Global COE Hi-Stat Discussion Paper Series 273

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Pricing Nikkei 225 Options Using Realized Volatility

Hi-Stat Discussion Paper

Masato Ubukata
Toshiaki Watanabe

January 2013

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Pricing Nikkei 225 Options Using Realized Volatility
M ASATO U BUKATA
Department of Economics, Kushiro Public University of Economics

T OSHIAKI WATANABE
Institute of Economic Research, Hitotsubashi University

Abstract
This article examines option pricing performance using realized volatilities with or without
handling microstructure noise, non-trading hours and large jumps. The dynamics of realized
volatility is specified by ARFIMA(X) and HAR(X) models. Main results using put options on the
Nikkei 225 index are: (1) ARFIMAX model performs best, (2) the Hansen and Lunde (2005a)
adjustment for non-trading hours improves the performance, (3) methods for reducing microstructure noise-induced bias yield better performance, while if the Hansen-Lunde adjustment is used,
the other methods are not necessarily needed and (4) the performance is unaffected by removing
large jumps from realized volatility.
JEL Classification Numbers: C13, C22, C52.
Keywords: microstructure noise, Nikkei 225 stock index, non-trading hours, option pricing, realized volatility.

 This is a revised version of Ubukata and Watanabe (2011). Financial support from the Ministry of Education, Culture,

Sports, Science and Technology of the Japanese Government through Grant-in-Aid for Scientific Research (No.18203901;
21243018; 22243021; 23730301) and the Global COE program “Research Unit for Statistical and Empirical Analysis in
Social Sciences” at Hitotsubashi University and the Joint Usage and Research Center, Institute of Economic Research,
Hitotsubashi University (IERPK1109, 1206) is gratefully acknowledged. We would also like to thank seminar participantsat the Bank of Japan, Tohoku University, Hiroshima University of Economics and Nihon University and the staff of theInstitute for Monetary and Economic Studies (IMES), the Bank of Japan, for their useful comments. All remaining errors
are solely our responsibility.

1 Introduction
One of the most important variables in option pricing is the volatility of the underlying asset. While
the well-known Black and Scholes (1973) model assumes that the volatility is constant, few would
dispute the fact that the volatility changes over time. Many time series models are now available
to describe the dynamics of volatility. One of the most widely used is the ARCH (autoregressive
conditional heteroskedasticity) family including ARCH model by Engle (1982), GARCH (generalized
ARCH) model by Bollerslev (1986) and their extensions.
The problem of using these models is that we must specify the model before estimating the volatility and the estimate of volatility depends on the specification of volatility dynamics. Recently, realized
volatility has attracted the attentions of financial econometricians as an accurate estimator of volatility. Realized volatility is independent of the specification of volatility dynamics because it is simply
the sum of squared intraday returns.
ARCH type models have already been applied to option pricing by Duan (1995), Bollerslev andMikkelsen (1999), Heston and Nandi (2000) and Barone-Adesi et al. (2008). More recently, someauthors have applied realized volatility to option pricing as well as the applications to volatility forecasting (Koopman et al. 2005; Andersen et al. 2007) and Value-at-Risk (Giot and Laurent, 2004;
Clements et al., 2008; Watanabe, 2012). Stentoft (2008) examine an option pricing model with realized volatility using inverse Gaussian distribution. Christoffersen et al. (2010) propose generalized
expected realized volatility model to incorporate realized volatility into the Heston-Nandi (2000)
GARCH model and derive closed-form option valuation formulas. Corsi et al. (2011) develop the
HAR model (heterogeneous interval autoregressive) by Corsi (2009) and applies it to option pricing.
This article investigates how different option pricing performance would be if we take account ofsome practical issues involved in calculating realized volatility. The high-frequency financial literature has developed to measure realized volatility associated with microstructure noise-induced bias,
the presence of non-trading hours and separate estimates of the continuous and discontinuous (jump)components of the volatility process. This raises the question of what calculation method for realized
volatility is reasonably useful in option pricing. As far as we know, there are few that have tackled
this problem. One exception is Bandi et al. (2008), which apply different realized volatilities to thepricing of S&P 500 index options and compare their option pricing performance. Their method is,
however, different from ours as follows. First, they compare the profits from the straddle trading strategy obtained by substituting the volatility forecasts from a time-series model for realized volatility
into the Black-Scholes option pricing formula. We employ a realized volatility option pricing model
without the Black-Scholes formula. Second, they compare the performance of realized volatilities
with or without taking account of microstructure noise, while our analysis further includes realized
volatilities with or without taking account of non-trading hours and asset price jumps and comparison
with ARCH-type models.
There are several problems in calculating realized volatility. First, realized volatility is influenced
by market microstructure noise induced by various market frictions such as bid-ask spread and non-1

synchronous trading (Campbell et al., 1997). There are some methods available for mitigating theeffect of microstructure noise on realized volatility (Aı̈t-Sahalia et al., 2005; Bandi and Russell, 2006,
2008, 2011; Barndorff-Nielsen et al., 2004b, 2008; Hansen and Lunde, 2006; Jacod et al., 2009; Kunitomo and Sato 2008; Oya 2011; Zhang, 2006; Zhang et al., 2005; Zhou 1996). It is worthwhile
applying these methods and comparing the results. We use several different methods for mitigating
the effect of microstructure noise on realized volatility. We analyze whether using these methods
may improve the performance of option pricing of Nikkei 225 stock index options traded at Osaka
Securities Exchange. Second, the Tokyo stock exchange, where the 225 stocks that constitute the
Nikkei 225 stock index are traded, opens only for 9:00–11:00 and 12:30–15:00. We cannot obtainhigh-frequency returns during the period when the market is closed. Adding the squares of overnight(15:00-9:00) and lunch-time (11:00-12:30) returns may make realized volatility noisy. Following
Hansen and Lunde (2005a), we calculate realized volatility without overnight and lunch-time returns
and multiply a constant such that the sample mean of daily realized volatility is equal to the sample
variance of daily returns. We examine whether this method is effective in option pricing by comparing with simply adding the squares of overnight and lunch-time returns. Third, financial markets
sometimes display asset price discontinuities, so-called jumps. Under jump-diffusion processes forthe underlying asset prices, the realized volatility includes variation due to jumps. We also constructrealized volatility removing significant large jumps and investigate its contribution to option pricing
performance.
Many authors have documented that realized volatility follows a long-memory process (Andersen
et al., 2001, 2003). We use the ARFIMA (autoregressive fractionally integrated moving average)
model and HAR model by Corsi (2009) to describe the dynamics of realized volatility. It is also wellknown in stock markets that today’s volatility is negatively correlated with yesterday’s return. We also
extend ARFIMA and HAR models to take account of this asymmetry in volatility. For ARCH typemodels, we use the simple GARCH model proposed by Bollerslev (1986), the EGARCH (exponential
GARCH) model by Nelson (1991) that may capture the asymmetry in volatility and the FIEGARCH(fractionally integrated EGARCH) model by Bollerslev and Mikkelsen (1996) that may also allow for
the long-memory property of volatility.
We calculate option prices under the assumption of risk neutrality for the examination of realized
volatility option pricing model, while it would be important to relax this assumption. However, adirect test for risk premium under some specifications of conditional expectation of the Nikkei 225
returns with realized volatility implies that the data used here may be insensitive to estimated parameters of the market price of risk. Thus, we only consider the case where the risk neutral and physicaldynamics of realized volatility are identical. Duan (1995) has developed a more general method for
pricing options in ARCH type models, which does not assume risk neutrality. We also calculate
option prices both by assuming the risk neutrality and by using the Duan (1995) method.
Main results using the Nikkei 225 stock index and its put options prices are: (1) ARFIMAX
model with daily realized volatility performs best, (2) the Hansen and Lunde (2005a) adjustment
without using overnight and lunch-time returns can improve the performance, (3) the performance is2

improved by mitigating the effect of microstructure noise on realized volatility, while if the Hansen
and Lunde (2005a), which also plays a role to remove the bias from the microstructure noise by setting
the sample mean of realized volatility equal to the sample variance of daily returns, is used, the othermethods for taking account of microstructure noise do not necessarily improve the performance, (4)the option-pricing performance is not necessarily improved by removing significant large jumps fromrealized volatility and (5) the Duan (1995) method does not improve the performance compared withassuming the risk neutrality.
The article proceeds as follows. Section explains several methods used in this article for calculating realized volatilities. Section  explains ARFIMA(X) and HAR(X) models to describe thedynamics of realized volatility and ARCH type models used in this article for comparison. Section 
explains how to calculate option prices using the ARFIMA(X) and HAR(X) models with daily realized volatility and ARCH type models with daily returns. Section  explains the data and Section 
compares the performance of option pricing. Section  concludes. The appendix provides a detailed
description of realized volatilities employed in this article.

2 Realized Volatility
We start with a brief review of realized volatility using the following diffusion process.

    (1)

where  is time,  is the log-price,   is a standard Brownian motion, and  and   are thedrift and the volatility respectively, which may be time-varying but are assumed to be independentof  . In this article, we call   or   volatility interchangeably although   is usuallycalled volatility in the finance literature. Then, the volatility for day  is defined as the integral of  over the interval   where and  represent the market closing time on day and respectively, i.e.,  (2)

which is called integrated volatility. The integrated volatility is unobservable, but if we have theintraday return data , we can estimate it as the sum of their squares   (3)which is called realized volatility. If the prices do not include any noise, realized volatility, i.e.,
provide a consistent estimate of

will

plim

(4)




3

There are two problems in calculating realized volatility under these settings. First, although therealized volatility is an accurate estimator of integrated volatility under the assumption of a continuous
stochastic model, it fails when there is market microstructure noise as seen in real high-frequency
data. The microstructure noise can be induced by various market frictions such as the discretenessof price changes, bid-ask bounces, and asymmetric information across traders, inter alia. 1 A growing
literature attempts to study an integrated volatility estimation from microstructure noise-contaminatedhigh-frequency data. In this article, we employ some influential integrated volatility estimators robust
to the microstructure noise.
Second, the Tokyo Stock Exchange is open only for 9:00–11:00 (morning session) and 12:30–
15:00 (afternoon session) except for the first and last trading days in every year, when it is open
only for 9:00-11:00. It is impossible to obtain high-frequency returns for 15:00–9:00 (overnight) and
11:00–12:30 (lunch-time). Since realized volatility obtained using high-frequency returns over 4.5hour trading period only captures the volatility during the part of the day that the market is open, we
need to extend the realized volatility to a measure of volatility for the full day. If we simply add thesquares of overnight and lunch-time returns, realized volatility may be subject to discretization error.
Hansen and Lunde (2005a) propose to calculate realized volatility only when the market is open,
which is denoted as, and multiply a constant  such that the sample mean of realized volatility
is equal to the sample variance of daily returns, i.e.,




















(5)

where       is the sample of daily returns and is the sample mean 2 .
In order to test the effects of taking into consideration the microstructure noise and the non-trading
hours on option pricing, we use as many as 34 daily realized volatilities listed in Table . Without
microstructure noise, it would be desirable to use intraday returns sampled at the highest frequencies.
Since the highest frequencies available for Nikkei 225 stock index is 1-minute, we first calculate realized volatility using 1-minute returns ( ). From the second to seventeenth methods in Tableare expected to correct the bias of the classical realized volatility and mitigate the variance increaseof the estimator induced by the microstructure noise. A more detailed description of the methods isprovided in the appendix. We apply the Hansen and Lunde (2005a) adjustment to the 17 kinds of realized volatilities, which are denoted as1min   ,5min  ,10min  ,15min ,20min ,  ,    ,   ,    ,    ,    ,      ,      ,       ,   
1

The literature on market microstructure provides important insights from early studies including Roll (1984), who
derives a simple estimator of the bid-ask spread based on the negative autocovariance of returns. Harris (1990) examines
the rounding effects emanating from the discreteness of transaction prices. In the recent literature on microstructure noise,
Meddahi (2002) and Hansen and Lunde (2006) examine the variance of microstructure noise as well as the correlation
between the microstructure noise and frictionless equilibrium price. Ubukata and Oya (2009) examine dependence of
microstructure noise.
2
See Martens (2002) and Hansen and Lunde (2005b) for the other methods.

4  ,     ,      . For comparison, we also calculate 17 kinds of daily realized volatilities constructed by adding the sqaures of overnight and lunch-time returns insteadof the Hansen and Lunde (2005a) adjustment, which are denoted as1min ,5min ,10min ,15min ,20min ,  ,    ,  ,   ,    ,     ,     ,     ,     ,     ,     ,      .

3 ARFIMA(X), HAR(X) and ARCH type Model
Many researchers have documented that realized volatility may follow a long-memory process. Let denote the -th order autocorrelation coefficient of variable  . Then,  follows a short-memory
process if     and a long-memory process if   . A stationary ARMA
model is a short-memory process. As  increases, the autocorrelation coefficient  of the longmemory process decays more slowly than that of the short-memory process. More specifically, theformer decays hyperbolically and the latter decays geometrically.
The most widely used for a long-memory process is ARFIMA(  ) model 3

!

 

"# #  NID   (6)!     !  and ""     "  arewhere  denotes the lag operator and !the -th and -th order lag polynomials assumed to have all roots outside the unit circle. The order ofintegration is allowed to take non-integer values. If, ARFIMA model collapses to stationary, it becomes non-stationary ARIMA model. If   ,  follows
ARMA model and if
a stationary long-memory process and if    ,  follows a non-stationary long-memory may be written as follows.
process. 









$

 

$



 

(7)

We assume that # follows an independent normal distribution with zero mean and variance  .
By setting and, which are selected by the Schwartz information criterion (SIC), and  where  is the unconditional mean of , we consider the following model.
 





#

"#



#  NID   (8)We estimate parameters ,  and " jointly using the approximate maximum likelihood method (Beran,1995), where it is assumed that    . We can estimate  as the sample
variance of residual.
We also employ HAR model by Corsi (2009) well-known as a simple approximate long-memory
3

See Beran (1994) for the details of long-memory and ARFIMA model.

5model of realized volatility. The model consists of three realized volatility components defined overdifferent time periods as follows.


%



% 

% 









% &&  NID   

(9)







where
 and
 are the average of the past realized




volatilities corresponding to time horizons of 5 trading days (one week) and 22 trading days (onemonth), respectively. We can estimate parameters % , % , % , % and  by applying simple linear
regression.
It is well-known that there is a negative correlation between today’s return and tomorrow’s volatility in stock markets. To take into account this phenomenon, we extend the above ARFIMA(0,d,1)
model (8) to the following ARFIMA(0,d,1)-X model 





 







'











#

"#

#  NID   



(10)

is negative and zero otherwise.
where '  is a dummy variable that takes one if the return on day 
We estimate parameters ,  ,  ,  , " and  using the same method as that for ARFIMA model.
If the estimate of  has a statistically significant positive value, it is consistent with a well-known
negative correlation between today’s return and tomorrow’s volatility in stock markets. The HAR
model (9) can be naturally extended to HAR-X model taking account of the asymmetry in volatility
as follows.




%

% 



% 





% 





%





%'









&  (11)&  NID   

We estimate parameters % , % , % , % , % , % and  using the same method as that for the HAR model.
The positive value of % indicates the negative correlation between today’s return and tomorrow’s
volatility.
Some researchers such as Barndorff-Nielsen et al. (2004a), Barndorff-Nielsen and Shephard(2001, 2002) and Nagakura and Watanabe (2011) have proposed a UC (unobserved components)
model4. Assuming that the asset price follows a contimuous-time model called square-root stochastic
variance model, they show that the realized volatility calculated using the discretely sampled data
follows an ARMA( , ) model. Since it is the realized volatility rather than its log that follows an
ARMA( , ) model and the distribution of the error term is unknown, the future volatility sampled for
option pricing may possibly be negative if we assume that the distribution of error term is normal.
Thus, we do not use this model in this article.

4

Nagakura and Watanabe (2011) consider microstructure noise while Barndorff-Nielsen et al. (2004a) and BarndorffNielsen and Shephard (2001, 2002) neglect it.

6

We also estimate ARCH type models using daily returns. We define daily return as 



 

(12)

where  is the closing price on day . We specify daily return as  

E )  )  NID  

( (

(13)

conditional on the information up to day 
and )where E    is the expectation ofis assumed to follow an independent standard normal distribution. Then,  is the variance ofconditional on the information up to day . We will explain how to specify E    later.
For volatility specification, we use three different ARCH type models. First is the GARCH model
proposed by Bollerslev (1986). Specifically, we use the GARCH(  ) model



*

%

+(



 % +   * ,(14)

where * , % and + are parameters, which are assumed to be non-negative to guarantee that volatility isalways positive. This model can capture the volatility clustering. Volatility is stationary if % +  ,and the speed for which the shock to volatility decays becomes slower as % + approaches to one.
As has already been mentioned, another well-known phenomenon in stock markets is volatility
asymmetry, which cannot be captured by the above GARCH model. To capture this phenomenon, we
also use the EGARCH model proposed by Nelson (1991). Specifically, we use the EGARCH(  )model *! 



*



")

- )







E )



 

! 

(15)

While the GARCH model specifies the process of  , the EGARCH model specifies that of its logarithm. Thus, it does not require non-negativity constraints for parameters. If "  , it is consistent
with the volatility asymmetry in stock markets. In this model, volatility is stationary if !  , andthe speed for which the shock to volatility decays becomes slower as ! approaches to one. Since ) is assumed to follow the standard normal distribution, E )  ./
Neither the GARCH nor EGARCH models allow volatility to have long-memory property. Hence,
we also use the FIEGARCH model proposed by Bollerslev and Mikkelsen (1996). Since this model isan extension of the above EGARCH model to allow the long-memory of volatility, it can also capture
the volatility asymmetry. We use the following FIEGARCH(   ) model.


!



  

*



")



- )





E )



 

! 

(16)

Similarly to the EGARCH model, it is consistent with the volatility asymmetry in stock markets if"  . As for , the same argument as that for ARFIMA model holds.
FIGARCH (Baillie et al., 1996) and FIAPGARCH (Tse, 1998) models can also take into account7

the possibility that the volatility follows a long-memory process. These models, however, have somedrawbacks. First, the variance of return will be infinite even though    (Schoffer, 2003).
Second, the parameter constraints to guarantee that the volatility is always positive are complicated(Conrad and Haag, 2006). Thus, we do not use these models in this article. We estimate parametersin the GARCH, EGARCH and FIEGARCH models using the maximum likelihood method 5.

4 Option Pricing
We first calculate option prices under the assumption of risk neutrality. If the traders are risk neutral,the expected return may be represented byE   (17)where and are continuously compounded risk-free rate and dividend rate. 6
The price of European option will be equal to the discounted present value of the expectation of
option prices on the expiration date. For example, the price of European put option with the exercise
price  and the maturity 0 is given by1     

0 E Max



(18)

where   is the price of the underlying asset on the expiration date  0 .
We cannot evaluate this expectation analytically if the volatility of the underlying asset follows
ARFIMA(X), HAR(X) or ARCH type models. We calculate option prices by simulating   from5

See Taylor (2001) for the estimation method for the FIEGARCH model.
It would be important to relax the assumption of risk neutrality if the risk is priced in the market. For example, theoption pricing models with realized volatility proposed by Christoffersen et al. (2010) and Corsi et al. (2011) allow for
more flexible specification of the expected return such that6EØ

 Ø ½     



 ¾
Ø 



 ¾ Øwhere  denotes the market price of risk and    corresponds to risk neutrality. Christoffersen et al. (2010) define  Ø¾ as¾¾
 Ø  Ø     Ø
a weighted average of two components by daily return innovation and realized volatility such that 
and Corsi et al. (2011) adopt an estimates of integrated volatility as 
 Ø¾ . As introduced in Stentoft (2008), this specification
allows us to directly test statistical significance of  using realized volatility, i.e.,ØØ    




Ø



Ø



 











   



Ø 

Ø

Ø

 
Ø Ø

Ø

We estimate the parameter  by the linear regression procedure with Newey and West (1987) standard errors and different
realized volatilities,
while its statistical significance is not obtained. Even in the other specification like Ø     ½        , the parameter  is not significant at the 5% level, although it is significant at the 10% level
Ø
Ø
Ø Ø
¾
in some regressions with different realized volatilities. These results imply that the data used here may be insensitive tothe market price of risk.

8



ARFIMA(X), HAR(X) or ARCH type models. Suppose that   (18) may be calculated as follows.

1  

0




2 



    are simulated. Then,   

Max

(19)

We set 2. For variance reduction, we used the control variate and the Empirical Martingale
Simulation proposed by Duan and Simonato (1998) jointly.
Duan (1995) relaxed the assumption of risk neutrality to derive option prices when the price ofunderlying asset follows ARCH type models. We also use this method. Following Duan (1995), wesetE   

3 
(20)
where 3 captures the risk premium.
Unless the traders are risk neutral, we must convert the physical measure 1 into the risk neutral measure 4 and evaluate the expectation in equation (18) under the risk neutral measure 4.
Duan (1995) makes the following assumptions on 4, called local risk-neutral valuation relationship(LRNVR).
1.

 follows a normal distribution under the risk neutral measure 4.

2. E 







3. Var 





Var .
  a.s.

Under assumptions and , daily returns under the risk neutral measure 4 must be represented by



5

5

6

6  NID  (21)Comparing equation (21) with equations (13) and (20) leads to

(

5

3 

(22)

)

6

3

(23)

Since assumption  means that volatilities are the same between 1 and 4, all we have to do forvolatility is to substitute equations (22) or (23) into ( in the GARCH volatility equation or ) in the
EGARCH and FIEGARCH volatility equations. For example, the GARCH(  ) volatility equation
will be
* %  +5  3    * ,  % + (24)Equations (21) and (24) constitute GARCH(  ) model under 4. Hence, we can evaluate the option
prices as follows.
[1 ] Estimate the parameters 3, * , % and + in GARCH(  ) model under 1 that consists of equations (13), (20) and (14).
9[2 ] Simulate   using GARCH(  ) model under 4 that consists of equations (21) and (24) bysetting the parameters 3, * , % and + equal to their estimates in [1].


[3 ] Substitute       simulated in [2] into equation (19) to obtain the option price.

Similarly, we can calculate the option price using the EGARCH and FIEGARCH models. The
EGARCH (  ) and FIEGARCH(   ) volatility equations under 4 will be 


*

!



! *  

*





"&



3

"&



3

&  3- &  3

-




./ 

(25)

./

(26)

For comparison, we also calculate option prices using the Black-Scholes formula with volatility
 as the standard deviation of daily returns over the past 20 days.

5 DataWe analyze the Nikkei 225 stock index options traded at the Osaka Securities Exchange. The underlying asset is the Nikkei 225 stock index, which is the average of the prices of 225 representative
stocks traded at the Tokyo Stock Exchange. The sample period is from May 29, 1996 to September
27, 2007. Following equation (12), we calculate the daily returns for the underlying asset as the logdifference of the closing prices of the Nikkei 225 index in consecutive days. Table 2 summarizes thedescriptive statistics of the daily returns (%) for the full sample. The mean is not significantly different from zero. While the skewness is not significantly different from zero, the kurtosis is significantly
above , indicating the well-known phenomenon that the distribution of the daily return is leptokurtic.
LB(10) is the Ljung-Box statistic adjusted for heteroskedasticity following Diebold (1988) to test thenull hypothesis of no autocorrelations up to lags. According to this statistic, the null hypothesis is
not rejected at the 1% significance level although it is rejected at the 5% level. We do not consider
autocorrelations in the daily return in the following analyses.
We calculate realized volatility using the Nikkei NEEDS-TICK data. This dataset includes the
Nikkei 225 stock index for every minute from 9:01 to 11:00 in the morning session and from 12:31
to 15:00 in the afternoon session. Sometimes, the time stamps for the closing prices in the morningand afternoon sessions are slightly after 11:00 and 15:00 because the recorded time shows when the
Nikkei 225 stock index is calculated. In such cases, we use all prices up to closing prices. Using theseprices, the 34 daily different realized volatilities listed in Table are calculated with or without using
the adjustment coefficient  defined by equation (5).
Before the computation of the 34 daily different realized volatilities, we provide the realized
volatility signature plots in Figure 1 to roughly gauge the impact of microstructure frictions containedin the high-frequency returns of the Nikkei 225 index. The signature plots are generated by the sample
mean of1min,2min,    ,20min based on equation (3) as a function of the samplingfrequency of the Nikkei 225 intraday returns.
(solid line) and(dotted line) correspond10

to the realized volatility with the squares of overnight and lunch-time returns or the Hansen and
Lunde (2005a) adjustment, respectively. If there is no severe microstructure noise, both plots shouldbe leveled off at the frequencies. However, we can find the large impact of microstructure noise, as
evidenced by a rapid decline in the plot ofwith short measurement intervals. On the other
hand,
applying the Hansen and Lunde (2005a) adjustment comparatively stabilizes for all
sampling frequencies, although there is a gradual increase in the range of 1-7 minutes. This result
implies that the Hansen and Lunde (2005a) adjustment, where the mean of realized volatility is equal
to the sample variance of daily returns, may play a role to partially offset the bias caused by the
microstructure noise as well as removing the discretization noise attributed to non-trading hours.
Figure 2 plots some kinds of realized volatilities and Table 3 summarizes the descriptive statistics of the 34 daily different realized volatilities. From
1min   to       are adjusted
such that the mean of realized volatility is equal to the sample variance of daily returns, but their meansare different because the adjustment coefficient  is calculated day by day using the past 1200 realized volatilities and daily returns. From
1min to      are not adjusted and theirmeans are much lower than those of the others. Among the 17 realized volatilities with the Hansen
and Lunde (2005a) adjustment,
1min   has the smallest standard deviation.
20min  has
the largest standard deviation of them as induced by the range from the minimum at
 to the
maximum at    . The standard deviation of  is the smallest of all. These results
are confirmed by Figure 2. Figure 2(a) shows that
15min  is more volatile than
1min and  , and Figure 2(b) shows that
1min is smaller on average and less volatile1min . The values of skewness and kurtosis indicate that the distributions of all realthanized volatilities are non-normal. LB(10) is so large that the null hypothesis of no autocorrelation isrejected. Table 3 (b) shows the descriptive statistics for log-realized volatilities. They are qualitatively the same as those of Table 3 (a) except skewness and kurtosis. While realized volatilities are
positively skewed, log-realized volatilities are negatively skewed at the 5% significant level except
10min , 15min , 20min  , 10min , 15min 
and 
20min . The kurtosis of log-realized volatilities is much smaller than those of realized volatilities. The kurtosis of 1min    and 1min  is not significantly above at the 5% level. The distributions of log-realized volatilities are much closer to the normal distribution
than those of realized volatilities. Thus, we use log-realized volatility as a dependent variable in the
ARFIMA model (8), HAR model (9), ARFIMAX model (10) and HARX model (11).
To measure the performance of option pricing, we also use prices of the Nikkei 225 stock index
options traded at the Osaka Securities Exchange. Nikkei 225 stock index options are European options
and their maturities are the trading days previous to the second Friday every month. Considering
theoretical option prices with respect to a risk neutral measure, we assess the performance of option
pricing using options which are most likely to be efficiently priced. For the Nikkei 225 stock index
options, put options are traded more heavily than call options. For the maturity, option trading seems
to be more active during the week following an expiration date and the options with the maturity morethan one month are not traded so much. Thus we concentrate on put options whose maturity is 30 days11

(29 days if the day when the maturity is 30 days is a weekend or holiday). On such days, we consider
put options with different exercise prices whose bid and ask prices are both available at the same time
between 14:00 and 15:00. For each option, we use the average of bid and ask prices at the same timeclosest to 15:00 as the market price at 15:00. The reason why we use the average of bid and ask prices
instead of transaction prices is that transaction prices are subject to market microstructure noise due
to bid-ask bounce (Campbell et al., 1997). We also exclude some kinds of put options which are notpriced at the theoretical range from the lower bound at 1 Max   exp 0   exp 0  tothe upper bound at 1  exp 0 .
Following Bakshi et al. (1997), we classify put options into five categories such as DITM (deep-inthe-money), ITM (in-the-money), ATM (at-the-money), OTM (out-of-the-money) and DOTM (deepout-of-the-money) using the moneyness which is the ratio of the underlying asset price over the exercise price. Table  shows this classification. We examine the performance in each category as wellas in total. Table  describes the put prices calculated as the average of bid and ask prices and theBlack-Scholes implied volatilities for each moneyness. The average put prices range from 21.30 yen
in DOTM to 2859.70 yen in DITM. Options in DOTM and OTM account for 41% and 16% of thetotal sample. The implied volatilities form a smile pattern where options in DOTM and DITM arecharacterized by higher volatilities more than 30% compared with 22.86% for options in ATM.
We estimate the ARFIMA(X) and HAR(X) models using 1200 daily realized volatilities up to
the day before the options whose maturity is one month are traded, where the adjustment coefficient defined by equation (5) is calculated using the same 1200 realized volatilities with 1200 daily returns. We also estimate ARCH type models using the same 1200 daily returns with risk-free rate anddividend. As mentioned, the daily returns are calculated as the log difference of closing prices. We
use CD rate as a risk-free rate and fix the annual dividend rate as  following Nishina and Nabil(1997). The first date when options whose maturity is one month are traded is April 11, 2001. We
first estimate the parameters in the ARFIMA(X), HAR(X) and ARCH type models using 1200 daily
realized volatilities and returns up to April 10, 2001, where we calculate the adjustment coefficient using the same 1200 daily realized volatilities and returns. Then, given the obtained parameter estimates, we calculate the put option prices on April 11, 2001 using CD rate and the Nikkei 225 index
at 15:00 on that date. The next date when options whose maturity is one month are traded is May 9,
2001. We first estimate the parameters in the ARFIMA(X), HAR(X) and ARCH type models using
1200 daily realized volatilities and returns up to May 8, 2001, where we calculate the adjustment
coefficient  using the same 1200 daily realized volatilities and returns. Then, given the obtained
parameter estimates, we calculate the put option prices on May 9, 2001 using CD rate and the Nikkei
225 index at 15:00 on that date. We repeat this procedure up to September 2007.
Figure 3 plots the estimates of all parameters in all models for each of the above 78 iterations.
Figure 3 (a) and (b) plot the estimates of parameters in the ARFIMA and ARFIMAX models using
15min . The estimates of in the ARFIMA and ARFIMAX models move around 0.5 andare above 0.5 in the latter half, indicating the long-memory and the possibility of non-stationarity oflog-realized volatility. The estimates of  in the ARFIMAX model are positive for all periods, indi12

cating the well-known phenomenon of a negative correlation between today’s return and tomorrow’s
volatility. Figure 3 (c) and (d) plot the estimates of parameters in the HAR and HARX models using
15min . The positive estimates of % , % and % in the HAR and HARX models for all periods
are consistent with the empirical results using S&P500 in Corsi (2009). The estimates of % in the
HARX model are positive, indicating the asymmetry in volatility. Figure 3 (e), (f) and (g) plot theestimates of parameters in ARCH type models using daily returns. The sum of the estimates of % and+ in the GARCH model and the estimates of ! in the EGARCH model are close to 1 for all periods,
indicating the well-phenomenon of volatility clustering. These models, however, do not allow for the
long-memory of volatility. The estimates of in the FIEGARCH model are more volatile than thoseof the ARFIMA(X) model. They move around 0.2 in the first half while they move up to 0.54 anddown to 0 in the latter half. These results provide evidence that a structural change may occur during
our sample period, but we leave it for future research. The estimates of " in the EGARCH and FIEGARCH models are negative for all periods, indicating a negative correlation between today’s returnand tomorrow’s volatility.

6 Results
To measure the performance of option pricing, we use four loss functions, MAE (Mean Absolute
Error), RMSE (Root Mean Square Error), MAPE (Mean Absolute Percentage Error) and RMSPE(Root Mean Square Percentage Error) defined as

MAE

MAPE




 

1



1

 

1 

1
1



RMSE





RMSPE








1

1




1

1

 

 





1

where  is the number of put options used for evaluating the performance, 1 is the price of the
7th put option calculated by each model and 1  is its market put price calculated as the average ofbid and ask prices at the same time closest to 15:00. From the fact that the lowest market put priceamounts to 1.5 yen which is calculated as the mid-point of the ask price at 2 yen and the bid price at
1 yen, any price 1 less than the lowest price is approximated at 1.5 yen. MAE and RMSE, which are
the absolute metrics, assign a lot of weight to options with high valuations such as DITM and ITM.
For MAPE and RMSPE as the relative metrics, much more weight may be put on DOTM and OTM
options with valuations close to zero.
Table  shows the values of loss functions for ARCH type models with daily returns, the ARFIMA(X) and HAR(X) models with
15min  and the BS model. In total, the ARFIMAX model performs best except for RMSE, while the HARX model performs best for RMSE. In DOTM, ARFIMAX
model performs best for RMSPE and MAPE while the FIEGARCH model performs best for the other13

loss functions. In OTM and ATM, the ARFIMAX model performs best except for RMSE and it performs best for RMSPE and MAPE in ITM. In DITM, the GARCH model performs best for all loss
functions. Although there are some exceptions depending on moneyness and loss function, we may
conclude that the ARFIMAX model performs best.
Tables  and  show the values of loss functions for the ARFIMAX model with 34 different realized volatilities. Table  shows the result for the realized volatilities calculated simply by adding
the squares of overnight and lunch-time returns instead of using the Hansen and Lunde (2005a) ad1min , which does not take acjustment. In total and all moneyness, the loss functions ofcount of microstructure noise at all, have larger values than those of the other realized volatilities
except  . This result is intuitive because1min is seriously affected by the
microstructure noise as shown by the realized volatility signature plot in Figure 1. Thus we conclude that the option pricing performance is improved by applying methods to remove microstructurenoise-induced bias in realized volatility.
Table  shows the result for the realized volatilities calculated using the Hansen and Lunde (2005a)
adjustment instead of adding the squares of overnight and lunch-time returns. In total, most loss
functions in Table  are smaller than those in Table  regardless of realized volatilities with or without
taking account of microstructure noise. The improvement could be seen for all loss functions in
DOTM and DITM and for RMSE and MAE in OTM. These results indicate that the Hansen and
Lunde (2005a) adjustment may improve the performance of option pricing. It is also noteworthy
that the performance of1min is no longer bad compared with the other realized volatilities,
15min  ,20min  and   perform best in total. This means
althoughthat the Hansen and Lunde (2005a) adjustment plays a role to remove not only the discretizationnoise included in the squares of the lunch-time and overnight returns but also partially offset the bias
caused by microstructure noise. This finding is also consistent with the result that the signature plotofin Figure 1 comparatively stabilizes for all sampling frequencies. Judging from the resultsin Tables  and , we conclude that: (1) the Hansen and Lunde (2005a) adjustment for removing thediscretization noise induced by non-trading hours improves the performance, (2) methods for reducing
microstructure noise-induced bias yield better performance, while if the Hansen and Lunde (2005a)
adjustment, which plays an additional role to partially offset the microstructure noise-induced bias, isused, they are not necessarily needed. 7 8
7

We have focused on put options whose maturity is 30 days so far. Following Barone-Adesi et al. (2008), the maturityof 30 days can be classified as short maturity. We also analyze put options whose maturity is 90 days classified as medium
maturity. The result, which is not reported in this article to save the space, also supports an evidence that the option pricing
performance is improved by taking account of microstructure noise but it does not necessarily improve the performance
when the Hansen and Lunde (2005a) adjustment is employed.
8Bandi et al. (2008) compare the option pricing performance of the realized volatilities of the S&P 500 index. Theirmethod is, however, different from ours as follows. (1) They compare the profits from the straddle trading strategy obtained by substituting the volatility forecasts from the ARFIMA model for realized volatility into the Black-Scholes option  ÀÄ, which are calculated using
pricing formula. (2) They only analyze the performance of  5min ÀÄ to
the Hansen and Lunde (2005a) adjustment, while we also analyze the performance of  1min ÀÄ and  1minËÊto  ËÊ, which are calculated by adding the lunch-time and overnight returns without using the Hansen and
Lunde (2005a) adjustment. (3) They do not analyze ARCH-type models.

14

So far, we assumed risk neutrality. As explained in Section , Duan (1995) has proposed a method
for GARCH option pricing relaxing this assumption. We also apply this method to the GARCH,
EGARCH and FIEGARCH models. Table  shows the result. The values of loss functions using
this method are not so much different from those assuming risk neutrality. This result means that theDuan (1995) method does not improve the performance of option pricing compared with assuming
risk neutrality.
Financial markets sometimes display asset price discontinuities, so-called jumps. If jump-diffusion
processes are used instead of equation (1), the realized volatility measures used in this article include
not only integrated volatility but also jump variation. Andersen et al. (2007) conclude that the performance of forecasting future realized volatility could be improved by using the HAR model withrealized volatility separately from significant jumps. In this subsection, the realized volatility without
significant jumps is constructed to assess its contribution to option pricing performance.
We employ the following procedure to remove significant jump variation. Barndorff-Nielsen andShephard (2004) propose a consistent estimator of the integrated volatility unaffected by jumps called
the realized bipower variation



















 



(27)



./. The bipower variation, which is originally defined as the sum of the products of
where adjacent absolute returns, is influenced by microstructure noise. To mitigate the noise-induced bias,
we construct the following realized bipower measure based on skip-one returns introduced in Huang
and Tauchen (2005) and Andersen et al. (2007)
 























(28)

We calculate   calculated from 15 minute high-frequency returns for which the bipower variation
signature plots in equation (28) are leveled off. Identification of the significant jumps is based on an
asymptotically normal test statistic using realized volatility and bipower variation













  Max   4   



(29)

where

 4
 

 








 





 






 






 


(30)

. .   4 is called the realized tripower quarticity based on skip-one returns which converges to the15integrated quarticity of continuous processes.
in equation (29) is replaced by15min during
the market open. Thus daily realized volatility without significant jumps (-8#2) is constructedby the realization of  and some critical value 

8#2

  ,       



(31)

where    denotes the indicator function. The proportion of day with significant jumps based on
significance levels +  takes 5.2%, 13.1%, 20.2% and 26.0% for thefull sample, respectively. We estimate the ARFIMAX model and simulating option prices using the-8#2 based on +, which is the same setting as Andersen et al. (2007).
summarizes the values of loss functions for15min  and-8#2 with the
Table
Hansen and Lunde (2005a) adjustment. The values of loss functions for-8#2 are not so muchdifferent from those for15min . Judging from the result in Table , we may conclude thatthe option-pricing performance is not necessarily improved by removing significant large jumps fromrealized volatility.

7 Conclusions
This article analyzes whether realized volatility is useful for option pricing. Different realized volatilities are calculated with or without taking account of microstructure noise, with or without using
overnight and lunch-time returns and with or without separating significant large jumps from realized volatility. This article compares the performance of option pricing among the ARFIMA(X) andHAR(X) models with daily realized volatility and the ARCH models with daily returns. Main results
using the Nikkei 225 stock index and its put options prices are: (1) ARFIMAX model with daily realized volatility performs best, (2) the Hansen and Lunde (2005a) adjustment without using overnight
and lunch-time returns can improve the performance, (3) the performance is improved by removing
the bias from microstructure noise, while if the Hansen and Lunde (2005a), which also plays a role toremove the microstructure noise-induced bias by setting the sample mean of realized volatility equal
to the sample variance of daily returns, is used, the other methods for taking account of microstructurenoise do not necessarily improve the performance, (4) the performance is not necessarily improvedby removing significant large jumps from realized volatility and (5) the Duan (1995) method does not
improve the performance compared with assuming the risk neutrality.
Several extensions are possible. First, Jacod et al. (2009) propose an alternative realized volatility
using preaveraging approach robust to microstructure noise. Andersen et al. (2012) propose two
new jump-robust estimators of integrated volatility called the minimum or median realized volatility.
It is interesting whether the performance of option pricing will also be improved by applying theirestimators. Second, Hansen et al. (2012) and Takahashi et al. (2009) have proposed to model daily
returns and realized volatility jointly. They extend ARCH type models and the stochastic volatility
model respectively. It is also interesting to apply their methods to option pricing.
16

Appendix Integrated volatility estimators with microstructure
noise
Here, we give a detailed review of various realized volatilities using the high-frequency returns employed in our analysis. Assume the 7-th intraday return for day  contaminates with microstructure noise as follows


where :

 



7.



7

.

9 



7.



7

.

:

9 

7.

9 

7

7.

9 

7

 

.

(A.1)

. and 9 represents microstructure noise.

Realized volatility with 1-, 5-, 10-, 15- and 20-minute returns,
10min,15min and20min.

1min,5min,Without microstructure noise, it would be desirable to use intraday returns sampled at the highest
frequencies. Since the highest frequencies available for the Nikkei 225 stock index is 1-minute, we ), which is denoted as1min.
first calculate realized volatility using 1-minute returns (However, it may fail to satisfy the consistency condition when there is market microstructure noise
as usually documented in real high-frequency data. Another classical approach is to use realized
volatility constructed from intraday returns sampled at moderate frequencies rather than at the highest
frequencies. This approach can partially offset the bias of the microstructure effect. In practice,researchers are necessarily forced to select a moderate sampling frequency. For example, it may be
regarded as around those frequencies for which realized volatility signature plots under alternative
sampling frequencies are leveled off. We provide the realized volatility signature plots in Figure 1
to roughly gauge the impact of microstructure frictions contained the high-frequency returns of the
Nikkei 225 index. In addition, evidence from previous studies suggests that it may be optimal to use
5 to 30-minute return data. Hence, we employ
5min,10min,15min and20minwhich are equal to the sum of squared 5-, 10-, 15- and 20-minute returns (   and ),
respectively.
Optimally-sampled realized volatility, .

The selection of a moderate sampling frequency is important to get an accurate estimate of the integrated volatility because the noise-induced bias at high sampling frequencies can be traded off withthe variance reduction obtained by high-frequency sampling. To take this trade off between the bias
and variance into account, Bandi and Russell (2008) provide a theoretical justification for the choiceof optimal sampling frequency in terms of the mean squared error (MSE) criterion. They derive thefollowing approximated optimal number of observations   based on the minimization of MSE in a17

finite sample

 



4
E: 





(A.2)



  ).
where 4 represents an integrated quarticity of the equilibrium price process ( 4It is estimated by 4 (realized quarticity) with low frequency returns such as 15
minute returns. Following the consistent estimator of noise moment as shown by Bandi and Russell : 
(2008), E:  can be estimated by E at the highest frequencies. Thus, the
optimally-sampled realized volatility, , is equal to the realized volatility with the optimal



4.E
 : 
number of observations calculated as .
The Bartlett-type kernel estimator in Barndorff-Nielsen et al. (2004b) with a finite sample
optimal number of autocovariances proposed by Bandi and Russell (2011),   .
1min,5min,15min and  have the obvious drawback that they do not incorporate all data and whereby information is lost. The methods introduced here take advantage ofthe rich sources in all high-frequency data. The problem of estimating the integrated volatility under
microstructure noise is similar to the autocorrelation corrections that are used in the long-run variance estimation in stationary time-series (Newey and West, 1987; Andrews, 1991). So it is natural
to consider kernel-based estimators of integrated volatility under microstructure noise. The literatureincludes the earlier study by Zhou (1996) who proposes a particular kernel estimator which incorporates the first-order autocovariance. Barndorff-Nielsen et al. (2004b) derive kernel-based estimatorsthat are far more precise than that of Zhou (1996). They examine the Bartlett-type kernel estimatordefined as











-



 









- 

(A.3)




where 
  is the -th autocovariance of intraday returns and - is equal
to realized volatility using returns sampled at the highest frequencies. This estimator weights therealized volatility and the  -th return autocovariances by Bartlett weights. The optimal number of
autocovariances is given by the minimization of MSE of the estimator in finite sample (see equation
7 to 10 in Bandi and Russell, 2011 for exact MSE minimization expressions). There is a convenient
rule-of-thumb for choosing  in practice as proposed in Bandi and Russell (2011). The expression is
obtained as

 




 4





(A.4)

denotes integrated volatility.
and 4 are estimated using realized volatility and realized
wherequarticity with lower frequency returns such as 15-minute returns. Hence,  with a finite sample
18

optimal number of autocovariances   leads to   .
The two-scale estimator with an asymptotically optimal number of subsamples proposed byZhang et al. (2005), .
Zhang et al. (2005) propose a two-scale or subsampling estimator in the spirit of the estimation of
the long-rum variance studied by Carlstein (1986). Denote the original grid of observation times as nonoverlapping subgrids,

 
. 
.   . Consider  is partitioned into  
 , for example, the first sub-grid starts at  th arrival time , ; 
and takes every 



(


.

.
), and the second sub-grid starts at . and th arrival time ( .  .
takes every 

. 




).

 
Then, the realized volatility for the subgrid   is defined as
 


where
and 




  





(A.5)

 .
;
7
 .. The two-scale estimator in Zhang et al. (2005) is given by   is subsampling return between transaction prices at times 

;

7




 .




 


 


.



(A.6)


 and

 
.
is the realized volatility for the full grid . The second term
where    derived
corrects the bias in the first term. The asymptotic optimal number of subsamples 
by minimizing the estimator’s asymptotic variance is given by  
 E: 
4


 

(A.7)



 :   
4 and E:  are estimated by realized quarticity with 15-minute returns and E





at the highest frequencies, respectively. Thus,  is equal to  with  .

The two-scale estimator in Zhang et al. (2005) with a finite sample optimal number of subsamples proposed by Bandi and Russell (2011),  .
Barndorff-Nielsen et al. (2004b) show that  in (A.6) can be written as follows











 



-







-



" 

(A.8)

where ", and " "              for    . The third term guarantees consistency of  and differentiates  from the inconsistent19 . This equation implies the two-scale estimator in Zhang et al. (2005) is almost identical to themodified Bartlett kernel estimator. Bandi and Russell (2011) additionally show that the finite sample
MSEs of  and  are very similar in practice. Hence, the  with   in (A.4)
corresponds to  .
The bias-corrected two-scale estimator in Zhang et al. (2005) with an asymptotically optimal
number of subsamples proposed by Zhang et al. (2005),   .
The two-scale estimator  has a finite sample bias as shown in Zhang et al. (2005) who provide
the approximate correction for this bias. On the other hand, Bandi and Russell (2011) report theexact bias-correction form. Following a suggestion by Bandi and Russell (2011), the bias-corrected
estimator is defined as 
 



 















(A.9)

 

Since   is asymptotically equivalent to , the asymptotically optimal number of sub  . Thus,   with    can be described by  
samples is given by .
The bias-corrected two-scale estimator in Zhang et al. (2005) with a finite sample optimal
number of subsamples proposed by Bandi and Russell (2011),    .
Since   is unbiased in a finite sample, the optimal number of subsamples is provided byminimizing the finite sample variance of  . Bandi and Russell (2008, 2011) show that the
optimal number of subsamples is defined as   

 ! "# Var   .,where, if .

Var

 ! "#

 




 4








 








 4












4























20









Var (A.10)





 


4


 4





 



4














4



 









4






 4






4


























(A.11)where  represents a variance of microstructure noise 9 and is estimated by     leads to    .
at the highest frequencies. Hence,   with The flat-top Bartlett kernel estimator with an asymptotically optimal number of autocovariances
proposed by Barndorff-Nielsen et al. (2008),    .
Barndorff-Nielsen et al. (2008) examine the following unbiased flat-top kernel type estimator (called
the realized kernel)



-




$ < -- 

(A.12)      and the non-stochastic $<   
where 
  with 


for <
is a weight function. The flat-top Bartlett kernel estimator is equivalent to  in<. For this class of kernels, Barndorff-Nielsen et al. (2008) show that thecase where $ <asymptotic distribution of is mixed normal with zero mean and rate of convergence  
when    where  is a constant. Then, the asymptotically optimal value of  which minimizes
the asymptotic variance is given by 
where =

 .

4. Hence,  with $<=  

(A.13)< and    corresponds to    .

The flat-top cubic kernel estimator and the flat-top modified Tukey-Hanning kernel estimatorwith an asymptotically optimal number of autocovariances proposed by Barndorff-Nielsen etal. (2008),     and     .
The estimators based on the cubic kernel and the modified Tukey-Hanning kernel are equivalent to with $<<< and $<
cos/ 
< . , respectively. When  = , for this class of kernels is consistent at the rate of convergence   as shown in Barndorff-Nielsen
et al. (2008). The asymptotically optimal value of  is expressed as  
 $

$







$ $ $ 

(A.14)

. 4, $$< <, $$ < < and $$ < <, where thewhere primes represent derivatives. The values of ($  $  $  ) amount to $  $  $     
21 for the cubic kernel and $  $  $        for the modified Tukey-Hanningkernel. We define     and      as  with  = at $<<< and $<
cos/ 
< . .

The flat-top Bartlett kernel estimator, the flat-top cubic kernel estimator and the flat-top modified Tukey-Hanning kernel estimator with a finite sample optimal number of autocovariances
proposed by Bandi and Russell (2011),    ,     and     .
Bandi and Russell (2011) provide an alternative way to choose the number of autocovariances in finite
samples. Denote  as Æ with  Æ  . The optimal value of Æ is defined in Theorem 3 of Bandiand Russell (2011) as follows

where bias  Æ ! "# bias  4
* $ *
 * $ * Var   

Æ

and

Var   * $ * 

 





$   



$ 

;

;



$   $   $   

;

$ 

;



and zeros everywhere else. For ;  Æ

$ 



;

$  

;

$ ;

;

$ ;

;

$ 

; ; $   
;


;

. 






* $ *  Æ$   $ ; ;

(A.16)) square matrices.



(A.17), the matrices $ and $ are defined as$   $   



;



$ 

;

;



$   $   

 

;



$ 

;

;

;

$ 

; ; $        are (Æwith *
  $ Æand $ >     $ ÆÆ For ;  Æ, the matrices $ and $ are defined as$   (A.15)$   $ $ ;

;


;
;

. 

;

$ 

; ; 

;

(A.18)and zeros everywhere else. Thus,  with  Æ   for the Bartlett kernel, cubic kernel and modified
Tukey-Hanning kernel leads to    ,     and     , respectively.

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27

28

Methods of calculating realized volatility
Realized volatility with returns sampled at the highest frequencies
Realized volatility with 5-minute returns
Realized volatility with 10-minute returns
Realized volatility with 15-minute returns
Realized volatility with 20-minute returns
Optimally-sampled realized volatility as proposed in Bandi and Russell (2008)
Bartlett kernel estimator in Barndorff-Nielsen et al. (2004b) with a finite
sample optimal number of autocovariances in Bandi and Russell (2011)
Two-scale estimator with an asymptotically optimal number ofsubsamples proposed by Zhang et al. (2005)
Two-scale estimator with a finite sample optimal number ofsubsamples proposed by Bandi and Russell (2011)
Bias-corrected two-scale estimator with an asymptotically optimal number ofsubsamples proposed by Zhang et al. (2005)
Bias-corrected two-scale estimator with a finite sample optimal number ofsubsamples proposed by Bandi and Russell (2011)
Flat-top Bartlett kernel estimator with an asymptotically optimal number of
autocovariances proposed by Barndorff-Nielsen et al. (2008)
Flat-top Bartlett kernel estimator with a finite sample optimal number of
autocovariances proposed by Bandi and Russell (2011)
Flat-top cubic kernel estimator with an asymptotically optimal number of
autocovariances proposed by Barndorff-Nielsen et al. (2008)
Flat-top cubic kernel estimator with a finite sample optimal number of
autocovariances proposed by Bandi and Russell (2011)
Flat-top modified Tukey-Hanning kernel estimator with an asymptotically optimal
number of autocovariances proposed by Barndorff-Nielsen et al. (2008)
Flat-top modified Tukey-Hanning kernel estimator with a finite sample
optimal number of autocovariances Bandi and Russell (2011)










 



      







 





 



      

              

 

 

 







      

Methods of daily adjustment
Hansen and Lunde
Squares of overnight(2005a) adjustment
and lunch-time returns
 1min 1min 5min 5min 10min 10min 15min 15min 20min 20min

Table 1: List of daily realized volatilities

Table 2: Descriptive statistics of daily returns

 )
 
 
   ( ) (  )
 

Mean

(
Std.
Min
Max
Skewness
Kurtosis
LB(10)The numbers in parentheses are standard errors.
LB(10) is the Ljung-Box statistic adjusted for heteroskedasticity following Diebold (1988) to test thenull hypothesis of no autocorrelations up to  lags.

29

30

ÀÄ

ÀÄ

ÀÄ

















ËÊ











 



 



  



 











 







(  )(  )
( 
)
( 
)
(  )(  )(  )( )(  )(   )(  )(  )(  )(  )(  )(  )(  )              

 





 

 















 

 



 







          

 



Min
 



































 







 

 













 



Max



















 













 



Skewness
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)























 



 

 





 

  











Kurtosis(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )


















  

 

 





  

   





 






 





 

 

LB()(  )     


 
( 
)  (  )   ( )    



( 
) 
(  ) ËÊ(  )    

( 
)
 
(  )  ËÊ (  )  
 
  

( 
)
  
(  )
 
ËÊ


(  )       

( 
)
 
(  )


ËÊ
 
( )  
 


 
( 
)   (  ) ËÊ  ( )       ( )  (  )   ËÊ (  )        ( )  (  )  ËÊ  (  )     ( )   (  )   ËÊ ( )   ( )    (  )  ËÊ ( )        
 
( 
)   (  )  ËÊ(  )      

 
( 
) 
(  )  ËÊ(  )       ( )    (  ) ËÊ( )       


( 
)  (  ) ËÊ(  )    


 
( 
)    (  )  ËÊ(  )     ( )  (  ) ËÊ(  )       
 
( 
) 

(  )   
The numbers in parentheses are standard errors. LB(10) is the Ljung-Box statistic adjusted for heteroskedasticity following Diebold(1988) to test the null hypothesis of no autocorrelations up to  lags.

ËÊ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

 1min 5min 10min 15min 20min                           
 1min 5min 10min 15min 20min                          
 

Std.

Table 3: Descriptive statistics of daily realized volatilities(a) Daily realized volatilities
Mean

31

ËÊ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ

ÀÄ


 

 





 





 





































  







Mean
(  )(  )
(  )(  )(  )(  )
(  )
(  )
(  )
(  )
(  )
(  )
(  )
(  )(  )
(  )(  )


 

 











 







 



 



  
 









 





 

 





 

 



Min
          



 





 



  

Std.










 







 

 

 



 









 

Max





 

 

 







   







 

 

 

 

 

 







 

 

 

 

Skewness
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
( 
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)









 

 

 

 











 



 



 

Kurtosis(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )(  )




















 

 







 




 



 









 

LB()ËÊ (  )      ( )(  ) (  )  


 
 

( 
)

(  ) ËÊ (  )     ( )(  ) ËÊ  (  )        ( )(  )ËÊ (  )   
 ( ) (  )ËÊ(  )     
 
 

( 
)

(  )ËÊ  (  )    

 
( 
)
 
(  )ËÊ(  )     ( )(  )ËÊ (  )     


 
( 
)

(  ) ËÊ(  )      
 
 

( 
)

(  )   ËÊ (  )         ( )(  )ËÊ (  )    


 

( 
)

(  )
 
ËÊ
 
(  )   


 

( 
)

(  ) ËÊ(  )    ( ) (  )
 
 
ËÊ
 
(  )        ( )(  ) ËÊ (  )        ( ) (  )
 
ËÊ
 
(  )        ( )(  )The numbers in parentheses are standard errors. LB(10) is the Ljung-Box statistic adjusted for heteroskedasticity following Diebold (1988) totest the null hypothesis of no autocorrelations up to  lags.

 1min  5min  10min  15min  20min                                          

 1min  5min  10min  15min  20min                                          

(b) Log daily realized volatilities

Table 4: Moneyness of put options
 
 



.
.
.
.
.











deep-in-the-money (DITM)
in-the-money (ITM)
at-the-money (ATM)out-of-the-money (OTM)
deep-out-of-the-money (DOTM)  price of underlying asset and   exercise price.

32

Table 5: Options data description

Sample size
Put price (yen)
Mean
Std.
Implied volatility (%)
Mean
Std.

DOTM
268

ITM
92

DITM
71

21.30 108.44 339.61 888.34
35.01 78.73 147.60 314.48

2859.701298.70

31.53
8.91

OTM
101

24.83
8.20

ATM
115

22.86
7.98

22.55
8.52

39.79
14.11

Put price and implied volatility are the average of the bid and ask prices and the BlackScholes implied volatility.

33

Table 6: Put option pricing performance using different models

Sample size
RMSE
GARCH
EGARCHFIEGARCH
ARFIMA
ARFIMAX
HAR
HARX
BS
MAE
GARCH
EGARCHFIEGARCH
ARFIMA
ARFIMAX
HAR
HARX
BS
RMSPE
GARCH
EGARCHFIEGARCH
ARFIMA
ARFIMAX
HAR
HARX
BS
MAPE
GARCH
EGARCHFIEGARCH
ARFIMA
ARFIMAX
HAR
HARX
BS

DOTM


OTM

ATM


ITM


DITM


Total


 
  

                                                            
 


 
 

                                            
 



 
             

 
 
 

 


 

 


 



 

 
 


 









 

 
 
 








 




 






 
 
 




 


 
 
 





 
 





 


 
 












 

 
 

 

 

The values of loss functions for the ARFIMA(X) and HAR(X) models are calculated using* indicates the best model which minimizes the loss function.

34

 15minÀÄ.

Table 7: Put option pricing performance using different realized volatilities without the Hansen and
Lunde (2005a) adjustment

Sample size
RMSE
1min
5min
10min15min20min   
                          MAE
1min
5min
10min15min20min   
                          

DOTM


OTM

ATM


ITM


DITM


Total


                                                                                                                           
            
 
  
 
 
  
 
  
 
 


 
  
 
 
                     
 




              
 
 
 
 
                                           
 

 


              This is calculated using the ARFIMAX model. * indicates the best model which minimizes the loss function.

35

Table 7: (Continued) Put option pricing performance using different realized volatilities without the
Hansen and Lunde (2005a) adjustment

Sample size
RMSPE
1min
5min
10min15min20min   
                          
MAPE
1min
5min
10min15min20min   
                          

DOTM


OTM




 


 
 

 




 

 


 
 
 
 
 

 
 





 





 

 
 
 
 
 

 
 
 
 
 
 
 

 

 






 






 

ATM


ITM


DITM


Total






 


 


 
 
 



 







 


 
















 


 




 
 
 
 
 
 
 

 


















 
















 
















 





 
 






 















This is calculated using the ARFIMAX model. * indicates the best model which minimizes the loss function.

36

Table 8: Put option pricing performance using different realized volatilities with the Hansen and
Lunde (2005a) adjustment

Sample size
RMSE
1min
5min
10min15min20min   
                            MAE
1min
5min
10min15min20min   
                            

DOTM


OTM

ATM


ITM


DITM


Total


 
  
 
 
 
               
             
  
 


                                                                                               

 

 






 
 

 
      
 
 
 
 
                          
 
 
  
             
 
 
  
 
          
 

 
                                  
 
 
 
 
          

This is calculated using the ARFIMAX model. * indicates the best model which minimizes the loss function.

37

Table 8: (Continued) Put option pricing performance using different realized volatilities with the
Hansen and Lunde (2005a) adjustment

Sample size
RMSPE
1min
5min
10min 15min 20min                                 
MAPE
1min
5min
10min 15min 20min                                 

DOTM


OTM

ATM



 

 
 
 
 
 
 
 
 


 
 
 







 
 
 













 
 
 
 
 
 
 
 
 

 

 
 
 
 



 
 







 

 




ITM


DITM


Total


 
 



 













 



 

 

























 


 






 





 





 
 








 

 


 

 































 
 

 





 


 


 

This is calculated using the ARFIMAX model. * indicates the best model which minimizes the loss function.

38

Table 9: Put option pricing performance of ARCH type models assuming the risk-neutrality and using
the Duan (1995) method

Sample size
RMSE
GARCH
Risk neutral
Duan
EGARCH
Risk neutral
Duan
FIEGARCH
Risk neutral
Duan
MAE
GARCH
Risk neutral
Duan
EGARCH
Risk neutral
Duan
FIEGARCH
Risk neutral
Duan
RMSPE
GARCH
Risk neutral
Duan
EGARCH
Risk neutral
Duan
FIEGARCH
Risk neutral
Duan
MAPE
GARCH
Risk neutral
Duan
EGARCH
Risk neutral
Duan
FIEGARCH
Risk neutral
Duan

DOTM


OTM

ATM


 
 

  

DITM


Total


                                    

                   

               

 
 


  
                  





 
           

ITM


 
 

 
 

 
 

 





 
 

 





 
 










 
 







 
























 
 




 





 


















 




 
 

“Risk neutral” shows the results assuming the risk-neutrality, which are the same as thosein Table 5. “Duan” shows the ones using the Duan (1995) method without assuming therisk-neutrality.

39

Table 10: Put option pricing performance with and without removing significant jumps from realized
volatility

Sample size
RMSE
RV
RV-Jump
MAE
RV
RV-Jump
RMSPE
RV
RV-Jump
MAPE
RV
RV-Jump

DOTM


OTM

ATM


        
  

ITM


DITM


Total


       

 
 

       


 

 
 

   

 



 







 
 








 

   









RV is  15minÀÄ . RV-Jump is the difference between RV and significant jump, which isdetected using the test statistic in (29) at the 0.1% significance level.

40Figure 1: Realized volatility signature plots
2.0

RVSR
RVHL

1.9
Average RVSR and RVHL

1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
1

3

5

7

9

11

13

Sampling frequency in minutes

41

15

17

19

Figure 2: Realized volatility(a) Some realized volatilities with the Hansen and Lunde (2005a) adjustment
RV(1min)HL
RV(15min)HL
RV(BR)HL

35

30

25

20

15

10

5

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

(b) With or without the Hansen and Lunde (2005a) adjustment
RV(1min)HL
RV(1min)SR

25

20

15

10

5

0

250

500

750

1000

1250

1500

42

1750

2000

2250

2500

2750

Figure 3: Parameter estimates
1.50(a) ARFIMA
d
μ
θ
σ2

1.25

1.00

0.75

0.50

0.25

0.00

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

(b) ARFIMAX
d
μ0
μ1
μ2
θ
σ2

1.00

0.75

0.50

0.25

0.00

-0.25
0

5

10

15

20

25

30

35

40

43

45

50

55

60

65

70

75

80

(c) HAR

β0
β1
β2
β3
σv2

0.4

0.3

0.2

0.1

0.0

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

β0
β1
β2
β3
β4
β5
σv2

(d) HARX
0.5

0.4

0.3

0.2

0.1

0.0

0

5

10

15

20

25

30

35

40

44

45

50

55

60

65

70

75

80

(e) GARCH
ω
β
α
β+ α

1.2

1.0

0.8

0.6

0.4

0.2

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

(f) EGARCH
1.0
ω
φ
θ
γ

0.8

0.6

0.4

0.2

0.0

0

5

10

15

20

25

30

35

40

45

45

50

55

60

65

70

75

80

1.2

(g) FIEGARCH
d
ω
φ
θ
γ

1.0

0.8

0.6

0.4

0.2

0.0
0

5

10

15

20

25

30

35

40

46

45

50

55

60

65

70

75

80