analysis of realized volatility in the two trading ... · einstein’s special theory of relativity...

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

Outline Introduction

Realized Volatility

Mixture of Distributions Hypothesis

Empirical Results

Monte Carlo Simulations

Fitting Results

Autocorrelation of Standardized Returns

Conclusion

Future work

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

Gopikrishnan et. al(1999)

Gopikrishnan et. al(1999)

Tsallis and Anteneodo(2003)

Student-t distribution

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)

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

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.

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.

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

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

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?

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

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

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.

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

)()()(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

Problems in calculating RV

Microstructure noise

Non-trading hours issue

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

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

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

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

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)

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

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

)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

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.

Lunch break

Overnight break

Morning Session

Afternoon session

Return time series in the different time zones for Mitsubishi Co.

Volatility signature plot for Mitsubishi Co.

t

ARVtRV 1)(

14% bias at 5min14% bias at 5min

32% bias at 5min32% bias at 5min

min

Variance of standardized returns

dta /1

2

Noise contribution Noise contribution

Sampling frequency

Afternoon session

Morning session

Mitsubishi Co.

Kurtosis of standardized returns

Sampling period

Rapid increaseRapid increase

Linear decreaseLinear decrease

Afternoon session

Morning session

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.

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

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

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

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

Constant volatility case

Time-varying volatility case

Gaussian time series with constant variance

Spin model simulation

Volatility dynamics is not known.

Monte Carlo Simulations

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

Volatility signature plot

)1/( 10 dtBB

Without noiseWithout noise

With noiseWith noise

Fitting formula

Variance of standardized returns

210000

210

dt

A

10000 length

Sampling frequency

40000 length

240000

210

dt

A

Slope depends on lengthSlope depends on length

No difference is seen in kurtosisNo difference is seen in kurtosis

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.)

)(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

)(1

)( tSn

tMj

jL=100 beta=2 alpha=20

2/)1()()( tMtMtr

Return time series

Return distributions

Cumulative return distributions

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

Realized volatility dt=1

t

tr

tr

Kurtosis：43.2

Std. dev.：0.00059

Kurtosis：2.92

Std. dev.：0.996

Return distribution Sampling frequency dt=1

dt

Variance of Standardized returns

2/)125*125(

210

dtA

Kurtosis of standardized returns

Fitting Results

2/

21

1

0dtA

A

Mitsubishi Co.

Morning session

Afternoon session

Mitsubishi Co.

Morning session + Afternoon session

Nomura

2/

21

1

0dtA

A

Afternoon session

Morning session

Nomura

Morning session + Afternoon session

dta /1

2

Mitsubishi Co.

Nomura

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

Nomura

Variance of standardized return without HL adjustmentVariance of standardized return without HL adjustment

VarianceHL

Morning session + Afternoon session

dt sampling period min

Mitsubishi Co.

Nomura

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

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

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

Morning session

Afternoon session

Mitsubishi Co.

t

tr

tr

ｔ

Spin model

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

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?

Volatility Distributions

Morning session

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

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

?

2NV

Volume and transactions

2NV