EURANDOM 6-8 March 2006

1. EURANDOM 6-8 March 2006 Analysis of financial data on different timescales - and a comparison with turbulence Robert Stresing Andreas Nawroth Joachim Peinke

2. Overview • Scale dependent analysis of financial and turbulence data by using a Fokker-Planck equation • Method for reconstruction of stochastic equations directly from given data • A new approach for very small timescales without Markov properties is presented • Existence of a special Small Timescale Regime for financial data and influence on risk

3. Analysis of financial data - stocks, FX data: • given prices s(t) • of interest:time dynamics of price changes over a periodt • Analysis of turbulence data: • given velocity s(t) • of interest:time dynamics of velocity changes over a scalet • increment: Q(t,t) = s(t + t) - s(t) • return: Q(t,t) = [s(t + t) - s(t)] / s(t) • log return: Q(t,t) = log[s(t + t)] - log[s(t)] Scale dependent analysis

4. 5 h 4 min 1 h • scale dependent analysis of Q(t,t): • distribution / pdf on scale t: p(Q,t) • how does the pdf change with the timescale? • more complete characterization: • N scale statistics • may be given by a stochastic equation:Fokker-Planck equation Scale dependent analysis t t= p(Q) p(Q) p(Q) Q in a.u. Q in a.u. Q in a.u.

5. p(Q, t2) p(Q,t0) scale t p(Q, t1) Q Q Q Q2 (t0,t2) Q1 (t0,t1) Q0 (t0,t0) Method to estimate the stochastic process Question: how are Q(t,t) and Q(t,t') connected for different scales t and t' ? => stochastic equations for: Fokker-Planck equation Langevin equation

6. Kramers-Moyal Expansion: Method to estimate the stochastic process with coefficients: Pawula’s Theorem: One obtains the Fokker-Planck equation: For trajectories the Langevin equation:

7. p(Q, t2) p(Q,t0) scale t p(Q, t1) Q Q Q Q2 (t0,t2) Q1 (t0,t1) Q0 (t0,t0) Method to estimate the stochastic process Langevin eq.: Fokker-Planck eq.:

8. Method: Kramers Moyal Coefficients Example: Volkswagen, t = 10 min

9. Functional form of the coefficients D(1) and D(2) is presented Method: The reconstructed Fokker-Planck eq. Example: Volkswagen, t = 10 min

10. Q [a.u.] Q [a.u.] Turbulence: pdfs for different scales t Financial data: pdfs for different scales t Turbulence and financial data scale t p(Q,t) [a.u.] p(Q,t) [a.u.]

11. Q [a.u.] Q [a.u.] Turbulence: pdfs for different scales t Financial data: pdfs for different scalest Method: Verification p(Q,t) [a.u.] scale t p(Q,t) [a.u.]

12. General multiscale approach: Method: Markov Property Is a simplification possible? witht1 < t2 < ... <tn Exemplary verification of Markov properties. Similar results are obtained for different parameters Black: conditional probability first order Red: conditional probability second order

13. Numerical Solution for the Fokker-Planck equation Method: Markov Property Journal of Fluid Mechanics 433 (2001) Markov

14. Numerical solution of the Fokker-Planck equation for the coefficients D(1) and D(2), which were directly obtained from the data. General view No Markov properties Numerical solution of the Fokker-Planck equation ? 4 min 1 h 5 h

15. Empiricism - What is beyond? finance: increasing intermittence Num. solution of the Fokker-Planck eq. turbulence: back to Gaussian 4 min 1 h

16. measure of distance d timescalet New approach for small scales Question: How does the shape of the distribution change with timescale? considered distribution reference distribution

17. Distance measures Kullback-Leibler-Entropy: Weighted mean square error in logarithmic space: Chi-square distance:

18. 1 s Distance measure: financial data Small Timescale Regime. Non Markov Fokker-Planck Regime. Markov process Small timescales are special! Example: Volkswagen

19. Financial and turbulence data smallest t finance turbulence

20. Dependence on the reference distribution Is the range of the small timescale regime dependent on the reference timescale? 1 s 10 s

21. Gaussian Distribution Financial and turbulence data Markov Markov turbulence finance

22. 1 s Dependence on the distance measure Are the results dependent on the special distance measure?

23. 1 s The Small Timescale Regime - Nontrivial

24. Autocorrelation Small Timescale Regime due to correlation in time? Q(x,t) |Q(x,t)|

25. 1 s The influence on risk Percentage of events beyond 10 s Volkswagen Allianz

26. Summary turbulence: back to Gaussian Markov process - Fokker-Planck equation finance: new universal feature? - Better understanding of dynamics in finance - Influence on risk - Method to reconstruct stochastic equations directly from given data. - Applications: turbulence, financial data, chaotic systems, trembling... http://www.physik.uni-oldenburg.de/hydro/

27. The End Thank you for your attention! Cooperation with St. Barth, F. Böttcher, Ch. Renner, M. Siefert, R. Friedrich (Münster)

28. Method scale dependence ofQ(x, t) : cascade like structure Q(x, t) ==> Q(x, t*) idea of fully developed turbulence cascade dynamics descibed by Langevin equation or by Kolmogorov equation

29. Method : Reconstruction of stochastic equations Derivation of the Kramers-Moyal expansion: H.Risken, Springer From the definition of the transition probability:

30. Method : Reconstruction of stochastic equations Taking only linear terms: Kramers Moyal Expansion:

31. DAX DAX