The Ising model of a magnet Focus on spin I: Sees local force field, y i, due to other spins {s j } plus external field, h I h

Agents and forces

Forces in people -agents buy Sell Hold

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 Three states buy, sell, hold, do nothing friction T. Lux and M. Marchesi, Nature , G Iori, Applications of Physics in Financial Analysis, EPS Abs, 23E

Auto Correlation Functions and Probability Density Numerical…but how can we understand what is going on?

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  =k λ / ( λ  -  ) 2

Risk aversion induced crashes ?

Speculative Bubbles and Illiquid Markets

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 B4 (2001) 523 P Richmond and S Solomon Int J Mod Phys 12 (3)

Generalised Langevin Equations

PDF fit to HSI

Generalised Lotka-Volterra wealth dynamics with Sorin Solomon, Hebrew University a – subsidy/taxation/ ‘minimum wage’ c – measure of competition w – total wealth in economy =w(t) D – ‘economic temperature’

Mean field approximation

Lower bound on poverty drives wealth distribution! w min w max

Why has Pareto exponent remained roughly constant and ~1.6-7 Min wealth to survive is ~K Average family has L members Family needs KL or become violent, strike etc is by definition min since prices adjust to it ie KL~ So poorest people who have no family will seek to ensure they receive /L Thus w min ~ /L  = 1/(1- x m ) ~ L/(L-1). If L=3 then  =1.5 ; L=4  =1.33; L>inf 

UK Income Distributions UK Badger 1980 Montroll & Shlesinger 1980 Cranshaw 2001 Souma 2002 ?Boltzman distribution

Problems No clustered volatility Not quite right shape around peak

Random walks Time

20 th century maths Fractional derivatives

19 th century Irish Stock Exchange Deals done 'matched bargain basis' members of exchange bring buyers and sellers together Essentially same as today albeit electronic trading Today, many more buyers and sellers. Recent studies of 19 th century markets find they were well integrated (Globalization and History, Kevin H. O'Rourke & Jeffrey G. Williamson, MIT Press 1999) Dublin traded international shares Not solely a regional market. World trends reflected in the Irish market No exchange controls. From 1801 to 1922 Ireland was part of UK Largest shares: Banks and key railways – Quality investments for UK investors Also traded in London.

Irish Stock Market data. Fractional time derivatives Lorenzo Sabatelli, Shane Keating Jonathan Dudley

Entropy Energy is about what is possible Entropy is about the probabilities of those possibilities happening A measure of number of possibilities or states available, W

Boltzmann-Gibbs Entropy Long history of application to equilibrium but near critical points….????

Issues  Self-organised critical systems?  Power laws?  Fractal behaviour?  Non extensive behaviour?

Tsallis ~1990 Weighting of rare events.

Power laws and non-extensivity Tsallis Cond-mat Mixing in many body systems Complexity

Applied to stock returns Michael & Johnson cond-mat/ S&P returns 1 minute data q~1.4

Simulation Long term buy hold Noise trader Fundamental trader …….

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…….

Alternative possibilities Higher moments automatically scale as sums of power laws with different slopes (Bouchaud et al Asyptotically dominant power law has exponent q But for smaller values of τ, another power law whose exponent is non- linear function of q might dominate Apparent multi-fractal behaviour, even though the process is a simple fractal with all moments determined by scaling of a single moment. For short data set, simple fractal may seem like multifractal due to slow convergence.

Heavy Tails of Buy/ Sell Order Volumes & Market Returns Distributions In most real markets, trades take place by matching pairs of buy and sell orders with compatible prices. Volume of each trade equals smallest volume of matched pairs Probability for each of 2 matched orders to exceed (or equal) a certain volume v is P(>v) ~ v - α Probability that both have a volume (equal or) larger than v is product P(>v)P(>v)~v -2 α Prediction of GLV for such market measurements is that trades volumes and trade-by-trade returns follow power law with exponent γ ~ 2 α ~ 3

Fokker Planck equation and ‘Hamiltonian’