Scaling Behaviors in Economics Time Series :Korean Stock Index and Firm Bankruptcy

1
Scaling Behaviors in Economics Time
SeriesKorean Stock Index andFirm Bankruptcy

• Jae Woo Lee, Kyoung Eun Lee,
• Jun Kyung Hwang
• Department of Physics,
• Inha University, Korea

2
Outline

• Price Index and Return
• Probability Distribution of Return and Volatility
• Autocorrelation Function
• Recurrence Time Distribution
• Scaling in Trade Volume
• Scaling in Firm Bankruptcy

3
Return and Volatility in stock market model
Standard model of stock market(EMH/Bachelier)

• Stock prices - a random walk superimposed on a
  constant drift
• Stochastic differential equation

where
4

• Volatility of log price changes of financial
  asset is a time dependent stochastic process.
• ARCH(Autoregressive conditional
  heteroscedasticity)
• - a stochastic process which is locally
  nonstationary but asymptotically stationary
• - empirically motivated discrete-time stochastic
  models for which the variance at time t depends
  conditionally on some past values of the square
  value of the random signal itself.

5
ARCH(p) model
GARCH(p,q) (Generalized ARCH)
where
control parameters
6
Numerical simulation of an ARCH(1) process
By Mantegna Stanley
7
Korea Composite Stock Price Index(KOSPI)
1997.11(IMF)
1992.3
1999.11
8
Return of KOSPI
Logarithmic return
Normalized return
1992.04
1999.12
9
1. Probability Distribution Function
normalized pdf of return
Central part of pdf is well fitted by Lorentzian
function.
10
Skewness and Kurtosis of pdf for return
Leptokurtic peaked and fatter tails
Asymmetry of pdf
11
Fat Tail and Power law of pdf for return
12
Exponents of pdf for return
13
Volatility
Volatility standard deviation at a
nonoverlapping time window of length T or
absolute return.
t
0
2T
3T
4T
5T
T
14
Volatility (T30min)
Volatility clustering
IMF
15
Volatility (T300min)
16
Probability density function of volatility
Central parts of pdf are well fitted by
lognormal function.
17
Cumulated pdf of volatility
18
Exponents of volatility
Inverse cubic law is questionable (Stanley et
al.)!
19
Effects of Asian Financial Crisis
Before IMF
After IMF
Korean government submitted bailouts to
international monetary fund (IMF) at 21 November
1997.
20
2. Autocorrelation Function

• Short time correlation of return
• Exponential decay at early time
• Characteristic time

21
Autocorrelation function of absolute return
for
for
Cf.
for S P 500
22
3. Recurrence Time Distribution of Volatility
Volatility
23
Recurrence Time Distribution (RTD)
24
Rescaled RTD
Rescaled RTD by average recurrence time T
25
Relation between Average recurrence time and
threshold
PDF for volatility
26
(No Transcript)
27
Summary of RTD

• Power law of RTD means the long time correlation
  of the rare events
• A long time memory exists in the recurrence time.
• RTD is a quantity characterizing nonlinear time
  series such as volatility of stock market index.

28
4. Scaling in Trade Volume
1992.3
1999.11
29
PDF of Trade Volume

• Asian financial crisis greatly influences to PDF
  of trade volume

KOSPI
KOSDAQ
Korean government submitted bailouts to
international monetary fund (IMF) at 21 November
1997.
30
Fat tail for PDF of trading volume
31
PDF of volume change
Semilogarithmic plot of pdf for the normalized
volume changes
32
Scaling of volume changes

• PDF of trade volume changes also follows a
  power-law

fat tail of pdf for volume changes
pdf for volume changes
33
Fat tails in volume change
Negative tail
Positive tail
34
Exponents for volume change
35
5. Power Law in Firm Bankruptcy

• Is there a power-law in the number of firms
  bankrupted?
• The distribution of firms debt showed power-law
  Fujiwara 2004.
• Income distribution in Japanese companies shows
  Zipf law with Pareto exponent -1 Okuyama
  Takayasu 1999
• We consider firms bankrupted in Korea in the
  period from 1 August 2002 to 28 October 2003.
• We also consider firms bankrupted in USA (Chapter
  11 Chapter 7) in the period 1 July 1986 to 29
  January 2007.

36
Firm Bankruptcy in Korea
The daily number of firms bankrupted against day
in Korea from 1 August to 2002 to 28 October 2003.
37
Cumulative pdf for the number of firms bankrupted
Korea
Log-Log plot of the cumulative probability
distribution for the number of firms bankrupted
versus the number of bankrupted firm.
38
Firm Bankruptcy in USA
Jul. 1986
Jan. 2007
The number of firms bankrupted per month
39
pdf for the number of bankrupted firms
USA
The pdf and cumulative pdf for the number of
bankrupted firms versus the number of bankrupted
firm.
40
asset and employee
asset
employee
Asset and the number of employee in bankrupted
firms per day.
41
Cumulative PDF of asset and employee
asset
employee
Asset and the number of employee in bankrupted
firms shows power-laws.
42
Summary

• We observe power law of pdf for return and
  volatility.
• Scaling exponents depend on the time lag.
• We observe short-range correlation of return and
  long-range correlation of volatility.
• PDF of recurrence time distribution shows
  power-law.
• PDF of the number of bankrupted firms, asset, and
  the number employee also show the power-law.
• We need models explaining fat tail and central
  parts of the distribution function.