Get Folder Network Neighbourhood Tcd.ie •Ntserver-usr •Get • richmond 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 Bankers Gamblers Pascal, Bernoulli Actuaries Newton Gauss 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 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 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 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 Herding – large number of agents coordinate their action Direct influence between traders through exchange of information Feedback of price changes onto themselves 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 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 Basic concepts of stocks and investors Stochastic dynamics Topics • 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, 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)2/22) All moments exist For symmetric distribution m=0; m2n+1= 0 and m2n = (2n-1)(2n-3)…. 2n Note for Gaussian: m4=34 =3m22 m4 is ‘kurtosis’ Some other Distributions Log normal PLN(x) ≡ (1/(2π)½ xσ) exp(-log2(x/x0)/22) mn = x0nexp(n22/2) 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? 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)