Download Cooperative phenomena Theory of Social Imitation

The Ising model of a magnet
Focus on spin I:
Sees local force field,
yi, due to other spins {sj}
plus external field, h
I
yi   J ij s j  h
h
j
 yi
e e
si  (1) p(up)  (1) p(down)   yi yi
e e
yi
Agents and forces
s (t  1)  s (t )  f (t )
Forces in people -agents
buy
Hold
Sell
Cooperative phenomena
Theory of Social Imitation Callen & Shapiro Physics Today July 1974
Profiting from Chaos Tonis Vaga McGraw Hill 1994
si (t  1)  si (t )  sgn[ fi (t )   i ]  D (t )
fi (t )  J i  s j (t )
j
 i  ( )
Time series and clustered volatility
Three states buy, sell, hold, do nothing
• friction


T. Lux and M. Marchesi, Nature 397 1999, 498-500
G Iori, Applications of Physics in Financial Analysis, EPS Abs,
23E
Auto Correlation Functions and Probability Density
Numerical…but how can we understand
what is going on?
Langevin
Models


s(t )   (t ) /  
Tonis Vaga Profiting from Chaos McGraw Hill 1994
J-P Bouchaud and R Cont, Langevin Approach to
Stock Market Fluctuations and Crashes Euro Phys J B6
(1998) 543
Instantaneous Re turn = demand/ liquidity
   |M  |T  |F  |News
 |M  
 |T   s   s 2
 |F   k ( p  p0 )
 (t )  0
 (t ) (t ')  2 D (t  t ')
A Differential Equation for stock
movements?
d 2x
ds
 2    f ( s )  k ( p  p0 )   (t )
dt
dt
f ( s )  (   ) s   s 2
(ln p  x)
Risk Neutral,(β=0);
Liquid market, (λ-)>0)
Two relaxation times
 1 = (λ-)~ minutes
 2 = 1 / ~ year
 =kλ/ (λ-)2
p  p0
 2  2D12
Risk aversion induced
crashes
V ( s)
ds
V ( s )

  (t )
dt
s
[f  - V s]
s
?
Speculative
  0 Bubbles and
Illiquid Markets
V (s)
s
*
V   s / 6
*
*2
 p / tB
tB
k p/
 / k
1
Fat Tails - How do we obtain P(s)  1/ s
?
Over-optimistic;
over-pessimistic;
ds
 f ( s, t )
dt
f ( s, t )  f ( s)  g ( s) (t )
g (s)  s
•

R Gilbrat, Les Inegalities Economiques, Sirey, Paris 1931
O Biham, O Malcai, M Levy and S Solomon,
• Generic emergence of power law distributions and Levy-stable fluctuations in
discrete logistic systems
• Phys Rev E 58 (1998) 1352


P Richmond Eur J Phys B4 (2001) 523
P Richmond and S Solomon Int J Mod Phys 12 (3) 2001 1
Generalised Langevin Equations
s  f (s)  sˆ2  ˆ1
f (s)  a1s  a2 s 2
  ( s  )

;    ( s, t | ˆ1 ,ˆ2 )
t
s
P( s, t )   ( s, t | ˆ )  ,
1
2
P
 
2P 

 D2  s ( sP)   D1 2  ( fP)
t
s  s
s
s

2
p ( x) 
1
a2 x dx
exp{  2
}
D1 ( x  D0 / D1 )
[ x  D0 / D1 ]
2
a
1
(1 1 )
2
D1
PDF fit to HSI
Generalised Lotka-Volterra
wealth dynamics
with Sorin Solomon, Hebrew University
ˆ(t )ˆ(t )  D (t  t ')
wi (t  1)  wi  wiˆ(t )  awi  aw  cwi w;
1
w
N
a – subsidy/taxation/ ‘minimum wage’
c – measure of competition
w – total wealth in economy =w(t)
D – ‘economic temperature’
w
j
j
Mean field approximation
ww w
ˆ(t )ˆ(t )  D (t  t ')
dwi
 wiˆ(t )  a ( w  wi )
dt
w   wj
j
aw
exp{
}
Dwi
P( wi ) 
wi[1 ]
  1 a / D
Lower bound on poverty drives wealth
distribution!
wmax
wmin
w  wi
w P ( w )dw


 P(w )dw
i
i
i
 ~ 1/(1  wmin / w )
i
i
Why has Pareto exponent remained roughly
constant and ~1.6-7
Min wealth to survive is ~K
Average family has L members
Family needs KL or become violent, strike etc
<w> is by definition min since prices adjust to it ie
KL~<w>
So poorest people who have no family will seek
to ensure they receive <w>/L
Thus wmin~<w>/L
 = 1/(1- x m) ~ L/(L-1).
If L=3 then  =1.5 ; L=4  =1.33; L>inf  >1
?Boltzman distribution
UK Income
Distributions
Badger 1980
Montroll & Shlesinger 1980
Cranshaw 2001
Souma 2002
UK
Problems
No clustered volatility
Not quite right shape around peak
Random walks
Time
Time
20th century maths
Fractional derivatives

 p (t )
t 

 p( x)

| x|
19th century Irish Stock Exchange
Deals done 'matched bargain basis'
members of exchange bring buyers and sellers together


Essentially same as today albeit electronic trading
Today, many more buyers and sellers.
Recent studies of 19th century markets find they were well
integrated
(Globalization and History, Kevin H. O'Rourke & Jeffrey G. Williamson,
MIT Press 1999)
Dublin traded international shares

Not solely a regional market.

No exchange controls.
From 1801 to 1922 Ireland was part of UK
World trends reflected in the Irish market

Largest shares: Banks and key railways –


Quality investments for UK investors
Also traded in London.
Irish Stock Market data.
Fractional time derivatives
Lorenzo Sabatelli, Shane Keating Jonathan Dudley
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
Entropy
Energy is about what is possible
Entropy is about the probabilities of
those possibilities happening

A measure of number of possibilities or
states available, W
Boltzmann-Gibbs Entropy
S   k  pi ln pi
i
1   pi
E      i pi
i
pi  exp   i / kT  / Z
Z   exp   i / kT 
i
Long history of application
to equilibrium but near
critical points….????
Issues




Self-organised critical
systems?
Power laws?
Fractal behaviour?
Non extensive behaviour?
Tsallis ~1990
Weighting of rare events.
1   pi
q
Sq  k

Lt q 1
i
1 q
 k  pi ln pi
i
1   pi
i

q
   i piq
i
Power laws and non-extensivity
pi
1  (1  q )   


q i
1
1 q
Zq
q 

Z
1 q
q
Z q   1  (1  q )  q i 



Tsallis Cond-mat 0010150
Mixing in many body systems
Complexity
1
1 q
i
S A B  S A  SB  (1  q)S ASB
Applied to stock returns
Michael & Johnson cond-mat/0108017

S&P returns 1 minute data
q~1.4
Simulation
Long term
buy hold
Noise trader
Fundamental
trader
…….
And finally.. Chance to dream
(by courtesy of Doyne Farmer, 1999)
$1 invested from 1926 to 1996 in US bonds
 $14
•$1 invested in S&P index
 $1370
$1 switched between the two routes
to get the best return…….
 $2,296,183,456 !!
Alternative possibilities
C2 ( ) 
S 2 (t   ) S 2 (t )
2
S (t )
2
1
1  
0.2    0.6
Higher moments automatically scale as sums of power laws with
different slopes (Bouchaud et al
Asyptotically dominant power law has exponent q
But for smaller values of τ, another power law whose exponent is nonlinear function of q might dominate
Apparent multi-fractal behaviour, even though the process is a simple
fractal with all moments determined by scaling of a single moment.
For short data set, simple fractal may seem like multifractal due to slow
convergence.
Heavy Tails of Buy/ Sell Order Volumes &
Market Returns Distributions
In most real markets, trades take place by matching pairs of buy
and sell orders with compatible prices.
Volume of each trade equals smallest volume of matched pairs
Probability for each of 2 matched orders to exceed (or equal) a
certain volume v is P(>v) ~ v-α
Probability that both have a volume (equal or) larger than v is
product P(>v)P(>v)~v-2α
Prediction of GLV for such market measurements is that trades
volumes and trade-by-trade returns follow power law with
exponent γ ~ 2α ~ 3
Fokker Planck
equation and
‘Hamiltonian’
P


1 H ({wi })

D( wi )[

]
t wi
wi D( wi ) wi
D( wi )  Dw
2
i
1
H ( wi )  ln D( wi ) 
N

j
K ( wi , w j ) wi w j
D( wi )
 ( D  cw) ln wi  aw / wi
dwi