why on earth are physicists working in ‘economics’? trinity finance workshop september 26 2000
DESCRIPTION
Summary Introduction A brief look at some data, stylised facts Rationale for interest of physicists Agent models Minority game and simulations Lotka Volterra Peer pressure models Crashes. Why on earth are physicists working in ‘economics’? Trinity Finance Workshop September 26 2000. - PowerPoint PPT PresentationTRANSCRIPT
Why on earth are physicists working in ‘economics’?Trinity Finance Workshop September 26 2000
Summary Introduction
• A brief look at some data, stylised facts Rationale for interest of physicists Agent models
• Minority game and simulations• Lotka Volterra• Peer pressure models
Crashes
Peter RichmondDepartment of PhysicsTrinity College Dublin
Systems
Fluctuations: S(t, ) = ln[P(t+ )/P(t)]
Price P(t)
Time t
~8% pa
~15% pa
FT All share index 1800-2001
FTA Index
0
500
1000
1500
2000
2500
3000
3500
1750 1800 1850 1900 1950 2000 2050
Ln FTA: 1800-1950;1950-2001
Ln(FTA) = 0.0043T - 4.7942
R2 = 0.426
1.7
2.2
2.7
3.2
3.7
4.2
4.7
1750 1800 1850 1900 1950 2000
Ln
FTA
Ln FTA = 0.0767t - 145.66
R2 = 0.955
0
1
2
3
4
5
6
7
8
9
1930 1950 1970 1990 2010
Ln
FTA
Dow Jones 1896-2001Dow Jones 1896-2001
0
2000
4000
6000
8000
10000
12000
14000
1880 1900 1920 1940 1960 1980 2000 2020
Ln FTA 1800-2001 Ln DJ 1896-2001
Ln DJ~ 0.061t - 114
R2 = 0.93
Ln FTA = 0.065t - 122.3
R2 = 0.92
Ln FTA = 0.004t - 4.5
R2 = 0.33
2
3
4
5
6
7
8
9
10
1780 1830 1880 1930 1980
Z,R,S If P(t+Δ)~P(t) or Δ« t then S(t) = Ln[P(t+Δ)/ P(t)] ~R(t)
-1
-0.5
0
0.5
1
1.5
1780 1880 1980
R(t)
S(t)
-600
-400
-200
0
200
400
600
800
1780 1880 1980
Z(t)
R(t)
S(t)
FTA (Annual Z Return-mean)
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1800 1850 1900 1950 2000
FTA Annual volatility
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1800 1850 1900 1950 2000
Average 0.024
Return fluctuations Cumulative Distribution
-0.5
0
0.5
-1.00 -0.50 0.00 0.50 1.00
Volatility Cumulative Distribution
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8
Brownian or random walks
(see TCD schools web site)
Time
Distance
Bachelier (1900) (pre-dated Einstein’s application of Brownian
motion to motion of large particles in ‘colloids’)• Theorie de la Speculation Gauthiers-Villars, Paris
ˆ ˆ
( ) ( )
ˆ(
( ) ( ) 2 δ
) (
(
( )
0
)
ˆ )t
s t r g t
g t
t D t t
t
t
0
Gaussian tails
Example: D1/2 = 0.178 r = 0.087
FTSE100 Daily data
FTSE fluctuations - Auto correlation function
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1 16 31 46 61 76 91 106
121
136
151
166
181
196
Detail beyond day 1
y = -2E -21 x - 0.0009 R2 = 3E -34
-0.20
-0.10
0.00
0.10
( ) ( ) ( )C S t S t
Led to…
Efficient market hypothesis, capital asset pricing model Markowitz 1965
Black-Scholes equation for option pricing 1973
Nobel Prize for Economics 1992But did it work?
…the ultimate in mega-disasters!
Caveat emptor… even withNobel Prize winners!!
Overthrow of economic dogma
Martingale <xt+1>=xt
Independent, identical differences - iid Valid only for t >> * BUT * is comparable with timescales of
importance ….where tails in pdf are important From observation tails are NOT Gaussian Tails are much fatter!
( ) ( ) ( )C S t S t
PDF is not Gaussian
( ) exp( | |) as | |C S S S
Discontinuity ..(cusp in pdf)..?
‘Near’ tails and ‘far’ tails
stable Levy may not even be valid for near tails
Volatility persistence and anomalous decay of kurtosis
Volatility is positively correlated• Over weeks or months
Anomalous decay of kurtosis
2 2
2 22
( ) ( )( ) 1
( )
1
0.2 0.6
S t S tC
S t
1
( ) : fluctuations converge
to Gaussian distributio
-but anomalously slo
n
wly!
N
ii
S t
Bounded Rationality and Minority Games – the ‘El Farol’ problem
Agents and forces
( 1) ( ) ( )s t s t f t
Forces in people -agents
buy
Sell
Hold
The Ising model of a magneta Prototype model of Statistical physics
Focus on spin I. This sees:a)local force field from other spins
b)external field, h
I
Mean Field tanh[ ]
i ij j
i i
j
s y
y J s h
h
Cooperative phenomenaTheory of Social Imitation Callen & Shapiro Physics Today July 1974
Profiting from Chaos Tonis Vaga McGraw Hill 1994
( ) ( )
sgn ( ) ( 1) ( )() [ ]( )
i i jj
i i ii i Ds t s
f t J s t
f t tt
Time series and clustered volatility
T. Lux and M. Marchesi, Nature 397 1999, 498-500 G Iori, Applications of Physics in Financial Analysis, EPS Abs, 23E A Ponzi,
Auto Correlation Functions and Probability Density
Langevin Models
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
( ) ( ) /s t t
0
2
|
| (
||
)
|
|
|
M T
M
NewF s
F
T
k p p
s s
( ) 0
( ) ( ') 2 ( ')
t
t t D t t
Instantaneous Re turn = demand/ liquidity
A Differential Equation for stock movements?
Risk Neutral,(β=0); Liquid market, (λ-)>0) Two relaxation times
1 = (λ-)~ minutes
2 = 1 / ~ year
=kλ/ (λ-)2
2
2
2 0
(ln
( )
(
))
)
)
(
( )
(k
p x
td x ds
dtf s
f s
dp
t
s s
p
2 21
0
2D
p p
(( )
)V s
s
ds
dtt
s
( ) [ - ]V s f V s
Risk aversion induced crashes
?
/Bt k
Speculative Bubbles
( )V s
* *2 / 6V s
/ /Bp t k p
*s
0
1How do we obtaiFat Tails - n ( ) 1/ ?P s s
Over-optimistic; over-pessimistic;
• 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 B In 2001• P Richmond and S Solomon cond-mat, Int J Phys
( , )ds
f s tdt
( ) ( )
(
( , ) (
)
)f f g s t
s
s
s
s
g
t
Generalised Langevin Equations
1 2
1 2
,
2 22
2 12 2
( )ˆ ˆ; ( , | , )
ˆ( , ) ( , | )
( )
ss t
t sP s t s t
P PD s P D fP
t s s s
1
2
1(1
2
21 2
2
2
)2
22
1(1
ex/
p{ }
( )
[ ]
)
/a
D
a x dxx D DD
p x
x D D
21 2( )f s a s a s
2 1ˆ ˆ( )f sd
ts
s
d
PDF fit to HIS
Generalised Lotka-Volterra wealth dynamics Solomon et al
1 1
( ) ( 1)
( )
i i
N Ni
i i j j ii i
w t w t
cwaw aw w w R t w
N N
a – tax rate a/NΣw – minimum wage w – total wealth in economy at t c – measure of competition
GLV solution Mean field
Relative wealth
And Ito
1
1( ) ( )
N
jj
w t w t w wN
1
exp{ /(2 )}( )
1 / 2
a DxP x
xa D
/ 1i ix w w x
Lower bound on poverty drives wealth distribution!
12
1
1
1
M
M m
m
xDa
x x
x
Why is ~1.5?
1+2 or 2+4 dependents
1+3 dependents
…. 1+9
~ 1/ 4 ~ 1.33mx
~ 1/ 3 ~ 1.5mx
~ 1/ 9 ~ 1.1mx
(4 / ) /(1 / )
/
Finite size effects
/1 ; ~
1
If then 1
Wealth can fall into hands of a few
a D a D
D a
a D KK N
K
N e
Generalised Langevin models
( )ij i
j i
dxf x x
dt
2( / )
What to choose for ?
Assymetric in unlike molecular forces
1
02 ( ) ?
( ) ~ ?
3 ( ?
)
0
~
x
x
x
f
x
a
f x xe
f x
e xf x
ae x
Choose simple exponential:
f(x1+x2) ~ f(x1)f(x2)
Link to Marsili and Solomon (almost)
( ) / ( ) /j i i jx x x xi
j i j i
dxa e a e
dt
Autocatalytic term of GLV
Leads to Marsili within mean field approximation:P(x1,x2|t)=P(x1|t)P(x2|t)
/
2
Get GLV (almost) via transform: i
i
xi
ii j
j i j i j
e w
wdww a w a
dt w
Scale time t/ζ -> t
Discrete time & Maps 1 ( )n nx f x
Logistic map f is analytic
2( ) ( )f x x x
Lorentz Cauchy
( ) ( 1/ ) / 2f x x x
2
1( )
(1 )p x
x
Singular termCorresponds to autocatalytic term in GLV
Levy like map
1/1
| || |
( ) sgn[ 1/ ]2
xx
f x x x
1
2( )[1 ]
xp x
x
Stock Exchange Crashes
Analogy with earthquakes and failure of materials
Scale invarianceAllegre
Continuous Power law
Discrete Log periodic
solutions
Include Log periodic corrections
Log periodic Oscillations DJ 1921-1930
How much longer and deeper?
We predict: Bearish phase with rallies rising near end 2002 / early 2003 followed by new strong descent and a bottom ~20 Jan 2004 after which recovery..we think!Sornette and Zhou cond-mat/0209065 3 Sep 2002
After a crash…beyond Coppock?
Interest RateCorrelation with stock price –0.72
Interest Rate SpreadCorrelation with stockPrice –0.86
And finally.. Chance to dream(by courtesy of Doyne Farmer, 1999)
$1 invested from 1926 to 1996 in US bonds $14
$2,296,183,456 !!
•$1 invested in S&P index
$1370
$1 switched between the two routes
to get the best return…….