GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 298

Model-free Implied Volatility Index of Japanese Stock Market

Nattapol TAKKABUTR

March 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Model-free Implied Volatility Index of Japanese Stock Market*

Nattapol TAKKABUTR†
March 2013

Abstract
This paper empirically studied the model-free implied volatility indices constructed from options
prices of the Nikkei 225 index during 2005-2010. The concept of corridor volatility index is comparedand contrasted with the methodology of the famous VIX index developed by the Chicago Board Options
Exchange (CBOE). The relative corridor widths are found to be able to explain relative variations
between volatility indices with different corridors’ widths. Also, the corridor volatility index is found tobe the better predictor of the future realized volatility of the underlying stock index.

JEL Classification Code: G01, G100, G130
Key words: Volatility Index, Crisis, Financial Crisis, Option, Option-implied*

I would like to thank Kosuke Oya, Graduate School of Economics, and Center for the Study of Finance and Insurance (CSFI),
Osaka University, for providing comments as well as supporting information regarding the VXJ index.
†

Graduate School of Economics, Osaka University, Japan, jgm042tn@mail2.econ.osaka-u.ac.jp

1. Introduction
A stock market volatility index is an index of stock market returns volatility. Returns
volatility of the stock market is a measure of its riskiness and uncertainty. In the simplest form,
the volatility of stock is a standard deviation of stock’s returns during the period of interest.
Data of stock’s volatility is used in portfolio construction and optimization and is one of theearliest notions learnt by any persons taking an introduction to finance course. It is also one of
the important parameters that financial risk managers must pay careful attention to in managing
their portfolio. Volatility calculated from historical data of stock’s returns is called realized
volatility. As most financial market participants may have learnt, history might not repeat itself.
It is more important to be able to calculate or predict volatility of stock’s returns in the future.
Chicago Board Options Exchange’s VIX Index
The most referred stock market volatility index is the Chicago Board Options Exchange’s
VIX index (CBOE’s VIX), which was first introduced in 1993. It is designed to measure themarket’s expectation of the next 30-days’ volatility as implied by the at-the-money S&P100
Index option price.1 The index is also known as “the Investor Fear Gauge” because high levelsof VIX are coincident with a high degree of market turmoil.2 However, because the returns’
volatilities used in the calculation of the index are derived from the Black-Scholes model, theindex is model-dependent and is subject to all the underlying assumptions of the Black-Scholesmodel. Though the VIX index became well-known and widely accepted as a benchmark for
U.S. stock market volatility, it has been criticized for its practicality from both academia andindustry practitioners.3
In 2003, the Chicago Board Options Exchange and Goldman Sachs revised the calculation
method of the index. The old VIX became VXO and the new one is currently the world’s mostreferenced stock market volatility index. The VIX index is now calculated based on S&P500

1
2
3

See The CBOE Volatility Index – VIX or VIX White Paper for more details.
Whaley (2000) explained in details regarding the origin of this name.
See more comparison of the old VIX and new VIX in Carr and Wu (2006).

-1-

Index option data. The interpretation of the index is unchanged; it measures the market’s
expectation of 30-day volatility implicit in the S&P500 index option.
The major improvement in the new VIX index is that it no longer depends on the BlackScholes model; in other words, it is now a model-free implied volatility index. Also, because
the calculation method of the index is applied from variance swap contracts, the VIX index is
replicable. 4 With the possibility of replication, the Chicago Board Options Exchange has
introduced both VIX Futures and VIX Options enabling not only institutional investors but also
retail investors to easily assess volatility exposure.
The new calculation method is so effective that it does not calculate only the volatility index
of the S&P500; volatility indexes of Dow Jones, NASDAQ100, and Russell 2000 are calculated
as well. Moreover, the Chicago Board Options Exchange calculates a volatility index of crudeoil, gold, and Euro currency. In addition, exchanges around the world adopt CBOE’s VIX
calculation method in constructing their own volatility indices. Andersen et al. (2011) provides
a summary and comparison of the calculation methods adopted by international exchanges.
Volatility Index Japan
Despite being one of the world’s leading capital markets, Japan lags behind the U.S. in
introducing a stock market volatility index. It was not until recent years that the Center for the
Study of Finance and Insurance of Osaka University (CSFI), with the cooperation of the Osaka
Securities Exchange, introduced the Volatility Index Japan (VXJ).
VXJ index is calculated from Nikkei 225 options prices. It reflects the market’s expectationof volatility of the Nikkei 225 over the coming 30 days as implicit in the option prices. The
calculation method is similar to the method of VIX index with differences in underlying options
data.
In addition, CSFI developed its own version of volatility index, called the CSFI-VXJ. The
methodology of CSFI-VXJ uses a spline function to interpolate the implied variances from4

For more details on volatility swaps contract, refer to Demeterfi et al. (1999).

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observed options prices. The new methodology aims to reduce the approximation error
associated with the methodology of CBOE’s VIX, detailed in Fukusawa et al. (2011). VXJ andCSFI-VXJ share many characteristics with its counterpart, the VIX index, such as the statistical
relationship between the volatility index and its underlying stock market index.5
However, unlike the VIX index, the VXJ and CSFI-VXJ indices are far from being thebenchmark volatility index of Japan’s stock market. They are constructed with academic
purposes in mind. Therefore, there are no derivative securities on VXJ or CSFI-VXJ indices. It
is possible that trading and hedging volatility in Japan is not well developed as the U.S. market
or industry practitioners still feel more comfortable undertaking volatility transactions in an
over-the-counter style.
Corridor Implied Volatility Index
Corridor implied volatility is a kind of model free implied volatility based on the industrypracticed corridor variance swap concept. A corridor variance swap contract is a conditionalvariance swap contract that accumulates variance only when the underlying asset’s prices fallwithin the predetermined range.6
Andersen and Bondarenko (2007) proposed that a volatility index constructed from a
corridor implied volatility concept is more accurate in reflecting the true realized variance,compared to the current method employed by CBOE. The main argument against the VIX
index is that the random truncation of options used in calculation creates inaccuracy. The
calculation concept of VIX index demands the use of options with strike prices ranging fromzero to infinity to support the whole range of return distribution, but in practice, only limited
and finite strikes of options are traded. CBOE uses random truncation in screening the option
data to be used in the calculation of the VIX index, which is a source of inaccuracy.

5

The Center for the Study of Finance and Insurance , Osaka University provided details of VXJ on its website, http://wwwcsfi.sigmath.es.osaka-u.ac.jp/en/activity/vxj/vxj.pdf
6
For more documents regarding Corridor Volatility, see Allen et al. (2006), Carr & Madan (1998), Lieberman & Omprakash(2007)-3-

Instead of random truncation, Andersen and Bondarenko (2007) constructed corridor
implied volatility indexes based on options on S&P500 futures. The corridor widths were
determined according to percentile of estimated risk-neutral density. They found that thevolatility indexes created under this method added more information in a realized volatility
prediction and produced lower forecasting root-mean-square errors when compared to otherbenchmark volatility measures: realized volatility itself, Black-Scholes implied volatility, and
VIX index.
Nevertheless, Andersen et al. (2011) revised the Andersen and Bondarenko (2007) method,and pointed out potential drawbacks in determining corridor widths based on percentile of riskneutral density (RND), which may distort the estimated volatility index. The revised method
based on the findings of Andersen and Bondarenko (2010) that option prices will reflect tailmoments of RND. The cut off strike price is a function of RND’s tail moments which can becalculated directly from option prices.
In this study, corridor volatility indices for the Japanese market estimated under the revised
method of Andersen et al. (2011) are compared to the volatility index calculated under CBOE’s
VIX method, the VXJ, and the CSFI-VXJ. In particular, the forecast performance of estimatedvolatility indices is compared against the realized volatility.

2. CBOE’s VIX methodology
In this section, the method of CBOE’s VIX methodology is discussed in detail. It will beapplied to options on the Nikkei 225 index to construct a volatility index for the Japanesemarket. Though the CSFI constructs the VXJ based on the CBOE’s VIX methodology, the
VXJ’s method is slightly different from the CBOE’s VIX in various aspects. The exact methodof CBOE’s VIX will be used to construct a benchmark volatility index for the Japanese stock
market.

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The CBOE’s VIX index is essentially a square-root of a model-free implied variance basedon out-of-the-money (OTM) options prices. A model-free implied variance is an integrated
quadratic return variation under a risk-neutral expectation. Assuming that the log price of theunderlying asset,, follows the continuous Brownian semi-martingale diffusion process;,denotes a standard Brownian motion.

where

(1)

is the drift parameter andis the varianceparameter of (1) process without specific parametric assumption. Demeterfi et al.(1999) andAndersen et al. (2010) showed that the theoretical realized variance under (1) process during atime interval 0,

is an integrated variance or a total quadratic return variation during 0,,

which can be expressed by:.

(2)

The expectation of realized variance in (2) under the risk-neutral measure can be expressed as;
1

1

,

(3)

∙ denotes the expectation under risk-neutral measure.

where

Demeterfi et al. (1999) and Andersen et al. (2011) showed that the model-free implied
variance in (3) from the current time for the next period can be derived from option prices
by:
2

,

,

,

,

,

2

,

,

, (4)

,

where

,

is a risk-free interest rate and,is the forward price of an underlying asset atcurrent time for the next period. is the remaining time-to-maturity of options measured inyears.

,

and,are the prices of European put and call options with strike-5-

and the,

remaining time-to-maturity .
, ,,denotes the price of an OTM option at strike. For

is the price of an OTM put option and vice versa for an OTM call option. For, , both put and call options are said to be at-the-money (ATM).

Nevertheless, in reality, the number of option strikes is limited and discrete. Consequently,
CBOE approximates the implied variance2,,

under (4) by the following formula;

∆

,

,

⋯

where 0

⋯

,

1

,

1

/2.
ATM level,

,

,

(5)

represents observed strikes and, . ∆first strike directly below the forward underlying asset priceobserved strikes calculated as ∆,,∆is an interval between, and for 1

is a mid-quote price of OTM options with ais the,∆strike. Fororis an average of put mid-quote and call mid-quote.

The CBOE’s VIX is a square root of a linear combination of (5) calculated from options
with two nearest maturities to a 30-day horizon, excluding options with a time-to-maturity lessthan seven days.

100

where

and

,

so that

,

365
,
30

1. CBOE measures(6)

in minutes

as a fraction of years. For more details regarding the methodology of CBOE’s VIX, refer to
CBOE’s VIX whitepaper, which is publicly available for download from CBOE’s website7.

3. The corridor volatility index
A corridor volatility index is based on the concept of corridor variance swaps and can be
regarded as a restricted version of the CBOE’s VIX. As Andersen et al. (2011) pointed out, the7

http://www.cboe.com/micro/VIX/vixintro.aspx-6-need for a corridor volatility index arises from the problematic random cut-off rule to exclude
options from the VIX calculation. The random cut-off rule is the source of spurious breaks andartificial jumps in the VIX index unrelated to the variation of the underlying return series. Dueto the proliferation of derivative securities with the VIX index as an underlying asset, Andersen
et al. (2011) stressed the need for a less erroneous benchmark volatility index.
The current cut-off rule of CBOE indicates that all OTM options with positive bid prices are
eligible to be included in the calculation of the VIX. Additional rules state that, starting from thetoward the lowest strike, when two-consecutive zero bid prices of OTM puts are observed,

the OTM puts beyond that point will be excluded from the calculation, even if they havetoward the maximum strike

positive bid prices. Similarly, starting from, OTM calls

beyond observed two-consecutive zero bid prices will be excluded. This random truncationinduces a time-varying effective strike range for the VIX index.
A corridor volatility index alleviates us of the spurious break arising from random
truncation by pre-determining and controlling the effective strike range in a time-consistent,manner. Define an indicator function
0

for a given pair of barriers

andsuch that, specifically;,in other words,
within,,1,

takes the value of one only when the underlying asset price at is. Carr and Madan (1998) show that an expected integrated variance under risk-

neutral measure during a time interval 0,
barriers1,

derived from a corridor variance swaps withcan be expressed as;

,

1

,

-7-

2

,

, (7)

where∙ denotes the expectation under risk-neutral measures and

OTM option at strike,is the price of theas previously defined. It is easy to recognize that (7) is simply a

restricted version of (4). The implied corridor variance can be estimated using (5). Then, in thesame manner with the CBOE’s VIX, the corridor volatility index for the exact 30-day horizon
can be derived using (6).

4. Defining corridors
Andersen et al. (2011) pointed out that a corridor of volatility index should provide broad
coverage of strikes but should not include excessive extrapolation of noisy or non-existing
quotes for deep OTM options. A desired corridor should be automatically or endogenously
adjusted over a time horizon, provide a direct and transparent linkage to the option prices, andcenter on the forward price of the underlying asset, .

Andersen and Bondarenko (2007) proposed defining a corridor based on a percentile of
RND. However, Andersen et al. (2011) pointed out that the corridors defined by a percentile of
RND are not centered on the forward price,and that the estimation of RND requires

additional assumptions regarding the density model, which complicates the calculation andmake it less transparent. Instead, Andersen et al. (2011) proposed a new method based on thework of Andersen and Bondarenko (2010), which stated that option prices reflect tail momentsof RND. The left and right tail moments of a positive random variable, , with strictly positive
density

for allare given by;

and
where

∙ and

,

∙ stand for left tail and right tail moments at a given point

step, Andersen et al. (2011) defined a ratio statistic,
within the support of ,-8-

. In the next

, as an indicator of the location of.
can be regarded as a cumulative density function (CDF) because it is strictly increasing on
0, ∞ , with0equals to

0 and

∞

1. In addition,is centered on the mean of , which. For any given percentile

in the range offunction,is defined as;,
In the context of RND of forward prices,

∈ 0,1

.

(8)

can be expressed in terms of OTM put andcall options prices as;.
Andersen et al. (2011) showed that the(9)ratio provides direct linkage to the option prices, isindependent of RND functional form, and is centered on the forward price. Using (8) and (9),strikes associated with left and right truncation points at the required percentile of RND can beeasily obtained.

5. Data
Option data on the Nikkei 225 index from December 2005 to December 2010, totaling
1,227 observation dates, obtained from NEEDS-FinancialQUEST2.0, are used to estimate
volatility indices as defined in section 2 and 3. Nikkei 225 options are European-style options
that can be exercised on the second Friday of the expiration month. The settlement price isbased on special quotation (SQ) calculated from the total opening prices of each component
stock of the Nikkei 225 on the business day following the last trading day. Strike prices are
multiples of ¥500 intervals based on the Nikkei 225, but for the nearest three expiration months-9-the strike prices are multiples of ¥250 intervals. On any trading day, strike prices were set suchthat 17 strike prices are available for any maturity month, eight below and eight above the atthe-money (ATM) strike price.
The screening process began by eliminating options with maturities less than seven days.
Then, for each trading day, the pair of ATM call and put options with the least difference in theclosing mid-quotes was identified for each maturity. If a pair could not be identified, the optionsin that maturity were entirely disregarded. During this step, an implied forward index level,,,

was calculated based on the closing mid-quotes of ATM call and put options from therelationship,

where

,

,

,

,

,

,

(10)

is the risk-free rate, which is an average of both buying and selling rates of newissues of three-month certificate of deposits (CDs). is the remaining time-to-maturity of theoptions, andandare the prices of ATM call and put options, respectively.

is theATM strike price.
For maturities where ATM options were identified, only options with non-zero bid prices
were retained. Next, mid-quotes were calculated for all options. The estimated volatility indiceswere based on OTM options that have more liquidity than in-the-money options and, hence, areless subject to pricing error.
Table 1 shows summary statistics of the options data for 1,227 trading days, from the end of
December 2005 to the end of December 2010. The total number of options available during thesampling period is 618,686 options for all maturities, or an average of 504 options per
observation day. For the near-term calculation, 66,188 put and call options were available
before the screening process. Only 30% or 19,982 OTM put and call options passed thescreening process and were used in index calculation. For the next term calculation, there were60,254 options in total before the screening process. Around 42% or 25,145 OTM put and call-10-

options were included in the index calculation. A sum of OTM puts and calls, minus one, is thenumber of available strikes used in the estimation of volatility indices. On average, 17 and 21
strikes were available, respectively, for the near-term and the next-term calculation of indices.
[Table 1: Summary statistic of Nikkei 225 options]

It should be noted that an additional filtering rule was applied to the CBOE’s methodology.
At least three OTM options must exist after the screening process; otherwise, the integrated
variance of the previous trading day, calculated from (5) and (7), will be used instead.

6. Empirical results
Table 2 shows summary statistic of volatility indices calculated using methodology
described in section 2 and 3, as well as, the VXJ and CSFI-VXJ obtained from the CSFI’s
website. CBVX denotes the volatility index calculated under CBOE’s methodology described in
section 2. CX99, CX97, and CX95 denote the corridor volatility indices calculated under
Andersen et al. (2011)’s method described in section 3. The number behind ‘CX’ indicates theupper percentile range offunction in (9). CX99 corresponds to the widest corridor

volatility index calculated from the options within [1, 99] percentile range of, while CX95

corresponds to the narrowest corridor volatility index calculated from options within [5,95]
percentile range of.

MFIV is a special case of corridor implied volatility index, in which the corridor width is set
to∞, ∞ . In other words, MFIV represents a full-blown version of the CBOE’s VIX, whichuses all the available options data residing deep in both tails of RND, or deep-out-of-the-money
options.
RV denotes realized volatility for the next 30-day period from the observation date of avolatility index. In particular, RV is calculated from daily closing spot prices of the Nikkei 225
index from to

30:

-11-

,

100

where is the target time-to-maturity of options, in this case, equal to 30 days.

(11)is the daily

closing price of Nikkei 225 index. RV is normalized into a percentage for ease of comparison to
the volatility indices.
During the sampling period from December 2005 to December 2010, there were a total of
1,277 observation dates. However, there existed times when the integrated variance under (5) or(7) could not be obtained due to the unavailability of valid options data. When the implied
forward index level in (10) cannot be obtained, ATM strike cannot be determined and all theoptions data are disregarded. Instead, the integrated variance of the near-term or the next-term
was carried over from the previous trading day. CSFI and CBOE have similar procedures toensure the availability of volatility indices. However, the carry-over procedure may cause
artificial correlations or characteristics of the volatility indices that are unrelated to differencesin the methodology. Consequently, the observation dates, in which the integrated variance ofany volatility indices could not be calculated by normal procedures, were excluded from further
analysis. Under this policy, the final observations were reduced to 1,071 days, a loss of around
16% of total observations.

[Table 2: Summary Statistics of 30-day Implied Volatility Indices]

According to Table 2, all the volatility indices, including RV, are positively skewed as
expected. The standard deviation of volatility indices are in compliance with the width ofcorridors. The smallest corridor index shows the smallest standard deviation. CSFI-VXJ, whichis calculated using different approach from the CBOE’s methodology, shows the largest
standard deviation among the implied volatility indices, even larger than the full-blown MFIV
index.

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The statistics of MFIV and CBVX are nearly identical. It is possible that the deep OTM
options data may hardly exist so that the CBOE’s truncation rule is enough to cover all theoptions quoted within reasonable strikes range.

7. Volatility indices over time
Figure 1 shows the movement of volatility indices as well as RV over time. As seen from
Table 2, volatility indices normally fluctuate between 20-30 levels; however, volatility indicesspiked in response to major financial events. The first spike observed in August 2007 may be a
response to major financial assets sell-off in the U.S. as a result of the subprime loan crisis. The
volatility indices peaked in October 2008 when the subprime loan crisis hit the Japanese market
the hardest. In Japan, the financial crisis during this time is called the Lehman shock. After thebankruptcy of Lehman Brothers Holding Inc. and a series of attempts of the U.S. Government tobail out distressed major financial institutions in September 2008, the Japanese market finallycould not hold on. The Daiwa Life Assurance declared bankruptcy and the Nikkei 225 index fell
10% during the day on October 10th, 2008. The spike during May 2010 may be a response to theStandard & Poor’s reduction of Greece’s sovereign debt rating to a junk status, a part ofEuropean debt crisis that has been lingering until now.

[Figure 1: Daily volatility indices and realized volatility during December 2005 to December 2010.]

The shaded areas in Figure 1 indicate the periods that are magnified and depicted in Figure
2. The left shaded area is the second half of 2006, which is a period just before the breakout of
the subprime loan crisis in the U.S.. The shaded area in the middle represents the second half of2008 when the subprime loan crisis in the U.S. became Global financial crisis and severely
affected the Japanese market. The right shaded area denotes the second half of 2010 period, apost-crisis period when the Japanese market already recovered from the financial slump.

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During the pre-crisis and the post-crisis period depicted in Panel a) and c) of Figure 2, theRV seemed to fluctuate below all the implied volatility indices, in line with the negative
variance risk premium found in recent financial literatures, for example, Carr and Wu (2009)
and Bollerslev et al. (2009). The relationship between implied volatility indices and RV
reversed during August to October 2008.
The MFIV is expected to cap all other volatility indices as it uses all the available options
data. This relationship is true for CBVX, CX99, CX97, and CX95 which are all based on thefiltering rules of the CBOE. The relationship can be confirmed from statistics presented in Table
1, in which means and standard deviations of the narrower corridors volatility indices are lessthan those of the wider corridors volatility indices. However, the VXJ and CSFI-VXJ
occasionally exceeded the MFIV, especially during the crisis period in October 2008.
According to the VXJ documentation8, VXJ adopted the same methodology as CBOE’s
VIX but the determination of ATM options were based on transaction prices rather than midquotes. In addition, CSFI applies carry-over procedures or uses previous trading day data, when
the integrated variance in (5) cannot be estimated due to the unavailability of options data.
These discrepancies may cause the break in the relationship between the MFIV and the VXJ.
On the contrary, CSFI-VXJ uses a different approach in estimation of integrated variance.
Consequently, the relationship between MFIV and other narrower corridors volatility indicesdoes not apply to CSFI-VXJ.

[Figure 2: Closed-up of implied volatility indices and realized volatility by observation periods. ]8

http://www-csfi.sigmath.es.osaka-u.ac.jp/en/activity/vxj_download.php?

-14-8. Effective strike range of corridor volatility indices
According to Andersen et al. (2011), an effective range, or ER, measures the coverage ofoptions data used in calculation of implied volatility indices, and is defined as;

,

where

,

,

√

,

,

√

,

(12)

is the implied forward index level in (10), is the remaining maturity of options, andis the Black-Scholes implied volatility estimated from ATM options.

anddenotes thelowest and the highest strikes used in calculation of the integrated variance in (5) or (7). On
each observation date, effective ranges were calculated for both the near-term and next-term
integrated variance estimation.
Figure 3 shows effective strike ranges for both the near-term and the next-term options dataon each observation date. Only effective ranges of MFIV, the widest corridor index, and CX95,
the narrowest corridor index, were presented for ease of comparison. Effective ranges of CBVX
are similar to those of MFIV. Effective ranges of CX99 and CX97 fluctuates between those ofMFIV and CX95. Ranges of MFIV were plotted as line graphs, while ranges of CX95 wereshown as area. The coverage above the zero line is the coverage of OTM call options, while thecoverage below the zero line represents coverage of OTM put options.
As pointed out by Andersen et al. (2011), shifts in ER can cause artificial jumps in theimplied volatility index unrelated to the fluctuation in the underlying asset. From Figure 3, it isobvious that strike ranges of CX95 are more stable than the ranges of MFIV whether in thenear-term or the next-term options. The width of MFIV fluctuated between 2.43 and 10.50, thedifference over 4 times as much between the widest and narrowest range observed. The width of
CX95 varied from 0.89 to 2.44. MFIV is likely to produce artificial jumps in the volatility index
more than the CX95. In particular, during October 2008, the Lehman Shock crisis period, thecoverage of MFIV exhibited a sudden shrinkage and expansion on day-to-day basis. The-15-

asymmetry in the coverage of OTM puts and calls found in Andersen et al. (2011), whichreflects the relative importance of OTM puts over OTM calls for pricing volatility, can also be
observed from Figure 3.
[Figure 3: Daily effective strike range of MFIV and CX95.]
To systematically investigate the effect of the differences in strike ranges between MFIV
and CX95, the following simple regression test as used by Andersen et al. (2011) is applied;
,
where

/

95

1.

and

(13)

are regression coefficients to be estimated. On eachobservation date , the corridor widths of the near-term and the next-term options were linearly
combined, separately for MFIV and CX95, using a similar weighting method to the index
calculation defined in (6).

is then calculated by taking differences between the widths ofcorridors of MFIV and CX95. The test in (13) essentially aims to measure how much variations
in relative values of MFIV and CX95 can be explained by differences in effective strike ranges
between the two indices.

[Table 3: MFIV/CX95 relative strikes ranges regression results.]

Using OLS estimation, the regression results show that both coefficients in (13) are
statistically different from zero. The differences in strike ranges between MFIV and CX95
contribute to relative variations between the two indices. Though the adjusted R-squared foundin this study is less than what documented in Andersen et al. (2011), the adjusted R-squared
nearly reaches the 50% level. The sample standard deviation ofequals 1.17, so theregression results imply that an increase in the effective range of MFIV by one standard
deviation will inflate a relative value of MFIV to CX95 by an additional 1.67% (1.17 x 1.43%).

-16-

The finding that effective ranges has significant explanatory power for the relative size of
implied volatility indices is similar to the results of Andersen et al. (2011).

9. Correlations of volatility indices
The summary statistics presented in Table 2 suggested that volatility indices are very similarin properties. Figure 1 and Figure 2 also showed that indices fluctuate closely to each other.
Table 4 shows a correlations matrix of volatility indices as well as RV, Nikkei 225 returns forthe next 30-days, and Nikkei 225 returns for the next day. The future-period returns of Nikkei
225 were input to see if the volatility indices have potentials to predict the future returns of theirunderlying asset. RV is the 30-days-ahead realized volatility calculated from (11). Since implied
volatility indices are the expectation of future realized volatility under risk-neutral measure, it isinteresting to test the correlation between the predictive measures and the realized measure.

[Table 4: Correlations of volatility indices and volatility index returns.]

The correlations between levels of the implied volatility indices are high and positive. They
are all statistically different from zero at 5% significant level. The correlation results confirm
the properties found in summary statistics as well as Figure 1 and Figure 2. The correlations ofvolatility index returns share the same characteristics and are all statistically different from zero.
The correlations between volatility indices and future realized volatility, RV, are statisticallydifferent from zero and are all nearly reaching the 70% level. The high correlations between
volatility indices and the realized volatility for the next 30 days period are expected, because
volatility indices are expectations of future volatility under a risk-neutral measure.
The correlations between volatility indices and future returns of Nikkei 225 are notstatistically different from zero at 5% significant level, implying that volatility indices may not
be a good predictor of future-period returns. The correlations between daily returns of volatility-17-indices, defined as/

1, and the future-period returns of the Nikkei 225 index areneither statistically different from zero.
Nevertheless, on separate tests not presented here, correlations between implied volatility
indices and daily return of the Nikkei 225 index on the same trading day are negative andsignificantly different from zero. Similarly, correlations between daily returns of implied
volatility indices and daily returns of Nikkei 225 index are all above 60% and are statisticallydifferent from zero at the 5% significant level. This fact is in accordance with recent financial
literatures that found negative correlations between stock returns and volatility.
To investigate if the width of corridors and different observation periods affect the negative
relationship between stock returns and implied volatility index returns, Figure 4 shows scatterplots between Nikkei 225 returns and MFIV returns, as well as, Nikkei 225 returns and CX95
returns. MFIV is considered a special case as corridor volatility index with the widest corridor,
which uses all the available options strikes. CX95, the narrowest corridor volatility index, iscalculated using only options strikes which fell within [5, 95] percentile ofdefined in (9).

The Nikkei 225 returns and implied volatility index returns were standardized by de-meaningand dividing by respective standard deviations. Observation periods are divided into the precrisis, crisis, and post-crisis periods, corresponding to the periods between 2005/12-2007/06,
2007/07-2009/03, and 2009/04-2010/12, respectively.

[Figure 4: Volatility index returns vs. Nikkei 225 index returns.]

The negative relationship between Nikkei 225 returns and returns of both volatility indicesheld regardless of observation periods and the width of corridors as evidenced from thedownward sloping regression lines in Figure 4. To confirm the finding, a simple regression test
using OLS estimation with, was performed for each observation period.

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is thedaily returns of the Nikkei 225 index andor CX95. The estimated coefficientsis the daily returns or percentage change of MFIV

were all negative and significant at 5% level.

It should be noted that Andersen et al. (2011) found that the use of a corridor implied
volatility index assists in preserving the relationship between volatility index returns and theunderlying stock market index returns when there is a sudden shrinkage or expansion of theeffective strike ranges. Though the effect of corridor widths was found to be important in
explaining variations in implied volatility indices as discussed in section 8, the corridor widths
did not affect the relationship shown in Figure 4.

[Figure 5: Volatility index returns vs. Nikkei 225 index returns in October 2008 and January 2009.]

To confirm if the corridor width does not affect the relationship between stock index returns
and volatility index returns, scatter plots similar to Figure 4 were reproduced using data in
October 2008 and January 2009. In October 2008, the effective strike ranges of MFIV shrink
dramatically, while in January 2009, the opposite occurred. Figure 5 shows that different
corridor widths may affect the relationship between the underlying stock index returns and thevolatility index returns but the impact was minimal, unlike the findings documented inAndersen et al. (2011). The drastic impact of corridor differences may become apparent under
intraday observations.

10. Implied volatility index as predictor of realized volatility
Predictive performance of volatility indices toward the realized volatility for the next 30days is compared in this section. By construction, an implied volatility index estimated from
options data with a 30-day remaining maturity is an expectation of future realized volatility forthe same horizon under risk-neutral measure. In section 8, it is evidenced that the choice of-19-corridors affects the characteristic of the volatility index. This section tests wheter variations incorridors affect predictive performance of the volatility index.
The parameters of following a simple regression between the future-realized volatility andthe volatility index are obtained using an OLS estimation.
,
where

(14)

denotes the implied volatility indices, MFIV, CBVX, CX99, CX97, CX95, VXJ, andCSFI-VXJ. Though

and

has the same subscript ,volatility calculated from the current time todenotes the annualized realized

30 using (11). It should be noted that the

regression was estimated on a monthly basis using non-overlapping series ofto prevent

autocorrelation.
The estimation window for regression in (14) was set between December 2005 to December
2009, a total of 49 observations. The data during 2010 will be used in testing the forecast
performance of the volatility indices against the realized volatility. All the coefficients ofregression results in Table 5 are statistically significant at 5% level. The estimated coefficientsand adjusted R-squared are similar across alternative implied volatility measures. However, thenarrow corridor volatility indices CX97 and CX95 performed slightly better than others basedon adjusted R-squared statistics.

[Table 5: Regression results of realized volatility]

As a robustness test, the same regression was performed using all the observations from
December 2005 to December 2010, a total of 61 observations. The adjusted R-squared measure
still indicates that CX97 performed the best in explaining variations in the future-realized
volatility, followed by CX95. However, if the estimation window were set such that 21
sampling months during the subprime crisis period, between July 2007 to March 2009, were-20-

excluded from the total observations, the CSFI-VXJ performs the best with adjusted R-squared
around 42%, followed by VXJ with adjusted R-squared around 41%. However, the intercept ofregression for all the volatility measures became statistically insignificant.
In the next step, the estimated coefficients in Table 5 are used to calculate a root meansquared error (RMSE) of out-of-sample forecast performance during 2010. In particular, theRMSE is calculated from;
,

∑
where

(15)

is the forecasted future realized volatility based on the estimated coefficients of (14).

is the total number of forecast periods. It should be noted here that, though the estimation ofcoefficients used non-overlapping monthly data, the forecast process used all the available daily
data. Each forecast is regarded as an independent forecast for the next 30-days’ realized
volatility.
Table 6 shows RMSE results of 238 daily forecasts from January to December 2010. The
forecast performance is in line with the adjusted R-squared criteria presented in Table 5. The
CX97 corridor volatility index has the best forecast performance with the lowest RMSE of 4.24,
followed by CX95 index with RMSE equaling 4.26. When the non-overlapping data of all 61
observation months were used to estimate coefficients of (14), the RMSE criteria of in-sample
daily forecasts during 2010 confirm that CX97 has the best forecast performance with thelowest RMSE of 4.12. Nevertheless, when the forecast is conducted using estimated parametersfrom the estimation window that excluded the 21 months during the crisis period, the MFIV and
CBVX indices gave the lowest RMSE of 4.05.
With exception to the third estimation window showed in Table 6, CX97, the corridor
volatility index with a moderate corridor width, seems to outperform other volatility measures
as a predictor of realized volatility over the next 30 days.

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[Table 6: RMSE of realized volatility forecast.]

11. Conclusions
In this paper, various implied volatility measures were introduced, compared, andcontrasted. The methodology of the well-known CBOE’s VIX index was applied to construct avolatility index of the Japanese stock market using Nikkei 225 options data. A new approach
proposed by Andersen et al. (2011) to overcome the random truncation of the CBOE’s VIX
method was applied and tested against the CBOE’s method.
Volatility indices calculated under CBOE’s method and Andersen et al. (2011) shared very
similar properties. The corridor volatility indices fluctuated to a lesser degree than the CBOE’s
VIX and the index that used all options strikes, MFIV, during the observation period. The
negative volatility risk premium, which dictated that a realized volatility should be less than its
expectation under a risk-neutral measure, can be observed when the realized volatility with amatching horizon was plotted with volatility indices. However, a large positive premium can beobserved during the Lehman shock crisis in 2008.
The corridor widths have explanatory power against the relative variations between twovolatility indices with different strikes coverage and corridor volatility indices seemed toperform better in predicting future realized volatility. However, the merit of using corridor
volatility index was not fully realized as the differences in performance were minimal. Further
study using high-frequency intraday data may help convey the strengths of corridor volatility
indices as documented by Andersen et al. (2011).

-22-

References
Allen, P., Einchcomb, S. and Granger, N. (2006). “Conditional Variance Swap - Product Note.”
JPMorgan, European Equity Derivatives Strategy.
Andersen, T. G. and Bondarenko, O. (2007), “Construction and Interpretation of Model-Free
Implied Volatility,” CREATS Research Paper 2007-24, Department of Economics andBusiness, Aarhus University.
Andersen, T. G., and Bondarenko, O. (2010), “Dissecting the Pricing of Equity Index Volatility,”
Working paper, Northwestern University and University of Illinois at Chicago.
Andersen, T. G., Bondarenko, O. and Gonzalez-Perez, M. T. (2011), “Coherent Model-Free
Implied Volatility: A Corridor Fix for High-Frequency VIX,” CREATS Research
Paper 2007-24, Department of Economics and Business, Aarhus University.
Bollerslev, T., Tauchen, G. and Zhou H. (2009), “Expected Stock Returns and Variance Risk
Premia,” The Review of Financial Studies 22 (11), pp. 4463-4492.
Carr, P., and Madan, D. (1998), “Towards a Theory of Volatility Trading.” pp. 417-427.
Volatility, Edited by Risk Publications, R. Jarrow.
Carr, P., & Wu, L. (2006), “A Tale of Two Indices,” The Journal of Derivatives 13 (3), pp. 1329.
Carr, P. and Wu, L. (2009), “Variance Risk Premiums,” The Review of Financial Studies 22 (3),pp.1311-1341.
Demeterfi, K., Derman, E., Kamal, M. and Zou, J. (1999). “More Than You Ever Wanted To
Know About Volatility Swaps,” Quantitative Strategies Research Notes. Goldman
Sachs.
Fukusawa, M., Ishida, I., Maghrebi, N., Oya, K., Ubukata, M. and Yamazaki, K. (2011),“Model-free Implied Volatility: From Surface to Index,” International Journal of
Theoretical and Applied Finance 14 (4), pp. 433-463.

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Lieberman, P. K. and Omprakash, A. (2007), “Corridor Variance Swap: A Cheaper Way to Buy
Volatility?” BNP PARIBAS, Corporate and Investment Banking, U.S. Equities &Derivatives Strategy.
Maghrebi, N. VXJ Guide. Retrieved January 13, 2010, from http://www-csfi.sigmath.es.osakau.ac.jp/en/activity/vxj/vxj.pdf.
Osaka Securities Exchange. Nikkei 225 Option. Retrieved January 13, 2010, fromhttp://www.ose.or.jp/e/futures/ind_opo2.html.
THE

CBOE

VOLATILITY

INDEX

-

VIX.

(2009).

Retrievedfrom

http://www.cboe.com/micro/VIX/vixwhite.pdf.
Whaley, R. E. (2000), “The Investor Fear Gauge – Explication of the CBOE VIX,” The Journalof Portfolio Management 26 (3), pp.12-17.

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Table 1: Summary statistic of Nikkei 225 options.
Out-of-the-money is abbreviated as OTM. Total denotes total number of options observed under each category
during the whole 1,227 observation dates. Average refers to average number of options per day across allobservation dates. Min and Max are the minimum and maximum number of options observed during the sampling
period.

Options before screening process (all maturities)

618,686

Average options before screening per day

504

Total

AverageMin

Max

Near term options before screening

66,18854

28

98

OTM put after screening

10,7069

322

OTM call after screening

9,276

8

3

29

Next term options before screening

60,25452

22

96

OTM put after screening

12,94711

3

21

OTM call after screening

12,19811

4

34

Table 2: Summary Statistics of 30-day Implied Volatility Indices.
From left to right, the corridor widths were sorted from the widest to the narrowest, with MFIV representing thewidest corridor and CX95 representing the narrowest corridor implied volatility indices. VXJ, CSFI-VXJ, and RV arepresented for comparison. Q25 to Q75 denotes the 25th to 75th percentile of the total observations. There were a totalof 1,227 observation dates during December 2005 to December 2010. However, after the elimination of incalculable
sampling dates, 1,071 observations remain.
MFIV

CBVX

Mean

29.93

29.93

Std Dev
Skewness
Kurtosis
min
max
Q25
Q50
Q75

12.66
2.10
8.07
14.32
93.53
22.31
26.60
32.51

12.65
2.10
8.04
14.32
93.53
22.31
26.60
32.51

CX99

29.55

CSFI
VXJ
29.35

24.88

12.49
2.09
7.98
14.12
91.45
21.99
26.34
32.08

13.03
2.23
8.87
13.47
97.27
21.65
25.88
31.89

14.76
3.33
17.01
8.50
113.90
17.33
22.25
27.12

CX97

CX95

VXJ

29.17

28.20

27.43

12.48
2.19
8.54
14.12
93.05
21.64
25.85
31.54

12.15
2.25
8.92
13.77
92.12
20.91
24.97
30.27

11.84
2.28
9.09
13.77
89.12
20.40
24.44
29.33

-25-

RV

Table 3: MFIV/CX95 relative strikes ranges regression results.
The simple regression test specified in (13) was conducted to test if relative differences incorridor widths contribute to the relative variations between MFIV and CX95. After eliminationof outliers and invalid data, a total of 1,067 observations were used in the regression under OLS
estimation. Standard errors of coefficients were Newey-West HAC standard errors calculatedusing 20 lags. P-values for both coefficients were less than 1e-6.

Coefficient
Std. Error
t-Statistic
p-value

0.03360.00714.74
00.01430.00149.88
0= 0.466

Table 4: Correlations of volatility indices and volatility index returns.
RV denotes 30-days-ahead realized volatility calculated using (11). RET30 is the 30-days-ahead log-returns of the
Nikkei 225 index. RETd is the daily return of Nikkei 225 index in the following trading day. The volatility index
return is/-1, where denotes each volatility index.
Panel a) Correlations of volatility indices
MFIV

CBVX

CX99

CX97

CX95

MFIV

1

CBVX

1.00001

CX99

0.99950.9994

1

CX97

0.99850.99850.99951

CX95

0.99770.99770.99900.9994

1

VXJ

CSFI
VXJ

RV

VXJ

0.99850.99850.9981

0.9971

0.9963

1

CSFI VXJ

0.99650.9964

0.99680.99680.99650.99781RV

0.65810.65820.66540.67230.67490.6646

0.6763

1

RET30

RETd

RET30

0.03800.03790.0356

0.03320.0336

0.03700.0357-0.4218

1

RETd-0.0116-0.0115-0.0120-0.0113-0.0092-0.0110-0.0048-0.1597

0.2425

1

CX95

VXJ

CSFI
VXJ

RV

RET30

RETd

Panel b) Correlations of volatility index returns
MFIV

CBVX

CX99

CX97

MFIV

1

CBVX

0.99941

CX99

0.99580.9951

1

CX97

0.9821

0.98150.9812

1

CX95

0.96440.96390.96380.9471

1

VXJ

0.94030.93970.93720.91850.9008

1

CSFI VXJ

0.9422

0.94120.93810.92410.90350.9577

1RV-0.0293-0.0284-0.0293-0.0270-0.0268-0.0255-0.02621

RET30-0.0297-0.0297-0.0280-0.0272-0.0246-0.0255-0.0264-0.2332

1

RETd

0.01740.01890.01820.01510.01440.05090.0425-0.0667

0.2423-26-

1

Table 5: Regression results of realized volatility
Regression equation (14) is estimated using an OLS estimation with non-overlapping 30-days-ahead realized
volatility,, as the dependent variable and each series of implied volatility index,
as the predictor variable. The estimation window is from the end of December 2005 to the end of December
2009, totaling of 49 observations. Standard errors of coefficients are Newey-West HAC standard errors with one
lag, automatically determined by AIC criteria.

MFIV

Coefficient
Std. Error

4.869(2.164)0.668(0.086)

0.360

CBVX

Coefficient
Std. Error

4.869(2.164)0.668(0.086)

0.360

CX99

Coefficient
Std. Error

4.847(2.171)

0.685(0.082)

0.368

CX97

Coefficient
Std. Error

4.886(2.169)0.706(0.081)

0.373

CX95

Coefficient
Std. Error

4.853(2.259)0.728(0.066)

0.371

VXJ

Coefficient
Std. Error

4.670(2.168)0.684(0.091)

0.364

CSFI-VXJ

Coefficient
Std. Error

5.603(2.105)0.656(0.079)0.368

Table 6: RMSE of realized volatility forecast.
RMSEs of 238 daily forecast of realized volatility over the next 30-days period during January to December 2010 are
calculated using (15). The coefficients estimated from regression in (14) with different estimation windows were used
to derive the RMSE. Lower RMSE indicates better forecast performance.

MFIV
CBVX
CX99
CX97
CX95
VXJ
CSFI VXJ

Estimation window 1(2005/12 – 2009/12)
49 observations

Estimation window 2(2005/12 – 2010/12)
61 observations

4.347
4.347
4.283
4.248
4.260
4.383
4.360

4.172
4.172
4.134
4.115
4.134
4.211
4.182

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Estimation window 3(2005/12 –2007/06 :2009/04-2010/12)
39 observations
4.050
4.050
4.066
4.085
4.125
4.096
4.069

Figuree 1: Daily volatility indices and realized vo latility during December 200
05 to Decembeer 2010.
The shhaded areas are magnified in Figure
F
2. RV plootted on each observationodatee is the realizedd volatility for the next
30-dayy period.

-28-

Figuree 2: Closed-up of implied vollatility indices and realized volatiliyv
by obsservation perioods.
The voolatility indicess plotted in Figu
ure 1 for each ssub-sampling periods are magnnified. Panel a)) shows indicess before
the subbprimeloan crissis began to em
merge in the U.S
S. Panel b) show
ws indices durin
ng Lheman Shoock crisis hit Jaapanesemarket. Panel c) show
ws the indices duringdthe last hhalf of 2010 wh
hen the Nikkei 225
2 index alreaddy recovered frrom theslump.
a)P
Pre-crisis: July – December 20062b) Crrisis: July – Deecember 2008c) Poost-crisis: July – December 20102-29-

nd CX95.
Figuree 3: Daily effecctive strike ran
nge of MFIV an
The poositive region shows the upper bound coveerage for OTM call options, whilew
the negattive region sho
ows thelower bound coveragge for OTM pu
ut options. The ranges of MFIIV were plotted as line, whilee the ranges off CX95
were pplotted as shadeed area. Panel a)a and Panel b) sshow, respectiv
vely, ER for thee near-term andd the next-term options
data.
Panel aa) Near-term maturitymPanel bb) Next-term maturitym-30-

Figuree 4: Volatility indexireturns vs.
v Nikkei 225 index returns.
Pre-criisis, crisis, and post-crisis refeer to 2005/12–22007/06, 2007/0
07–2009/03, andd 2009/04– 20110/12, respectiv
vely. Yaxis shhows daily retuurns of volatility indices and X-axis is dailyy returns of thee Nikkei 225 inindex. Both retturns ofvolatillity indices and the Nikkei 225
5 index were dee-meaned and sttandardized by their standard ddeviations.

-31-

Figuree 5: Volatility indexireturns vs.
v Nikkei 225 index returns in October 20008 and Januarry 2009.
MFIV is the volatilitty index with th
he widest corriddor, and CX95
5 is the volatilitty index with thhe narrowest corridor.
The Y
Y-axis shows daaily returns of volatility indicces and the X-aaxis is daily returns of the Niikkei 225 index
x. Bothreturnss of volatility inndices and the Nikkei
N
225 indeex were de-meaaned and standaardized by theirr standard deviaations.

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