ex post estimate of realized
volatility is calculated as the standard deviation of log returns. This measure is based on arolling 30-day sample period, which is consistent with the fixed time-to-expiration of the
new VIX index’s hypothetical option

σ r ,t =1Tt∑ (r − r )
Tt

s =1

2

t ,s

t

(1)

where Tt is the number of calendar days until maturity, rt , s is the return on a given day s until
expiry, and rt = T1t ∑st=1 rt , s denotes the average return over the monthly sample period. For
T

comparative purposes, this measure of realized volatility is annualized.
These daily measures of realized volatility are plotted against implied volatility
for the US and Japanese markets in Figures 2 and 3 respectively. Implied volatility does not11constitute a perfect forecast of realized volatility but the relationship between these ex anteand ex post measures of volatility tends to be positive. There is evidence from both stock
markets that the relationship appears to weaken as the level of volatility increases. The
conventional approach adopted by several studies that examine the significance of thisrelationship is to estimate the following linear regression model

σ r ,t = γ 0 + γ 1 σ it + ξ t(2)where σ r ,t and σ it denote the daily estimates of realized and implied volatility, respectively.
Based on the sign and magnitude of the regression coefficients, it is possible to assess theextent of the informational content of implied volatility. In the case where γ 0 = 0 and γ 1 = 1 ,implied volatility is an unbiased forecast of realized volatility. It is also an efficient estimateof realized volatility if the residuals ξ t are white noise.
Theestimation

results

reported

in

Table

3

are

based

on

white

heteroskedasticity-consistent standard errors under the efficiency hypothesis. The slope
coefficient γ 1 is found to be positive and significant suggesting that implied volatility
contains some information about future volatility. However, it seems to be a biased estimateof subsequent volatility since the F-test rejects the joint hypothesis that the intercept γ 0 andslope γ 1 are significantly different from zero and unity, respectively. There is also evidenceof serially correlated residuals, which suggests that implied volatility does not constitute anefficient estimate of actual volatility. Although the relationship between implied andrealized volatility remains positive, the extent of unbiasedness and efficiency differ acrossmarkets. The regression line lies below the 45 o line characteristic of unbiased forecasts for
US market, and it is rather flatter for the Japanese market. As noted by Christensen andPrabhala (1998), the negative intercept for the US market may be at least partially, inducedby the errors-in-variables problem associated with the implied volatility index.
12

The positive relationship may be the artifact of the measurement process resulting
in overlapping observations of volatility. The error terms ξt are indeed likely to show signsof heteroskedasticity due to the serial dependence of the regression variables. In order toreduce measurement errors and inconsistent regression estimators from daily observations,the regression model is reexamined using nonoverlapping observations. The autoregressive
process equation (3) which describes the behavior of monthly realized volatility includes
implied volatility as an exogenous variable.

σ r ,t = ∑ k γ k σ r , t − k + γ i σ i , t + ξ t(3)

The two time-series of monthly measures of realized volatility until option expiry
and implied volatility one-month ahead are obtained from all maturities of the S&P 500 and
Nikkei 225 options from February 1990 to January 2005. The expiration dates differ acrossthese options markets, but the sampling procedure results in 180 nonoverlapping
observations for each volatility series.
The estimation results for monthly expirations reported in Table 3 indicate that theinclusion of additional lags in order to correct for serial correlation in the residuals induce a
marginal improvement of the explanatory power of the regression models. For both markets,
the parameters γ 1 for lagged values of realized volatility are positive but statisticallyinsignificant whereas the γ i coefficients for implied volatility are significant. These findings
are consistent with early evidence that implied volatility does contain some useful
information about subsequent volatility.
However, the F-statistics for Wald tests reject the joint hypothesis that γ k = 0 and

γ i = 1 at the 1% level in the US market. Similar test results are obtained for the Japanesemarket at the 5% level with two lags of realized volatility included into the autoregressive
process. Thus, there remains evidence of implied volatility being a biased forecast of future
13

volatility. Compared to the regression results with overlapping observations, there ishowever weaker evidence against the efficiency of implied volatility judging from the testsof serial correlation.
5. Forecasting performance of implied volatility against alternative GARCH models
5.1. In-sample forecasting performance
The empirical results reported thus far suggest that implied volatility does contain some
information about future volatility but it remains biased and to some extent inefficient. It is
important at this point to determine whether alternative GARCH models of market volatility,
which have the merit of capturing the clustering properties of market volatility and its
persistence over time, do also contain useful information that is not reflected by implied
volatility. The conditional mean and variance equations for the standard GARCH-in-mean
model can be written as follows

rt = λ0 + λ 1σ t2−1 + ε t

σ t2 = w0 + α ε t2−1 + β σ t2−1(4)where the conditional mean, variance and error terms ε t are conditional on the information
set ℑt −1 available at time t − 1 . In this standard GARCH-in-mean specification (model (1)),
the conditional variance is positive definite and stationary when β > 0 and α + β < 1 . The
persistence of the volatility process increases as approaches unity. It is noted that thevariance term is included as a measure of risk into the return generating process in order toaccount for possible risk premium in stock markets.
Following Day and Lewis (1992), and Lamoureux and Lastrapes (1993), it ispossible to assess the additional informational content of implied volatility by including
implied variance into the conditional variance equation (model (2)).

σ t2 = w0 + α ε t2−1 + β σ t2−1 + π 1 σ i2,t −1(5)14

The test of significance of π provides evidence on whether implied volatility contains
useful information on the dynamics of future volatility beyond that reflected by past
volatility levels and innovations. In order to avoid underestimating the significance of
implied volatility, a stronger test of information efficiency proposed by Amin and Ng (1997),
can be performed by including both the contemporaneous and past implied variance into theconditional variance equation (model (3))σ t2 = w0 + α ε t2−1 + β σ t2−1 + π 0 σ i2,t + π 1σ i2,t −1(6)

The test of the informational content of implied volatility relative to GARCH modelling isstraightforward in the sense that if the GARCH terms α and β are both found to bestatistically insignificant while π 0 and π 1 = −π 0 (α + β ) are, then implied volatility is
sufficient to convey all past and current information about future volatility.
In addition to volatility persistence, it is important given the growing literature onleverage effects, to account for the asymmetric response of volatility to negative news. The
GARCH-in-mean model proposed by Glosten, Jagannathan and Runkle (1993) allows forthe volatility dynamics to depend not only on the magnitude of innovations but on their sign
as well. This is achieved by estimating the following conditional variance equation (model(4)).

σ t2 = w0 + α ε t2−1 + δ d t −1 ε t2−1 + β σ t2−1(7)where dt −1 is an indicator variable which equals unity when ε t −1 < 0 and zero otherwise.
Following this GJR approach, it is also possible to examine the hypothesis of whether
implied volatility contains useful information beyond that conveyed by past returns by
including past implied variance into the conditional variance equation (model (5)).

σ t2 = w0 + α ε t2−1 + δ d t −1 ε t2−1 + β σ t2−1 + π 1 σ i2,t −1(8)

The significance of π 1 can be interpreted as evidence of the implied volatility conveying
15

useful information about future volatility beyond that contained in the history of stock
returns even upon accounting for the asymmetric impact of shocks on the return-generatingprocess.
The five competing models differ only in their specification of the conditionalvariance. The models (2), (3) and (5) include implied volatility as explanatory variable inorder to assess its informational content relative to referring to the standard GARCH-M
model (1) and GJR model (4). Table 4 presents the estimation results of these models withdaily sample observations using the quasi-maximum likelihood method. The GARCH terms
for the standard model (1) are positive and significant except for the negative parameter

α for the Japanese stock market index. The sum of α and β parameters is close to unity,
suggesting volatility clustering in both markets. There is however no evidence of asignificant risk premium over the sample period.
The parameter π 1 associated with past implied volatility in model (2) is found to bepositive and significant for both the US and Japanese markets. This evidence suggests that
implied volatility provides incremental information about future volatility that is notcontained in the history of returns as described by various GARCH-M models. The GARCH
terms α and β are statistically insignificant for the US market whereas the magnitude of theparameter β decreases for the Japanese market. This evidence on information efficiency is
consistent with the empirical findings by Day and Lewis (1992), Harvey and Whaley,(1992) and Lamoureux and Lastrapes (1993), inter alia. In order to avoid underestimating
the significance of implied volatility, model (3) further includes the contemporaneous
observations of implied variance as explanatory variable. The estimation results indicatethat the parameter π 0 for current implied variances is positive and significant but theparameter π 1 for the lagged observations of implied variances becomes negative for bothmarkets. It is interesting to note that GARCH effects remain significant in the presence of16

past and contemporaneous implied volatility. These results suggest that implied volatility isreflective of some but not all information on future volatility contained in historical returns.
There is evidence based on the sign and magnitude of the δ parameters in theestimation of model (4), which accounts for the asymmetric effects of bad news on market
volatility, that the leverage effects are significant in the US stock market but insignificant
for Japanese equity. The incorporation by model (5) of past implied variances as
explanatory variable has again the effect of altering the significance of GARCH terms. The
significant parameter π 1 for past implied volatility is indeed associated with lower degree ofvolatility persistence for both markets. However, the asymmetric effects of bad news arefound to be significant in the presence of past implied volatility. These empirical results
suggest that even after allowing for leverage effects, implied volatility does still reveal some
useful information about future volatility that is not reflected by historical returns. It does
not reflect all information contained in the historical series though.
Judging from the estimates of the log-likelihood functions, it is the GARCH-M
model (3) including past and current implied variances that performs better than alternativemodels for both markets. But among the estimation models, which do not include implied
volatility, it is the GJR model accounting for potential leverage effects that performs
marginally better than the standard GARCH-M model.4
5.2. Out-of-sample forecasting performance
The results reported so far imply within-sample forecasts which assume that parameters
4

The models of volatility dynamics were also estimated using monthly time-series for options
expirations. The results, which are not reported here for the sake of brevity, are qualitatively
similar to the estimation results from daily data, though the GJR models are no longer associatedwith an excess log-likelihood over GARCH-in-mean models for both the US and Japanesemarkets. This is possibly due to the lower monthly data frequency, which does not fully capture
the impact of leverage effects on the volatility dynamics.
17

estimates are stable over time. The economic usefulness of the implied volatility index inforecasting future market volatility may be judged in light of out-of-sample tests ofrobustness. Furthermore, the forecasting performance of implied volatility can be assessedagainst alternative models describing volatility dynamics. With the focus being restricted to
GARCH models where implied volatility is not included, it is the GJR model that is used forcomparative purposes given its higher log-likelihood function relative to the standard
GARCH-M model. It is noted that the forecasting performance of GARCH models may beaffected when the underlying assumption of variance stationarity is violated. It can be also
sensitive to small sample periods. Hence, the out-of-sample forecasting of volatility over the
five-year period from 2000 to 2004 is performed using implied volatility as well as GJR
model estimates over the entire 1990s decade.
There is evidence from forecasts based on implied volatility and GJR models
shown in Figures 4 and 5 for the US and Japanese markets respectively, that both forecast
series represent close approximations of realized volatility. There is also a tendency for
implied volatility in the US market to remain at levels above realized volatility. This result is
consistent with evidence from Fleming (1998) that the implied volatility from S&P 100
options is upwardly biased. The tendency for volatility in the US market since 2003 toremain below the 20% upper boundary is suggestive of regime switches in market volatility.
The sharp increase in realized volatility associated with the Latin American debt crisis in
2002 is also reflected in the forecast series, albeit with some time lag. This constitutes
evidence that implied volatility is able to capture not only the impact of domestic shocks but
the volatility spillover from international stock markets as well.
Figure 6 shows the time-series of forecast errors relative to realized volatility, withpositive errors indicating the overestimation of market volatility. There is evidence fromboth markets that the implied volatility and GJR forecasts fluctuate around zero, with a18

more obvious upward bias for the US market since 2003. While the forecast errors are rather
close for the US market, they seem to depart from each other for the Japanese series. The
spread in forecast errors seems to be more significant when both series overestimate future
volatility than when they underestimate it. Judging by the sharper increases in GJR forecasts,
there is thus evidence that inference from GARCH models, which assume variancestationarity, are more sensitive to large deviations of stock returns from the long-run mean.
In contrast, implied volatility seems to constitute a smoother forecast of future volatility. It
is more reflective of patterns in realized volatility than the GJR conditional volatility model.
In order to evaluate the predictive accuracy of implied volatility relative to GJR
forecasts, it is possible to compute the proportion of explained variability P following Blair,
Poon and Taylor (2001)∑ (σ r ,t − ν t )2

P = 1−∑ (σ r ,t − σ r )2(9)whereν t denotes the time-series of implied volatility or GJR forecasts depending on themodel being used, and σ r represents the mean of realized volatility over the forecast period.
This measure of predictive power estimates the sum of squared forecast errors relative tovariations in actual volatility. It approaches unity when forecast errors are small and cantake the negative sign when forecast errors have greater variability than actual volatility.
The estimated P measures of explained variability for the US market amount to0.566 for implied volatility and 0.520 for GJR forecasts. For the Japanese market however,these measures take the lower estimates of 0.297 for implied volatility and -0.040 for GJR
forecasts. This negative P value suggests that inferences from the GJR model are likely toshow more variability than the actual fluctuations of market volatility. This is not a desirable
feature of a forecasting model, and it can be due to the stronger sensitivity of GJR modellingto large shocks to equilibrium returns. The lower P-values for the Japanese market relative19

to the US estimates may be related to the stronger tendency for options mispricing following
the inception of trading on Osaka Securities Exchange. The existence of arbitrage
opportunities has the potential of introducing a degree of bias in implied volatility that
ultimately affects its forecasting performance. Inference from GJR models based on the
shorter estimation period from 1995 to1999 where both VIX indices tend to converge isassociated with a lower P statistic of 0.462 for the US market but a positive estimate 0.038
for the Japanese market. The discrepancies in predictive power may also result fromstructural shifts in the level of uncertainty associated with the Japanese economic recovery.
Judging by the higher P measures of VIX indices relative to GJR models for bothmarkets, the implied volatility index offers more accurate forecasts of future market
volatility. This empirical evidence is also consistent with Blair, Poon and Taylor (2001)
results based on the implied volatility from S&P 100 index options. Part of the explanation
for the higher predictive performance of implied volatility may lie in, inter alia, the absenceof restrictions on variance stationarity and the model-free methodological approach applied
to aggregate expectations about future volatility. It is equally plausible that this evidence
pertains to the forward-looking nature of implied volatility, the formation of expectations
and the information efficiency of options markets. Addressing these issues is important asthey relate to the determinants of implied volatility, but they lie beyond the scope of thispaper.
6. CONCLUSION

The principal objective of this study was to examine from an international
perspective, the stochastic properties and forecasting performance of stock market volatility
implied in options prices. For this aim, we developed an index of implied volatility
heretofore unavailable for the Japanese equity market. The Nikkei 225 implied volatility
index was constructed following the methodology underlying the calculation of the new20

VIX index for S&P 500 benchmark. These implied volatility indices are useful in providing
a volatility measure with data frequency comparable to that of benchmark stock prices. Thisestimate of volatility may not be as accurate as the level of stock prices due to thecomplementary of these parameters. Such measurement problems are however akin to thedifficulties in quantum mechanics of ascertaining with accuracy both the location andvelocity of objects.
The empirical analysis provided some insights into the stochastic dynamics andthe relationship between implied volatility and realized volatility. Despite its upward bias,
the implied volatility index is found to be reflective of changes in actual volatility. There isalso evidence that implied volatility reveals some but not all information on future volatility
contained in historical returns. The tests of out-of-sample performance indicate that implied
volatility represents a smoother forecast of future volatility. This higher performance
contrasts with inference from GARCH models, which assume variance stationarity and are
more sensitive to large deviations of returns from the long-run mean. Judging from themeasures of predictive power, the implied volatility index provides more accurate forecastsof future market volatility.
The empirical evidence and the development of the new implied volatility index
for the Japanese equity market have some bearing for market regulation, policy-making andrisk-hedging decisions and open interesting avenues for future research. The implied
volatility index can be helpful in examining the impact of the release of macroeconomic
information on the dynamics of implied volatility using for instance, event-study
methodology. These studies can contribute to a better understanding of market sentiment,
investor confidence, and the perceptions by market participants of economic uncertainty.
Drawing parallel with the leverage effects in stock market returns, future research may also
explore the asymmetric effect of bad news on the implied volatility index. Furthermore, the21

economic significance of implied volatility forecasts can be also assessed in the VaR context
of risk-hedging. Finally, by providing a model-free measure of uncertainty, the implied
volatility indices can throw light on the debate over what constitutes excessive market
volatility. It can thereby benefit research on the impact of financial crises and help bring abroader perspective to the issues and controversies related to margin regulation and stock
market volatility.
REFERENCES
Ait-Sahalia, Y., and A. W. Lo, 1998, “Nonparametric estimation of state-price densities implicit infinancial asset prices,” Journal of Finance, 53, 499-547.
Amin, K., and V. Ng, 1997, “Inferring future volatility from the information in implied volatility inEurodollar options: a new approach”, Review of Financial Studies, 10, pp. 333-367.
Ball, C., and A. Roma, 1994, “Stochastic volatility option pricing,” Journal of Financial andQuantitative Analysis, 29, 589-607.

Black, F., and M. Scholes, 1973, “The valuation of options and corporate liabilities”, Journal of
Political Economy, 81, pp. 637-654.
Blair, B. J., S. Poon, and S. J. Taylor, 2001, “Forecasting S&P100 volatility: the incremental
information content of implied volatilities and high frequency index returns,” Journal of
Econometrics, 105, 5-26.

Britten-Jones, M., and A. Neuberger, 2000, “option prices, implied price processes and stochastic
volatility, Journal of Finance, 55, 839-866.
Carr, P., and L. Wu, 2006, “A tale of two indices”, Journal of Derivatives, forthcoming.
Canina, L., and S. Figlewski, 1993, “The information content of implied volatility,” Review of
Financial Studies, 6, 659-681.

Chambers, D. R., and S. K. Nawalkha, 2001, “An improved approach to computing implied
volatility,” Financial Review, 38, 89-100.
Christensen, B. J., and N. R. Prabhala, 1998, “The relation between implied and realized volatility”,
Journal of Financial Economics, 50, 125-150.

Corrado, C. J., and T. W. J. Miller, 1996, “A note on a simple, accurate formula to compute implied
standard deviations”, Journal of Banking and Finance, 20, pp. 595-603.
Day, T. E., and C. M. Lewis, 1992, “Stock market volatility and the information content of stockindex options,” Journal of Econometrics, 52, pp. 289-311.
Duan, J., G. Gauthier, and J. Simonato, 1999, “An empirical approximation for the GARCH option
pricing model,” Journal of Computational Finance, 2, pp. 75-116.
22

Dumas, B., J. Fleming, and R. E. Whaley, 1998, “Implied volatility functions: empirical tests,”
Journal of Finance, 53, pp. 2059-2106.

Fleming, J. 1998, “The quality of market volatility forecasts implied by S&P 100 index option
prices,” Journal of Empirical Finance, 5, pp. 317-345.
Fornari, F., 2004, “Macroeconomic announcements and implied volatility in swaption markets”, BIS
Quarterly Review, pp. 79-86.

Giot, P., 2005, “Implied volatility indexes and daily value at risk models”, The Journal of
Derivatives, pp. 54-64.

Glosten L.R., R. Jagannathan and D.E. Runkle, 1993, “On the relation between the expected value
and the volatility of the nominal excess return on stocks”, Journal of Finance, 48, pp.
1779-1801.
Harvey, C. R., and R. E. Whaley, 1992, “Market volatility prediction and the efficiency of the S&P
100 index option market,” Journal of Financial Economics, 31, pp. 43-73.
Hentschel, L., 2003, “Errors in implied volatility estimation”, Journal of Financial and Quantitative
Analysis, 38, pp. 779-810.

Heston, S., 1993, “A closed-form solution for options with stochastic volatility,” Review of
Financial Studies, 6, pp. 327-344.

Hull, J. C., and A. White, 1987, “The pricing of options with stochastic volatilities,” Journal of
Finance, 42, pp. 281-300.

Jorion, P., 1995, “Predicting volatility in the foreign exchange market,” Journal of Finance, 50,
507-528.
Lamoureux, C. B., and W. D. Lastrapes, 1993, “Forecasting stock return variance: toward anunderstanding of stochastic implied volatilities,” Review of Financial Studies, 6, 293-326.
Neely, C. J., 2005, “Using implied volatility to measure uncertainty about interest rates”, Federal
Reserve Bank of St Louis Review, May/June, pp. 407-425.

Pena, I., G. Rubio, and G. Serno, 1999, “Why do we smile? On the determinants of the implied
volatility function,” Journal of Banking and Finance, 23, pp. 1151-1179.
Poon, S. H., and C. W. J., Granger, 2003, “Forecasting volatility in financial markets: a review,”
Journal of Economic Literature, 41, pp. 478-539.

Ritchken, P., and R. Trevor, 1999, “Pricing options under generalized GARCH and stochastic
volatility processes,” Journal of Finance, 54, 377-402.
Whaley, R. E., 2000, “The investor fear gauge,” Journal of Portfolio Management, Spring, 12-17.

23

TABLESTABLE 1. Distributional properties of implied volatility and stock market returns
S&P 500 series
Nikkei 225 series
Stock
ImpliedLog IV
Stock
ImpliedLog VIX
returns
volatility differences
returns
volatility differences
Mean
0.00030.1988-0.0001-0.00030.24760.0002
Std. Dev.
0.0102
0.06350.05500.01480.06470.0642
Minimum-0.0711
0.0931-0.2751-0.0723
0.0769-0.6597
Maximum
0.05570.4574
0.41690.12430.6376
0.9382
Skewness-0.10310.90630.60820.19610.65282.0064
Kurtosis6.89863.72156.86476.35304.117140.1397
Jarque-Bera2485.06
620.702676.43
1858.13481.53 227517.89
Notes: The sample period extends from January 2, 1990 to December 31, 2004. JB is thestandard Jarque-Bera normality test distributed as χ 2 on the null. The series ofstock market returns and changes in implied volatility are calculated with log
differences.

24

Table 2. Autocorrelation and stationarity tests of implied volatility and stock returns
Autocorrelation function

ϕ1

ϕ2

ϕ3

ϕ10

ϕ 25

Stationarity tests
ADF
LB

S&P500 index
Stock returns-0.003 -0.019-0.0350.022
0.003-37.904b
0.222
Implied volatility 0.981
0.964
0.950
0.889
0.782
-4.220a
0.401
Changes in VIX-0.059 -0.074-0.0690.050
0.048-23.007c
1.000
Nikkei 225 index
Stock returns-0.020 -0.051
0.001
0.007
0.032
-46.979c
0.790bImplied volatility 0.966
0.943
0.923
0.823
0.669
-5.9370.121
Changes in VIX-0.166 -0.044-0.045-0.002
0.053-28.187c
0.290
Notes: The sample period extends from January 2, 1990 to December 31, 2004. ADF are theAugmented Dickey-Fuller test statistics, using Schwarz information criterion withadditional lags included to eliminate any remaining ARCH effects. The 1% criticalvalues are -3.966, -3.435 and -2.566 for unit root tests with (a) trend and intercept, (b) withintercept only, and (c) with neither. LB refers to the p-values for Ljung-Box Q-statistics
distributed as χ 2 on the null of no autocorrelation up to the 10th order.

25

Table 3．Modelling the relationship between realized and implied volatility

γ0

γi

γ1

γ2

Wald
DW LB(1) LB(4) Adjusted
Test
Test
R2
Daily observations
S&P 500-0.0209 0.9916
526.882 0.091 3566
12310
0.583
(0.000) (0.000)(0.000)(0.000) (0.000)
Nikkei 225
0.0640 0.8070
108.689 0.068 3654
13196
0.266
(0.000) (0.000)(0.000)(0.000) (0.000)
Monthly expirations
S&P 500-0.0205 0.9999
22.489 1.738 3.027
8.358
0.596
(0.196) (0.000)(0.000)(0.082) (0.079)-0.0191 0.9282 0.0733
15.071 1.868 0.765
4.097
0.598
(0.243) (0.000) (0.470)(0.000)(0.382) (0.393)
Nikkei 225
0.0401 0.8808
3.072 1.472 12.450 13.222
0.364(0.087) (0.000)(0.049)(0.000) (0.010)
0.0472 0.9002 0.1446 -0.1938 2.839 1.779 1.036
3.644
0.394
(0.039) (0.000) (0.164) (0.013) (0.026)(0.309) (0.456)
Notes: Estimation of the regression model with white-heterostkedasticity consistent standard
Regression
parameters

errors. The sample of daily observations includes rolling volatility series over the period
January 1990 to December 2004. The sample of monthly observations includes 180
non-overlapping expirations covering the period from February 1990 to January 2005.
Asymptotic p-values for regression coefficients reported in parentheses. F-statistics for
Wald Test of the null of (γ 0 = 0, γ i = 1) and for regression models with additional lags, thenull of (γ k = 0, γ i = 1) with the associated p-values reported in parentheses. DW is theDurbin-Watson statistic for the test of first-order serial autocorrelation. LB(k) denotes
Ljung-Box Q-statistics distributed as χ 2 on the null of no serial correlation up to lag k,with corresponding asymptotic p-values reported in parentheses.

26

TABLE 4. GARCH modelling of stock market volatility in the US and Japanese markets
Model
Parameters

S&P 500 stock index

Nikkei 225 stock index

Model (1)Model (2)Model (3) Model (4) Model (5) Model (1)Model (2)Model (3) Model (4) Model (5)λ0λ10.02600.02890.2221a0.01330.8448-0.0776c -0.1002b 0.0684c -0.0921b-0.1047a

3.77340.1575 -19.7920

1.83910.01974.6164c

5.0157b -4.0209c

2.95883.2893c

w00.0005-0.0109a -0.0010c

0.0010a-0.00130.0061a0.00520.0008c

0.0050a0.00050.05380.9421ab0.00590.9347aa-0.0355
0.7231a

a

a

c

b

0.02200.9016a-0.0236a0.7089a0.1020a0.1743a0.0005a0.10690.2130a0.0007a

α
β
π0
δ
π1

-0.0076-0.00540.01650.8856a0.0038a0.0025a -0.0035a

0.08330.8899a

0.09740.7136a0.0096
0.9761a0.0038a0.0005a -0.0037a

LB

0.225
0.338
0.507
0.279
0.290
0.956
0.837
0.878
0.967
0.781
LM
0.723
0.268
0.326
0.832
0.489
0.790
0.225
0.050
0.479
0.280
Log-L
12913.33 12994.84 13133.58 12955.37 13009.80 11200.90 11226.22 11304.84 11242.72 11280.27
Excess log-L
81.51
220.25
42.04
96.47
25.32
103.9441.82
79.37
Notes: The sample period of daily observations extends from January 1990 to December 2004.
The model (1) refers to the GARCH-M model equation (4), model (2) to the GARCH-M
with past implied volatility equation (5), model (3) to the GARCH-M with current and past
implied volatility equation (6), model (4) to the GJR-GARCH-M equation (7) and model(5) to the GJR-GARCH-M with past implied volatility equation (8). The various GARCH
models are estimated with Bollerslev-Woodridge robust standard errors. The parameters
estimates λ0 and w0 are multiplied by 10 and 10 , respectively. Significance at the 1, 5 and2

3

10% level is denoted by a, b and c, respectively. LB refers to the p-values associated with
Ljung-Box test for serial correlation in squared residuals up to the 10th order. LM refers to
p-values of the Lagrange-Multiplier test for neglected ARCH effects. Both tests are
distributed as χ 2 on the null. LogL is the estimated log-likelihood function. Excess log-l isthe excess log-likelihood with respect to model (1).

27

FIGURESFIGURE 1. The behavior of implied volatility in the Japanese and US markets.7

Implied volatility index.6.5

Nikkei 225 implied volatility

.4

.3

.2

.1
S&P 500 implied volatility
.0
1990

1992

1994

1996

1998

2000

2002

2004

28

FIGURE 2. The relation between implied and realized volatilities in the US market.6

S&P 500 realized volatility

.5

.4

.3

.2

.1

.0
.0

.1

.2

.3

.4

.5

S&P 500 implied volatility

FIGURE 3. The relation between implied and realized volatilities in the Japanese market.8

Nikkei 225 realized volatility

.7
.6
.5
.4
.3
.2
.1
.0
.0

.1

.2

.3

.4

.5

.6

.7

Nikkei 225 implied volatility

29

FIGURE 4. Out-of-sample forecasting of the US stock market volatility.6
Implied volatility

GJR volatility

S&P 500 volatility

.5

.4

.3

.2

.1
Realized volatility
.0
2000

2001

2002

2003

2004

FIGURE 5. Out-of-sample forecasting of the Japanese stock market volatility.7
Implied volatility

GJR volatility.6

Nikkei 225 volatility

.5

.4

.3

.2
Realized volatility
.1
2000

2001

2002

2003

2004

30

FIGURE 6. Out-of-sample forecast errors of stock market volatility
120

40
150
S&P 500 Forecast Error (%)

0
100
-40
50
-80
0
-50

Nikkei 225 GJR modelS&P 500 GJR model
Implied volatility index

-100
2000

2001

2002

2003

2004

31

Nikkei 225 Forecast Error (%)

80