Transcript WAITING TIME DISTRIBUTIONS FOR FINANCIAL MARKETS

WAITING TIME DISTRIBUTIONS
FOR FINANCIAL MARKETS
Lorenzo Sabatelli1,2, Shane Keating1, Jonathan Dudley1
and
Peter Richmond1
1 Department
of Physics, Trinity College Dublin 2, Ireland
and
2 Hibernian Investment Managers, IFSC, Dublin 1, Ireland
The authors acknowledge support from the EU via Marie Curie Industrial
Fellowship MCFH-1999-00026
Objective
• waiting time distribution (WTD) for the Irish
stock market 1850 to 1854.
– 10 stocks out of a database of 60 are examined.
– waiting time distributions vary from a day to some
months are
• compare with WTD for Japanese yen currency
returns 1989-1998
– waiting times vary from a minute to over an hour
19th century Irish Stock Exchange
• Deals done 'matched bargain basis'
• members of exchange bring buyers and sellers together
– Essentially same as today
– Today, many more buyers and sellers.
• Recent studies of 19th century markets find they were well
integrated
• Dublin traded international shares
– Not solely a regional market.
• World trends reflected in the Irish market
– No exchange controls.
– From 1801 to 1922 Ireland was part of UK
• Largest shares: Banks and key railways –
– Quality investments for UK investors
– Also traded in London.
Random walks
Time
Time
Markovian Random walk

p( x, t  1)    ( x  x ') p( x ', t )dx '

p( x, 0)   ( x)
Continuous time random walk
Montroll & Weiss 1965


p( x, t )   ( x)(t )    (t  t ')   ( x  x ') p( x ', t ')dx ' dt '
 

0
t

t
t
0
 (t )    (t ')dt '  1    (t ')dt '
Fourier Laplace Transform

 

( s) spˆ (k , s)  1  ˆ (k )  1 pˆ (k , s)
1  1  


s
s
Choice for memory function Ф
t 
, t  0, 0    1
 (t ) 
(1   )
1
 ( s )  1- 
s
 Mittag Leffer function:

t n
n
 (1) (n   1) , t  0
 (t )  E (t  )   n 0
 ~ sin( )( ) , t  

t
Results: Irish Stock Market data
Irish Data:Ensemble of 10 Stocks(1850-4)
1
Survival Probability Distribution
1
10
100
1000
Intermediate Regime
Characteristice time ~ 10 days
0.1
Power Law
y = x^-0.4
0.01
Exponential
y=e^-0.2x
0.001
Time (days)
Sharp drop-off
Results: Yen Currency Market data
Yen Data
1
Survival Probability Distribution
1
0.1
10
100
1000
Exponent of -1.87 from 4 to 31 minutes
0.01
Hump
0.001
Exponent of -0.93
Hump
0.0001
0.00001
Time (minutes)
Conclusions:
•
Irish data,
– outside the cut off regime, survival time distribution exhibits two clear regions
– can be well fitted by Mittag Leffler function
– power law tail has exponent of magnitude less than unity (~ 0.4)
•
Japanese yen
– short waiting times (1 to 30 minutes) fits power law over large range
– but exponent greater than unity (~ 1.9)
– larger values of time shows a smaller power law regime having an exponent
between 0.9 and 1.1 that is
• at the border of the regime that can be fitted with a Mittag Leffler function.
– for larger waiting times, data exhibit two ‘humps’.
• The characteristic time could be associated with opening and closing of the major global trading
centres.