Fractional Brownian motion and applications
Part I: fractional Brownian motion in Finance
INTRODUCTION
The fBm is an extension of the classical Brownianmotion that allows its disjoint increments to be
correlated.
Motivated by empirical studies, several authors havestudied financial models driven by the fBm.
Fractional stochasticvolatility models
Fractional Black-Scholes model
INTRODUCTION
Fractional stochastic volatility models (see Comte and Renauld(1998) or Comte, Coutin and Renault (2003) explain better the
long-time behaviour of the implied volatility.
Nevertheless, the fBm (and then the volatility) are notMarkovian, and this becomes a strong difficulty to study and toput these models into practice (the usual techniques assume the
Markov property).
INTRODUCTION
The introduction of the fractional Black-Scholes model, where theBrownian motion in the classical Black-Scholes model is replaced by
a fBm, have been motivated by empirical studies (see for exampleMandelbrot (1997), Shiryaev (1999) or Willinger (1999)).
Unfortunately, they allow for arbitrage opportunities (see for exampleCheridito (2003) and Sottinen (2001)). This cashm between theoryand practice have been the motivation of several works that have
tried to preserve the fBm approach at the same time they exclude thearbitrage opportunities:
INTRODUCTION
Elliot and Van der Hoek (2003) or Hu and Oksendal (2003) suggested models where the classical integrals were substituted by
integrals in the Wick sense. These models have not arbitrageopportunities but, as it was proved in Bjork and Hult (2005), they
have no natural economic interpretation.
Cheridito (2003) proves that the arbitrage opportunitiesdisappearby introducing a minimal ammount of time between transactions. Guasoni (2005) proves that they also disappear under transaction
costs. These papers open a very interesting field of research.
THE FRACTIONAL BROWNIAN MOTION
( )HHHH stststR
222
2
1),( −−+=
( ) function covariance thehasit if 1,0parameter
Hurstwith (fBm)motion Brownian fractional a
called is processGaussian centeredA
∈H
BH
.0 that assumed isit Usually 0 =HB
THE FRACTIONAL BROWNIAN MOTION
Basic properties
motionBrownian standard a is,2/1 If 2/1BH =
galesemimartin anot is ,2/1 If HBH ≠
Ht
Hat
-H
B
Baa
of law theas same theis
of law the,0for
:similar-self isIt
>
( )[ ] ( ) HHs
Ht stBBE 22 −=−
THE FRACTIONAL BROWNIAN MOTION
H<λλ every for continous,Hölder -
( )( )[ ]
( )( )[ ] 0
correlated negatively incrementsdisjoint
1/2 If
0
:correlated positively incrementsdisjoint
2/1 If
<−−
⇒
<
>−−
⇒
>
Hr
Hs
Hs
Ht
Hr
Hs
Hs
Ht
BBBBE
H
BBBBE
H
THE FRACTIONAL BROWNIAN MOTION
Simulation of a typical path of fBm:
(from Cheridito (2001))
H=0.2
H=0.5
H=0.8
THE FRACTIONAL BROWNIAN MOTION
H=0.2
H=0.5
H=0.8
(from Dieker (2004))
THE FRACTIONAL BROWNIAN MOTION
Representations
Mandelbrot and Van Ness (1968):
( )( ) ( )( )
( )2
1
0
2
1
2
1
1
2
1
2
1
1
2
11)( where
,)(
1
+
−+=
−−−=
∫
∫
∞ −−
−+−+
HdsssHC
dWsstHC
B
HH
sR
HHHt
THE FRACTIONAL BROWNIAN MOTION
Other representations (see for example Nualart (2003))
( ) ,,0 s
t
HHt dWstKB ∫=
( )
( )2
1
2
1
2
32
1
21
,22
12
where
,),(
2/1 Case
−−
−=
−=⇒
>
∫−−−
HH
HHc
duususcstK
H
H
t
s
HHH
HH
β
THE FRACTIONAL BROWNIAN MOTION
( )
( )
( )
2
1
2
12
3
2
1
2
12
1
21
,2121
2
where
2
1
),(
2/1 Case
+−−=
−
−−
−
=⇒
<
−−−
−−
∫
HHH
Hc
dusuusH
sts
tcstK
H
H
Ht
s
HH
HH
HH
β
THE FRACTIONAL BROWNIAN MOTION
Some works (as Alòs, Mazet and Nualart (2001) or Comte andRenault (1998)) deal with the following truncated version of the
fractional Brownian motion:
( )∫−−=
t
sHH
t dWstW0
2
1
This process is not a fBm, but it has a simpler representationwhile it preserves most of the basic properties of the fBm.
STOCHASTIC CALCULUS WITH RESPECT TO THE FRACTIONAL BROWNIAN MOTION
calculus sItô' classicalapply not can We
galesemimartin anot is 2/1
⇒
⇒≠ HBH
Possible approaches
Pathwise techniques
(Zähle (1998))
Malliavin calculustechniques
(Carmona, Coutin andMontseny (2003), Alòs,
Mazet and Nualart (2000))
Integration of deterministic functions
),(1,1 ],0[],0[ stRHHst =
We denote by H the Hilbert space with scalarproduct defined by
[ ]
( )( ).isometry thisdenote We. with associated
H spaceGaussian theand Hbetween isometry
an toextended becan 1 mapping The
1
,0
ϕϕ HH
H
Htt
BB
B
B
→
→
STOCHASTIC CALCULUS WITH RESPECT TO THE FRACTIONAL BROWNIAN MOTION
STOCHASTIC CALCULUS WITH RESPECT TO THE FRACTIONAL BROWNIAN MOTION
( )
( )∫ ∫
∫ ∫
−
−
−−=
⇓
−−=
>
T T
ur
H
t H
H
dudrurHH
dudrurHHstR
H
0 0
22
0
2
0
22
12,
12),(
:2/1
ψϕψϕ
Then we deduce the representation
( ) ( ) ( )∫ ∫
∂∂=
T
s
T
s
HH dWdrrsrr
KB
0, ϕϕ
STOCHASTIC CALCULUS WITH RESPECT TO THE FRACTIONAL BROWNIAN MOTION
In the case H<1/2, similar arguments give us that
( ) ( ) ( )[
( ) ( ) ( )( ) s
T
s
H
T
HH
dWdrsrsrr
K
ssTKB
−∂
∂+
=
∫
∫
ϕϕ
ϕϕ
,
,0
Pathwise integrals in the case H>1/2
exists integral Stieltjes-Riemann Then the
1. with , and orders of functions
continousHölder are , that Suppose
∫
>+
fdg
gf
βαβα
( ) ( ) ( )
( )∫
∫
∫
∂∂+
∂∂+=
>
t Hs
Hs
t Hs
Ht
Hss
H
dBBsx
F
dsBsx
FFBtF
dB)F(BFH
0
0
,
,0,0,
Moreover sense). Stieltjes-Riemann (in the
exists enough,regular is and 2/1 If
STOCHASTIC CALCULUS WITH RESPECT TO THE FRACTIONAL BROWNIAN MOTION
APPLICATIONS IN FINANCE
Models driven by the fBm: the arbitrage problem
Consider the fractional Black-Scholes model for a bond (Xt) and a stock (Yt) (H>1/2):
( )[ ]Htt
t
BtrYY
rtX
σν ++=
=
exp
)exp(
0
The introduction of this model has beenmotivated by empirical studies (see for example
Willinger et al. (1999))
APPLICATIONS IN FINANCE
This model gives arbitrage opportunities. Forexample, we can take
( )[ ]( )[ ]122exp2:
22exp1:
01
00
−+=
+−=Htt
Htt
Btc
BtcY
σνϑσνϑ
Then, Itô’s formula gives us that
( ){ }
( ) strategy financing-self arbitragean is ,
12exp)exp(
10
2
0
0
1
0
00
100
00
10
ϑϑ
σν
ϑϑϑϑ
ϑϑ
⇓
−+=
+++=
+
∫∫Ht
t
uu
t
uu
tttt
BtrtcY
dYdXYX
YX
APPLICATIONS IN FINANCE
Cheridito (2003) proved that, even the market allowsfor arbitrage strategies, these strategies cannot be
constructed in practice. In fact, he proved that if thereis a mimimum ammount of time between transactions, the arbitrage opportunities disappear. The main idea is
the following:
( ))exp~
then (and
0 assume we,simplicity of sake For the
0tHtBYY =
=ν
actualized value
APPLICATIONS IN FINANCE
Consider the strategy defined by
{ } ( ]
h
gg
ii
n
ii ii
>−
+=
+
−
=∑ +
ττ
ϑ ττ
1
1
1,00
1
where
111
actualized value
( )
( ) ( ) ( )( )∑ −=⋅+=
⇓
+
HHiT ii
BBgYVV ττϑ
ϑϑ
expexp~~~
financing-self is ,
1
10
10
APPLICATIONS IN FINANCE
Assume that this strategy allows for arbitrage and let k be the first moment l such that
( ) ( )( ) ..0expexp1
1saBBg
l
i
HHi ii
>−∑=
+ ττ
( ) ( )( )
( ) ( )( )( ) ( )( )HH
k
k
i
HHi
k
i
HHi
kk
ii
ii
BBg
BBg
BBg
ττ
ττ
ττ
expexp
expexp
expexp
1
1
1
1
1
1
−+
−=
−
+
+
+
∑
∑−
=
=
Notice that
0≤
It can be <0!!
It can be <0!!
Con
trad
ictio
n!!!!
APPLICATIONS IN FINANCE
Guasoni (2006) proved that the arbitrage opportunities also disappear undertransaction costs. To achieve an arbitrage, at some pointt0 we have to starttrading. This decision generates a transaction cost which must be recoveredat a latter time, and this is possible only if the asset price moves enough in the future. Hence, if at all times there is a remote possibility of arbitrary
small price changes, then downside risk cannot be eliminated,and arbitrageis impossible.
The above results by Cheridito (2003) and Guasoni (2006) open a newscenario, where the fBm can be an appropiate for stock price modelling ifwe assume that the non-existence of arbitrage strategies isnot due to the
market, but to the existence of restrictions on the tradingstrategies.
APPLICATIONS IN FINANCE
Long-memory stochastic volatility models
Stochastic volatility models:
ttttt dWSdtrSdS σ+=
Stochastic process
(see for example Heston (1993), Hull and White(1987), Stein and Stein (1991) or Scott (1987))
If the volatility is not correlated with W, thesemodels deal to a symmetric implied volatility smile
(see Renault and Touzi (1996))
A asymmetric implied volatility skew can be explained by the existence of a negative correlation
between W and the volatility process.
APPLICATIONS IN FINANCE
Nevertheless, the dependence of the implied volatility on time tomaturity (term structure) is not well explained by classical stochastic
volatility models.
In practice, de decreasing of the smile amplitude when time to maturityincreases turns out to be much slower than it goes according to
stochastic volatility models.
With this aim, Comte and Renault (1998) and Comte, Coutin andRenault (2003) have proposed stochastic volatility models based on the
fBm. These models allows us to explain the observed long-time behaviour of the implied volatility.
APPLICATIONS IN FINANCE
In Comte and Renault (1998) the volatility process isgiven by
( )( ) ( ) H
s
t sttt
tt
dBeemYmY
Yf
∫−−− +−+=
=
00
where,
αα β
σ
uncorrelated with W H>1/2
In this context, the classical Hull and Whiteformula gives us that call option prices can be
written as
APPLICATIONS IN FINANCE
−= ∫ t
T
t stBSQt FdstT
StCEV 21;, σ
Classical Black-Scholes formula
Risk-neutral probability
Then, the authors state that the dynamics of theimplied volatilty are directly related to the dynamics of
∫−=
T
t st dstT
u 21: σ
∞→= −+ hhOuuCov H
htt ),(),( that Notice 22
(this does not vanish at the exponential rate, but at thehyperbolic rate, which explains the long-time behaviour
of stochastic volatilities)
APPLICATIONS IN FINANCE
A recent paper of Comte, Coutin and Renault (2003) deal with a stochastic volatility process of the form :
( ) ( )
processroot square a is ~ where
,~10
212
s
t
st dsst
σ
σβ
σ β∫
−−Γ
=
view.ofpoint nalcomputatio thefromsimpler becomes
modelmemory -long thisMarkovian, is ~As sσ
APPLICATIONS IN FINANCE
In resume, fractional stochastic volatility models allow us to explainthe long-time behaviour of the implied volatility, but they are more
complex and new technical difficulties arise.
BIBLIOGRAPHY
E. Alòs, O. Mazet and D. Nualart (2001): Stochastic calculus withrespect to Gaussian processes. Annals of Probability 29, 766-801.
E. Alòs and D. Nualart: Stochastic integration with respect to thefractional Brownian motion. Stochastics and Stochastic Reports 75, 129-152.
C. A. Ball and A. Roma (1994): Stochastic volatility option pricing. Journal of Financial and Quantitative Analysis 29, 589-607.
P. Carmona, L. Coutin and G. Montseny (2003): Stochasticintegration with respect to the fractional Brownian motion. Ann. Institut Henri Poincaré 39 (1), 27-68.
P. Cheridito (2001): Regularizing fractional Brownian motion with a view towards stock price modelling. PhD Dissertation.
P. Cheridito (2003): Arbitrage in fractional Brownian motion models. Finance and Stochastics 7 (4), 533-553.
F. Comte and E. Renault (1998): Long-memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291-323.
F. Comte, L. Coutin and E. Renault (2003): Affine fractional stochasticvolatility models with application to option pricing. Preprint.
P. Guasoni (2006): No arbitrage under transaction costs, withfractional Brownian motion and beyond. Mathematical Finance 16 (3), 569-582.
S. L. Heston (1993): A closed-form solution for options with stochasticvolatility withe applications to bond and currency options. The Reviewof Financial Studies 6, 327-343.
J. C. Hull and A. White (1987): The pricing of options on assets withstochastic volatilities. Journal of Finance 42, 281-300.
B. B. Mandelbrot (1997): Fractals and scaling in finance, discontinuity, concentration, risk. Springer.
B. B. Mandelbrot and J. W. Van Ness (1968): Fractional Brownianmotion, fractional noises and applications. SIAM Review 10, 422-437.
D. Nualart (2003): Stochastic calculus with respect to the fractionalBrownian motion and applications. Contemporary Mathematics336, 3-39.
B. Oksendal (2004): Fractional Brownian motion in Finance. Preprint.
E. Renault and N. Touzi (1996): Option hedging and implicit volatilitiesin stochastic volatilty models. Mathematical Finance 6, 279-302.
L. C. G. Rogers (1997): Arbitrage with fractional Brownian motion. Math. Finance 7, 95-105.
A. N. Shiryaev: Essentials of stochastic finance: facts, models, theory. World Scientific (1999).
E. M. Stein and J. C. Stein (1991): Stock price distributions withstochastic volatility: An analytic approach. The Review of FinancialStudies 4, 727-752.
L. O. Scott (1987): Option pricing when the variance changesrandomly: theory, estimation and application. Journal of Financialand Quantitative Analysis 22, 419-438.
R. Schöbel and J. Zhu (1999): Stochastic volatility with an Ornstein-Uhlenbeck process: an extension. European Finance Review 3, 23-46.
Fractional Brownian motion and applications
Part II: Applications to surface growth modelling
Most of our life takes place on the surface of something:
INTERFACES IN NATURE
Interesting questions:
formation, growth and dynamics
SOME EXAMPLES (I)
0
2
4
6
8
10
-5 -4 -3 -2 -1 1 2x
)(xh
x
SOME EXAMPLES (II)
combustion particle deposition
SOME EXAMPLES (III)
Radial symmetry tumor growth
(Bru et al., Biophysical Journal 2003)
BASIC SCALING NOTIONS (I)
Roughness:
( )∑=
=L
i
tihL
th1
,1
)(
Ballistic deposition
Mean height
Interface width (roughness) ( ) ( ) ( )[ ]∑=
−=L
i
thtihL
tLw1
2,
1,
BASIC SCALING NOTIONS (II)
( ) βttLw =, ( ) αLtLw =,
Lzt
Lt
x
zx
lnln ≈≈
A typical plot of the time evolution of the surface width
( )
≈=
xt
tfLtLwz α
βα
,;
(saturation due by correlation
( )tLw ,ln
tln
NOTIONS ON FRACTAL GEOMETRY (I)
Fractal dimension
( ))/1ln(
lnlim 0 l
lNd lf →=
NOCIONS DE GEOMETRIA FRACTAL (II)
Self-affinity (exact or statistical)
( ) )(bxhbxh α−≈
Fractal dimension and self-affinity (exact or statistical)
( )lxx
lxhxhl
≡−
≈−≡∆
21
21 )()( α
and then α−= 2fd
NOTIONS ON MODELLING (I)
Random deposition
2/1
,
=
+=∂∂
β
xtdWFt
h
NOTIONS ON MODELLING (II)
Random depositionwith surface relaxation
2,4/1,2/1
,2
2
===
+∂∂+=
∂∂
z
dWx
hF
t
hxt
βα(Edward-Wilkinson)
NOTIONS ON MODELLING (III)
Molecular beamepitaxy (MBE)
4,8/3,2/3
,4
4
===
+∂∂+=
∂∂
z
dWx
hF
t
hxt
βα(MBE)
CORRELATED NOISE (FBM)
( ) ( ) ( )'',,,1
ttxxxtxt −−≈ − δηη ϕ
BIBLIOGRAPHY
Barabási et al.: Fractal concepts in surface growth.