Get Folder

1. Get Folder • Network Neighbourhood • Tcd.ie • Ntserver-usr • Get • richmond

2. 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

3. Physics and Finance-history • Bankers • Newton • Gauss • Gamblers • Pascal, Bernoulli • Actuaries • Halley • Speculators, investors • Bachelier • Black Scholes >Nobel prize for economics

4. 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

5. Books – Financial math • Options, Futures & Other Derivatives • JC Hull • Mainly concerned with solution of Black Scholes equation • Applied math (HPC, DCU, UCD)

6. 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

7. 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/

8. Perhaps you want to become an actuary. Or perhaps you want to learn about investing? Motivation What happened next?

11. 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!

12. Why physics? • Statistical physics • Describes complex behaviour in terms of basic elements and interaction laws • Complexity • Arises from interaction • Disorder & order • Cooperation & competition

13. Financial Markets • Elements = agents (investors) • Interaction laws = forces governing investment decisions • (buy sell do nothing) • Trading is increasingly automated using computers

14. Social Imitation Theory of Social Imitation Callen & Shapiro Physics Today July 1974Profiting from Chaos Tonis Vaga McGraw Hill 1994 buy Hold Sell

15. 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 h

16. Mean Field theory • Gibbsian statistical mechanics

17. Jij=J>0 Total alignment (Ferromagnet) • Look for solutions <σi>= σ σ = tanh[(J σ + h)/kT] +1 -h/J σ*>0 y= tanh[(J σ+h)/kT] y= σ -1

18. Orientation as function of h y= tanh[(J σ+h)/kT] ~sgn [J σ+h] +1 Increasing h -1

19. Spontaneous orientation (h=0) below T=Tc T<Tc +1 T>Tc σ* Increasing T

20. Social imitation • Herding – large number of agents coordinate their action • Direct influence between traders through exchange of information • Feedback of price changes onto themselves

21. Cooperative phenomenaNon-linear complexity

22. Opinion changesK Dahmen and J P Sethna Phys Rev B53 1996 14872J-P Bouchaud Quantitative Finance 1 2001 105 • magnets si trader’s position φi (+ -?) • field h  time dependent random a priori opinion hi(t) • h>0 – propensity to buy • h<0 – propensity to sell • J – connectivity matrix

23. 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

24. Need mechanism for changing mind • Need a dynamics • Eg G Iori

25. Topics • Basic concepts of stocks and investors • Stochastic dynamics • 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

26. 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?

27. Normal v Log-normal distributions

28. 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 Probability distribution density functions p(x)

29. Cumulative probability function C(x) = Probability that x<X = -  x P(x)dx = P<(x) P>(x) = 1- P<(x) C() = 1; C(-) = 0

30. Average and expected values For string of values x1, x2…xN average or expected value of any function f(x) is In statistics & economics literature, often find E[ ] instead of

31. Moments and the ‘volatility’ m n < xn > =  p(x) xn dx Mean: m 1 = m 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.

32. Gaussian (Normal) distributions PG(x) ≡ (1/(2π)½σ) exp(-(x-m)2/22) All moments exist For symmetric distribution m=0; m2n+1= 0 and m2n = (2n-1)(2n-3)…. 2n Note for Gaussian: m4=34 =3m22 m4is ‘kurtosis’

33. Some other Distributions Log normal PLN(x) ≡ (1/(2π)½ xσ) exp(-log2(x/x0)/22) mn = x0nexp(n22/2) Cauchy PC (x) ≡ /{1/(2 +x2)} Power law tail (Variance diverges)

34. Levy distributions NB Bouchaud uses  instead of  Curves that have narrower peaks and fatter tails than Gaussians are said to exhibit ‘Leptokurtosis’

35. 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]2/22) • Answer required is • P(x>400) = 400 dx p(x) ~ 400 dx 1/((2)½) exp(-x2/22) • = 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)