:Korean Stock Index and
Firm Bankruptcy
Jae Woo Lee , Kyoung Eun Lee,
Jun Kyung Hwang
Department of Physics,
Inha University, Korea

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

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 dS
  dt
  dz
S where dz
 
( t ) dt :Wiener process
P (

)

1
2
 exp(
 
2
/ 2 )
 
( t )

( t
'
)
 
( t
 t
'
)

• 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.

ARCH(p) model
 t
2   o
 
1 x t
2

1
    p x t
2
 p
, S ( t )
 i i

 t

1 x i where
 o
,

1
,  ,

: control parameters p x :random variables with zero mean and variance t
 t
2
GARCH(p,q) (Generalized ARCH)
 t
2   o
 
1 x t
2

1
    p x t
2
 p
 
1
 t
2

1
    q
 t
2
 q where
 o
,

1
,  ,
 p
,

1
, 
 q
:control parameters

Numerical simulation of an ARCH(1) process
 o

0 .
45 ,

1

0 .
55
By Mantegna & Stanley

Korea Composite Stock Price Index(KOSPI)
1997.11(IMF)
1992.3
1999.11

1992.04
 t

1 min
Return of KOSPI
Logarithmic return x
 t
( t )
 log i ( t
  t )
 log i ( t )
Normalized return g
 t
( t )
 x
 t
( t )



(
 t x
 t
)
( t )

1999.12

1. Probability Distribution Function normalized pdf of return
 t

10 min
 t
Central part of pdf is well fitted by Lorentzian function.

Skewness and Kurtosis of pdf for return skewness
 x
 
 x
 3 kurtosis
 x
 
 x
 4

3
Asymmetry of pdf
Leptokurtic: peaked and fatter tails

Fat Tail and Power law of pdf for return
 t

10 min p ( x )
 x

(
 
1 )
P

( x )
   x p ( x
'
) dx
'  x
 


Positive tail Negative tail
KOSPI
2.46( =10min)
2.71( =30min)
2.87( =60min)
2.4( =1min) DAX
2.56( =10min)
2.73( =30min)
3.03( =60min)
2.6( =1min)
Hang-Seng
S & P 500
2.9( =10min)
3.5( =60min)
2.32( =1min)
3.05( =1day)
2.95( =1min)
2.69( =16min)
2.83( =128min)
3.34( =1day)
3.05( =1day) Nikkei
2.75( =1min)
4.0( =1day)
2.75( =1min)

Volatility = standard deviation at a nonoverlapping time window of length T or absolute return.
V
T
( t )

1 n t t

' n 

1
 t
( g ( t
'
)
  g

)
2
V
T
( t )

1 n t t

' n 

1
 t g ( t
'
) t
0 T 2T 3T 4T 5T

Volatility (T=30min)
Volatility clustering
IMF

Volatility (T=300min)

Probability density function of volatility y
 y
0

2
A
 wx exp(
 x
(ln x c
2 w
2
)
2
)
Central parts of pdf are well fitted by lognormal function.

Cumulated pdf of volatility
P

( V
T
) ~ V
T
 
P ( V
T
) ~ V
T

(
 
1 )

P

( V
T
) ~ V
T
 
KOSPI S & P 500
T=10min 2.29(5) T=32min 3.10(8)
T=30min 2.27(6) T=64min 3.19(10)
T=60min 2.29(8) T=128min 3.30(15)
T=300min 2.5(1)
T=600min 2.4(2)
Inverse cubic law is questionable (Stanley et al.)!

Effects of Asian Financial Crisis
Before IMF After IMF
Korean government submitted bailouts to international monetary fund (IMF) at 21 November 1997.

C
 t
(

)
 x
 t
( t ) x
 t
( t
 
)
   x
 t
( t )
 2
• Short time correlation of return
• Exponential decay at early time:
• Characteristic time :
  c
5 .
9
C
 t min
(

) ~ e
 
/
 c

Autocorrelation function of absolute return
C
| x |
(

)

| x
 t
( t ) x
 t
( t
 
) |
  
| x
 t
( t ) |
 2
C
| x |
(

) ~
  
1 ,

1

0 .
52 ( 2 ) for
  
1

5 .
9 min
C
| x |
(

) ~
  
2 ,

2

0 .
25 ( 2 ) for

1
   
2

100 min
Cf.

2

0 .
30 ( 8 ) for S & P 500

3. Recurrence Time Distribution of Volatility
Volatility: r
 t
( t )
 x
 t
( t )



(
 t x
 t
)
( t )

T
1
T
2
T
3
T
4 r c

Recurrence Time Distribution (RTD) p f
( t )
 t
 

Rescaled RTD
Rescaled RTD by average recurrence time T p f
( t / T )

1
T f ( t / T ) f ( x )
 x
 

Relation between Average recurrence time and threshold t

N
 t

N e
T
P ( r )
 t
T


N e
N

 r c r
  dr


 r c
P ( r ) dr
 r c
  
1
T
 r c
 
1 r
PDF for volatility r c

T
 r c
 
1

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.

1992.3
1999.11

 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.

Fat tail for PDF of trading volume

PDF of volume change
V r

V ( t

T )

V ( t )
Semilogarithmic plot of pdf for the normalized volume changes


PDF of trade volume changes also follows a power-law
T

1 min
T

1 min pdf for volume changes fat tail of pdf for volume changes

Fat tails in volume change
Positive tail Negative tail
P ( V r
)

V r

( 1
 
Vr
)

Exponents for volume change
Time lag Positive Tail Negative Tail
1min
10min
30min
60min
600min
0.96(2)
0.88(1)
0.86(9)
0.901(7)
0.989(4)
1.02(2)
0.83(1)
0.81(1)
0.917(9)
0.993(1)






Is there a power-law in the number of firms bankrupted?
The distribution of firm’s 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.

Firm Bankruptcy in Korea
The daily number of firms bankrupted against day in Korea from 1 August to 2002 to 28 October 2003.

Cumulative pdf for the number of firms bankrupted
P ( x
 n )
 n
 
,
 
0 .
91 ( 2 )
Korea
Log-Log plot of the cumulative probability distribution for the number of firms bankrupted versus the number of bankrupted firm.

Jul. 1986
Jan. 2007
The number of firms bankrupted per month

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.

asset asset and employee employee
Asset and the number of employee in bankrupted firms per day.

Cumulative PDF of asset and employee asset employee
Asset and the number of employee in bankrupted firms shows power-laws.







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 powerlaw.
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.