1. Realized Volatility Distribution in Japanese Stock Market Tetsuya Takaishi Hiroshima University of Economics To be published in Evolutionary and Institutional Economic Review

2. Outline Introduction Realized Volatility Mixture of Gaussian Distributions Stock Data & Results Distribution of Realized Volatility Conclusions

3. Introduction 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

4. 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

5. GARCH(1,1) model Bollerslev(1986) QGARCH model Engle,Ng(1993) Sentana(1995) Question: which model should we use? The value estimated may depends on the model we use.

6. 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).

7. Integrated volatility (IV) IV Realized Volatility Andersen, 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 Sampling frequency

8. morning sessionafternoon session How to deal with the intraday returns during the breaks? Hansen 、 Lunde(2005) RV without returns in the breaks break Correct RV so that the average of RV matches the variance of the daily returns break Domestic stock trade at the Tokyo stock exchange 09:0011:0012:3015:00 underestimated average variance T: trading days A problem in calculating RV

9. Mixture of Gaussian Distributions 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 Two time scales

10. Probability distribution of return in a shot time scale 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

11. 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)

12. Daily return Hitachi

13. Nippon Steel Daily return

14. Realized volatility

15. Nippon Steel Realized volatility

16. Hitachi Gaussian?

17. Nippon Steel

18. ToyotaSonyNomuraHitachiDaiwaMizuho var. 4.8322.5883.9774.84792.67035.75582.192 kurt. 1.63242.3692.0720.48151.74290.97044.809 var. 0.9160.9220.9901.13550.93481.04591.051 kurt. -0.1800.405 0.74080.21200.03500.548

19. Return distribution of 7 stocks

20. Distribution of r/sigma from 7 stocks Kurtosis 0.5488 fitting to Gaussian dist.

21. 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

22. Distribution of RV Hitachi

23. Distribution of RV Mizuho

24. Distribution of RV Nippon Steel

25. Distribution of RV Toyota

26. HitachiNippon Steel MizuhoToyota Gamma 0.02010.0190.0280.017 Lognormal 0.0140.01670.0230.00997 IGamma 0.0098001470.0180.00493 RMS of residuals IGamma>Lognormal>Gamma

27. 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.