analysis of financial data on different timescales - and a comparison with turbulence robert...
Post on 19-Dec-2015
218 views
TRANSCRIPT
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