Why on earth are physicists working in ‘economics’? Trinity Finance Workshop September 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 Richmond Department of Physics Trinity College Dublin

Systems

Fluctuations: S(t, ) = ln[P(t+ )/P(t)] Price P(t) Time t

~8% pa ~15% pa

FT All share index

Ln FTA: ;

Dow Jones

Z,R,S If P(t+Δ)~P(t) or Δ« t then S(t) = Ln[P(t+Δ)/ P(t)] ~R(t)

Average 0.024

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 0 Gaussian tails

Example: D 1/2 = r = 0.087

FTSE100 Daily data

Led to…  Efficient market hypothesis, capital asset pricing model Markowitz 1965  Black-Scholes equation for option pricing 1973  Nobel Prize for Economics 1992  But did it work?

…the ultimate in mega- disasters! Caveat emptor… even with Nobel Prize winners!!

Overthrow of economic dogma Martingale =x t 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!

PDF is not Gaussian 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

Bounded Rationality and Minority Games – the ‘El Farol’ problem

Agents and forces

Forces in people -agents buy Sell Hold

The Ising model of a magnet a Prototype model of Statistical physics Focus on spin I. This sees: a)local force field from other spins b)external field, h I h

Cooperative phenomena Theory of Social Imitation Callen & Shapiro Physics Today July 1974 Profiting from Chaos Tonis Vaga McGraw Hill 1994

Time series and clustered volatility T. Lux and M. Marchesi, Nature , 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

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

Risk aversion induced crashes ?

Speculative Bubbles

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

Generalised Langevin Equations

PDF fit to HIS

Generalised Lotka-Volterra wealth dynamics Solomon et al 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

Lower bound on poverty drives wealth distribution!

Why is ~1.5? 1+2 or 2+4 dependents 1+3 dependents …. 1+9

Generalised Langevin models Choose simple exponential: f(x 1 +x 2 ) ~ f(x 1 )f(x 2 )

Link to Marsili and Solomon (almost) Autocatalytic term of GLV Leads to Marsili within mean field approximation: P(x 1,x 2 |t)=P(x 1 |t)P(x 2 |t) Scale time t/ζ -> t

Discrete time & Maps Logistic map f is analytic

Lorentz Cauchy Singular term Corresponds to autocatalytic term in GLV

Levy like map

Stock Exchange Crashes

Analogy with earthquakes and failure of materials

Scale invariance Allegre Continuous Power law Discrete Log periodic solutions

Include Log periodic corrections

Log periodic Oscillations DJ

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/ Sep 2002

After a crash…beyond Coppock? Interest Rate Correlation with stock price –0.72 Interest Rate Spread Correlation with stock Price –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…….