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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 h j yi e e si (1) p(up) (1) p(down) yi yi e e yi Agents and forces s (t 1) s (t ) f (t ) Forces in people -agents buy Hold Sell Cooperative phenomena Theory of Social Imitation Callen & Shapiro Physics Today July 1974 Profiting from Chaos Tonis Vaga McGraw Hill 1994 si (t 1) si (t ) sgn[ fi (t ) i ] D (t ) fi (t ) J i s j (t ) j i ( ) Time series and clustered volatility Three states buy, sell, hold, do nothing • friction T. Lux and M. Marchesi, Nature 397 1999, 498-500 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 s(t ) (t ) / 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 Instantaneous Re turn = demand/ liquidity |M |T |F |News |M |T s s 2 |F k ( p p0 ) (t ) 0 (t ) (t ') 2 D (t t ') A Differential Equation for stock movements? d 2x ds 2 f ( s ) k ( p p0 ) (t ) dt dt f ( s ) ( ) s s 2 (ln p x) Risk Neutral,(β=0); Liquid market, (λ-)>0) Two relaxation times 1 = (λ-)~ minutes 2 = 1 / ~ year =kλ/ (λ-)2 p p0 2 2D12 Risk aversion induced crashes V ( s) ds V ( s ) (t ) dt s [f - V s] s ? Speculative 0 Bubbles and Illiquid Markets V (s) s * V s / 6 * *2 p / tB tB k p/ / k 1 Fat Tails - How do we obtain P(s) 1/ s ? Over-optimistic; over-pessimistic; ds f ( s, t ) dt f ( s, t ) f ( s) g ( s) (t ) g (s) s • 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) 2001 1 Generalised Langevin Equations s f (s) sˆ2 ˆ1 f (s) a1s a2 s 2 ( s ) ; ( s, t | ˆ1 ,ˆ2 ) t s P( s, t ) ( s, t | ˆ ) , 1 2 P 2P D2 s ( sP) D1 2 ( fP) t s s s s 2 p ( x) 1 a2 x dx exp{ 2 } D1 ( x D0 / D1 ) [ x D0 / D1 ] 2 a 1 (1 1 ) 2 D1 PDF fit to HSI Generalised Lotka-Volterra wealth dynamics with Sorin Solomon, Hebrew University ˆ(t )ˆ(t ) D (t t ') wi (t 1) wi wiˆ(t ) awi aw cwi w; 1 w N a – subsidy/taxation/ ‘minimum wage’ c – measure of competition w – total wealth in economy =w(t) D – ‘economic temperature’ w j j Mean field approximation ww w ˆ(t )ˆ(t ) D (t t ') dwi wiˆ(t ) a ( w wi ) dt w wj j aw exp{ } Dwi P( wi ) wi[1 ] 1 a / D Lower bound on poverty drives wealth distribution! wmax wmin w wi w P ( w )dw P(w )dw i i i ~ 1/(1 wmin / w ) i i 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 <w> is by definition min since prices adjust to it ie KL~<w> So poorest people who have no family will seek to ensure they receive <w>/L Thus wmin~<w>/L = 1/(1- x m) ~ L/(L-1). If L=3 then =1.5 ; L=4 =1.33; L>inf >1 ?Boltzman distribution UK Income Distributions Badger 1980 Montroll & Shlesinger 1980 Cranshaw 2001 Souma 2002 UK Problems No clustered volatility Not quite right shape around peak Random walks Time Time 20th century maths Fractional derivatives p (t ) t p( x) | x| 19th 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 19th 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. No exchange controls. From 1801 to 1922 Ireland was part of UK World trends reflected in the Irish market 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 Irish Data:Ensemble of 10 Stocks(1850-4) 1 Survival Probability Distribution 1 10 100 1000 Intermediate Regime Characteristice time ~ 10 days 0.1 Power Law y = x^-0.4 0.01 Exponential y=e^-0.2x 0.001 Time (days) Sharp drop-off 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 S k pi ln pi i 1 pi E i pi i pi exp i / kT / Z Z exp i / kT i 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. 1 pi q Sq k Lt q 1 i 1 q k pi ln pi i 1 pi i q i piq i Power laws and non-extensivity pi 1 (1 q ) q i 1 1 q Zq q Z 1 q q Z q 1 (1 q ) q i Tsallis Cond-mat 0010150 Mixing in many body systems Complexity 1 1 q i S A B S A SB (1 q)S ASB Applied to stock returns Michael & Johnson cond-mat/0108017 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 •$1 invested in S&P index $1370 $1 switched between the two routes to get the best return……. $2,296,183,456 !! Alternative possibilities C2 ( ) S 2 (t ) S 2 (t ) 2 S (t ) 2 1 1 0.2 0.6 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 nonlinear 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-2α Prediction of GLV for such market measurements is that trades volumes and trade-by-trade returns follow power law with exponent γ ~ 2α ~ 3 Fokker Planck equation and ‘Hamiltonian’ P 1 H ({wi }) D( wi )[ ] t wi wi D( wi ) wi D( wi ) Dw 2 i 1 H ( wi ) ln D( wi ) N j K ( wi , w j ) wi w j D( wi ) ( D cw) ln wi aw / wi dwi