Statistical laws observed in Japanese Stock Markets and a Stochastic Model

1. Statistical laws observed in Japanese Stock Markets and a Stochastic Model Taisei Kaizoji and Michiyo Kaizoji E-mail: kaizoji@icu.ac.jp Homepage: http://subsite.icu.ac.jp/people/kaizoji/ Taipei, Taiwan December 20, 2002

2. The aims: (i) To show statistical properties in Japanese stock markets (ii) To model the observed statistical properties Data: a database covering securities from the Tokyo Stock Exchange (TSE). The database provides daily data which covers the period between 4 January, 1975 and 4 January, 2002.

3. The statistical properties of the Nikkei Average log-return: r(t) =ln p(t) – ln p(t-1)

4. Mean 0.0 S.D. 0.01 Kurtosis 8.2 Skewness -0.08

5. The cumulative distribution of log- return for Nikkei 225

6. The cumulative distribution of log-return for Nikkei 225 in semi-log plot An exponential distribution:

7. The cumulative distribution of return volatility for Nikkei 225: F(|r|>x) F(|r|>x) x return volatility: |r(t)|=|ln p(t) – ln p(t)|

8. The cumulative distribution of return volatility for Nikkei 225 in semi-log plot ln F(|r|>x) x An exponential distribution:

9. Intraday volatility for Nikkei Average Intraday volatility: h(t): the highest price in a trading day t l(t): the lowest price in a trading day t

10. Histogram of intraday volatility for Nikkei Average f(G) G

11. Histogram of intraday volatility on semi-log plot f(G) ln G(t) A log-normal distribution:

12. Autocorrelation function (ACF) of return volatility for Nikkei Average ACF lag ACF ln(lag)

13. Autocorrelation function (ACF) of intraday volatility for Nikkei Average ACF lag ACF ln(lag)

14. A Calm Time Interval Distribution for return volatility The threshold value for excess volatility is equal to 0.8.

15. A Calm Time Interval Distribution for return volatility Histogram Time interval Semi-log plot Time interval

16. The Poisson process on excess volatility: k : the number of occurrences of an excess volatility in a time interval The exponential distribution of the calm time interval k=0 :

17. A Calm Time Interval Distribution for intraday volatility

18. The Empirical Facts on the Nikkei 225 Index: • The return distribution follows an exponential distribution (Laplace distribution). • 2. The return-volatility distribution follows an exponential distribution. • 3. The intrady-volatility distribution follows a log-normal distribution. • 4. Both of the return volatility and the intraday volatility have long memories. • 5. The distribution of the calm time interval for volatility follows an exponential distribution (Poisson process).

19. Previous works on return volatility: Cizeau, et.al. (Physica A 245 (1997) 441) study the return volatility of the S&P500 stock index from 1984 to 1996 and find the return-volatility distribution can be well described by a log-normal function.

20. Statistical Properties of shares listed in TSE. Data: The data ofthe stock price and the daily trading volume for Nittetsu Mining for the 26-year period from Jan. 4, 1975 to Jan. 9, 2002 . Nittetsu Mining is one of the largest mining companies in Japan.

24. Zipf Distribution: Reference: T. Kaizoji and M. Nuki, Scaling Law for the Distribution of Fluctuations of Share Volume (2002)

25. The cumulative distribution of log-return for Nittetsu Mining in semi-log plot

26. The distributions of return volatility for Nittetsu Mining -Semilog plot-

28. A Calm Time Interval Distribution for Nittetsu Mining

29. A Calm Time Interval Distribution for Nittetsu Mining Reference: T. Kaizoji, M. Kaizoji “Scaling law for time interval distributions in stock makrets” (2002)

30. A stochastic model of stock market Interacting Traders Fundamentalists m n Fundamentalists’ buy or sell Interacting Trader’s buy or sell Aggregate excess demand by Interacting Traders Aggregate buy or sell by Fundamentalists

31. Decision-making of Interacting Traders: The transition probabilities: : from a seller to a buyer : from a buyer to a seller The average opinion: The stochastic differential equation:

32. Fundamentalists Interacting Traders buy or sell buy and sell The stock exchange Market maker Demand –supply balance: The market price: The trading volume: The relative price change:

33. Explanations of the empirical facts: The market price: The trading volume: ln P(t) ln P*(t) 0 M(t) Bear market Bull market The relative price change: Stochastic fundamental news Dynamics in the stochastic model

34. The results of simulation: • (i) On-off intermittent dynamics of the trading-volume changes (|M(t)|-|M(t-1)|) • A Zipf distribution of the trading-volume changes • An exponential distribution of log-returns

35. A time series of the volume change A distribution of the volume change

36. The cumulative distribution of volume change in log-log plot

37. A time series of the log-return The cumulative distribution of log-return in semi-log plot

38. Future works: • How to control the excess volatility of shares and the Stock price index. • Statistical Properties of the mutualcorrelations between shares.