1. 28-Sep-04SS 40131 Get Folder Network Neighbourhood –Tcd.ie Ntserver-usr –Get »richmond

2. 28-Sep-04SS 40132 Books – Econophysics Statistical Mechanics of Financial Markets –J Voit Springer ISBN 3 540 41409 6 Patterns of Speculation; A study in Observational Econophysics –BM Roehner Cambridge Introduction to Econophysics –HE Stanley and R Mantegna Cambridge –Cambridge ISBN 0 521 62008 2 Theory of Financial Risk: From Statistical Physics to Risk Management –JP Bouchaud & M Potters Cambridge Financial Market Complexity –Johnson, Jefferies & Minh Hui Oxford

3. 28-Sep-04SS 40133 Econophysics SS-4013 Syllabus What is a stock? –Reading the press; fundamental and noise traders –Price formation; market and limit orders; order book –Historical data; indices and price time series; price returns; volatility; fat tails Distribution functions; –joint distributions; –Bayes’ theorem; conditional and unconditional distributions –Characteristic functions; normal distributions; Levy distributions; kurtosis; –combining random variables;central limit theorem; stable distributions; Markov process; Chapman Kolmogorov; Bachelier’s approach to price fluctuations; Additive noise, Gaussian and Wiener random walks; multiplicative walks Stochastic differential equations –Langevin Equations

4. 28-Sep-04SS 40134 Breakdown of Bachelier and Gaussian – Empirical or Stylized facts Distribution function for Returns Scaling of financial data (price returns) –Mandelbrot; Stable Levy distributions –‘Stanley’ data analysis for high frequency returns Autocorrelation functions –Price returns –Clustered volatility

5. 28-Sep-04SS 40135 Stock portfolios Risk –Distribution functions; cumulative distributions –Gaussian v power laws Minimizing risk –Markowicz’ theory and efficient market theory –Correlations –Stock taxonomy –Minimal spanning trees

6. 28-Sep-04SS 40136 Options Futures; calls and puts Black Scholes

7. 28-Sep-04SS 40137 Minority Game El Farol problem Bounded rationality

8. 28-Sep-04SS 40138 Simple agent models –Model of Bouchaud Cont Noise traders Fundamental traders Non linear effects: crashes and bubbles –Peer pressure and Lotka Volterra models –Can agents be modelled as molecules?

9. 28-Sep-04SS 40139 2002 Exam 1 A) How is a 1st order Markov process defined? [1] –Consider random variable x that takes values x1, x2, x3…..etc at times t1, t2, t3….etc –The probability for x1,t1 given the earlier sequence is –p(x1,t1|x2,t2, x3,t3…etc). –A first order Markov process is one where this probability distribution only depends on the previous value: ie p = p(x1,t1|x2,t2) –Generally just called a Markov process

10. 28-Sep-04SS 401310 2 B) The Chapman Kolmogorov equation is an integral equation for the conditional probability –p(x1,t1|x2,t2) =  -   p(x1,t1|x3,t3) p(x3,t3|x2,t2)dx3 Explain how Bachelier used this equation to obtain a Gaussian distribution for stock price returns stating the assumptions used. You may ignore any complications due to Ito corrections. –Bachelier assumed – p(x1,t1|x2,t2)  p(x1-x2,t1-t2|x2,t2) –He further assumed the fluctuations were independent. I.e. p(x1,t1|x2,t2)  p(x1,t1) –CK equation now reduces to –p(x1,t1) =  -   p(x1-x3,t1-t3) p(x3,t3)dx3 –And can be solved with ansatz p(x,t) = p0(t)exp[-p02(t)x2) –This yields p02(t1+t2) = p02(t1) p02(t2)/ [ p02(t1) + p02(t2)] –  p0(t) = H/  t –Substitute  2 = t/2  H2 –Obtain P(x,t) = exp[-x2/2  2(t)] / /  2  (t)

11. 28-Sep-04SS 401311 2 continued Bachelier then identifies the random variable with the log of the asset price, S. If this follows a random walk we have S –S 0 ~ (rt+  )Sie lnS/S 0 ~ rt +  Thus in expression for Gaussian –x = ln S/S 0 – rt and  (t) =  t This ignores the correction of Ito. With this correction included the correct expression is X = ln S/S 0 – {r -  2 /2}t Hence distribution function for stock market returns of time horizon, t follows Gaussian distribution for all t.

12. 28-Sep-04SS 401312 3 How does distribution observed for stock price returns deviate from Gaussian? –Stock returns exhibit fat tailed distribution function. Modelled by Mandelbrot as Levy distribution with tail exponent of ~1.7. More recently Stanley et al have shown high frequency returns follow an exponent of ~4 (cumulative distribution ~3)

13. 28-Sep-04SS 401313 4 Bachelier assumes volatility, , to be constant. Sketch out how the volatility looks in practise Volatility is |return| or an average of |return| over time. Illustrative sketch for annual volatilities for FTA index over 19 th and 20 th centuries. Red line is constant assumed by Bachelier.

14. 28-Sep-04SS 401314 5 What is a market order? [1] –Market orders are executed immediately when a matching order(s) arrives irrespective of the stock price –The price may change during the waiting time What is a limit order? [1] –A limit order is triggered when the market price reaches a predetermined threshold –Used to protect against unlimited losses or buying at too high a price –No guarantee exists that the order will be executed at or even close to the threshold

15. 28-Sep-04SS 401315 6 The pictures below are part of a market maker’s order book. Annotate the pictures explaining their meaning and show how the order is completed 162.2

16. 28-Sep-04SS 401316 7 How is the order book modified by the presence of a limit order?

17. 28-Sep-04SS 401317 8 Suppose orders arrive sequentially at random each with a 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,  ~ 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 (7.10 3 )} = ½ {1 – Erf (7.10 3 )} –Information given: 2/  *  z  dy exp(-y 2 ) = 1-Erf (x) –and tables of functions containing values for Erf(x) and or Erfc(x)