Discussion Papers In Economics
And Business

ARE VOLATILITY EXPECTATIONS CHARACTERIZED BY REGIME SHIFTS?
EVIDENCE FROM IMPLIED VOLATILITY INDICES

Kazuhiko Nishina
Nabil Maghrebi
Mark J. Holmes

Discussion Paper 06-20

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANARE VOLATILITY EXPECTATIONS CHARACTERIZED BY REGIME SHIFTS?
EVIDENCE FROM IMPLIED VOLATILITY INDICES

Kazuhiko Nishina
Nabil Maghrebi
Mark J. Holmes

Discussion Paper 06-20

July 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ARE VOLATILITY EXPECTATIONS CHARACTERIZED BY REGIME SHIFTS?
EVIDENCE FROM IMPLIED VOLATILITY INDICES*

Kazuhiko Nishina
Osaka University, Japan
Nabil Maghrebi **Wakayama University, JapanMark J. Holmes
Waikato University, New Zealand

ABSTRACTThis paper examines nonlinearities in the dynamics of volatility
expectations using benchmarks of implied volatility for the US andJapanese markets. The evidence from Markov regime-switching models
suggests that volatility expectations are likely to be governed by regimes
featuring a long memory process and significant leverage effects. Market
volatility is expected to increase in bear periods and decrease in bull
periods. Leverage effects constitute thus an important source of
nonlinearities in volatility expectations. There is no evidence of longswings associated with financial crises, which do not have the potential ofshifting volatility expectations from one regime to another for longprotracted periods.

JEL Classification: C32, C51, G13, G15
Keywords: Markov Regime Switching, Implied Volatility Index, Nonlinear Modelling.

*

Acknowledgement: The authors acknowledge with thanks the Grants-in-Aid for Scientific
Research No. 18330069 by the Japan Society for the Promotion of Science and theprovision of options database by Osaka Securities Exchange.
** Corresponding author: Nabil Maghrebi, Graduate School of Economics, Wakayama
University, Sakaedani 930 Wakayama 640-8510, Japan. Tel: +81-73-457-7658, Fax:
+81-73-457-7659. Email: nebilmg@eco.wakayama-u.ac.jp

1. INTRODUCTION
The behavior of asset prices and their reaction to economic information was theimpetus behind the development of a rich literature on market efficiency. But, changes in
speculative prices do not only provide evidence on the validity of equilibrium models ofasset pricing and have implications for investment management and risk hedging. Market
volatility has also strong bearing for financial regulation, monetary policymaking andinternational market integration as well, given the growing evidence of volatility spillovers
across markets and countries, particularly during financial crises. Forecasting market
volatility is however a difficult exercise in quantifying uncertainty, which further gains incomplexity when perceptions of periodic economic reports and theoretical relationships bymarket participants are not consistent over time.
Arguably, anticipations of market volatility may shed light on how new
information interacts with investors’ beliefs to produce changes in asset prices. As such, exante measures of volatility can be reflective of investors’ perception of market risk.
Traditionally, the time-series of historical returns allows for the modelling of market
volatility in order to capture important features such as shock persistence, volatility
clustering and leverage effects reflected by the asymmetric impact of news. The present
study focuses rather on ex ante measures of short-term market volatility implied by stockindex option prices. The empirical analysis is aimed at providing evidence on mean
reversion and regime shifts in implied volatility dynamics, and thus investors’ anticipationsof price fluctuations.
The importance of these empirical issues is evident for instance, in the explicit
reference to the implied functions from options markets in the Bank of England’s monetary
policy meetings. This growing interest from policymakers is consistent with evidence frommany recent studies such as Carr and Wu (2006), who suggest that the S&P 500 new implied1

volatility index is reflective of the increasing uncertainty that precedes monetary policy
decisions about the Fed Fund Target rate. Similar evidence is obtained from alternative
markets including the results by Neely (2005) which indicate that changes in implied
volatility from Eurodollar interest rates coincide with major events related to monetary
policy decisions, real economy and equity markets. Also, Fornari (2004) suggests that
implied volatility from swaptions markets is reflective of market participants’ reaction tomacroeconomic announcements about the release of US economic indicators.
This close connection between monetary policy decisions and changes in implied
volatility adds to the early evidence on the informational content of implied volatility, whichis rather mixed. Indeed, the empirical studies by Day and Lewis (1992) based on S&P 100
index options, and by Lamoureux and Lastrapes (1993) based on individual stock options,suggest that implied volatility is a biased and inefficient estimate of market volatility.
Canina and Figlewski (1993) provide also evidence of insignificant correlation between
S&P 100 implied volatility and future market volatility. However, other empirical results are
more supportive of the informational content of implied volatility. For instance, Harvey andWhaley (1992) suggest that implied volatility provides an efficient forecast of future
volatility. The empirical evidence from Fleming (1998) also suggests that despite theupward bias, the forecast errors associated with S&P 100 implied volatility are orthogonal
to parameters included in ARCH models. Moreover, Christensen and Prabhala (1998) findthat the forecasting performance of implied volatility is higher than historical volatility.
Furthermore, Blair, Poon and Taylor (2001) find that the S&P 100 implied volatility index ismore accurate for out-of-sample forecasting than realized volatility, irrespective of data
frequency and the forecasting horizon. The empirical tests by Becker, Clemens and White(2006) suggest that the S&P 500 implied volatility index is efficient with respect to some but
not all the information set available for forecasting purposes. More recently, there is2

evidence from Nishina, Maghrebi and Kim (2006) of higher out-of sample forecasting
power for the S&P 500 and Nikkei 225 implied volatility indices.1
The informational content of implied volatility has been traditionally examinedusing conventional regression analysis and GARCH modelling. The present study examines
the nonlinear dynamics of the relationship between implied volatility and realized volatility
using Markov regime-switching models introduced by Hamilton (1989). The economic
motivation for modelling the dynamics of implied volatility with regime-switchingprocesses lies in the changing likelihood for market volatility to fluctuate between regimesof higher versus lower volatility and/or slower against faster mean reversion. Much like thereturn-generating process, the dynamics of anticipated volatility can be also governed bydifferent regimes characteristic of periods of bearish or bullish markets and associated withthe troughs and peaks of economic cycles.
In contrast, the literature on implied volatility provides little empirical evidence
based on regime-switching dynamics. This study constitutes, to the best knowledge of theauthors, the first attempt to test for the existence of Markov regime switches in anticipationsof stock market volatility. 2 The regime switching approach is rather widely applied toaccount for structural breaks in a range of financial variables including interest rates, equity
1

There is evidence from Jorion (1995) for instance that implied volatility provides efficientestimates of future volatility in currency markets as well.

2

Guo and Wohar (2006) identify regime changes in stock market volatility implied by S&P
100 and S&P 500 options prices using Bai and Perron (1998) method to determine
structural breaks in the mean level of market volatility. Their approach differs howeverfrom Markov regime-switching modelling proposed by Hamilton (1989), which is appliedin the present paper. Furthermore, this model allows for regime-dependent speed ofadjustment and tests for leverage effects and the impact of past realized volatility.
3and foreign exchange markets. Using a bivariate regime-switching model, Hamilton and
Lin (1996) find evidence of higher stock market volatility during periods of economic
recession. Also, Engel and Hakkio (1996) provide evidence based on regime-switchingmodels that periods of high volatility in the bilateral exchange rates in the European
Monetary System are associated with speculative attacks and subsequent realignment.
Dahlquist and Gray (2000) investigate also the effect of currency target zones in the EMS on
mean reversion and speed of adjustment of short-term interest rates.
The numerical derivation of volatility implied by option prices faces however
impediments stemming from measurement errors, as well as theoretical difficulties
associated with the option pricing model. There is indeed evidence of volatility smiles andsmirks, where different estimates of Black-Scholes implied volatility are derived for options
with different exercise prices and same maturity. This empirical evidence is inconsistent
with the assumption of constant volatility underlying the option pricing theory byBlack-Scholes (1973).3 These difficulties were conducive to the development of a richliterature of model-free approaches to the estimation of implied volatility. These include
inter alia, the methodology suggested by Ait-Sahalia and Lo (1998) based on polynomials
and splines smoothing, and by Britten-Jones and Neuberger (2000) based on the adjustmentof the volatility process to option prices in the same way that interest rates are fitted to bondprices.
3

The need to reconcile theoretical assumptions with empirical observations was partly theimpetus behind the development of alternative option valuation models including Hulland White (1987) and Heston (1993), where the volatility parameter is substituted by theentire joint probability distribution of returns and volatility changes. More recent studies
by Duan, Gauthier and Simonato (1999) and Ritchken and Trevor (1999) examine option
pricing under GARCH processes.
4Thus, in order to reduce measurement errors and avoid problems of numericalconvergence, this study uses two indices of implied volatility, namely the new VIX index on
the S&P 500 benchmark disseminated by the Chicago Board of Options Exchange and a
new index similarly computed from Nikkei 225 stock average options traded on Osaka
Securities Exchange. The model-free approach used in estimating these implied volatility
indices takes into account the term structure of implied volatility and possible nonlinearities
such as the volatility smiles. It aggregates information across different options maturities
and exercise prices and expectations about future volatility across market participants
including hedgers, arbitrageurs and speculators.
This study differs from previous studies on several accounts. It provides new
evidence on regime shifts in expectations of stock market volatility implied by options
prices. It allows for the identification of regimes characterized by expectations of lower
volatility and bursts of turbulence, the speed of mean reversion and the length of thememory process. Furthermore, it tests for the existence of leverage effects in options
markets in the sense that good and bad news from the stock market exert asymmetric effects
on expectations of short-term volatility. It also addresses the important question of whether
the regimes of expected volatility are reflective of a feedback process through whichanticipations of future market volatility adapt to changes in realized volatility. Thus, newevidence from regime switching models is provided on the relationship between implied
volatility and realized volatility, which is examined using conventional regression analysis.
Furthermore, it provides empirical results from an international perspective as it
uses implied volatility indices for the US and Japanese stock markets. In the absence of
implied volatility benchmark readily available for the Japanese equity market, this study
uses the Nikkei 225 implied volatility index reported in Nishina, Maghrebi and Kim (2006).
The construction of an implied volatility index for the Japanese equity market is important5

since the Nikkei 225 index constitutes the underlying asset of financial derivatives traded on
two other major Asia Pacific derivatives markets, namely the Chicago Mercantile Exchange,
and Singapore Exchange Derivatives Trading Division. The regime-switching models are
estimated over a sample period that covers important events such as the Asian financial
crisis, the Russian debt crisis, the Long Term Credit Management crisis, the burst of theinformation technology bubble, and the Japanese economic recession, among others.
The remainder of the paper is structured as follows. The next section describes theMarkov regime-switching models used to examine the nonlinearities in implied volatility
dynamics. Section 3 presents the sample data and distributional properties of implied
volatility indices. Section 4 discusses the empirical results for the Japanese and US markets.
The regime-switching models include various conditioning variables and are estimatedusing both the levels and first differences in implied volatility. Section 5 concludes thepaper.

2. REGIME-SWITCHING MODELLING OF IMPLIED VOLATILITY
In the absence of perfect knowledge of when structural breaks in implied volatility
can take place, regime shifts are incorporated in the volatility-generating process following
the Markov regime-switching model by Hamilton (1989). The two-regime Markov processused in the present study accounts for latent states of the relationship between implied
volatility and the set of past information. The sign and significance of model parameterswhich describe the dynamics of implied volatility are driven by discrete switches in theindicator variable. This unobservable variable takes the value of z t = i, for i = 1, 2 , anddetermines which regime governs the volatility dynamics at time t . With each observation
being drawn from a distribution conditional on the prevailing regime, the model parameters
are likely to differ in sign and/or magnitude across regimes. These regime-dependent
6

parameters allow for the characterization of each state with various features such as higher
or lower volatility, slower or faster reversion to the long-run mean, strong or insignificant
leverage effects.
The regime-switching models described below the behavior of implied volatility functionof past returns and other conditioning variables. The following model (1) tests for regimesof volatility expectations depending on the long-run mean and leverage effects
vt = wi + β i rt −1 + ζ t(1)

where the error terms are distributed as ζ t ~ i.i.d . N (0, σ ζ2 ) . This model defines regimes interms of higher and lower implied volatility judging from the magnitude of drifts and thesign and significance of slope coefficients. A negative coefficient suggests that the implied
volatility index tends to increase during bearish markets relationship, providing a measure
following Whaley (2000) of investors’ fear and anxiety. It is also possible to test for mean
reversion in expectations of future volatility by accounting for information contained in pastobservations of implied volatility.
vt = wi + δ i vt −1 + β i rt −1 + ζ t(2)

According to model equation (2), the dynamics of implied volatility can be driven bypast levels of implied volatility as well as past returns. An extension of model (1) to include
the squared returns provides a more appropriate test for asymmetric effects of news on
implied volatility as model (3) accounts for both the sign and magnitude of shocks to thereturn-generating process.

vt = wi + β i rt −1 + γ i rt 2−1 + ζ t(3)

It is also possible to integrate models (2) Model (3) to allow for long-run mean
reversion in implied volatility as well as leverage effects. Model (4) allows for the presenceof nonlinearities in the relationship between expected volatility and market returns across7

regimes and examines the issue of whether the presence of leverage effects affects thesignificance of mean reversion and/or the length of the memory process.

vt = wi + δ i vt −1 + β i rt −1 + γ i rt 2−1 + ζ t(4)

The model parameters are assumed to evolve according to a first-order Markov
process, which is not path-dependent in the sense that the current regime z t depends only on
regime z t −1 prevailing over the preceding period. This process is governed by the following
transition probability conditional on past information.
Prob( z t = 1 | z t −1 = 1) = p11(5-1)

Prob( z t = 2 | z t −1 = 1) = p12(5-2)

Prob( z t = 1 | z t −1 = 2) = p 21

(5-3)

Prob( z t = 2 | z t −1 = 2) = p 22

(5-4)

The typical transition probability is denoted by pij = Prob( z t = j | z t −1 = i ) with ∑ p ij = 1 .

The probability pii in the transition matrix denotes the likelihood that implied volatility
remains in regime z = i given that the same regime prevailed at time t − 1 . The probabilityof a switch from regime z = i at time t − 1 to regime z = j at time t is pij = 1 − pii . The

Markov model allows for multiple switches between regimes and the dynamics of theseshifts depend on the conditional transition probabilities, with the average duration of a given
regime i expressed as (1 − pii ) −1 . The stochastic process z t can be shown to follow an
autoregressive processz t = (1 − p11 ) + π z t −1 + ς t(6)where π = p11 + p 22 − 1 and ς t denotes innovation terms which are assumed to be
uncorrelated with lagged values of the state variable z . In the model specifications (1) to (4),implied volatility is defined as a function of the history of stock market returns and realized
8

volatility. It is not assumed that the explanatory variables follow Markov regime-switchingprocesses. Under the assumption that the error terms ζ t in implied volatility models arenormally distributed conditional upon the history ℑt −1 , the cumulative density function
depends on the regime indicator z . Given the above specifications of implied volatility, thedensity function depends on the conditioning variables of past returns and realized volatility
as well. This conditional density of implied volatility can be obtained from the joint densityof implied volatility and state variable as follows.
f (vt | z t ; ℑt −1 , ϑ ) = ∑ f (vt | z t = i; ℑt −1 , ϑ ) ⋅ Prob( z t = i | ℑt −1 , ϑ )(7)where ϑ represents the vector of model parameters. The unknown parameters in the implied
volatility models are estimated using maximum likelihood. Hamilton (1990) shows that the
maximum likelihood estimates of the transition probabilities can be expressed aspˆ ij =

∑ Prob(z = j, z = i | ℑ ;ϑˆ )∑ Prob(z = i | ℑ ;ϑˆ )
t =2

t −1

t

t =2

t −1

T

(8)

T

where ϑ̂ denotes the maximum likelihood estimates of model parameters. This estimation
procedure is applied to each model specification for the levels and first differences in
implied volatility in the Japanese and US stock markets.

3. INDEX DESCRIPTION AND DISTRIBUTIONAL PROPERTIESThe empirical evidence on the dynamics of implied volatility is based on
regime-switching tests using the new implied volatility index disseminated by the CBOE
and a similar index for Nikkei 225 index options traded on Osaka Securities Exchange.
Whereas the new VIX index is available from CBOE database, this study uses the Nikkei
implied volatility index introduced in Nishina, Maghrebi and Kim (2006) to measure
volatility expectations in the Japanese market. The new VIX index gathers consensus
information on options market’s expectations about future stock market volatility, without
9

relying on any theoretical model of option pricing. The methodology provides anapproximate measure of stock market volatility from a hypothetical option with exercise
price equal to the futures price and with thirty days remaining to maturity. As such, the new
VIX index provides an estimation of a thirty-day return variance swap rate from a portfolioof options spanning the nearest two maturities. The contribution of each option to theimplied volatility index is an increasing (decreasing) function of the exercise price for put(call) options.4
The empirical analysis is based on the daily time-series of the implied volatility
indices for a sample period extending from January 1990 to December 2004, and spanning180 options maturities. Figure 1 describes the behavior of spot prices and implied volatility
indices for the U.S. and Japanese markets. There is a tendency for the S&P 500 index toincrease monotonously until the burst of the information technology bubble. This patterncontrasts with the tendency for Nikkei 225 index to decrease from its height in early 1990,
reflecting the persistent recession of the Japanese economy during the 1990s. Although
there appears a tendency for implied volatility across markets to converge in more recent
years, the Nikkei 225 implied volatility seems to remain typically higher than expectationsof US market volatility.
There are instances of sharp increases in implied volatility in both markets. The
occasional spikes in implied volatility tend to be associated with sharp decreases in stock
market prices. These events are seemingly associated with significant economic events such
4

The model-free methodology for the calculation of S&P 500 implied volatility index isthoroughly explained in CBOE documentation. It follows the original VIX index based on
S&P100 American options calculated using Black-Scholes pricing model. The rationale
underlying the calculation of these indices and their major differences are discussed in
Carr and Wu (2006).
10

as the onset of the Asian financial crisis in 1997, Russian debt default and LTCM crisis in
1998. The increase in implied volatility associated with the burst of the IT bubble in 2000 isless pronounced than that related to the Latin American debt crisis in 2002. These implied
volatility patterns may be indicative of a negative relationship between market volatility andreturns, which can be formally examined with Markov regime-switching modelling.
The distributional properties of stock returns and implied volatility are describedby Table 1. There is evidence that the Japanese market tends to be associated with lower
mean returns and higher volatility. The average implied volatility, as well as first differences,
is found to be higher than comparable statistics in the US market. Based on unit-root tests
following Augmented Dickey-Fuller and Phillipps-Perron tests, the time-series of stock
market returns and volatility are also found to be stationary in both markets.

4. EMPIRICAL EVIDENCE
4.1. REGIME SWITCHING AND THE LEVEL OF IMPLIED VOLATILITY

The estimation results of the various first-order Markov regime-switching models
with respect to implied volatility levels are reported in Table 2 and 3 for the US andJapanese markets, respectively. Model 1 allows for regime-dependency in the relationshipbetween expected volatility levels and stock market returns. There is evidence that the slope
coefficients in both regimes are negative for the US market, but insignificant for theJapanese market. Evidence of negative slope suggests that market volatility is expected toincrease in bear periods and decrease in bull periods. Given the acceptance of the null of
equal slopes for both markets, it is the magnitude of drifts which defines regimes of high andlow expected levels of volatility.
Model 2 expresses the relationship between past returns and volatility upon the
inclusion of autoregressive terms. There is a significant increase in the log-likelihood

11

function compared to the previous Model 1 for both markets. For the US market, tworegimes of implied volatility are identified, one with longer memory and negative
correlation with stock market returns (regime 1) and another with shorter memory andpositive correlation with returns (regime 2). Again, there is no evidence of a negative
relationship between implied volatility and stock market returns for the Japanese market.
The regime of low expected volatility is characterized by shorter memory and positive
correlation with returns.
Testing for the asymmetric impact of news on market volatility should take into
account both the sign and magnitude of returns. Thus, Model 3 describes the level of
implied volatility as a function of past return levels as well as squared innovations, while
excluding autoregressive terms. There is evidence of significant leverage effects since boththe sign and magnitude of returns are likely to affect volatility expectations. However, thenegative relationship with returns is not found to be regime-dependent in both markets since
the null of equal β parameters across regimes cannot be rejected. Given thepositive γ coefficients, shocks to the return-generating process, are irrespective of their sign,
conducive to expectations of higher volatility. Thus, the evidence of leverage effects isfound to be regime-dependent in both markets.
The estimation of Model 4, which is comparable to Model 3 but inclusive ofautoregressive terms, reveals the existence of two regimes featuring equal intercepts andpositive serial correlation for the US market. The two regimes differ only with respect to therelationship of volatility expectations with market returns. The presence of leverage effects
is manifested by the significance of both β and γ parameters, with the level of expectedvolatility in regime 1 being less responsive to variations in market returns. In contrast, theestimation results for the Japanese market suggest that drift terms and autoregressive
parameters are regime-dependent. Regime 1 is identified with expectations of low volatility,
12

long memory and significant leverage effects while regime 2 is associated with expectationsof high volatility, and significant mean reversion in the absence of leverage effects.
The inferred probabilities of regime 1 from Model 4 are reported in Figures 2 and 3
for the US and Japanese markets, respectively. These figures suggest that in the absence oflong swings, the process of volatility expectations tends to switch randomly and abruptly
between regimes. It is clear from Figure 2 that there is a stronger likelihood for volatility
expectations to be governed by regime 1, which features positive serial correlation andrelatively less significant leverage effects. There are more frequent regime shifts in S&P 500
implied volatility index, particularly over the period associated with the onset of the Russiandebt default and LTCM crises in 1998 until the Latin American debt crisis in 2002. The
higher frequency of regime switches may be reflective of the increased uncertainty
generated by financial crises. Such events have the potential of increasing the likelihood of
the alternative regime, typically characterized by stronger leverage effects.
With respect to the Japanese market, it appears from Figure 3 that regime 1 tends toprevail over the sample period, except for very short-lived switches to the alternative regime.
The predominant regime is characterized by expectations of low volatility levels, longer
memory, and significant leverage effects. Significant events such as the Asian financial
crisis seem to trigger shifts towards the volatility regime featuring expectations of higher
volatility, and significant mean reversion. Judging from the inferred probabilities reported
for both markets, it seems that financial crises are not associated with long swings in thesense that they do not have the potential of shifting volatility expectations from one regime
to another for long protracted periods.

4.2. REGIME SHIFTS AND CHANGES IN VOLATILITY EXPECTATIONS

The dynamics of volatility expectations are also estimated with respect to firstdifferences in implied volatility. While retaining the features of previous regime-switching13

models in terms of mean reversion and asymmetric impact of returns (leverage effects withrespect to changes in expected volatility), it is also possible to test for nonlinearities in therelationship between changes in implied volatility and the dynamics of realized volatility.
∆vt = wi + δ i ∆vt −1 + β i rt −1 + γ i rt 2−1 + ϕ i ∆σ r ,t −1 + ζ t(9)Given the definition of the new implied volatility index as the approximation ofvolatility implicit in a hypothetical option with thirty days remaining to maturity, realized
volatility is defined at time t as the ex post annualized measure of standard deviation of
returns until option expiration, from t + 1 to t + 30 . The sign and significance ofparameters ϕ in Model (9) across regimes can provide evidence on the adjustment processthat governs the formation of expectations about market volatility. It allows for theexamination of the important issue of whether implied volatility rises following a marginalincrease in realized volatility.
Tables 4 and 5 report the estimation results for Models (1) to (5) with respect tochanges in implied volatility for the US and Japanese markets, respectively. Judging fromthe LR test, the dynamics of expected volatility are better described according to Model 5
based on all conditioning variables, including past changes in realized volatility. The degreeof mean reversion does not differ across regimes in the US market. In the Japanese market
instead, it is the sensitivity to the magnitude, as opposed to the sign, of shocks in the
return-generating process that is hardly different across regimes.
There is at least, one regime of expectations for decreasing volatility in both markets(regime 2). Anticipations of decreasing volatility are characterized by the asymmetricimpact of news and positive relationship with changes in realized volatility. This evidence
suggests that marginal decreases in realized volatility are conducive to expectations oflower volatility. With respect to the Japanese market, this regime is also associated withslower mean reversion. The alternative regime featuring anticipations of increasing
14

volatility is characterized by faster mean reversion, symmetric impact of news and negative
relationship with past changes in realized volatility.
Judging from the inferred probabilities exhibited by Figure 4 for the US market,
regime 1 prevails with insignificant drift, significant mean reversion, weaker asymmetricimpact of news and weaker adjustment to the dynamics of realized volatility. Similar topatterns revealed by Figure 2, the frequency of regime shifts seems to increase since the mid
1990s, in response to more turbulent periods including the Russian debt default and LTCM
debt crises. The onset of financial crises increases the likelihood of the alternative regime of
expectations for significant decreases in market volatility, with slower mean reversion,stronger asymmetric reaction to news, and significant positive adjustment to the dynamics
of realized volatility.
As illustrated by Figure 5, the inferred probabilities for the Japanese market suggest
also that regime 2 is more likely to prevail with expectations of decreasing volatility, slower
mean reversion, asymmetric impact of news and significant positive adjustment to changes
in realized volatility. Compared to the US market results, there are less frequent switches
toward the alternative regime featuring expectations of increasing volatility, with stronger
mean reversion, significant though not asymmetric impact of news, and inverse adjustment
to changes in realized volatility. These regime changes are seemingly associated withsignificant events such as the Asian financial crisis and the burst of the Japanese bubble,
which heralded a decade-long period of mounting bad debts, deflationary pressures, andeconomic recession. Arguably, the results suggest that the worsening economic prospects
may have been conducive, particularly in the early 1990s, to expectations of increasing
volatility even in the presence of positive returns and marginal decreases in market
volatility.

155. CONCLUSION

This study examines nonlinearities in the dynamics of market volatility implied in
options prices in the Japanese and US markets. The analysis tests for regime switches involatility expectations using the CBOE new VIX index and a similarly computed index
from Nikkei 225 options prices, not available in economic databases. The characterizationof these regimes is based on a set of conditioning variables, which includes past returns andrealized volatility. The first-order Markov regime-switching models test also for theasymmetric impact of news and the presence of an adjustment mechanism through which
volatility expectations respond to the dynamics of realized volatility. The testing approachallows also for the examination of regime switches in volatility expectations in association
with financial crises.
The empirical evidence with respect to implied volatility levels suggests that theregime governing volatility expectations in the US market features a long memory process,and relatively less significant leverage effects. The prevailing regime in the Japanese market
is characterized by expectations of low volatility levels, longer memory and significant
leverage effects. These results indicate that market volatility is expected to be higher in bear
periods and lower in bull periods and that leverage effects constitute an important source of
nonlinearities in expectations of market volatility. Given the evidence of positive serial
correlation, the expected level of volatility in both markets is not likely to be so much
whittled down by mean reversion as by leverage effects.
There is also evidence from the first differences in implied volatility that theprevailing regime for the rate of change in expected volatility in the US market is likely to
be characterized by significant mean reversion, weaker asymmetric impact of news, andpositive, albeit less significant, adjustment to changes in realized volatility. The dynamics ofvolatility expectations in the Japanese market are likely to be driven by a regime of16

anticipated decreases in volatility, with slower mean reversion, asymmetric relationshipwith returns and positive adjustment to changes in realized volatility.
The onset of financial crises does not have the effect of shifting the implied
volatility process from one regime to another for long protracted periods. For the Japanesemarket however, major events such as the burst of the asset bubble and the Asian financial
crises seem to trigger the regime characterized by expectations of significant increments involatility, stronger mean reversion and negative correlation with changes in realized
volatility. The immediate reversal to the pre-crisis regime, featuring expectations ofdecreasing volatility and strong mean reversion, suggests that the impact of financial crises
on the dynamics of volatility expectations is short-lived. With respect to the US market, it
increases the likelihood of regimes characterized by expectations of decreasing volatility,
significant mean reversion, stronger asymmetric impact of news and more significant
adjustment process.
Evidence of regime shifts in volatility expectations has some implications forrisk-hedging, policymaking and future research in financial economics. It offers new
avenues for research on such important issues as to whether market consensus expectations
are consistent with rational expectations and whether financial crises may be induced byregimes consistent with self-fulfilling expectations and speculative bubbles. Empiricalstudies may also shed light on the relationship between the leverage effects and the speed ofmean reversion or length of the memory process across regimes. Future research inbehavioral finance can benefit from tests of the stochastic properties of investors’ attitudes
towards risk, regime-dependencies in investor confidence and irrational exuberance. From
the risk-hedging perspective, regime shifts imply nonlinear serial dependence in volatility
expectations. It is thus important to examine to what extent the quantification of risk
exposure is affected, the estimation of dynamic hedge ratios is further complicated, and the17composition of asset portfolios is altered.
Finally, the empirical results have important implications for policymaking
market regulation. Given the evidence that the release of new information is usuallyfollowing by rapid changes in asset prices and significant increases in trading activity, theregime-dependencies in market expectations can have some bearing on the scheduledannouncement times of economic reports. It is thus interesting to examine the issue of theannouncement of monetary policy shifts or changes in margin lending regulation are
preceded or followed with regime switches in volatility expectations.

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20

TABLESTABLE 1. Distributional moments and unit-root test results
Mean

Std.Dev. Skewness Kurtosis

JB

ADF

PP

S&P 500

0.03100.0102-0.10316.89862485.06 -62.665c -62.877c

Nikkei 225-0.03120.01480.19616.35301858.13 -46.979c -63.862c

S&P 500

0.19880.06350.90633.7215620.70-3.863b -5.379a

Nikkei 225

0.2476

0.06470.65284.1171481.53-6.143b -6.907 b

Changes in
Implied Volatility
S&P 500-0.00100.01220.56909.26276605.84 -22.530c -73.542c

Nikkei 225

0.00260.01650.645039.4066 216373.60 -42.551c -84.201c

Stock Returns

Implied Volatility

Notes: The sample period extends from January 2, 1990 to December 31, 2004. JB refers
to Jarque-Bera statistics for normality tests. ADF refers to Augmented
Dickey-Fuller tests using Schwarz information criterion.

PP refers toPhillips-Perron tests with Newey-West bandwidth using Bartlett kernel. The
superscripts a, b and c refer to unit root tests with trend and intercept, with intercept
only, with neither trend nor intercept, respectively. The means of return series andvolatility first differences are scaled by 102.

21

Table 2. Regime-switching modelling of implied volatility(S&P 500 index)
Model 1Model 2Model 3Model 4Model parametersa0.2664-0.0129a0.2581a0.0033aw101612a0.0093a0.1561a0.0035aw2
1.1093a0.9810aδ10.9393a0.9796aδ2-0.5693a-0.2328a-0.6094a-0.6226aβ1-0.3991a
0.0632a-0.5627a-1.4508aβ240.4928a
7.0033aγ190.7546a
9.9678aγ2

ϕ1
ϕ2

Hypothesis tests
6074.104122.558a 5423.406a
585.366a
1.588
45.094a0.118
50.754a0.0377
w1 = w20.0798
δ1 = δ 2
1184.4911a
β1 = β 212.8244aγ1 = γ 2
ϕ1 = ϕ 2
2.022
50.034a2.626
71.5032ap11 = p 22
LL
7166.36
11997.22
7369.75
13491.68
Notes: Significance at the 1, 5 and 10 % level is denoted by a, b andacrespectively. The estimated Markov regime-switching models

arerepresentedbyvt = wi + β i rt −1 + ζ t forvt = wi + δ i vt −1 + β i rt −1 + ζ t

forvt = wi + β i rt −1 + γ i rt 2−1 + ζ t

for

Model

Model
Model

3,

1,
2,
and

vt = wi + δ i vt −1 + β i rt −1 + γ i rt 2−1 + ζ t for Model 4. The null

hypothesis tests are distributed as χ 2 (1) . LL is the log maximum
likelihood function.

22

Table 3. Regime-switching modelling of implied volatility(Nikkei 225 index)
Model 1Model 2Model 3Model 4Model parametersa0.31060.1522a0.3030a0.0048aw1
0.2080a0.0088a0.2057a0.3494aw2
0.7995a0.9747aδ1
0.9611a-0.0693aδ2-0.03390.5529a-0.1489a-0.3231a
β1
0.00440.0072-0.09910.8789aβ220.5833a
5.7630aγ111.6824a-24.6102a
γ2

ϕ1
ϕ2

Hypothesis tests
4590.3206 1764.6474a 3844.3931a 10957.0160a
225.7556a
9901.6532a
0.163985.26211a
0.3771430.3168a28.2025a432.8464a

w1 = w2
δ1 = δ 2
β1 = β 2γ1 = γ 2
ϕ1 = ϕ 24.5052b20.8045a
5.4429b175.5199a
p11 = p 22
LL6822.32
11007.45
6883.55
11386.28
Notes: Significance at the 1, 5 and 10 % level is denoted by a, b andacrespectively. The estimated Markov regime-switching models

arerepresented

by vt = wi + β i rt −1 + ζ t forvt = wi + δ i vt −1 + β i rt −1 + ζ t

forvt = wi + β i rt −1 + γ i rt 2−1 + ζ t

for

Model

Model
Model

3,

1,
2,
and

vt = wi + δ i vt −1 + β i rt −1 + γ i rt 2−1 + ζ t for Model 4. The null

hypothesis tests are distributed as χ 2 (1) . LL is the log maximum
likelihood function.

23

Table 4. Regime-switching modelling for changes in implied volatility(S&P 500 index)
Model 1Model 2Model 3Model 4Model 5Model Parametersa-0.0793-0.0788a-0.0080-0.1138a-0.0146w1aaa4.33114.3243-0.1302-0.0133-0.1106aw2-0.0437a-0.0358a-0.0723aδ1-0.1076b-0.0769a-0.0479a
δ2
0.0965a0.0592a-0.6216a-1.4887a-0.6243a
β10.20780.1097-1.5029a-0.6162a-1.5016aβ2
5.5455a8.9329a5.8918aγ19.21184a
5.7441a9.5776aγ20.0184c
ϕ10.0640a
ϕ2
Hypothesis Tests
912.1114a 898.6265a
8.6952a6.1134b5.3364bw1 = w2
1.9562
6.9142a2.3351δ1 = δ 2aa0.67350.11771306.9559 1364.9120
1298.9449a
β1 = β 219.6618a15.0836a
18.5454aγ1 = γ 2
5.2107bϕ1 = ϕ 2
16.6083a13.5024a87.9058a
79.8012a84.8625ap11 = p 22
LL
11893.48
11895.73
13453.9313470.09
13473.32
a bcNotes: Significance at the 1, 5 and 10 % level is denoted by , and respectively.

The estimated Markov regime-switching models are represented by∆vt = wi + β i rt −1 + ζ t for Model 1, ∆vt = wi + δ i ∆vt −1 + β i rt −1 + ζ t for
Model

2,

∆vt = wi + β i rt −1 + γ i rt 2−1 + ζ t∆vt = wi + δ i ∆vt −1 + β i rt −1 + γ i rt 2−1 + ζ t

for

for

Model

3,

Model

4,

and

∆vt = wi + δ i ∆vt −1 + β i rt −1 + γ i rt 2−1 + ϕ i ∆σ r ,t −1 + ζ t for Model 5. The model

parameters w and σ e are scaled by 102. The null hypothesis tests are
distributed as χ 2 (1) . LL refers to the log maximum likelihood function.

24

Table 5. Table 4. Regime-switching modelling for changes in implied volatility(Nikkei 225 index)
Model 1Model 2Model 3Model 4Model 5Model Parameters0.0047-0.3498a-0.1257a
1.2607a2.0495aw1-3.4197a-0.0030
6.9409a-0.1317a-0.1290aw2-0.6558a-1.4394a-1.7631aδ1-0.1233a-0.1352a-0.1145aδ2-0.0112-6.3270a-0.3312a4.5286a5.6178aβ1-7.2362a-0.0434a
6.8867a-0.3435a-0.3132aβ2
3.9584a61.5712a12.1322aγ1-164.8804a
4.9217a4.4534aγ2-2.7179a
ϕ10.0274a
ϕ2
Hypothesis Testsa1248.3519
14.4029a
5711.6400a
295.9666a
338.6405a
w1 = w2
318.7542a
2294.9900a 2630.6941a
δ1 = δ 29520.5489a 6397.2327a 13819.9404a 6756.7345a 7119.3876a
β1 = β 24639.8593a
523.5600a
3.1972cγ1 = γ 2
1574.2915a
ϕ1 = ϕ 2
2.074943.6718a
33.1734a
162.7422a
102.2636a
p11 = p 22
LL10855.9310930.50
11133.5411183.11
11209.71a bcNotes: Significance at the 1, 5 and 10 % level is denoted by , and respectively.

TheestimatedMarkovregime-switchingmodelsarerepresented

by ∆vt = wi + β i rt −1 + ζ t for Model 1, ∆vt = wi + δ i ∆vt −1 + β i rt −1 + ζ t for
Model

2,

∆vt = wi + β i rt −1 + γ i rt 2−1 + ζ t∆vt = wi + δ i ∆vt −1 + β i rt −1 + γ i rt 2−1 + ζ t

for

for

Model

3,

Model

4,

and

∆vt = wi + δ i ∆vt −1 + β i rt −1 + γ i rt 2−1 + ϕ i ∆σ r ,t −1 + ζ t for Model 5. The model

parameters w and σ e are scaled by 102. The null hypothesis tests are
distributed as χ 2 (1) . LL refers to the log maximum likelihood function.

25

FIGURESFIGURE 1. The behavior of stock prices benchmarks and implied volatility indices(Stock price indices rebased to 100 as of January 2, 1990).6
Nikkei 225
Implied volatility.4500

Stock price index

400
300
200

.2
S&P 500
Implied volatility.0

S&P 500
Price index

100

Nikkei 225
Price index

0
90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

26

Implied volatility index.8

FIGURE 2. Inferred probabilities of regime1 for S&P 500 implied volatility index(Model 4 of expected volatility levels)
1.0

0.8

0.6

0.4

0.2

0.0
1990

1992

1994

1996

1998

2000

2002

2004

FIGURE 3. Inferred probabilities of regime1 for Nikkei 225 implied volatility index(Model 4 of expected volatility levels)
1.0

0.8

0.6

0.4

0.2

0.0
1990

1992

1994

1996

27

1998

2000

2002

2004

FIGURE 4. Inferred probabilities of regime1 for S&P 500 implied volatility index(Model 5 of first differences in expected volatility)
1.0

0.8

0.6

0.4

0.2

0.0
1990

1992

1994

1996

1998

2000

2002

2004

FIGURE 5. Inferred probabilities of regime1 for Nikkei 225 implied volatility index(Model 5 of first differences in expected volatility)
1.0

0.8

0.6

0.4

0.2

0.0
1990

1992

1994

1996

28

1998

2000

2002

2004