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Analysis of Realized Volatility in the two trading sessions of the Tokyo Stock Exchange
The Second International Conference “High-frequency Data
Analysis in Financial Markets” 28 Oct. 2011
Tetsuya Takaishi
Hiroshima University of Economics
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Outline Introduction
Realized Volatility
Mixture of Distributions Hypothesis
Empirical Results
Monte Carlo Simulations
Fitting Results
Autocorrelation of Standardized Returns
Conclusion
Future work
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Stylized properties of asset returns
Absence of autocorrelations
Slow decay of autocorrelation in absolute
returns
Fat-tailed (heavy tail) distributions
Volatility clustering
Leverage effect
Volume/volatility correlation
Aggregational Gaussianity
…..
Introduction
ARCH
GARCH
EGARCH
QGARCH
GJR-GARCH
SV model
…
Modeling time series
New stylized fact New stylized fact New modelNew model
Better predictability
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Gopikrishnan et. al(1999)
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Gopikrishnan et. al(1999)
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Tsallis and Anteneodo(2003)
Student-t distribution
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daily returns for 49 largest stock of the National Stock Exchange (NSE)
in India over the period Nov 1994—June 2002.
Matia, Pal, Stanley, Salunka(2003)
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Empirical property Empirical property Modeling Forecasting Modeling Forecasting
Computational resources Computational resources
In econometric and finance
ExperimentsObservations ExperimentsObservations
TheoryTheory
In science
New factsNew theory
New factsNew model
New method
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23 September 2011The OPERA experiment announced that neutrinos could be faster than light.
23 September 2011The OPERA experiment announced that neutrinos could be faster than light.
Faster-than-light neutrinos
The OPERA result is based on the observation of over 15000 neutrino events measured at Gran Sasso, and appears to indicate that the neutrinos travel at a velocity 20 parts per million above the speed of light.
The OPERA result is based on the observation of over 15000 neutrino events measured at Gran Sasso, and appears to indicate that the neutrinos travel at a velocity 20 parts per million above the speed of light.
They observe a neutrino beam from CERN 730 km away
at Italy’s INFN Gran Sasso Laboratory.
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Einstein’s special theory of relativity says
Anything having mass can not be faster than light(in vacuum).Anything having mass can not be faster than light(in vacuum).
Neutrino mass has been established.Neutrino mass has been established.
A faster-than-light particle is the particle traveling in the past.A faster-than-light particle is the particle traveling in the past.
If true, it is the biggest discovery in science.
This might be a violation of special relativity?
Same scientists claim that
Any other new theory behind that?
The experimental result has not been confirmed yet.
We need more evidence.
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In 2010, Tokyo Stock Exchange launched “arrowhead”, the next generation trading system In 2010, Tokyo Stock Exchange launched “arrowhead”, the next generation trading system
Speed of trading system(1) 5 millisecond Order Response(2) 3 millisecond Information Distribution
Speed of trading system(1) 5 millisecond Order Response(2) 3 millisecond Information Distribution
Further speed up is scheduled next year!
What is the consequence of “faster-than-light neutrinos in finance?
There is more:
Main feature
Co-location area
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How can we reduce the transmission time?
Only way is to reduce the physical distance Only way is to reduce the physical distance Limitation from the special relativity theoryLimitation from the special relativity theory
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Neutrino network
Maybe science fiction,…but
If we believe the particle traveling in the past, you could send today’s price data to you in the past.If we believe the particle traveling in the past, you could send today’s price data to you in the past.
Forecasting and modeling no more needed?Forecasting and modeling no more needed? Causality problem?
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It is well-known that return distributions show heavy tail distributions.
Source of the heavy tail distributions?
One explanation of this evidence is the mixture of distributions hypothesis
Clark(1976)
tttR N(0,1) ~ t
Returns are described by Gaussian with time-varying volatility
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tttR
t
t
tR
Standardized returns will be Gaussian variables with mean 0 and variance 1
N(0,1) ~ t
Volatility is unobserved in the markets.
Volatility is estimated by using high-frequency data.
Realized volatility: sum of squared returns
t
t
t
RV
R
2/1
Main purposeMain purpose
Normality of standardized returns
Validity of Mixture of distributions hypothesis
Variance =1 Kurtosis=3
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T.G. Andersen, T. Bollerslev, F.X. Diebold and P. Labys, 2000, “Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian”, Multinational Finance Journal 4 (2000), 159–179.
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T.G. Andersen, T. Bollerslev, F.X. Diebold and H. Ebens, 2001, “The distribution ofrealized stock return volatility,” Journal of Financial Economics 61, 43–76
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)()()(ln tdWttpd
N
i
kiTtt rRV1
2
*
dsstt
TtT )()( 22 Integrated volatility (IV) for T period
IV
Realized Volatility
)(ln)(ln kipipri
Andersen, Bollerslev (1998)
Let us assume that the logarithmic price process follows
a stochastic diffusion as
Realized volatility is defined by a sum of
squared finely sampled returns.
return calculated using high-frequency data
k
TN k: sampling period
)0( k
W(t): Standard Brownian motion
volatility spot:)(t
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Problems in calculating RV
Microstructure noise
Non-trading hours issue
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morning session afternoon session
How to deal with the intraday returns during the breaks?
break breakbreak
Domestic stock trade at the Tokyo stock exchangeDomestic stock trade at the Tokyo stock exchange
09:00 11:00 12:30 15:00
Let us consider daily volatility
Usually stock exchange markets are not open for a whole day.
start end
Non-trading hours issue
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Hansen and Lunde (2005) introduced an adjustment factor
RV without including returns in the breaks
T
t
t
T
t
t
RV
RR
c
1
0
1
2)(
Correct RV so that the average of RV matches the variance of the daily returnsCorrect RV so that the average of RV matches the variance of the daily returns
0
tt cRVRV
underestimated
Average of original realized volatilities
Variance of daily returns
T: trading daysc: adjustment factor
Original realized volatility
For standardized returns this changes the value of variance but not kurtosis For standardized returns this changes the value of variance but not kurtosis
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In order to avoid non-trading hours issue we calculate RV in the two trading sessions separately.
: morning session RV :afternoon session RV
break09:00 11:00 12:30
start end
tMSRV , tASRV ,
t
tMS
tMS
RV
R
2/1
,
,
Close
tMS
Open
tMStMS PPR ,,, lnln Close
tAS
Open
tAStAS PPR ,,, lnln
t
tAS
tAS
RV
R
2/1
,
,
1.1. 2.2.
Open
tMSP ,Close
tMSP ,
Open
tASP ,
Close
tASP ,
Morning return Afternoon return
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break09:00 11:00 12:30
start end
Open
tMSP ,
Close
tASP ,
t
tAStMS
tIntra
RVRV
R
2/1
,,
,
)(
Close
tAS
Open
tMStIntra PPR ,,, lnln
3.3. Morning session + Afternoon session
This could be underestimated
Larger variance is expectedLarger variance is expected
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Microstructure noise
)()()( ttrtr
),0(:)( 2 WNt
true noise
Observed returns are also contaminated by noise
Price discreteness, bid-ask spreads, etc.
N
i
N
i
iii
N
i
iii
N
i
rrrrRV1 1
2
1
22
1
2 2)(
Noise terms
)()(ln)(ln ttPtP
Observed prices are contaminated by microstructure noise
)()()( tttt
In the presence of noise RV is calculated as follows
22 N
Zhou(1996)
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RV
NRVNRVRV
22 2
12
t
T
RVRVRV
221
t
TN
Sampling frequency (interval)
When N is large, the contribution of the noise terms becomes large.
5-min frequency is often used for RV construction
Common practice: do not use very high-frequency returns
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3.Mixture of Distributions Hypothesis(MDH)
Clark( 1973)
tttR
Unconditional distributions of asset returns show fat-tailed distributions.
Asset returns are described by Gaussian variables with the time-varying volatility
Volatility varies in time
t
t
tR
From MDH returns standardized by their standard deviations are
expected to be Gaussian random variables
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)2
exp()2()|(2
22/122
t
tt
rrP
22
0
2 )|()()( ttt drPPrP
)2/()(ln 221)(
th
t
t eh
hP
th
tt ehhP/1
)(
Unconditional distributions
Lognormal distribution
Inverse gamma distribution
2
tth Clark(1973)
Paretz(1972)
Volatility distribution
Student-t distribution
Unconditional distributions of asset returns are given by the superposition of the Gaussian and volatility distributions.
We do not know the form of volatility distributionsWe do not know the form of volatility distributions
No consensus is made for the functional form of volatility distributions No consensus is made for the functional form of volatility distributions
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Empirical ResultsOur analysis is based on 5 stocks on the Tokyo Stock Exchange
1:Mitsubishi Co.
2:Nomura Holdings Inc.
3:Nippon Steel Co.
4.Panasonic Co.
5.Sony Co.
Our data set begins June 3, 2006 and ends December 30, 2009.
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Lunch break
Overnight break
Morning Session
Afternoon session
Return time series in the different time zones for Mitsubishi Co.
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Volatility signature plot for Mitsubishi Co.
t
ARVtRV 1)(
14% bias at 5min14% bias at 5min
32% bias at 5min32% bias at 5min
min
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Variance of standardized returns
dta /1
2
Noise contribution Noise contribution
Sampling frequency
Afternoon session
Morning session
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Mitsubishi Co.
Kurtosis of standardized returns
Sampling period
Rapid increaseRapid increase
Linear decreaseLinear decrease
Afternoon session
Morning session
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1t
ttR
Ry
Lowest frequency
Variance 1Kurtosis 1
2
tRRV
Kurtosis could be frequency-dependentKurtosis could be frequency-dependent
We sample only one return which is largest.
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tt NR
ttr
jkjk
Nr ,
Assume that returns consist of N sub returns and each sub return is just given as a Gaussian variable with a constant variance
We calculate RV from this series by sampling k returns.
Each sampled return contain N/k sub returns
Each k-sampled return is described as
N/k N/k N/k
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22
11
2
, j
k
j
k
j
kjkk
NrRV
2/1
k
ttRV
Ry
In this case RV is given by
Standardized returnStandardized return
1
1
22
222
k
j
j
tt
k
N
NEyE
Variance
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2
213
)1(3
3
2
424
2
42
1
224
2
244
2
2
442
2
1
22
4424
k
k
kN
k
N
NE
k
N
k
N
NE
k
N
NEyE
k
j ml
mlj
t
k
j
j
tt
Kurtosis
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2
13
2
213
2
213
22
4
N
t
t
N
kyE
yE
t
t
t
Nk
1t
At high sampling frequency
Sampling period
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Constant volatility case
Time-varying volatility case
Gaussian time series with constant variance
Spin model simulation
Volatility dynamics is not known.
Monte Carlo Simulations
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Calculate RV at various sampling frequencies
Repeat the process 5000 times
t
Constant volatility case
)04.0,0( 2N
T=10000
Make T=10000 time series
We also introduce artificial microstructure noise
ttr
)04.0,0( 2N
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Volatility signature plot
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)1/( 10 dtBB
Without noiseWithout noise
With noiseWith noise
Fitting formula
Variance of standardized returns
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210000
210
dt
A
10000 length
Sampling frequency
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40000 length
240000
210
dt
A
Slope depends on lengthSlope depends on length
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No difference is seen in kurtosisNo difference is seen in kurtosis
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iS
jS
iS
S.Bornholdt, Int.J.Mod.Phys.C12(2001) 667
takes +1 or -1
Buy Sell
We may assign +1 state to “Buy order” and
-1 state to “Sell order”
Agents live at sites on an n-dimensional lattice
Each site has a spin.
Spin model
(In this study we use 2-dimensional lattice.)
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)(1
)( tSn
tMj
j
n
j
ijiji tMtStSJth1
|)(|)()()(
Local interaction
)))(2exp(1/(1 1)1( thptS ii
ptSi -1 1)1(
Spins are updated by the following probability
Global interaction
Local interaction: Majority effect
Global interaction: Minority effect
Difference between buy and sell orders
0
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)(1
)( tSn
tMj
jL=100 beta=2 alpha=20
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2/)1()()( tMtMtr
Return time series
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Return distributions
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Cumulative return distributions
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Realized volatility in Spin model
L=125×125, β=2.0, α=20
t=1 corresponds to one spin update.
T=125x125=15625
We define
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Realized volatility dt=1
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t
tr
tr
Kurtosis:43.2
Std. dev.:0.00059
Kurtosis:2.92
Std. dev.:0.996
Return distribution Sampling frequency dt=1
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dt
Variance of Standardized returns
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2/)125*125(
210
dtA
Kurtosis of standardized returns
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Fitting Results
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2/
21
1
0dtA
A
Mitsubishi Co.
Morning session
Afternoon session
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Mitsubishi Co.
Morning session + Afternoon session
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Nomura
2/
21
1
0dtA
A
Afternoon session
Morning session
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Nomura
Morning session + Afternoon session
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dta /1
2
Mitsubishi Co.
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Nomura
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Noise contributionsNoise contributions
HL adjustment factor also adjusts microstructure noise effects. HL adjustment factor also adjusts microstructure noise effects.
HL adjustment factor
Daily return
Morning session + Afternoon session
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Nomura
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Variance of standardized return without HL adjustmentVariance of standardized return without HL adjustment
VarianceHL
Morning session + Afternoon session
dt sampling period min
Mitsubishi Co.
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Nomura
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Mitsubishi Nomura Sony Nippon St. Panasonic
MS 1.07 0.995 1.03 1.03 1.03
AS 0.95 0.872 0.997 1.04 1.04
MS+AS 1.12 1.02 1.03 1.13 1.01
Variance
Fitting results
Kurtosis
Mitsubishi Nomura Sony Nippon St. Panasonic
MS 2.92 2.75 2.75 2.91 2.72
AS 3.27 3.31 3.28 3.13 3.01
MS+AS 2.72 2.79 2.95 2.73 2.83
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Mitsubishi Nomura Sony Nippon St. Panasonic
MS 120 147 143 128 138
AS 138 141 118 130 143
MS+AS
358 310 340 360 483
120min
150min
270min
2/
21
1
0dtA
A
Mitsubishi Co.
Trading time
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0 tttttttttt EEErrE
tttr
2))(( cEEcrcrE tttttt trEc
Autocorrelation of standardized returns
Autocorrelation of returns is insignificant Autocorrelation of returns is insignificant
We assume
Autocorrelation of absolute returns is not necessarily zero Autocorrelation of absolute returns is not necessarily zero
t
t
tr
For standardized returns
Autocorrelation is always zero not only for returns but also for absolute returnsAutocorrelation is always zero not only for returns but also for absolute returns
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Morning session
Afternoon session
Mitsubishi Co.
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t
tr
tr
t
Spin model
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ConclusionWe analyze the normality of standardized returns by using realized
volatilities in the two trading sessions of the Tokyo Stock Exchange.
Variance of standardized returns is largely affected by microstructure noise.
Kurtosis of standardized returns shows unexpected behavior: linear
dependence at lower frequency and rapid increase at high frequency
Linear dependence can be understood by
Finite size effect Discretization effect Finite size effect Discretization effect
Normality is recovered in the appropriate limit.
2
213
22
4
t
NyE
yE
t
t
kurtosis 0t
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Future Work
What is the rapid increase of the kurtosis at the high frequencies?
Same analysis for exchange rate
More clear results?
Other moments? ktyE
Deference between clock time and tick time?
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Volatility Distributions
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Morning session
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Realized volatility, transactions and volume
t tt
)(ln)(ln)(ln tptpttp
)(tp )( ttp )2( ttp )3( ttp
)(
1
. )(ln)(lntN
i
tran
t
t
iptp
Price change in Δt
# of transactions in Δt
Price change between i-th and i+1
th transactions.
Price change
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twt
tN
i
tran
t
ttN
iptpt
)()(
)(ln)(ln)(
1
.
2.2 ))(ln()( ipt tran
tw
Variance of price change at one transaction
Realized volatility and # of transactions
tRV ttp )()(ln
)()()( 22ttNtRV wtRV
Plerou et. al.(2000)
)()(
2 ttN
RVw
t
Diffusion process
N(t) and )(2 twAre there any correlations between
?
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2NV
Volume and transactions
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2NV