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EconophysicsPhysics and Finance (IOP UK)Socio-physics (GPS)

Molecules > peoplePhysics World October 2003http://www.helbing.org/

ComplexityArises from interactionDisorder & orderCooperation & competition

Stochastic ProcessesRandom movements

Statistical Physics cooperative phenomenaDescribes complex, random behaviour in terms of basic elements and

interactions

Physics and Finance-history Bankers

Newton Gauss

Gamblers Pascal, Bernoulli

Actuaries Halley

Speculators, investors Bachelier Black Scholes >Nobel prize for economics

Books – Econophysics

• Statistical Mechanics of Financial Markets • J Voit Springer

• Patterns of Speculation; A study in Observational Econophysics

• BM Roehner Cambridge• Introduction to Econophysics

• HE Stanley and R Mantegna Cambridge• Theory of Financial Risk: From Statistical

Physics to Risk Management • JP Bouchaud & M Potters Cambridge

• Financial Market Complexity• Johnson, Jefferies & Minh Hui Oxford

Books – Financial math

Options, Futures & Other Derivatives

• JC Hull

Mainly concerned with solution of Black Scholes equation

• Applied math (HPC, DCU, UCD)

Books – Statistical Physics

Stochastic Processes• Quantum Field Theory (Chapter 3) Zimm Justin• Langevin equations• Fokker Planck equations• Chapman Kolmogorov Schmulochowski• Weiner processes; diffusion• Gaussian & Levy distributions

Random Walks & Transport• Statistical Dynamics, chapter 12, R Balescu

Topics also discussed in Voit

Read the business press Financial Times Investors Chronicle General Business pages

Fundamental & technical analysis Web sites

• http://www.digitallook.com/ • http://www.fool.co.uk/

Motivation

What happened next?

Perhaps you want to become an actuary.

Or perhaps you want to learn about investing?

DJ Closing Price

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1200020/09/68

20/09/71

20/09/74

20/09/77

20/09/80

20/09/83

20/09/86

20/09/89

20/09/92

20/09/95

20/09/98

20/09/01

20/09/04

Volume of stock traded

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10000000

15000000

20000000

25000000

30000000

23/09/68

23/09/71

23/09/74

23/09/77

23/09/80

23/09/83

23/09/86

23/09/89

23/09/92

23/09/95

23/09/98

23/09/01

23/09/04

FTSE Closing Price

0

1000

2000

3000

4000

5000

6000

7000

8000

1990-05-07

1993-01-31

1995-10-28

1998-07-24

2001-04-19

2004-01-14

2006-10-10

Date

Questions Can we earn money during both upward and

downward moves?• Speculators

What statistical laws do changes obey? What is frequency, smoothness of jumps?

• Investors & physicists/mathematicians What is risk associated with investment? What factors determine moves in a market?

• Economists, politicians Can price changes (booms or crashes) be

predicted?• Almost everyone….but tough problem!

Why physics?

Statistical physics• Describes complex behaviour in terms of

basic elements and interaction laws

Complexity• Arises from interaction

• Disorder & order• Cooperation & competition

Financial Markets Elements = agents (investors) Interaction laws = forces governing

investment decisions • (buy sell do nothing)

Trading is increasingly automated using computers

Social Imitation Theory of Social Imitation Callen & Shapiro Physics Today July 1974

Profiting from Chaos Tonis Vaga McGraw Hill 1994

buy

Sell

Hold

Are there parallels with statistical physics? E.g. 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

sgn[ ]

( )

i ij jj

i i

i i i

y J s h

s y

V s y s

h

Mean Field theory Gibbsian statistical mechanics

,

/ /

/ /

( )

tanh

i i

i i

i i i is

y kT y kT

y kT y kT

ij jj

s s s p s

e e

e e

J s h

kT

sgn[ ] tanh[ ]

i ij jj

y J s h

x x

Jij=J>0 Total alignment (Ferromagnet)

Look for solutions <σi>= σ

σ = tanh[(J σ + h)/kT]

-1

+1

y= σ y= tanh[(J σ+h)/kT]

-h/J σ*>0

Orientation as function of h

y= tanh[(J σ+h)/kT] ~sgn [J σ+h]

-1

+1

Increasing h

Spontaneous orientation (h=0) below T=Tc

+1

3

1/ 2

Suppose ~ 0; / 1

tanh / 3! ...

6(1 )

~ [ ]

0 c c

c

h K J kT

x x x

K

K

T T T T

T T

σ*

T<Tc

T>Tc

Increasing T

Social imitation Herding – large number of agents

coordinate their action Direct influence between traders

through exchange of information Feedback of price changes onto

themselves

Cooperative phenomenaNon-linear complexity

Opinion changesK Dahmen and J P Sethna Phys Rev B53 1996 14872J-P Bouchaud Quantitative Finance 1 2001 105

magnets si trader’s position φi (+ -?) field h time dependent random

a priori opinion hi(t)

• h>0 – propensity to buy• h<0 – propensity to sell • J – connectivity matrix

Confidence? hi is random variable

<hi>=h(t); <[hi-h(t)]2>=Δ2

h(t) represents confidence• Economy strong: h(t)>0• expect recession: h(t)<0

• Leads to non zero average for pessimism or optimism

Need mechanism for changing mind

Need a dynamics• Eg G Iori

1

( 1) sgn[ ( ) ( )]?N

i i ij ji

t h t J t

Topics Basic concepts of stocks and investors Stochastic dynamics

• Langevin equations; Fokker Planck equations; Chapman, Kolmogorov, Schmulochowski; Weiner processes; diffusion

Bachelier’s model of stock price dynamics Options Risk Empirical and ‘stylised’ facts about stock

data• Non Gaussian• Levy distributions

The Minority Game• or how economists discovered the scientific method!

Some simple agent models• Booms and crashes

Stock portfolios• Correlations; taxonomy

Basic material

What is a stock?• Fundamentals; prices and value; • Nature of stock data• Price, returns & volatility

Empirical indicators used by ‘professionals’

How do investors behave?

Normal v Log-normal distributions

Probability distribution density functions p(x)

characterises occurrence of random variable, XFor all values of x:

p(x) is positive

p(x) is normalised, ie: -/0 p(x)dx =1

p(x)x is probability that x < X < x+x

a b p(x)dx is probability that x lies between a

and b

Cumulative probability function

C(x) = Probability that x<X

= - x P(x)dx

= P<(x)

P>(x) = 1- P<(x)

C() = 1; C(-) = 0

Average and expected values

For string of values x1, x2…xN

average or expected value of any function f(x) is

In statistics & economics literature, often find E[ ] instead of

1

1( ) ( ) ( )

( ) ( )

N

n

N n

f x f x f x p x dxN

f x dC x

Lt

xf xf

Moments and the ‘volatility’ m n < xn > = p(x) x n dx

Mean: m 1 = m

Standard deviation, Root mean square (RMS) variance or ‘volatility’ :

2 = < (x-m)2 > = p(x) (x-m)2dx

= m2 – m 2

NB For mn and hence to be meaningful, integrals have to converge and p(x) must decrease sufficiently rapidly for large values of x.

Gaussian (Normal) distributions

PG(x) ≡ (1/ (2π)½σ) exp(-(x-m)2/22)

All moments exist

For symmetric distribution m=0; m2n+1= 0 andm2n = (2n-1)(2n-3)…. 2n

Note for Gaussian: m4=34 =3m22

m4 is ‘kurtosis’

Some other Distributions

Log normalPLN(x) ≡ (1/(2π)½ xσ) exp(-log2(x/x0)/22)

mn = x0nexp(n22/2)

CauchyPC (x) ≡ /{1/(2 +x2)}

Power law tail(Variance diverges)

Levy distributions

NB Bouchaud uses instead of

Curves that have narrower peaksand fatter tailsthan Gaussians are said to exhibit‘Leptokurtosis’

Simple example

Suppose orders arrive sequentially at random with mean waiting time of 3 minutes and standard deviation of 2 minutes. Consider the waiting time for 100 orders to arrive. What is the approximate probability that this will be greater than 400 minutes?

Assume events are independent. For large number of events, use central limit theorem to obtain m and . Thus

• Mean waiting time, m, for 100 events is ~ 100*3 = 300 minutes• Average standard deviation for 100 events is ~ 2/100 = 0.2 minutes

Model distribution by Gaussian, p(x) = 1/[(2)½] exp(-[x-m]2/22) Answer required is

• P(x>400) = 400

dx p(x) ~ 400

dx 1/((2)½) exp(-x2/22)

• = 1/()½ z

dy exp(-y2)

• where z = 400/0.04*2 ~ 7*10+3

• =1/2{ Erfc (7.103)} = ½ {1 – Erf (7.103)}

  Information given: 2/ * z

dy exp(-y2) = 1-Erf (x) and tables of functions containing values for Erf(x) and or Erfc(x)