EURANDOM 6-8 March 2006
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Transcript of EURANDOM 6-8 March 2006
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Analysis of financial data ondifferent timescales
- and a comparison with turbulence
Robert StresingAndreas NawrothJoachim Peinke
EURANDOM 6-8 March 2006
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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
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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
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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
)
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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
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Method to estimate the stochastic process
p(Q,) Q
D(1)(Q, ) 2
Q2 D(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, ) 1n!
lim 0 ( Q Q) np Q , | Q, d Q with coefficients:
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Method to estimate the stochastic process
Q(x, )
D(1)(Q, ) D(2)(Q, ) ()
p(Q,) Q
D(1)(Q, ) 2
Q2 D(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.:
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Method: Kramers Moyal Coefficients
D(n )(Q, )lim 0 M (n )(Q,,)lim 01
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
,001
,
= 6
00s,
² )
Example: Volkswagen, = 10 min
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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
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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 h4 h1 h15 min4 min
-4 -2 0 2 410-4
10-2
100
102
104
L0,6L0,35L0,2L0,1L
Q [a.u.]
p(Q
,)
[a.u
.]
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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
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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
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Method: Markov Property
10 -6
10 -4
10-2
10 0
10 2
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)
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General view
Numerical solution of the Fokker-Planck equationfor the coefficients D(1) and D(2), which were directly obtained from the data.
-0.01 0.00 0.0110-310-210-1100101102103
-0.01 0.00 0.0110-310-2
10-1100
101102
-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
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Empiricism - What is beyond?
-0.01 0.00 0.0110-310-210-1100101102103
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
10-1100
101102
Q
1 h
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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
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Distance measures
Kullback-Leibler-Entropy:
dK (pN (Q,), pR ) pN (Q, )ln pN (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
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Distance measure: financial data
1 s
Small timescales are special! Example: Volkswagen
100 101 102 103 104 1050.0
0.2
0.4
0.6
timescale in sec
d KFokker-Planck Regime.Markov process
Small TimescaleRegime.Non Markov
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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
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Dependence on the reference distribution
Is the range of the small timescale regime dependent on the reference timescale?
100 101 102 103 104 1050.0
0.2
0.4
0.6
timescale in sec
d K
1 s2 s5 s10 s
1 s 10 s
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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
Markov Marko
v
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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 of
mea
sure
dKdMdC
1 s
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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
permutatedoriginal
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Autocorrelation
Small Timescale Regime due to correlation in time?
|Q(x,t)|Q(x,t)
101 102 103 104-0.2-0.10.00.10.20.30.40.5
Lag in sec
AC
F
BayerVWAllianz
101 102 103 104-0.2-0.10.00.10.20.30.40.5
Lag in sec
AC
F
BayerVWAllianz
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The influence on risk
100 101 102 103 104 1050
2.10-4
4.10-4
6.10-4
8.10-4
0.0
0.2
0.4
0.6
timescale in sec
Pro
babi
lity
d K
100 101 102 103 104 1050
5.10-4
1.10-3
2.10-3
2.10-3
0.0
0.2
0.4
timescale in sec
Pro
babi
lity
d K
Volkswagen Allianz
Percentage of events beyond 10
1 s
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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/
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Thank you for your attention!
Cooperation with
St. Barth, F. Böttcher, Ch. Renner, M. Siefert,
R. Friedrich (Münster)
The End
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Method
scale dependence of Q(x, ) : cascade like structure
Q(x, ) ==> Q(x, )
idea of fully developed turbulenceL
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
Q2 D(2)(Q, )
p(Q,)
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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
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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
nn
n
n
n
txpn
ttxMx
xdtxpttxMxxxn
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DAX
DAX