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Econophysics Physics and Finance (IOP UK) Socio-physics(GPS) Molecules > people Physics World October 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 Newton Gauss Gamblers Pascal, Bernoulli Actuaries 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

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

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 Sell Hold

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

Mean Field theory Gibbsian statistical mechanics

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

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

Spontaneous orientation (h=0) below T=Tc +1 σ* T<Tc T>Tc 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 B J-P Bouchaud Quantitative Finance magnets s i  trader’s position φ i (+ -?) field h  time dependent random a priori opinion h i (t) h>0 – propensity to buy h<0 – propensity to sell J – connectivity matrix

Confidence? h i is random variable =h(t); =Δ 2 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

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

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 x 1, x 2 …x N average or expected value of any function f(x) is In statistics & economics literature, often find E[ ] instead of

Moments and the ‘volatility’ m n  =  p(x) x n dx Mean: m 1 = m Standard deviation, Root mean square (RMS) variance or ‘volatility’ :  2 = =  p(x) (x-m) 2 dx = m 2 – m 2 NB For m n and hence  to be meaningful, integrals have to converge and p(x) must decrease sufficiently rapidly for large values of x.

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

Some other Distributions Log normal P LN (x) ≡ (1/(2π) ½ xσ) exp(-log 2 (x/x 0 )/2 2 ) m n = x 0 n exp(n 2  2 /2) Cauchy P C (x) ≡ /{1/( 2 +x 2 )} 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 /2  2 ) Answer required is P(x>400) =  400  dx p(x) ~  400  dx 1/((2  ) ½  ) exp(-x 2 /2  2 ) = 1/(  ) ½  z  dy exp(-y 2 ) where z = 400/0.04*  2 ~ 7*10 +3 =1/2{ Erfc ( )} = ½ {1 – Erf ( )} Information given: 2/ *  z  dy exp(-y 2 ) = 1-Erf (x) and tables of functions containing values for Erf(x) and or Erfc(x)