Analysis of financial data ondifferent timescales

- and a comparison with turbulence

Robert StresingAndreas NawrothJoachim 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

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

p(QN ,N ,...,Q1,1)

-0.01 0.00 0.0110-3

10-2

10-1

100

101

1025 h4 min 1 h

Q in a.u. Q in a.u. Q in a.u.

p(Q

)

p(Q

)

p(Q

)

Method to estimate the stochastic process

Q

p(Q,0)

Q

p(Q, 1)

Q

p(Q, 2)scale

Q0 (t0,0)

Q1 (t0,1)

Q2 (t0,2)

Question: how are Q(t,) and Q(t,')

connected for different scales and ' ?

=> stochastic equations for:

p(Q, )...

Q(t, )...

Fokker-Planck equation Langevin equation

Method to estimate the stochastic process

p(Q, ) Q

D(1)(Q, )2

Q2D(2)(Q, )

p(Q,)

One obtains the Fokker-Planck equation:

Q

D(1)(Q, ) D(2)(Q, ) ()

For trajectories the Langevin equation:

Pawula’s Theorem:

D(4 ) 0 D(k ) 0 k 2

p Q,

Q

n

D(n )(Q, )p Q, n1

Kramers-Moyal Expansion:

D(n )(Q, )1

n!lim 0 ( Q Q) n

p Q , | Q, d Q with coefficients:

Method to estimate the stochastic process

Q(x, )

D(1)(Q, ) D(2)(Q, ) ()

p(Q, ) Q

D(1)(Q, )2

Q2D(2)(Q, )

p(Q,)

Q

p(Q,0)

Q

p(Q, 1)

Q

p(Q, 2)scale

Q0 (t0,0)

Q1 (t0,1)

Q2 (t0,2)

Langevin eq.:

Fokker-Planck eq.:

Method: Kramers Moyal Coefficients

D(n )(Q, )lim 0 M (n )(Q,, )lim 0

1

n!( Q Q) n

p Q , | Q, d Q

0 5 10 15 20 25 300

4.10-4

8.10-4

1.10-3

2.10-3

2.10-3

²

M (1)

(Q=

0,00

1,

= 6

00s,

² )

Example: Volkswagen, = 10 min

Method: The reconstructed Fokker-Planck eq.

Functional form of the coefficients D(1) and D(2) is presented

p(Q, ) Q

D(1)(Q, )2

Q2 D(2)(Q, )

p(Q, )

Example: Volkswagen, = 10 min

-2.10-3 0 2.10-3-0.01

0.00

0.01

-1.10-3 0 1.10-30

2.10-7

4.10-7

Q

Q

Turbulence:

pdfs for different scales Financial data:

pdfs for different scales

Turbulence and financial data

Q [a.u.]

p(Q

,)

[a.u

.]

scal

e

-0.5 0.0 0.510-7

10-5

10-3

10-1

101

103

105

12 h

4 h

1 h

15 min

4 min

-4 -2 0 2 410-4

10-2

100

102

104

L0,6L0,35L0,2L0,1L

Q [a.u.]

p(Q

,)

[a.u

.]

Turbulence:

pdfs for different scales Financial data:

pdfs for different scales

Method: Verification

-4 -2 0 2 410-4

10-2

100

102

104

Q [a.u.]

p(Q

,)

[a.u

.]

Q [a.u.]

p(Q

,)

[a.u

.]

-0.5 0.0 0.510-7

10-5

10-3

10-1

101

103

105

scal

e

Method: Markov Property

General multiscale approach:

p(Q1,1 | Q2, 2;...;Qn , n )p(Q1,1 | Q2, 2)

Exemplary verification of Markov properties. Similar results are obtainedfor different parameters

Black: conditional probability first orderRed: conditional probability second order

p(Q1,1;...;Qn, n )p(Q1,1 | Q2, 2)...p(Qn 1, n 1 | Qn, n )p(Qn, n )

with 1 < 2 < ... < n

p(Q1,1;...;Qn, n )

Is a simplification possible?

-0.09 -0.045 0.0 0.045 0.09-0.09

-0.045

0.0

0.045

-0.09

Method: Markov Property

10 -6

10 -4

10-2

10 0

102

10 4

-4 -2 0 2 4u /

r

u0/-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

8

Journal of Fluid Mechanics 433 (2001)

Numerical Solution for the Fokker-Planck equation

p(Q1,1,...,QN ,N )Markov

p(Q1,1 | Q2, 2)

General view

Numerical solution of the Fokker-Planck equation

for the coefficients D(1) and D(2),

which were directly obtained from the data.

-0.01 0.00 0.0110-3

10-2

10-1

100

101

102

103

-0.01 0.00 0.0110-3

10-2

10-1

100

101

102

-0.01 0.00 0.0110-3

10-2

10-1

100

101

102

Q

Q

Q

?

4 min 1 h 5 h

Numerical solution of the Fokker-Planck equationNo Markov

properties

Empiricism - What is beyond?

-0.01 0.00 0.0110-3

10-2

10-1

100

101

102

103

Q

4 min

Num. solution of the Fokker-Planck eq.

finance:

increasing

intermittence

turbulence:

back to

Gaussian

-0.01 0.00 0.0110-3

10-2

10-1

100

101

102

Q

1 h

New approach for small scales

measure of distance d

1

2

timescale

Question: How does the shape of the distribution

change with timescale?

referencedistribution

considereddistribution

Distance measures

Kullback-Leibler-Entropy:

dK (pN (Q,), pR ) pN (Q, )lnpN (Q,)

pR

dQ

Weighted mean square error in logarithmic space:

dM (pN (Q,), pR )pR pN (Q,) ln pN (Q, ) ln pR

2

dQ

pR pN (Q,) ln2 pN (Q, ) ln2 pR

dQ

Chi-square distance:

dC (pN (Q,), pR )pN (Q,) pR

2

dQ

pR

dQ

Distance measure: financial data

1 s

Small timescales are special! Example: Volkswagen

100 101 102 103 104 105

0.0

0.2

0.4

0.6

timescale in sec

d KFokker-Planck

Regime.

Markov process

Small Timescale

Regime.

Non Markov

Financial and turbulence data

100 101 102 103 104 1050.0

0.2

0.4

timescale in sec

d K

Allianz

10-5 10-4 10-3 10-2 10-1 1000.00

0.01

0.02

0.03

timescale in sec

d K

WK2808_1

10-5 10-4 10-3 10-2 10-1 1000.00

0.01

0.02

0.03

0.04

0.05

timescale in sec

d K

WK2808_2

finance

turbulence

100 101 102 103 104 1050.0

0.2

0.4

0.6

timescale in sec

d K

VW

smallest

Dependence on the reference distribution

Is the range of the small timescale regime dependent on the reference timescale?

100 101 102 103 104 105

0.0

0.2

0.4

0.6

timescale in sec

d K

1 s

2 s

5 s

10 s

1 s 10 s

Financial and turbulence data

Gaussian Distribution

100 101 102 103 104 1050.0

0.2

0.4

0.6

0.8

timescale in sec

d K

VWAllianz

10-4 10-3 10-2 10-10.00

0.04

0.08

timescale in sec

d K

WK2808_1

WK2808_2

finance turbulence

MarkovMarko

v

Dependence on the distance measure

Are the results dependent on the special distance measure?

100 101 102 103 104 1050.0

0.5

1.0

timescale

valu

e o

f m

ea

sure

dK

dM

dC

1 s

The Small Timescale Regime - Nontrivial

1 s

100 101 102 103 104 1050.0

0.2

0.4

0.6

timescale in sec

d K

permutated

original

Autocorrelation

Small Timescale Regime due to correlation in time?

|Q(x,t)|Q(x,t)

101 102 103 104-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Lag in sec

AC

F

Bayer

VW

Allianz

101 102 103 104-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Lag in sec

AC

F

Bayer

VW

Allianz

The influence on risk

100 101 102 103 104 105

0

2.10-4

4.10-4

6.10-4

8.10-4

0.0

0.2

0.4

0.6

timescale in sec

Pro

ba

bili

ty

d K

100 101 102 103 104 105

0

5.10-4

1.10-3

2.10-3

2.10-3

0.0

0.2

0.4

timescale in sec

Pro

ba

bili

ty

d K

Volkswagen Allianz

Percentage of events beyond 10

1 s

Summary

Markov process - Fokker-Planck equation

finance:new

universalfeature?

- 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

http://www.physik.uni-oldenburg.de/hydro/

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

L

r2

r1

cascade dynamicsdescibed by Langevin equation

or by Kolmogorov equation

Q(x, )

D(1)(Q, ) D(2)(Q, ) ()

p(Q, ) Q

D(1)(Q, )2

Q2D(2)(Q, )

p(Q,)

Method : Reconstruction of stochastic equations

Derivation of the Kramers-Moyal expansion:

dytxttypxytxttxp

xdtxptxttxpttxp

,|,)(,|,

,,|,,

0

)(!

)()(

n

nn

xxxn

xyxy

xdtxpxxttxMxn

ttxp

xxdytxttypxyxn

txttxp

n

n

n

nn

n

,)(,,!

11,

)(,|,)(!

1,|,

1

0

From the definition of the transition probability:

H.Risken, Springer

Method : Reconstruction of stochastic equations

Taking only linear terms:

txpttxptOtt

txp,,)(

, 2

)(),(!/),,( 2)( tOttxDnttxM nn

1

)(2 ,),()(,

n

nn

txptxDx

tOt

txp

Kramers Moyal Expansion:

xdtxpxxttxMxn

ttxp n

n

n

,)(,,!

11,

1

1

1

,!

,,

,,,)(!

1

n

n

n

n

n

n

txpn

ttxM

x

xdtxpttxMxxxn

DAX

DAX