View
3
Download
0
Category
Preview:
Citation preview
logo1
logo2
Fractional Levy processes: capturing stylized facts ofempirical data
Jeannette WoernerTechnische Universitat Dortmund
partly joint work withSebastian Engelke and Alexander Schnurr
- Empirical evidence for correlation in financial data- Possibilities beyond Brownian-semimartingale models
- Fractional Brownian motion models- Fractional Levy models- Well-balanced Ornstein-Uhlenbeck models
J.H.C. Woerner (TU Dortmund) 1 / 40
logo1
logo2
Motivation
01.0
1.73
23.0
7.74
12.0
2.76
05.0
9.77
28.0
3.79
17.1
0.80
11.0
5.82
01.1
2.83
24.0
6.85
14.0
1.87
05.0
8.88
27.0
2.90
19.0
9.91
12.0
4.93
02.1
1.94
24.0
5.96
16.1
2.97
08.0
7.99
29.0
1.01
21.0
8.02
12.0
3.04
04.1
0.05
0
1000
2000
3000
4000
5000
6000
7000
8000
EWI104s (USA)
Time
Inde
x va
lue
J.H.C. Woerner (TU Dortmund) 2 / 40
logo1
logo2
0 50 100 150 200 250 300 350 400
0.00
50.00
100.00
150.00
200.00
250.00
EEX daily data 2007
day
pri
ce
J.H.C. Woerner (TU Dortmund) 3 / 40
logo1
logo2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
5,1150005,1175005,1200005,1225005,1250005,1275005,1300005,1325005,1350005,1375005,1400005,1425005,1450005,1475005,150000
SAP 2.1.2006
trades
log
pric
es
J.H.C. Woerner (TU Dortmund) 4 / 40
logo1
logo2
Classical Modelling
discrete data Xtn,0 , · · · ,Xtn,n
tn,n = t =fixed, ∆n,i → 0 as n→∞Classical stochastic volatility model
Xt =
∫ t
0µsds +
∫ t
0σsdBs + Jt
Question:
Do we have to go beyond Brownian semimartingales in some situations?
J.H.C. Woerner (TU Dortmund) 5 / 40
logo1
logo2
Empirical observation in financial data versusclassical modelTwo important characteristics of Brownian semimartingales:
Correlation between neighbouring increments is zero.
[nt]∑i=1
(X i+1n− X i
n)(X i
n− X i−1
n)→ 0
Not satisfied in many situations:- typically high frequency data and electricity data possess short rangedependence- some stock indices possess long range dependence
Quadratic variation is finite or realized volatility settles down to anon-trivial limit when the sampling frequency increases.Not satisfied in many situations:- typically for high frequency data it increases (market microstructure)for electricity it increases- for some stock indices it decreases
J.H.C. Woerner (TU Dortmund) 6 / 40
logo1
logo2
Fractional Brownian Motion
A fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1),BH = {BH
t , t ≥ 0} is a zero mean Gaussian process with the covariancefunction
E (BHt BH
s ) =1
2(t2H + s2H − |t − s|2H), s, t ≥ 0.
The fBm is a self-similar process, that is, for any constant a > 0, theprocesses{a−HBH
at , t ≥ 0} and {BHt , t ≥ 0} have the same distribution.
For H = 12 , BH coincides with the classical Brownian motion. For
H ∈ (12 , 1) the process possesses long memory and for H ∈ (0, 1
2) thebehaviour is chaotic.
J.H.C. Woerner (TU Dortmund) 7 / 40
logo1
logo2
Modelling of jumps
Xt Levy process with distribution function Pt .
Ee iuXt = Pt(u) = et(iµu−σ2
2u2+
R(e iuz−1−iuh(z))ν(dz))
Levy triplet (µ, σ2, ν(dx))
A measure for the activity of the jump component of a Levy process is theBlumenthal-Getoor index β,
β = inf{δ > 0 :
∫(1 ∧ |x |δ)ν(dx) <∞}.
This index ensures, that for p > β the sum of the p-th power of jumps willbe finite.
J.H.C. Woerner (TU Dortmund) 8 / 40
logo1
logo2
Estimation of Correlation:
Assuming a model of the type
Xt = Yt +
∫ t
0σsdBH
s
we look at the quantity:
Sn =
∑[nT ]−1i=1 (X i+1
n− X i
n)(X i
n− X i−1
n)∑[nT ]
i=1 (X in− X i−1
n)2
→ CH =1
2(22H − 2)
confidence interval for Brownian motion based model:−cγ
√√√√√∑[nt]
i=1 |X in− X i−1
n|4
3(∑[nt]
i=1 |X in− X i−1
n|2)2, cγ
√√√√√∑[nt]
i=1 |X in− X i−1
n|4
3(∑[nt]
i=1 |X in− X i−1
n|2)2
,where cγ denotes the γ-quantile of a N(0, 1)-distributed random variable.
J.H.C. Woerner (TU Dortmund) 9 / 40
logo1
logo2
Robust Bipower Version:
Rn =
∑[nt]−1i=1 (X i+1
n− X i
n)(X i
n− X i−1
n)∑[nt]−1
i=1 |X i+1n− X i
n||X i
n− X i−1
n|
p→ CH
KH,
where KH = E (|B2 − B1||B1 − B0|) = 2π (CH arcsin(CH) +
√1− C 2
H)CH
KH− cγ
√√√√√v2∑[nt]−2
i=1 |X i+2n− X i+1
n|4/3|X i+1
n− X i
n|4/3|X i
n− X i−1
n|4/3
µ34/3
(∑[nt]−1i=1 |X i+1
n− X i
n||X i
n− X i−1
n|)2
,
CH
KH+ cγ
√√√√√v2∑[nt]−2
i=1 |X i+2n− X i+1
n|4/3|X i+1
n− X i
n|4/3|X i
n− X i−1
n|4/3
µ34/3
(∑[nt]−1i=1 |X i+1
n− X i
n||X i
n− X i−1
n|)2
.
J.H.C. Woerner (TU Dortmund) 10 / 40
logo1
logo2
Comparison of Sn and Rn
Inverting SN and Rn we get estimates for H.If no jumps are present Sn and Rn lead to the same estimate for H
If jumps are present Rn leads to the true H, whereas for Sn thedenominator gets larger hence H is closer to 0.5 than the true one.
Hence we can use both to distinguish between pureBrownian-semimartingale and possible non-semimartingales and acomparison of both to detect jumps even in the non-smimartingale case.
J.H.C. Woerner (TU Dortmund) 11 / 40
logo1
logo2
J.H.C. Woerner (TU Dortmund) 12 / 40
logo1
logo2
●
●
● ●
● ●
● ●
● ●
●
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Estimators R_n inverted and S_n inverted for US
values omitted
estim
ated
H
●
●
● ●
● ●
● ●
● ●
●
J.H.C. Woerner (TU Dortmund) 13 / 40
logo1
logo2
J.H.C. Woerner (TU Dortmund) 14 / 40
logo1
logo2
2 4 6 8 10 12 14
0.2
0.3
0.4
0.5
0.6
high frequency data
sampling frequency
H
SnRnconfidence bounds Snconfidence bounds Rn
J.H.C. Woerner (TU Dortmund) 15 / 40
logo1
logo2
2 4 6 8 10 12 14
−0.
3−
0.2
−0.
10.
00.
10.
2
EEX hourly data
sampling frequency
H
SnRn
J.H.C. Woerner (TU Dortmund) 16 / 40
logo1
logo2
Results:
- We certainly have a correlation structure in all of the data sets, henceclassical Brownian semimartingales are not sufficient.- The index-data exhibits long range dependence with H about 0.6 andjumps in small economies.- The electricity data and high frequency data exhibits negativecorrelations.- It is not clear if a Brownian motion based component is present.
J.H.C. Woerner (TU Dortmund) 17 / 40
logo1
logo2
−1e−03 −5e−04 0e+00 5e−04 1e−03
010
0020
0030
0040
0050
00
high frequency data
log−returns
dens
ity
empirical ditributionnormal distribution
J.H.C. Woerner (TU Dortmund) 18 / 40
logo1
logo2
−0.0040 −0.0035 −0.0030 −0.0025 −0.0020 −0.0015 −0.0010
020
4060
80
high frequency data
log−returns
dens
ity
empirical ditributionnormal distribution
J.H.C. Woerner (TU Dortmund) 19 / 40
logo1
logo2
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Phelix Day Peak
log−returns
dens
ity
empirical distributionnormal distribution
J.H.C. Woerner (TU Dortmund) 20 / 40
logo1
logo2
−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03
020
4060
Borsenindex (1973−2006)
Differenz aufeinanderfolgender logarithmierter Werte
Dic
hte
empirische DichteDichte der Normalverteilung
J.H.C. Woerner (TU Dortmund) 21 / 40
logo1
logo2
Problems
- log-returns are normally distributed in fractional Brownian motionbased models.- Does it make sense to use non-semimartingales?In classical theory this would allow for arbitrage unless transaction costsor specific trading rules are considered.
Questions:- Do we indeed have a to go beyond semimartingales?- How can we include semi-heavy tails in the log-returns?- How can we manage to produce peaks like in the energy data?
J.H.C. Woerner (TU Dortmund) 22 / 40
logo1
logo2
Fractional Levy motion
Idea:Use the same moving average kernel as for fractional Brownian motion.
Linear fractional stable motion (Samorodnitsky and Taqqu (1994))
Lα,H(t) =
∫|t − s|H−1/α − |s|H−1/αSα(ds),
where Sα denotes an α-stable random measure α ∈ (0, 2) and H ∈ (0, 1).Moving average fractional Levy motion (Benassi et.al (2004),Marquardt (2006))
XH(t) =
∫|t − s|H−1/2 − |s|H−1/2L(ds),
where L denotes a Levy process with finite second moment, vanishing firstmoment and H ∈ (0, 1).
J.H.C. Woerner (TU Dortmund) 23 / 40
logo1
logo2
Moving average fractional Levy motion
Aim: Find suitable conditions for a unifying approach.Questions:- What kind of moment conditions do we need?- What is the correct exponent in the kernel function?In the following we look at
X (t) =
∫ ∞−∞
f (t, s)L(ds),
where L denotes a Levy process with Blumenthal-Getoor index β, finitep-th moment and characteristic exponent∫
(e iuz − 1− iuz1|z|≤1(z))ν(dz)
and f (t, s) = |t − s|d − |s|d .
J.H.C. Woerner (TU Dortmund) 24 / 40
logo1
logo2
Characteristic function
By Rajput and Rosinski (1989) the process is well defined if∫ ∫|xf (t, s)|2 ∧ 1ν(dx)ds <∞
∫ ∫|f (t, s)x(1|xf (t,s)|≤1 − 1|x |≤1)|ν(dx)ds <∞
then Xt is infinitely divisible with characteristic exponent
iu
∫ ∫f (t, s)x(1|xf (t,s)|≤1 − 1|x |≤1)ν(dx)ds
+
∫ ∫(e iuzf (t,s) − 1− iuzf (t, s)1|zf (t,s)|≤1(z))ν(dz)ds
Remark: If L is symmetric the drift vanishes.
J.H.C. Woerner (TU Dortmund) 25 / 40
logo1
logo2
Conditions on d
Checking the integrability conditions we need
− 1
β< d < 1− 1
q
where q = min(p, 2).This unifies the two previous approaches:- for the linear fractional stable motion d = H − 1/α corresponds to−1/α < d < 1− 1/α- for the other one d = H − 1/2 corresponds to −1/2 < d < 1/2 wherethe lower bound is not optimal since no assumption on theBlumenthal-Getoor index is made.
J.H.C. Woerner (TU Dortmund) 26 / 40
logo1
logo2
Correlation structure
Let L be square-integrable with vanishing first moment with−1
2 < d < 1/2, then the kernel function f is square integrable given by∫ ∞−∞‖f (t, s)‖2ds = C (d)|t|2d+1, ∀t ∈ R,
where
C (d) =
∫ ∞−∞
(|1− u|d − |u|d
)2du.
Moreover, the autocovariance function of X is given by
Cov(X (t),X (u)) =C (d)E [L(1)2]
2
[|t|2d+1 − |t − u|2d+1 + |u|2d+1
].
J.H.C. Woerner (TU Dortmund) 27 / 40
logo1
logo2
Continuity
Similarly we obtain
E (Xt − Xs)2 = C (t − s)2d+1
which leads to Holder continuous sample paths of the order γ for γ < d ifd > 0.If d < 0 we do not obtain continuous paths:The paths of the moving average process cannot be more regular then thekernel.
J.H.C. Woerner (TU Dortmund) 28 / 40
logo1
logo2
Are fractional Levy motion semimartingales?
Consider
X (t) =
∫|t − s|d − |s|dL(ds),
where L denotes a Levy process with finite second moment, vanishing firstmoment and Blumenthal-Getoor index β.X with d > 0 is a semimartingale if and only if
1
1− d> β.
(cf. Bender et.al (2010)) We can generalize this to finite p-th moment1 ≤ p ≤ 2 and obtain
max(0, 1− 1
β) < d < min(1− 1
p, 2).
J.H.C. Woerner (TU Dortmund) 29 / 40
logo1
logo2
Remark
To produce a semimartingale we need: β < 1 ≤ p
This means we can model long range dependence also bysemimartingales. The long range dependence component is of finitevariation, hence the generalized drift in the semimartingalecharacteristic.
(At , 0, 0)
J.H.C. Woerner (TU Dortmund) 30 / 40
logo1
logo2
Fractional Levy motion (two-sided exponential,d=0.2)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10
0
10
20
30
40
50
J.H.C. Woerner (TU Dortmund) 31 / 40
logo1
logo2
Fractional Levy motion (truncated stable, α = 0.5,d=0.2)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
J.H.C. Woerner (TU Dortmund) 32 / 40
logo1
logo2
Fractional Levy motion (truncated stable, α = 0.5,d=-0.4)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4
−3
−2
−1
0
1
2
3
4
5
J.H.C. Woerner (TU Dortmund) 33 / 40
logo1
logo2
Fractional Levy motion (Poisson, d=0.1)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−15
−10
−5
0
5
10
J.H.C. Woerner (TU Dortmund) 34 / 40
logo1
logo2
Fractional Levy motion (Poisson, d=-0.1)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10
−5
0
5
J.H.C. Woerner (TU Dortmund) 35 / 40
logo1
logo2
Further Questions
- Do we need stationary processes like Ornstein-Uhlenbeck processes orprocesses with stationary increments?- Is a polynomialy decaying correlation function good or do we need abehaviour which lies between polynomial and exponential decay?
A natural candidate is the well-balanced Levy drivenOrnstein-Uhlenbeck process.
Xt =
∫exp(−λ|t − s|)dLs
with λ > 0.
Xt =
∫ t
−∞exp(−λ(t − s))dLs +
∫ ∞t
exp(−λ(s − t))dLs .
J.H.C. Woerner (TU Dortmund) 36 / 40
logo1
logo2
Properties
- stationary process and semimartingale- Holder continuous of the order γ < 1/2.- Cov(Xt+h,Xt) = E (L1)2(exp(−λh)/λ+ h exp(−λh))
- Corr(Xk+1 − Xk ,X1 − X0) = exp(−λk)(12 + 1
21−exp(λ)+λ exp(λ)
1−exp(−λ)−λ exp(−λ)) +
λk exp(−λk)(12 + 1
21−exp(λ)+λ exp(−λ)
1−exp(−λ)−λ exp(−λ))- we can construct process with stationary increments and samecorrelation structure by
Xt =
∫ ∞−∞
exp(−λ|t − s|)− exp(−λ|s|)dLs .
J.H.C. Woerner (TU Dortmund) 37 / 40
logo1
logo2
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0 500 1000 1500 2000 2500
0.4
0.6
0.8
1.0
Borsenindex (1973−2006)
k
Kor
rela
tion
i) lambda=0.0003568 RSS=10.67ii) lambda=0.0009278 RSS= 1.631
J.H.C. Woerner (TU Dortmund) 38 / 40
logo1
logo2
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0 200 400 600 800 1000
0.6
0.7
0.8
0.9
1.0
SAP
k
Kor
rela
tion
i) lambda=0.0004528 RSS=1.076ii) lambda=0.001465 RSS= 0.097
J.H.C. Woerner (TU Dortmund) 39 / 40
logo1
logo2
Conclusion
- Various types of financial data exhibit a correlation structure.
- Fractional Brownian motion models may explain this behaviour, but lieoutside the class of semimartingales.
- Fractional Levy motions behave similarly, but additional allow for moreflexibility in the distributions and may reproduce peaks as in energy data.
- Fractional Levy motions might lead to semimartingales with a correlationstructure. - Well-balanced Ornstein-Uhlenbeck processes also lead to acorrelation structure, which is in between polynomial and exponentialdecay.
J.H.C. Woerner (TU Dortmund) 40 / 40
Recommended