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

 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 Overview

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

scale dependent analysis of Q(t,  ): – distribution / pdf on scale  : p(Q,  ) – how does the pdf change with the timescale? more complete characterization: – N scale statistics – may be given by a stochastic equation: Fokker-Planck equation 5 h 4 min1 h Q in a.u. p(Q)  

Method to estimate the stochastic process Q p(Q,  0 ) Q p(Q,  1 ) Q p(Q,  2 ) scale  Q 0 (t 0,  0 ) Q 1 (t 0,  1 ) Q 2 (t 0,  2 ) Question: how are Q(t,  ) and Q(t,  ') connected for different scales  and  ' ? => stochastic equations for: Fokker-Planck equation Langevin equation

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

Method to estimate the stochastic process Q p(Q,  0 ) Q p(Q,  1 ) Q p(Q,  2 ) scale  Q 0 (t 0,  0 ) Q 1 (t 0,  1 ) Q 2 (t 0,  2 ) Langevin eq.: Fokker-Planck eq.:

Method: Kramers Moyal Coefficients Example: Volkswagen,  = 10 min

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

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

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

Method: Markov Property General multiscale approach: Exemplary verification of Markov properties. Similar results are obtained for different parameters Black: conditional probability first order Red: conditional probability second order with  1 <  2 <... <  n Is a simplification possible?

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

General view Numerical solution of the Fokker-Planck equation for the coefficients D (1) and D (2), which were directly obtained from the data. ? 4 min1 h5 h Numerical solution of the Fokker-Planck equation No Markov properties

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

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

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

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

Financial and turbulence data finance turbulence smallest 

Dependence on the reference distribution Is the range of the small timescale regime dependent on the reference timescale? 1 s10 s

Financial and turbulence data Gaussian Distribution finance turbulence Markov

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

The Small Timescale Regime - Nontrivial 1 s

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

The influence on risk VolkswagenAllianz Percentage of events beyond 10  1 s

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

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

Method scale dependence of Q(x,  ) : cascade like structure Q(x,  ) ==> Q(x,    ) idea of fully developed turbulence cascade dynamics descibed by Langevin equation or by Kolmogorov equation

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

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

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