realized volatility distribution in japanese stock market tetsuya takaishi hiroshima university of...
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Realized Volatility Distribution in Japanese Stock Market
Tetsuya Takaishi
Hiroshima University of Economics
To be published in Evolutionary and Institutional Economic Review
Outline
Introduction Realized Volatility Mixture of Gaussian Distributions Stock Data & Results Distribution of Realized Volatility Conclusions
Introduction
N(0,1) ~ ttttr
In finance volatility is an important value for option pricing, portfolio selection, risk management, etc.
Volatility is not a direct observable from asset prices.
Price return
We need to estimate volatility by a certain method.
volatility
)(ln)(ln ttptprt
Model estimation of volatility
Make a model which captures the volatility properties observed in financial markets
•Volatility clustering
•Fat-tailed distribution
ARCH model Engle(1982)
GARCH model Bollerslev(1986)
QGARC model Engle, Ng(1993), Sentana(1995)
EGARCH model Nelson(1991)
GJR-GARCH model Glosten, Jagannathan, Runkle(1993)
etc.
Stylized facts of financial prices
,ttty
)1,0(~ Nt21
21
2 ttt y
GARCH(1,1) model Bollerslev(1986)
21
211
2 tttt yy
QGARCH model Engle,Ng(1993)
Sentana(1995)
Question: which model should we use?
The value estimated may depends on the model we use.
Realized volatility(RV) : a model-free estimate of volatility
RV is constructed using high frequency data.
1. We measure RV using high-frequency data of some stocks traded on the Tokyo stock exchange and analyze the distributions of RV.
2. We examine whether the price return distribution on the Tokyo stock exchange is considered to be a superposition of two distributions ( mixture of Gaussian distributions).
dWtdtttpd )()()(ln
n
initt rRV
1
2/1
dsst
tt
1
22 )( Integrated volatility (IV)
IV
Realized VolatilityAndersen, Bollerslev (1998)
Let us assume that the logarithmic price process follows a stochastic diffusion as
drift term daily volatility at day t
Realized volatility is defined by summing up n intraday returns.
intraday return calculated using high-frequency data)(ln)(ln sjpjprj
ns
1 Sampling frequency
n
morning session afternoon session
How to deal with the intraday returns during the breaks?
Hansen 、 Lunde(2005)
RV without returns in the breaks
T
tt
T
tt
RV
RRc
1
0
1
2)(
break break
Correct RV so that the average of RV matches the variance of the daily returns
0tt cRVRV
break
Domestic stock trade at the Tokyo stock exchange
09:00 11:00 12:30 15:00
underestimated
average
variance
T: trading days
A problem in calculating RV
Mixture of Gaussian Distributions
1T
The daily return distribution is a superposition of two distributions?
•short time scale: equilibrium with a Gaussian distribution with a constant volatility
•long time scale: volatility slowly changes
2T
Two time scales
)( 2tP
)2
exp()2()|(2
22/122
ttt
rrP
22
0
2 )|()()( ttt drPPrP
Probability distribution of return in a shot time scale
2tth
/1)( thtt ehhP
)2/()(ln 221)( th
tt e
hhP
thtt ehhP /1)(
Gamma distribution
Lognormal distribution
Inverse gamma distribution
Gaussian distribution with a constant volatility
Let us assume that in a long time scale the volatility slowly changes in time with a probability distribution
The unconditional probability distribution of return is given as a superposition of two distribution: Gaussian distribution and volatility distribution.
Beck, Cohen (2003)
Superstatistics
Stock Data & Results
7 stocks on the Tokyo stock exchange
from March 1, 2006 to February 28, 2008 (493 trading days)
1:Nippon Steel
2:Toyota Motor
3:Sony
4:Nomura Holdings
5:Hitachi
6:Daiwa Securities
7:Mizuho Financial Group
Each realized volatility is calculated using 5-min intraday returns.
(avoid micro-structure noise)
Daily return
Hitachi
Nippon Steel
Daily return
Realized volatility
Nippon Steel
Realized volatility
Hitachi
2/1)(
)()(
cRV
trtr
t
tttr )( tt
tr
)(Gaussian?
)(tr
Nippon Steel
2/1)(
)()(
cRV
trtr
t
)(tr
Nippon Steel
Toyota Sony Nomura Hitachi Daiwa Mizuho
var. 4.832 2.588 3.977 4.8479 2.6703 5.7558 2.192
kurt. 1.6324 2.369 2.072 0.4815 1.7429 0.9704 4.809
var. 0.916 0.922 0.990 1.1355 0.9348 1.0459 1.051
kurt. -0.180 0.405 0.405 0.7408 0.2120 0.0350 0.548t
tr
)(
)(tr
Return distribution of 7 stocks
Distribution of r/sigma
from 7 stocks
Kurtosis 0.5488
fitting to Gaussian dist.
Distribution of RV
What is the functional form of the distribution of RV?
Andersen et al.(2001) : lognormal distribution
Straeten and Beck(2009): lognormal or inverse gamma
Previously, lognormal or inverse gamma distributions are suggested.
Gerig et al.(2009): inverse gamma
Distribution of RV
Hitachi
Distribution of RV
Mizuho
Distribution of RV
Nippon Steel
Distribution of RV
Toyota
Hitachi Nippon Steel
Mizuho Toyota
Gamma 0.0201 0.019 0.028 0.017
Lognormal 0.014 0.0167 0.023 0.00997
IGamma 0.0098 00147 0.018 0.00493
RMS of residuals
IGamma>Lognormal>Gamma
Conclusions
We calculated RV for 7 stocks traded in the Tokyo stock exchange market.
The distribution of the daily return normalized by RV is close to a Gaussian distribution.
The best fit to RV is given by the inverse gamma distribution.
The distributions of returns on TSE can be viewed with a superposition of the inverse gamma and Gaussian distributions.