Transcript 4028-0-Introduction

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Econophysics
Physics and Finance
(IOP UK)
Socio-physics
(GPS)
Molecules > people
Physics World October 2003
http://www.helbing.org/
Complexity
Arises from interaction
Disorder & order
Cooperation & competition
Stochastic Processes
Random movements
Statistical Physics
cooperative phenomena
Describes complex, random behaviour in terms of basic elements and
interactions
Physics and Finance-history
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Bankers
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Gamblers
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Pascal, Bernoulli
Actuaries
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Newton
Gauss
Halley
Speculators, investors
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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
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Options, Futures & Other
Derivatives
• JC Hull
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Mainly concerned with solution of
Black Scholes equation
• Applied math (HPC, DCU, UCD)
Books –
Statistical Physics
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Stochastic Processes
•
•
•
•
•
•
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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
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Topics also discussed in Voit
Read the business press
Financial Times
 Investors Chronicle
 General Business pages
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Fundamental & technical analysis
 Web sites
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• http://www.digitallook.com/
• http://www.fool.co.uk/
Motivation
Perhaps you want to become an
actuary.
 Or perhaps you want to learn about
investing?

What
happened
next?
5000000
0
20/09/04
20/09/01
20/09/98
20/09/95
20/09/92
20/09/89
20/09/86
20/09/83
23/09/04
23/09/01
20/09/80
23/09/98
23/09/95
20/09/77
23/09/92
23/09/89
20/09/74
23/09/86
23/09/83
20/09/71
23/09/80
10000000
23/09/77
20/09/68
20000000
23/09/74
30000000
23/09/71
23/09/68
DJ Closing Price
12000
10000
8000
6000
Volume of stock traded
4000
25000000
2000
15000000
0
FTSE Closing Price
8000
7000
6000
5000
4000
3000
2000
1000
0
1990-05- 1993-01- 1995-10- 1998-07- 2001-04- 2004-01- 2006-1007
31
28
24
19
14
10
Date
Questions
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Can we earn money during both upward and
downward moves?
• Speculators
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What statistical laws do changes obey? What is
frequency, smoothness of jumps?
• Investors & physicists/mathematicians
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What is risk associated with investment?
What factors determine moves in a market?
• Economists, politicians
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Can price changes (booms or crashes) be
predicted?
• Almost everyone….but tough problem!
Why physics?
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Statistical physics
• Describes complex behaviour in terms of
basic elements and interaction laws
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Complexity
• Arises from interaction
• Disorder & order
• Cooperation & competition
Financial Markets
Elements = agents (investors)
 Interaction laws = forces governing
investment decisions
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• (buy sell do nothing)
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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
Hold
Sell
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
yi   J ij s j  h
j
si  sgn[ yi ]
V ( si )   yi si
h
Mean Field theory

Gibbsian statistical mechanics
si  si 

e
yi / kT
e
yi / kT
 tanh
 s p(s )
s  , 
e
i
i
 yi / kT
 yi / kT
e
 J ij s j  h
j
kT
yi   J ij s j  h
j
sgn[ x]  tanh[ x]
Jij=J>0 Total alignment
(Ferromagnet)

Look for solutions <σi>= σ
σ = tanh[(J σ + h)/kT]
+1
-h/J
y= tanh[(J σ+h)/kT]
σ*>0
y= σ
-1
Orientation as function of h
y= tanh[(J σ+h)/kT] ~sgn [J σ+h]
+1
Increasing h
-1
Spontaneous orientation (h=0)
below T=Tc
Suppose h ~ 0; K  J / kT
1
tanh x  x  x 3 / 3! ...
6(1  K )

K
 ~ [Tc  T ]1/ 2 T  Tc
T<Tc
+1
T>Tc
 0 T  Tc
σ*
Increasing T
Social imitation
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Herding – large number of agents
coordinate their action
Direct influence between traders
through exchange of information
 Feedback of price changes onto
themselves
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Cooperative phenomena
Non-linear complexity
Opinion changes
K Dahmen and J P Sethna Phys Rev B53 1996 14872
J-P Bouchaud Quantitative Finance 1 2001 105
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magnets si
field h
 trader’s position φi (+ -?)
 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
N
i (t  1)  sgn[hi (t )   J ij j (t )]?
i 1
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Basic concepts of stocks and investors
Stochastic dynamics
Topics
• Langevin equations; Fokker Planck equations; Chapman,
Kolmogorov, Schmulochowski; Weiner processes;
diffusion
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Bachelier’s model of stock price dynamics
Options
Risk
Empirical and ‘stylised’ facts about stock
data
• Non Gaussian
• Levy distributions
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The Minority Game
• or how economists discovered the scientific method!
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Some simple agent models
• Booms and crashes
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Stock portfolios
• Correlations; taxonomy
Basic material
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What is a stock?
• Fundamentals; prices and value;
• Nature of stock data
• Price, returns & volatility
Empirical indicators used by
‘professionals’
 How do investors behave?
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Normal v Log-normal
distributions
Probability distribution density
functions p(x)
characterises occurrence of random variable, X
For 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

N
1
f  x   Lt  f ( xn)   f ( x) p( x)dx
N  N n 1




f ( x)dC ( x)

In statistics & economics literature, often
find E[ f x ] instead of f x


Moments and the ‘volatility’
m
n > =  p(x) x

<
x
n
Mean: m 1 = m
n
dx
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)2/22)
All moments exist
For symmetric distribution
m=0; m2n+1= 0
and
m2n = (2n-1)(2n-3)…. 2n
Note for Gaussian: m4=34 =3m22
m4 is ‘kurtosis’
Some other Distributions
Log normal
PLN(x) ≡ (1/(2π)½ xσ) exp(-log2(x/x0)/22)
mn = x0nexp(n22/2)
Cauchy
PC (x) ≡ /{1/(2 +x2)}
Power law tail
(Variance diverges)
Levy distributions
NB Bouchaud uses 
instead of 
Curves that have
narrower peaks
and fatter tails
than 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?
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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
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Model distribution by Gaussian, p(x) = 1/[(2)½] exp(-[x-m]2/22)
Answer required is

•
•
•
•
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P(x>400) = 400 dx p(x) ~ 400 dx 1/((2)½) exp(-x2/22)
= 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)