Why on earth are physicists working in ‘economics’? Trinity Finance Workshop September 26 2000

1. 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 Why on earth are physicists working in ‘economics’?Trinity Finance Workshop September 26 2000 Peter Richmond Department of Physics Trinity College Dublin

2. Systems

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

5. ~15% pa ~8% pa

6. FT All share index 1800-2001

7. Ln FTA: 1800-1950;1950-2001

8. Dow Jones 1896-2001

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

11. Average 0.024

14. Brownian or random walks (see TCD schools web site) Distance Time

15. Bachelier (1900) (pre-dated Einstein’s application of Brownian motion to motion of large particles in ‘colloids’) • Theorie de la Speculation Gauthiers-Villars, Paris Gaussian tails 0

16. Example: D1/2 = 0.178 r = 0.087

17. FTSE100 Daily data

18. 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?

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

20. Overthrow of economic dogma • Martingale <xt+1>=xt • 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!

21. PDF is not Gaussian Discontinuity ..(cusp in pdf)..?

22. ‘Near’ tails and ‘far’ tails stable Levy may not even be valid for near tails

23. Volatility persistence and anomalous decay of kurtosis • Volatility is positively correlated • Over weeks or months

24. Anomalous decay of kurtosis

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

26. Agents and forces

27. Forces in people -agents buy Hold Sell

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

29. Cooperative phenomenaTheory of Social Imitation Callen & Shapiro Physics Today July 1974Profiting from Chaos Tonis Vaga McGraw Hill 1994

30. Time series and clustered volatility • T. Lux and M. Marchesi, Nature 397 1999, 498-500 • G Iori, Applications of Physics in Financial Analysis, EPS Abs, 23E • A Ponzi,

31. Auto Correlation Functions and Probability Density

32. 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

33. 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

34. Risk aversion induced crashes ?

35. Speculative Bubbles

37. 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 Over-optimistic; over-pessimistic;

38. Generalised Langevin Equations

39. PDF fit to HIS

40. 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

41. GLV solution • Mean field • Relative wealth • And Ito

42. Lower bound on poverty drives wealth distribution!

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

46. Generalised Langevin models Choose simple exponential: f(x1+x2) ~ f(x1)f(x2)

47. Link to Marsili and Solomon (almost) Leads to Marsili within mean field approximation: P(x1,x2|t)=P(x1|t)P(x2|t) Autocatalytic term of GLV Scale time t/ζ -> t

48. Logistic map f is analytic Discrete time & Maps

49. Lorentz Cauchy Singular term Corresponds to autocatalytic term in GLV

50. Levy like map