Fractional Brownian motion and applications · 2009-08-18 · STOCHASTIC CALCULUS WITH RESPECT TO...

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


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 −=−


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


Simulation of a typical path of fBm:

(from Cheridito (2001))

H=0.2

H=0.5

H=0.8


H=0.2

H=0.5

H=0.8

(from Dieker (2004))


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


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


( )

( )

( )

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


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

→

→



( )

( )∫ ∫

∫ ∫

−

−

−−=

⇓

−−=

>

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


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


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))


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


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


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


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!!!!


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.


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.


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.


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


−= ∫ 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)


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


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.

Fractional Brownian motion and applications · 2009-08-18 · STOCHASTIC CALCULUS WITH RESPECT TO...

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