Discussion Papers In Economics
And Business

The stochastic dynamics, forecasting ability
and risk factors in implied volatility
Kazuhiko Nishina
Nabil Maghrebi
Discussion Paper 02-12

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe stochastic dynamics, forecasting ability
and risk factors in implied volatility
Kazuhiko Nishina
Nabil Maghrebi
Discussion Paper 02-12
July 2002

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The stochastic dynamics, forecasting ability and risk factors in implied volatility¶

Kazuhiko Nishina
Graduate School of Economics, Osaka University, Japan
Phone: +81-66-850-5243 Fax: +81-66-850-5277
Email: nishina@econ.osaka-u.ac.jp

Nabil Maghrebi
Faculty of Economics, Wakayama University, Japan
Phone: +81-73-457-7658 Fax: +81-73-457-7659
Email: nebilmg@eco.wakayama-u.ac.jp¶ The Grant-in-Aid for Scientific Research by the Japanese Ministry of Education and Technology as

well as the research grant by Wakayama University are gratefully acknowledged. We thank the Osaka
Securities Exchange and the Institute for Economic and Econometric Research at Wakayama University,
for providing the financial database used in this study.

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The stochastic dynamics, forecasting ability and risk factors in implied volatility

ABSTRACTA new benchmark of volatility implicit in Nikkei 225 stock option prices
is constructed and its stochastic dynamics are examined in comparison
with the S&P implied volatility index. There is a tendency for the levelsof implied volatility to move in tandem during the late 1990s despite theidiosyncrasies of the booming U.S. economy and Japanese recession.
The long-term convergence dynamics of implied volatility are found tobe similar but the convergence of the transitory component towards zerois inherently different across markets. There is evidence from theestimation of asymmetric GARCH models that useful information onleverage effects is embedded in implied volatility. Albeit upward biased,
implied volatility is found to constitute a good approximate of futurelevels of realized volatility in the Japanese market. Based on a two-factor
model of risk components consistent with the intertemporal CAPM
theory, there is also evidence that implied volatility is function ofstochastic changes in the investment opportunity set proxied bymeasures of uncertainty about the domestic real economy and theinternational financial economy.
JEL Classification: C51, G13

1. Introduction

For plausible explanations of dramatic changes in stock market volatility, it is essential to understandhow and why market volatility does ebb and flow. Given the widespread perception that option
trading is akin to volatility trading, the empirical questions of whether the volatility implied in
option prices is redundant, or whether it constitutes a substitute or complement to alternative
measures of volatility, have gained in importance. In understanding the complexity of risk and the
dynamics of risk hedging, there is no doubt that the contribution of theoretical and methodologicaladvances in option pricing has been significant. But the limited empirical evidence on the

3

information content of implied volatility leaves large rooms for further analysis that may shed lights
on the properties of financial market volatility and the economics and efficiency of options markets
as well.
The present study is an attempt to enrich the existing literature with new perspectives
from alternative options markets and the construction of a new implied volatility index. In gathering
information contained in the term structure of implied volatility and volatility smiles, thisbenchmark allows for the aggregation of short-term volatility expectations across option investors,
hedgers and arbitrageurs. In contrast to the numerous indices of financial asset prices, volatility
benchmarks are rather rare. The Chicago Board of Options Exchange uses the real-time OEX
bid-ask quotes on S&P 100 American options to publish since 1993 the CBOE-VIX index of implied
volatility. We construct a similar index of implied volatility on the Nikkei 225 stock index, which isthe average price of representative stocks listed on the first section of Tokyo Stock Exchange.
Despite the fact that the Nikkei 225 stock average is the underlying asset of financial derivatives
traded on three major Asia-Pacific derivatives markets, albeit under different contract specifications,
there is to the knowledge of the authors, no volatility benchmark for Japanese equity markets.1
In constructing the implied volatility index from Nikkei 225 stock index options traded on
the Osaka Stock Exchange, there is an attempt to provide a useful addition to the list of financial
benchmarks, which can facilitate comparative analysis of volatility across international markets.
The replication exercise provides also the opportunity to examine four core empirical issues
addressed in sequence in the remainder of the paper, which is structured as follows. The next section
presents a review of literature on option pricing as far as theoretical and empirical issues in implied
volatility are concerned. Section 3 outlines the methodology underlying the construction of theimplied volatility index and the measurement problems encountered in its implementation withrespect to Japanese stock index options. The distributional and time-series properties of the implied
volatility index are discussed in section 4. Section 5 examines its relationship with conditional
volatility in asymmetric Generalized Autoregressive Conditional Heteroskedasticity (GARCH)

4

modeling of returns. It analyzes the information embedded on the asymmetric impacts of bad andgood news, or leverage effects, which derive from the increased risk of holding stocks when prices
are sharply decreasing, thereby discounting the value of equity relative to corporate debt andincreasing corporate leverage. The analysis in Section 6 focuses on the ability of implied volatility toforecast the realized levels of market volatility over options expirations. A structural examination of
its relation with risk factors related to the domestic real economy and the international financial
economy is made in Section 7. Section 8 concludes the paper.

2. Option pricing theory and implied volatility

The underlying asset price S is generally assumed to follow the diffusion process described inequation (1) where the drift µ ( S , t ) and volatility σ ( S , t ) terms is function of S and time t andwhere W is a standard Brownian motion.
dS (t ) = µ ( S , t ) dt + σ ( S , t ) S (t ) dW (t )(1)

The theoretical value of an European call according to Black and Scholes (1973) option pricing
model is function of the exercise price K , constant volatility σ ( S , t ) = σ over the option’s life, andrisk-free interest rate r in addition to the state variables, time to expiration τ and the underlying
asset price S . The option valuation model extended by Merton (1973a) to account for stochastic
dividends at the anticipated continuous yield q can be expressed as

C = C ( S , τ ; K , r , q, σ ) = Se − qτ N (d 1 ) − Ke − rτ N (d 2 )
d

1

-∞

2π

where N (d ) = ∫e − x / 2 dx
2(2)d 1 = {ln(S / K ) + (r − q + σ / 2)τ } (σ τ )2d 2 = d1 − σ τWhereas the mathematical advances in option valuation by Black and Scholes (1973)
spurred an extant theoretical literature on option pricing, the idea first introduced by Latane andRendleman (1976) of extracting market volatility from option premium added another dimension to5

the empirical research on derivatives markets. An estimate of the unobservable state variable market
volatility σ can be derived as a function of the option price and remaining parameters using a
numerical iterative process that reduces the error between the observed option price and its
theoretical value. Convergence problems arise when equating the observed and model values fails.
The numerical difficulties stem in part from the fact that option pricing models do not typicallyprovide closed-form solutions and that the volatility function cannot be reduced to a scalar.2
The rich literature on volatility implicit in option premium seems to evolve around thenumerical and estimation problems, distributional properties, informational content of implied
volatility and the characterization of its relationship with conditional and realized volatilities. The
empirical evidence, indisputable so far, is that daily estimates of implied volatility vary acrossexercise prices and exhibit a ‘volatility smile’ that is inconsistent with the theoretical foundations of
the parsimonious Black-Scholes valuation model, which assumes volatility to be constant,independent of exercise prices and time-invariant. Volatility smiles are suggestive of Black-Scholes
mispricing of deep in- and out-of-the-money options or options market inefficiency. They may
represent de facto empirical violations of the underlying assumptions including the normal
distribution of returns. Pena, Rubio and Serno (1999) attribute volatility smiles to transaction costs
and find a significant inverse relationship between the degree of curvature and time to expiration.
Even in the presence of ‘flat’ smile when Black-Scholes implied volatility is constant across exercise
prices, the square of implied volatility is shown by Britten-Jones and Neuberger (2000) to be equal
to the risk-neutral expectation of squared volatility and shown to be positively biased estimate ofrealized volatility. Unless volatility is constant, it can be shown using Jensen’s inequality,t +τ∞ C (t + τ , K ) − C ( t , K )E  ∫ σ (t ) 2 dt  ≤ 2∫
dK
2

 t
0
K

(3)

that under risk-neutral probability distribution and continuous time, it is the square of volatility
rather than volatility per se that is an unbiased estimate of realized volatility.
The examination of the properties of implied volatility constitutes a joint test of options

6

pricing model and market efficiency. To mitigate inherent inconsistencies in Black-Scholesmodeling, a wide array of competing approaches to recover unbiased estimates of implied volatility
and alternative option pricing models has been advocated.3 At one end of the implied volatility
spectrum, a “model-free” approach is advocated by Britten-Jones and Neuberger (2000) who adjust
arbitrary volatility processes to option prices, drawing upon the standard practice of fitting interest
rate processes to bond prices. Modeling the implied volatility function to depend on the exercise
price includes studies using polynomials, interpolation or splines smoothing of the pricing function
by Shimko (1993), Longstaff (1995), Malz (1997), and Ait-Sahalia and Lo (1998), inter alia.
GARCH option pricing including Duan, Gauthier and Simonato (1999) and Ritchken and Trevor(1999) attempt to capture the path-dependence dynamics of volatility and the negative correlation
between volatility and returns.
The stochastic properties of volatility are also addressed in earlier studies by Hull andWhite (1987), Stein and Stein (1991) and Heston (1993) among others, who develop models where
the constant volatility parameter is substituted with the entire joint probability distribution of returns
and changes in volatility. Although Ball and Roma (1994) find the modeling of stochastic volatility
to be consistent with implied volatility smiles, the approach is complicated with the estimation of themarket price of volatility risk. Assuming volatility to be a deterministic function of the underlying
asset price and time, as in models developed by Rubinstein (1994), Derman and Kani (1994) and
Dupire (1997), is also inconsistent with the findings by Buraschi and Jackwerth (2001). The
examination by Dumas, Fleming and Whaley (1998) of the predictive and hedging performances of
the ad hoc Black-Scholes model and quadratic-form modeling of the deterministic volatility
function σ ( K , t ) suggests that the latter outperforms the former on both accounts. There is evidence
that the estimated deterministic function is itself unstable over time and Black-Scholes hedge-ratios
are more reliable. As far as at-the-money options are concerned, there is also earlier evidence by
Ledoit and Santa-Clara (1998) that as expiration draws near, Black-Scholes implied volatility
converges to the asset’s instantaneous or stochastic volatility.

7

The final issue addressed in the literature concerns the informational content of implied
volatility, where early empirical evidence by Day and Lewis (1992) suggests that volatility implied
by S&P 100 index options is almost unbiased and constitutes an informative forecast but theconditional volatility based on GARCH modeling of returns does contain incremental information.
The significance of the informational content for forecasting purposes is also supported by evidence
from Lamoureux and Lastrapes (1993) using volatilities implicit in the price of individual stock
options. Albeit upwardly biased, implied volatility is also found by Fleming (1998) to constitute a
reliable forecast of ex post volatility, with forecast errors being orthogonal to parameters embeddedin ARCH models. Further evidence provided by Christensen and Prabhala (1998) explains away partof the bias in implied volatility with regime shifts in the pricing of index options around the October1987 stock market crashes. More recent evidence from Blair, Poon and Taylor (2001) indicates thatirrespective of data frequency and forecasting horizon, S&P 100 VIX index of implied volatility ismore accurate for out-of-sample forecasting than realized volatility.4 In sharp conflict with thismounting evidence, Canina and Figlewski (1993) suggest that S&P 100 implied volatility is neither
informative on realized volatility nor accurate in forecasting future volatility.
An international perspective on the informational content of implied volatility is added tothe empirical literature using the Nikkei 225 option prices and the parsimonious Black-Scholesmodel, which despite inherent inconsistencies, remains the cornerstone of theoretical and empiricalstudies in implied volatility. Before discussing the results of our findings, the methodology followedin the construction of the implied volatility benchmark is explained in the next section.

3. Implied volatility index: data and methodological issues

The volatility implicit in the daily closing prices of European options on Nikkei 225 stock index isestimated using the theoretical call valuation equation (2) and put price derived from put-call parity.
The Japanese index options expire on the second Friday of the contract month, covering a period of
144 maturities starting from the inception of options trading on OSE in June 1989 through June 2001.

8

The risk-free interest rate is approximated by the continuous yield on the three-month or one-month
Certificate of Deposit with the closest maturity to the expiration date. The dividend yield isestimated as a fraction of current stock index level, using the annual average yield on all stocks listed
on TSE. To facilitate comparison across international markets, we use the daily closes of S&P 100
implied volatility index based on average bid-ask option quotes, which are obtained from Thomson
Financial Datastream database. There are discrepancies in the sample sizes across markets due todifferences in trading holidays and focus is made on the newly constructed index of implied
volatility in the Japanese market.
The construction of the implied volatility index can be viewed as another variant of
implied volatility weighting schemes.5 The calculus of VIX, which is described in further detail in
Whaley (2000), involves a series of averaging and interpolation exercises and takes into accountboth the term structure of implied volatility and volatility smiles and sneers. Theoretically, the index
measures the market volatility implied by the price of a hypothetical option with 30 calendar days toexpiration and with exercise price exactly equal to the underlying asset price. The VIX benchmark is
constructed as a weighted average of volatilities implied by the prices of options in the at-the-money
neighborhood and nearest maturities. The implied volatility is calculated using the remaining time tomaturity expressed in calendar days τ c , and subsequently approximated using vt = vt τ c τ into
volatility over τ = τ c − 2 int(τ c / 7) trading days.6
To mitigate measurement problems deriving from short-lived options which are usuallyassociated with extreme values of implied volatility, options with less than eight calendar days toexpiration(i.e., approximately six trading days) are eliminated. Focus is made also on

near-the-money options, which are theoretically most sensitive to changes in volatility and unlike
deep in- or deep-out-of-the-money, they are associated with higher time premium. Upon identifying
the upper K = U and lower K = L exercise prices immediately above and below the daily stockindex price, the selection of the corresponding nearest-maturity m = 1 call (CU 1 , C L1 ) and put

9( PU 1 , PL1 ) options together with the second near-maturity m = 2 call (CU 2 , C L 2 ) and put( PU 2 , PL 2 ) options results in eight estimates that differ across option types, maturities and exercise
prices.
These volatility measures vt ( K , m) with exercise price K = U , L and maturity m = 1, 2
need to be averaged at each of the three levels to construct the VIX index. First, the simple averaging
vt ( K , m) = {vt (C , K , m) + v t ( P, K , m)} / 2 deals away with call and put notations. The remaining
two

couplesofimpliedvolatilities

from

the

nearest-maturity [v t (U ,1), v t ( L,1)] andsecond-nearest-maturity option [v t (U ,2), v t ( L,2)] are also used in the interpolation processdescribed by equation (4), which is aimed at adjusting for differences between the current stockindex level and exercise prices. This process results in the approximation of the implied volatility of
the hypothetical at-the-money option for each expiration month.
vt ( m) = vt (U , m)

U −S
S−L
+ v t ( L, m )
U −L
U −L

for m = 1, 2.

(4)

Finally, an extrapolation process is required to adjust for differences in periods of time τ n1
and τ n 2 remaining to the expiration of the nearby and second-nearest maturity options, respectively.
The implied volatility index is made to reflect the volatility implicit in the price of a single
theoretical option with 22 trading-days (approximately 30 calendar days or a month) remaining to
expiration.
vt = v t (1)τ 2 − 2222 − τ 1+ v t ( 2)
τ 2 − τ1
τ 2 − τ1

(5)

Apart from Black-Scholes misspecification and usual problems of non-synchronous
pricing of stock and options, the implementation of this methodology using data on the Japanese
options market poses some difficulties. The tendency for daily trading on the European-style Nikkei
225 stock index options on OSE to concentrate on near-maturity, second- and third-month
expirations should actually facilitate the implementation of VIX calculation. But, the likelihood of

10

non-convergence for the near-maturity near-the-money options is found to be higher with respect toearly periods of thin trading of Nikkei 225 options on OSE. The problem of non-convergence of
implied volatility numerical calculations, which may be attributed to option mispricing anderror-in-variables problems, can further complicate the implementation of VIX methodology by
undermining either the averaging or interpolation process.
The methodology does not allow for the upper or lower level of exercise prices on a given
trading day to differ across call and put options. In reconstructing the volatility index, it remains
silent on the appropriate way to deal with cases where no put option can be found to match exactly
the exercise price of calls at either the upper or lower levels. Substituting the non-converging option
price with that of the next-near-the-money option effectively compromises the simple averaging
across option types because of the discrepancy in exercise price boundaries across call and put
options. To minimize the loss of information, we adopt the simpler approach of referring in turn tothe opening, highest or lowest price when numerical convergence is achieved with respect to closing
prices.7 In the case where vt (1) = 0 (or vt ( 2) = 0 ), the value of VIX is approximated on the basis ofequation (5) and under the usual underlying assumption that volatility is related to the square root oftime, with the point estimate vt = v t ( 2) τ 2 22 (or vt = vt (1) τ 1 22 ). The resulting time-series ofdaily estimates of the implied volatility index provides an ex ante estimate of short-term volatility.
The next sections assess the distributional properties and usefulness of implied volatility inforecasting realized volatility.

4. Distributional and stochastic properties of implied volatility

This section examines the behavior of daily Nikkei 225 and S&P 100 VIX indices, as well as theirdistributional characteristics over monthly options expirations. Figure 1 exhibits the daily volatility
series, which reflect two distinct trends, apart from the short period following the inception of option
trading on OSE, during which implied volatility has been lower in the Japanese than in US options
markets. This period of rising stock prices and lower volatility precedes the burst of the Japanese

11stock market bubble by the end of the 1980s. The first pattern, which is obvious over the period
extending into the mid-1990s, reflects episodic surges and falls in Japanese market volatility, which
are in sharp contrast with the smoother and constant path of implied volatility in US markets. This
trend is reversed however when implied volatility in the US market increased and the alignment ofvolatility paths in both markets persisted until the end of the sample period. This parity in theimplied volatility levels is interesting is maintained despite the respective booms and slumps in the
US and Japanese economies.
From the distributional moments of price, return and implied volatility in both markets arereported in Panel A of Table 1, there is evidence that implied volatility is on average higher and morevolatile in Japanese stock markets than US markets. Upon conversion of annual implied volatility todaily basis (dividing by the square root of 250), the median of Nikkei 225 implied volatility equal to1.67% is found to be comparable to the sample standard deviation of returns 1.48%. The evidence,
which applies also to the US markets, suggests that implied volatility can constitute a good proxy ofrealized volatility.
Before examining the time series properties, determinants and forecasting ability of
implied volatility, it is noted from unit root test results, that with the exception of stock prices, alldaily and monthly return and volatility series are stationary. Judging from the distributional
moments and behavior of implied volatility over time, it seems that shocks to volatility are likely topersist. The stochastic dynamics of the implied volatility series can be examined using the class of
GARCH econometric models

v t = ϑ 0 + ∑ k =1ϑ k v t − k + etn

(6 )

χ t2 = α 0 + ∑i =1α i et2−i + ∑ j =1 β j χ t2− j
q

p

(7a)

As with the parsimonious GARCH(1,1) process, when the roots of φ1 = α 1 + β 1 lie outside the unit
circle (i.e.,), the conditional variance of the implied volatility index converges towards the long run
unconditional variance. By redefining the constant in the conditional variance equation in the

12

GARCH(1,1) process as function of the unconditional variance α 0 = ϖ (1 − φ1 ) , equation (8a) canbe rewritten as

χ t2 = ϖ + α 1 (et2−1 − ϖ ) + β 1 ( χ t2−1 − ϖ )(7b)

It is possible to allow for reversion to a long-term time varying ω t instead of a constant mean ϖ byexpressing the conditional variance in the component GARCH(1,1) model asχ t2 − ω t = α 1 (et2−1 − ω t −1 ) + β 1 ( χ t2−1 − ω t −1 )
ω t = ϖ + δ (ω t −1 − ϖ ) + γ (et2−1 − χ t2−1 )(7c)

In this nested setting, the long run component ω t defined in the second part of the system of
equations (7c) converges to ϖ with powers of δ while the transitory component χ t2 − ω t describedin the first equation converges to zero with powers of φ1 . The estimation results of the component
GARCH modeling of implied volatility in Japanese and US markets are reported in Panel A of Table
2. The diagnostic tests indicate that these models are not associated with serial correlation in thesquared standardized residuals and that there are no traces of ARCH effects. There is evidence thatthe value ϖ towards which the time-varying long-run volatility of the Japanese implied volatility
index converges is contrary to evidence from US markets, statistically insignificant. In both markets,convergence takes place at a slow rate determined with powers of δ , which is found to be typicallyclose to unity.
From the transitory equation, the insignificance of α 1 estimate for the Japanese market
suggests that differences between innovations in volatility and the permanent component do notaffect the short run movements of implied volatility. The influential force behind the convergence ofthe transitory component towards zero lies in past levels of implied volatility. In contrast, both pastlevels of volatility and past innovations in US implied volatility constitute significant determinantsof the transitory component. Judging from the magnitude of the transitory coefficients, the dynamics
of short-term convergence towards zero are inherently different across these markets. The reversion

13of the transitory component towards zero via powers of φ1 , tends to occur in the US market at fasterrates than in the Japanese market. The evidence is important for international portfolio hedging
purposes as it aids in understanding options investors’ formation of expectations about long-termlevels and short-term movements in stock market volatility.

5. Conditional variance, implied volatility and leverage effects

In light of reported evidence reported on the stochastic properties of implied volatility, it is important
to explore its relationship with alternative estimates of asset volatility. Judging from the comparative
behavior of daily returns and implied volatility estimates illustrated in Figure 2, it is likely that
periods of increased implied volatility coincide with episodes of higher variability of returns. This
adds to the graphical evidence from Figure 1 that these periods are also associated with declining
stock prices. To examine this stylized tendency for the volatility of equity returns to cluster andpersist and its asymmetric behavior during bullish and bearish markets, two variants of asymmetric
GARCH modeling are considered. The first variant represented by the threshold ARCH (TARCH)
model developed by Zakoian (1994), accounts for asymmetric shocks to volatility and describes themean equation and conditional variance function respectively as

rt = µ 0 + ∑k =1 µ k rt − k + ψ tn(8 )σ t2 = υ 0 + ∑i =1 (α iψ t2−i + γ iψ t2−i qt −i ) + ∑ j =1 β j σ t2− j ,
q

p

(9a)

where ψ t are normally distributed errors. The dummy variable q t is equal to unity in the case of goodnews (ψ t > 0) and zero otherwise. The news impact on volatility is function of the sign andmagnitude of γ . The impact is limited to α in the case of good news and amounts to α + γ withrespect to bad news. There is evidence of asymmetries when γ ≠ 0 and of leverage effects whenγ > 0 . When the leverage effect is assumed to be exponential, an alternative modeling of
non-asymmetric responses to shocks is considered. The exponential GARCH (EGARCH) model

14

proposed by Nelson (1991) expresses the variance equation aslog σ t2 = υ 0 + ∑i =1 (α iqψ t −iψp+ γ i t −i ) + ∑ j =1 β j log σ t2− j
σ t −iσ t −i(9b)

In this exponential mapping, the increase in volatility associated with bad news is higher than thatgenerated by good news when γ is negative and significant. Table 2 panel B reports Model I
estimates, which result from TARCH and EGARCH modeling of Japanese and US stock market
returns. Independent of the asymmetric GARCH variant, there is evidence of volatility persistence.
The results indicate also the existence of leverage effects as γ is found to be positive and significant
for TARCH models and negative and significant for EGARCH models. This constitutes evidence
that bad news is likely to generate a larger increase in market volatility than good news.
To examine its informational content of implicit volatility, Day and Lewis (1992) andLamoureux and Lastrapes (1993) propose its inclusion as an exogenous variable in the conditionalvariance equation. Instead of the GARCH in mean modeling used in the above studies, testing the
hypothesis that implied volatility does not contain information beyond that implied by realized
returns is performed using TARCH and EGARCH variance equations (10a) and (10b), respectively.

σ t2 = υ 0 + ∑i =1 (α iψ t2−i + γ iψ t2−i qt −i ) + ∑ j =1 β j σ t2− j + ξvt2−1qplog σ t2 = υ 0 + ∑i =1 (α i
q

(10a)

ψ t −iψp+ γ i t −i ) + ∑ j =1 β j log σ t2− j + ξ log v t2−1 (10b)σ t −iσ t −iThe significance of the coefficient associated with implied volatility ξ constitutes evidence
that this variable contains incremental information content on return volatility. Judging from theestimates of Model II reported in Table 2 Panel B, this coefficient is found to be positive andsignificant in both markets and irrespective of the TARCH or EGARCH modeling. As far as the US
market is concerned, the significance of ξ is accompanied by a significant change in the sign of
GARCH terms β 1 . Though the significance of leverage effects measured by the news impact
coefficient γ fades away, neither quadratic nor exponential estimates do change in sign. The log

15likelihood ratio tests suggest that irrespective of the market under examination, the inclusion of
implied volatility results in a significant improvement in model estimation. This is evidence that
implied volatility impounds useful information on the persistence of return volatility as well as on its
asymmetric response to bad and good news.
It is possible to visualize the leverage effect using the news impact curve, which reflects
the sensitivity of the conditional variance to innovations standardized by market volatility ψ / σ ,estimated around the median of the conditional variance. Figure 3 illustrates the impact functions of
the exponential modeling of Japanese and US market volatility with and without implied volatility. It
appears that increases in volatility differ in magnitude across markets, with the stronger impact
associated with Japanese market volatility. The slope of the curves with respect to bad news is
steeper than for good news. Modeling using implied volatility as regressor in the conditionalvariance equation aids in explaining part of the impact of bad news on Japanese volatility and theentire impact on US market volatility. Expectations of market declines may generate a buyingpressure on puts, thereby driving their prices up and increasing the level of implied volatility. The
evidence is important in understanding the important aspects of the relationship between implied
volatility and realized volatility, which is the subject of the next section.

6. Realized volatility and the forecasting ability of implied volatility

Having assessed the distributional properties and stochastic dynamics of the implied volatility, it is
the usefulness of this forward-looking measure of volatility in forecasting realized volatility that canbe now examined. The standard deviation of log returns provides an ex post estimate of realized
volatility at the end of the option maturity month m as

σ mh = τ m−1 ∑t =1 (rt − rm ) 2
t +τ mwhere rm = τ m−1∑

t +τ mt =1(11)

rt is the mean of spot returns over the maturity month and τ m refers to the16number of remaining days to expiration. It is possible to test as in Christensen and Prabhala (1998),
the significance of the relationship between implied and realized volatilities using the following
regression model.

σ mh = λ0 + λv v m + u m ,(12)where σ mh denotes historical volatility defined in equation (12) and v m is the annual implied volatility
estimated at the beginning of the expiration month (thirty calendar days before maturity). The
residuals u m are white noise when implied volatility constitutes an efficient estimate of observed
volatility. From Figure 4, it appears that there is a strong tendency for Nikkei 225 implied volatility
index to constitute an upper boundary for realized volatility on the spot market. The evidence isconsistent with results from Fleming (1998) that implied volatility is an upward biased volatility
forecast. Because implied volatility is shown by Jensen’s inequality (3) to be an upward biased
estimate of actual volatility, the intercept is likely to be also upwardly biased but the estimate of theslope coefficient does not necessarily suffer from such a bias. The slope of the estimated regression
model reported in Table 3 (Model I) is found to be positive and significant suggesting that useful
information on the future variability of returns is embedded in actual option prices.
Additional information is conveyed by the difference between lagged estimates of implied
volatility and realized volatility over the preceding expiration month. Model II reports the negativeand significant coefficient associated with this differential, which if viewed as the forecasting error,
suggests that changes in implied volatility are embedded in an ongoing adaptive process towards theobserved variability of returns. As suggested by the additional coefficient estimate in Model III,actual volatility is also negatively related with average spot returns, indicating a tendency to increaseduring periods of falling prices and increasing returns. This result is consistent with evidence onleverage effects analyzed in Section 5. Whereas the relation between market conditions and realized
volatility is found to be sample-independent, past deviations of implied volatility from historicallevels have become irrelevant as they decreased in magnitude over the late 1990s. Neither the sign

17

nor the significance of the implied volatility index in forecasting the future level of market volatility
seems to be altered. This evidence is inconsistent with results by Canina and Figlewski (1993) but inline with the conclusions by Day and Lewis (1992) and Lamoureux and Lastrapes (1993),
suggesting that implied volatility is significantly informative on the future variability of returns.

7. Risk factors and the predictability of implied volatility

In light of evidence on the stochastic dynamics of implied volatility and its relationship withleverage effects, it is important to examine the issue of whether changes in implied volatility are
predictable. Based on regression analysis, there is evidence by Harvey and Whaley (1992) that there
are predictable variations in daily fluctuations of implied volatility, the statistical significance of
which does not translate into economically profitable trading strategies. When predictability can beassessed on the basis of its relationship with various risk factors, a structural analysis of implied
volatility can be made drawing upon Merton’s (1973b) approach to intertemporal portfolio theory
and the intertemporal capital asset pricing model. In pricing securities in the dynamic market
framework, there are additional sources of risk stemming from uncertainties caused by shifts in theinvestment opportunity set. Equity risk is function of the stochastic investment opportunity set, i.e.,changes in expected returns and covariance of returns related to state variables such as the expected
inflation rate, the expected productivity of capital and proxies of the level of uncertainty about
economic activity.
Assuming that the state variables describing the instantaneous investment opportunities
follow a vector H of Markov diffusion process with drift µ H and volatility σ H vector components

dH = µ H dt + σ H dz H ,(13)with dz H denoting the standard Wiener process. The dynamics of the pricing kernel k suchthat E[d (kS )] = 0 , can also be described by a diffusion process that is a linear function of the statevariables dynamics

18

dk k = −r f ( H ) − ζ ( H )dz k ,(14)

where r f (H ) is the instantaneously riskless rate, ζ (H ) is the risk premium per unit of covariancewith the pricing kernel. Recognizing the asset volatility as σ S and the correlation with the pricing
kernel according to ρ Sk dt = dz S dz k , the expected returns can be expressed as8

E [dS S ] ≡ µ S dt = r f ( H )dt + ζ ( H ) ρ Sk σ S dt

(15)

When the pricing kernel is perfectly correlated with the market portfolio returns, the simple CAPM
ensues. The ICAPM derives from the expression of the pricing kernel innovations as a linear
function of market returns and stochastic changes in the opportunity set so that covariances with thestate variables also matter in the definition of systematic risk. Risk premia are thus determinedaccording to a multi-beta intertemporal capital asset-pricing model. As noted by Harvey and Whaley(1992), when implied volatility follows a random walk, its stochastic changes should be orthogonal
to any set of information variables, which includes innovations in the state variables under thepresent intertemporal setting.9 Whether fluctuations in implied volatility account for changes in thestochastic investment opportunity set is essentially an empirical issue. Consistent with theassumptions underlying the ICAPM, implied volatility can be expressed as a function of the vectorof stochastic state variablesv ≡ v(H )(16)

The vector of state variables includes in the present study, proxies for uncertainty in thedomestic real economy and the international financial economy. The leading diffusion index D ,which is compiled by the Economic and Social Research Institute in the Japanese Cabinet Officeaggregates in percentage terms, the directions of change in a selection of economic time series toanticipate turning points in the business cycle. It assesses the extent to which business fluctuations
spread across the economic sectors, generally approaching zero to signal recessions and 100 to
anticipate economic recoveries. As a measure of the dispersion of change across economic
indicators, it is widely believed to contain useful information on future structural changes in the real

19

economy. It averages a value of 50 (no turning points) over the sample period but as exhibited in
Figure 5, its behavior over time reflects cyclical patterns of anticipated fluctuations in businessconditions. Its decline in the early 1990s is consistent with the beginning of the prolonged economic
recession as it indicates the predominance of declining industries. Judging from the observed
patterns, the leading diffusion index is likely to be negatively correlated with estimates of implied
volatility.
The variations in the international financial investment opportunity set are, as noted by
Brennan, Wang and Xia (2001), not necessarily related to changes in the domestic output.
Uncertainty relates to unanticipated fluctuations in exchange rates, monetary and fiscal impulses,
and inflationary and deflationary pressures, among others. These additional sources of risk in theinternational financial economy are assessed hereafter using the volatility of the Morgan Stanley
Capital International world price index. An increase in the volatility of the international equity index
can signal investors’ perceptions of higher levels of uncertainty in international financial markets
that translates into an increase in implied volatility. The graphical evidence from Figure 5 suggeststhat implied volatility tends to move in tandem with the usually lower levels of MSCI world index
volatility.
To the extent that the relation between implied volatility in the Japanese options market
and the stochastic dynamics of these state variables is linear, the implied volatility function
expressed by equation (16) in an intertemporal setting is amenable to econometric estimationaccording to a two-factor model of risk components10v m = π 0 + π D Dm −1 + π M σ M ,m −1 + u m ,(17)where v is the ex ante estimate of implied volatility with one-month remaining to maturity, whereasu represents measurement and estimation errors. Whereas the implied volatility index is represented
by the point estimate at the beginning of expiration months, the standard deviation of MSCI worldindex is calculated using log returns until maturity. The test results of regression model (17) reported
in Table 4 Model I indicate that the estimates of uncertainty underlying the real and financial

20

economic conditions contain useful information on investors’ expectations of implied volatility. The
slope coefficients take the expected signs and are statistically significant, thereby confirming theevidence from the scatter diagrams and simple regression lines in Figure 5. There is evidence that
future levels of implied volatility are negatively related to the leading diffusion index level signaling
shifts in the prospects about the domestic real economy. Implied volatility is also found to increasesignificantly with rising levels of volatility in international equity markets.
There are however signs of serial correlation in the residuals suggesting that implied
volatility is not fully reflective of the dynamics of the stochastic investment opportunity set
described by these proxies of uncertainty in the real and financial economy. The estimation of an
autoregressive AR(1) model using nonlinear regression techniques implies Model II results, which
satisfy the stationarity condition as the inverted roots lie inside the unit circle. The expected signs
and statistical significance of the regression coefficients are confirmed. As a test of model stability,
the AR model is estimated with respect to two non-overlapping sample periods. There is evidence ofan increasing sensitivity of implied volatility to perceptions of uncertainty over the Japanese real
economic conditions. The prospects over the international financial economy continue to play asignificant role irrespective of sample periods. These results constitute evidence that expectations ofvolatility in options markets are not only function of perceptions of uncertainty about the domestic
economy but over the international financial economy as well.

8. Conclusions

This study made an attempt to examine the relationship between market volatility and its estimates
implicit in options prices and provides new evidence on many issues. First, a new benchmark of
implied volatility in the Japanese market was constructed following the methodology of the S&P
100 implied volatility index. The Nikkei 225 implied volatility index allows for the comparison of
the level of implied volatility across international markets. There is a tendency for closer alignmentof implied volatility levels over the late 1990s despite sharp contrasts between the booming US

21economy and slumping Japanese economy. Second, the new evidence on the stochastic dynamics ofimplied volatility sheds lights on options investors’ formation of expectations about long-term levels
and short-term movements in stock market volatility. Convergence towards the time-varying
long-run volatility component occurs at slow rates in both the US and Japanese markets. But the
dynamics of short-term convergence of implied volatility’s transitory component towards zero are
inherently different across markets as it tends to occur in the US market at faster rates than in theJapanese market.
Third, it is shown that using implied volatility in modeling the asymmetric behavior andclustering properties of market volatility implies significant improvements in model estimation.
Implied volatility adds useful insights into the leverage effects as its inclusion into the conditionalmodels of volatility explains part of the impact of bad news in Japanese markets and the entire
impact in US markets. Fourth, there is evidence that though upwardly biased, ex ante implied
volatility estimates contain useful information on future levels of realized volatility in the Japanesemarket. Fifth, the results indicate that volatility expectations in the Japanese options markets arefunction of uncertainty about the domestic real economy and international financial economy.
Implied volatility is found to be an increasing function of the level of volatility in international
equity markets and to be sensitive to potential shifts in the business cycle signaled by economic
indicators of dispersion of changes such as the leading diffusion index. These results add aninternational perspective to the limited literature on the informational content of implied volatility in
US markets. Whereas investors’ beliefs about future volatility are not necessarily based solely on theprice history, further analyses of the importance of market sentiment, and the impact of
non-quantifiable news on changing patterns of implied volatility are thus warranted.

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24

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25

TablesTable 1. Summary statistics and unit root test results
Mean
Panel A
Nikkei 225 Price

Descriptive Statistics
Median Std. Dev. Skewness Kurtosis

Unit root tests
ADF
Lag LB(1)statistic order20214.40 18975.10 5451.99

Daily observations
1.54905.1406 -2.156b

2- Return-0.0003-0.0004

0.01480.34946.8814 -41.531c

1- Volatility index

0.27180.26330.09070.57504.1256-6.548b2

S&P 100 Price359.08255.65209.740.83542.2288-1.917a3- Return

0.00050.00030.0101-0.37118.0955 -39.485c

1- Volatility index

0.19730.18630.06370.83003.7624-4.167a7

Panel B
VIX -30-day Forward

0.27540.26540.0919Monthly expirations
0.67243.6646 -4.405b

3

0.27220.26310.08330.24392.9852-4.102b0

Std. Dev. - Nikkei Spot

0.2166

0.19500.09491.21505.1522-6.902b0- MSCI world index

0.11230.10030.0482

1.24934.8342-4.811b13.77783.95510.6140-2.2172

12.0367 -4.270b

1- Average

Diffusion index

LB(k)0.0055(0.941)0.0000(0.995)0.0123(0.912)0.0019(0.965)0.0288(0.865)0.0007(0.978)0.6755(0.984)0.9392(0.967)6.458(0.264)1.4425(0.920)5.0802(0.406)0.0401(1.000)0.0198(0.888)0.8936(0.344)0.0476(0.827)0.0174(0.895)0.0005(0.981)

5.9433(0.919)6.9319(0.862)4.0825(0.982)
13.091(0.362)
11.217(0.510)Notes: The number of daily and monthly observations is 2953 and 144, respectively. ADF statistics refer tothe augmented Dickey-Fuller results for unit root tests. The appropriate lag order is determined using
Schwarz information criterion and additional lags are included to eliminate ARCH effects in theresiduals. For level tests of daily series, MacKinnon 1% critical values for rejection of the unit root
hypothesis are (a) –3.967 for tests with intercept and trend terms, (b) –3.436 for tests with intercept, and(c)–2.567 for tests with neither term. For level tests of monthly series with intercept, MacKinnon 1, 5

and 10% critical values for rejection of the unit root hypothesis are 3.477, -2.882 and –2.577,
respectively. LB(k) refers to Ljung-Box test of serial correlation in the residuals. LM(k) is the
Lagrange-multiplier test of heteroskedasticity up to order k. The tests are distributed as χ2(k) on thenull, with k equal to 5 and 12 for daily and monthly series, respectively. Statistics between brackets
represent probability values. For monthly expirations, standard deviations of returns on the Nikkei 225
spot and MSCI world price index are annualized statistics. The leading diffusion index is expressed inlogarithm.

26

Panel A

Table 2. Stochastic dynamics of implied volatility and leverage effects
Component GARCH modeling of implied volatilityγϖ
LB(10) LM(10) Adj. R2
δ
ϑ
ϑ
α
β
0

1

1

1

Nikkei 225 0.003 0.800 0.015 0.998 0.214 -0.030 -0.857 8.163 0.311
0.863(0.047) (0.000) (0.935) (0.000) (0.000) (0.161) (0.000) (0.613) (0.979)
S&P 100 0.003 0.867 0.000 0.978 0.053 0.142 0.687 7.745 0.093
0.954(0.000) (0.000) (0.003) (0.000) (0.209) (0.019) (0.000) (0.654) (1.000)
Panel B
Asymmetric GARCH modeling of return volatility
TARCH
LB(12) LM(12) LogL
ξ
µ0
υ0
α1
α2
β1
γ1
modelsModel I
Nikkei 225 0.000 0.000 0.023
0.901 0.134
8.779 0.395 8579.09(0.388) (0.000) (0.057)(0.000) (0.000)(0.722) (0.966)
S&P 100 0.000 0.000 -0.053 0.064 0.905 0.129
19.309 0.602 9729.22(0.027) (0.000) (0.007) (0.006) (0.000) (0.000)(0.081) (0.842)
Model II
Nikkei 225 0.000 0.000 -0.037
0.334 0.165 0.370 8.357 0.966 8668.31(0.962) (0.239) (0.000)(0.077) (0.000) (0.001) (0.757) (0.479)
S&P 100a 0.001 0.000 -0.029 0.039 -0.359 0.056 0.808 19.636 0.816 9906.74(0.000) (0.001) (0.121) (0.068) (0.183) (0.123) (0.000) (0.074) (0.634)
EGARCH
LB(12) LM(12) LogL
ξ
α1
β1
γ1
µ0
υ0
modelsModel I
Nikkei 225 0.000 -0.318 0.153
0.977 -0.099
9.453 0.471 8594.71(0.263) (0.000) (0.000)(0.000) (0.000)(0.664) (0.933)
S&P 100 0.001 -0.282 0.116
0.979 -0.084
19.188 0.539 9738.41(0.022) (0.000) (0.000)(0.000) (0.000)(0.084) (0.890)
Model II
Nikkei 225 0.000 -0.432 0.133
0.797 -0.146 0.174 9.895 0.809 8630.49(0.367) (0.005) (0.000)(0.000) (0.000) (0.070) (0.625) (0.641)
S&P 100b 0.001 2.808 -0.063-0.523 -0.028 1.954 20.128 0.736 9924.12(0.000) (0.000) (0.153)(0.042) (0.471) (0.000) (0.065) (0.717)
Notes: Component GARCH(1,1) model estimation using Bollerslev-Wooldrige robust standard errors andcovariance with lag order (7) determined using Schwartz information criterion. Model I refers toequations (9a) and (9b) which do not include implied volatility as independent variable in theconditional variance of TARCH ad EGARCH models, respectively. Model II includes implied
volatility as regressor in the conditional variance equations (10a) and (10b). Model estimation using
maximum likelihood ARCH (Marquardt) method and Bollerslev-Wooldridge robust standard errors &covariance. Model selection is based on the Schwartz information criterion and additional lags
included in the mean equation to eliminate ARCH effects. a (b) Lag order of three (four) in the meanequation. LB(k) refers to Ljung-Box test of serial correlation in the residuals. LM(k) is the
Lagrange-multiplier test of heteroskedasticity up to order k. The tests are distributed as χ2(k) on the
null. LogL refers to Log Likelihood statistics.

27

Table 3. Realized volatility and the forecasting ability of the implied volatility index
LB(12) LM(12) Adj. R2
λ
λ
λ
λ
0

v

v, h

S

Model I

0.032
0.670
14.944 0.723
0.416(0.094) (0.000)(0.245) (0.727)
Model II
0.054
0.630
-0.17912.018 0.713
0.427(0.011) (0.000)(0.025)(0.444) (0.735)
Model III
0.038
0.671
-0.151-7.881 15.104 0.989
0.491(0.059) (0.000)(0.044)(0.000) (0.236) (0.463)- Sub-period A
0.054
0.606
-0.169-9.334 9.646
0.769
0.480
1989/06~1995/06 (0.051) (0.000)(0.098)(0.002) (0.647) (0.678)- Sub-period B-0.015 0.858-0.061-7.208 9.725
0.670
0.515
1995/07~2001/06 (0.648) (0.000)(0.612)(0.002) (0.640) (0.771)
Notes: Number of monthly observations is 144. Estimation of the regression model (12),where λ v , λ v, h and λ S denote the coefficient estimates associated with the independent
variables v m , v m −1 − σ m −1 and rm , respectively. LB(k) refers to Ljung-Box test of serialhcorrelation in the residuals. LM(k) is the Lagrange-multiplier test of heteroskedasticity up
to order k. The tests are distributed as χ2(k) on the null.
Table 4. Structural analysis of implied volatility2πD
π M AR(1) Inverted LB(12) LM(12) Adj. R
π0
AR Roots
Model I
0.310 -0.037 0.947
72.815 0.887 0.355(0.000) (0.000) (0.000)(0.000) (0.562)
Model II
0.293 -0.026 0.743 0.519
0.52
10.358 1.490 0.529(0.000) (0.014) (0.000) (0.000)(0.499) (0.138)- Sub-period A
0.231 -0.017 1.040 0.547
0.55
9.812 1.187 0.571
1989/06~1995/06 (0.000) (0.236) (0.000) (0.000)(0.547) (0.321)- Sub-period B
0.395 -0.045 0.507 0.396
0.40
16.116 0.935 0.477
1995/07~2001/06 (0.000) (0.003) (0.001) (0.001)(0.137) (0.521)

Notes: Estimation of the regression model (17) where π D and π M represent the slope coefficientsassociated with past levels of the diffusion index, and of the standard deviation of MSCI
world index, respectively. LB(k) refers to Ljung-Box test of serial correlation in theresiduals. LM(k) is the Lagrange-multiplier test of heteroskedasticity up to order k.
Probability values reported in brackets. For AR models, the regression statistics are

based on the one-period ahead forecast errors.

28

Figures
.8

45000

.7

40000

.6

35000

__ Nikkei 225 VIX

.5

30000

.4

25000

.3

20000

.2

15000

.1

10000

S&P 100 VIX _|

.0

1990

1989

1991

1992

Nikkei 225 stock index

Annual impliedvolatility

Figure 1. The behavior of Nikkei 225 spot and implied volatility index in Japanese and US markets

Nikkei 225 spot
1993

1994

1995

1996

1997

1998

1999

2000 2001

5000

Figure 2. The time series of daily implied volatility index and return on Nikkei 225 stock index
.15
.10
.05
.00

Spot return

Return.05

Impliedvolatility

-.05
.04
-.10
.03

Volatility

.02
.01
.00

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

Figure 3. Asymmetric news impact functions from conventional and implied volatility-based TARCH
and EGARCH modeling of Nikkei 225 and S&P 100 stock index returns
Modeling without
implied volatility.0010

Nikkei 225 conditional variance.0005
S&P 100 volatility

.012

.0000

|_____ Modeling using|
implied volatility

.010
.008
.006
.004
.002
.000

Modeling without
implied volatility
-15

-10

Nikkei 225 volatility
-5

0

5

News term (standardized)

10

15

S&P 100 conditional variance

.0015

29

Figure 4. The behavior of realized and implied volatilities over monthly Nikkei 225 options expirations.7.6

Annualvolatilityestimates

Implied volatility
.5

.4

.3

.2

.1

.0
1989

Realized volatility

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

Monthly options maturity

Figure 5. The implied volatility in Japanese options market and its relation with the diffusion index andthe volatility of MSCI world price index.5Implied volatility
.4
.3
.2
.1

Diffusionindex

100

.0

MSCI world volatility

80

Implied and MSCI world volatilities.6

Diffusion index

60
40
20
0

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

.6

.6

.5

.5

.4

.4

Implied volatility

Implied volatility

Monthly expirations

.3

.2

.1

.3

.2

.1

VIX regreesion on log diffusion index.0

VIX regreesion on MSCI World volatility

.0

0

20

40

60

Diffusion index

80

100

.0

.1

.2

MSCI World standard deviation

.3

30

Footnotes
1

Nikkei 225 stock average futures are traded on the Singapore Exchange Derivatives Trading Division(formerly, on the Singapore International Monetary Exchange) since September 3, 1986, on Osaka
Securities Exchange since September 3, 1988 and on Chicago Mercantile Exchange together with futures
options since September 25, 1990.
2

In the absence of closed-form solutions, simple approximations of implied volatility values from option

premium can be made as in Corrado and Miller (1996) and Chambers and Nawalkha (2001), inter alia.
3This brief review of recent literature focuses on studies in implied volatility. A broader survey ofresearch in option pricing is thus beyond its scope but reference is made to useful reviews of tests ofoption pricing models by Bates (1996) and trends in options research by Chance (1999).
4

Additional empirical evidence by Jorion (1995) from currency options markets is indicative of higher

forecasting ability of implied volatility over GARCH conditional volatility.
5

These weighting schemes include inter alia, the equal weighting, the elasticity-based approach and theweighting on the basis of the value of vega or partial derivative with respect to volatility. The VIX
methodology is closer to the latter scheme as it relies on at- and near–the-money options, which tend also
to be the most liquid options.
6

The expression of implied volatility in trading days rather than calendar days is consistent with

empirical evidence in Davidson, Kim, Ors and Szakmary (2001) that volatility in financial markets ismore directly related to trading days for a wide variety of underlying assets.
7

The approach is based on the tacit assumption that investors’ expectations about volatility do not change

significantly during trading sessions. Its implementation results in as few as 5 and 4 cases, where theopening and highest prices are used for call and put options, respectively.
8

For further discussion on the mathematics and applications of the ICAPM, see Merton (1973b) and

Breeden (1979), inter alia.
9

The intertemporal approach is consistent with Black-Scholes option pricing, as the replication portfolio

remains riskless over a short period of time when the option and underlying asset are subject to the same
sources of uncertainty.
10

The sensitivity of implied volatility to risk factors can be also examined within the theoreticalframework of the arbitrage pricing model as in the study by Franks and Schwartz (1991) who regress
average implied volatility on economic variables such as real interest rates, oil prices and exchange rates.