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…….