fractional l evy processes: capturing stylized facts of ......logo1 logo2 fractional l evy...

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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

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Motivation

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5,1150005,1175005,1200005,1225005,1250005,1275005,1300005,1325005,1350005,1375005,1400005,1425005,1450005,1475005,150000

SAP 2.1.2006

trades

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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?


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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


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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.


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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.


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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.


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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

.


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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.


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sampling frequency

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SnRnconfidence bounds Snconfidence bounds Rn


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sampling frequency

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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.


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−1e−03 −5e−04 0e+00 5e−04 1e−03

010

0020

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high frequency data

log−returns

dens

ity

empirical ditributionnormal distribution


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−0.0040 −0.0035 −0.0030 −0.0025 −0.0020 −0.0015 −0.0010

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4060

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high frequency data

log−returns

dens

ity

empirical ditributionnormal distribution


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−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

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Phelix Day Peak

log−returns

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empirical distributionnormal distribution


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4060

Borsenindex (1973−2006)

Differenz aufeinanderfolgender logarithmierter Werte

Dic

hte

empirische DichteDichte der Normalverteilung


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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?


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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).


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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 .


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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.


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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

].


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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.


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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).


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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)


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Fractional Levy motion (two-sided exponential,d=0.2)

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Fractional Levy motion (truncated stable, α = 0.5,d=0.2)

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Fractional Levy motion (truncated stable, α = 0.5,d=-0.4)

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Fractional Levy motion (Poisson, d=0.1)

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Fractional Levy motion (Poisson, d=-0.1)

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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 .


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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 .


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Borsenindex (1973−2006)

k

Kor

rela

tion

i) lambda=0.0003568 RSS=10.67ii) lambda=0.0009278 RSS= 1.631


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0 200 400 600 800 1000

0.6

0.7

0.8

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1.0

SAP

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i) lambda=0.0004528 RSS=1.076ii) lambda=0.001465 RSS= 0.097


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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.


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