fractional brownian motion and applications

7/31/2019 Fractional Brownian Motion and Applications

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Uniwersytet JagiellonskiWydzial Matematyki i Informatyki

Instytut Matematyki

Fractional Brownian Motionand applications to financial

modelling

Ulamkowy proces Wienera i jego zastosowania

do modelowania finansowego

Marcin KrzywdaNumer indeksu: 1018034

Praca magisterska na kierunku matematyka (studia dzienne)

Opiekun: dr hab. Piotr Kobak

Krakow 2011


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Contents

Introduction iii

1 Fractional Brownian Motion 11.1 Definition and existence . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Long range dependence . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Sample path properties . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 p-variation and quadratic variation . . . . . . . . . . . . . . . . . . . 61.5 fBM is not a semimartingale . . . . . . . . . . . . . . . . . . . . . . . 8

2 Integration with respect to fBM 112.1 Pathwise integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Wick-Ito integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Spaces L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Wick exponential . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Analogue of Malliavin derivative . . . . . . . . . . . . . . . . 162.2.4 Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.5 Wick-Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.6 Wick-Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Conditional distribution of fBM . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Conditional Wick-Ito formula . . . . . . . . . . . . . . . . . . 24

3 fBM as a model in finance 253.1 Basics of financial markets modelling . . . . . . . . . . . . . . . . . . 253.2 Pathwise fractional Black-Scholes model . . . . . . . . . . . . . . . . 273.3 Wick-fractional Black-Scholes model . . . . . . . . . . . . . . . . . . 283.4 Other approaches to exclude arbitrage . . . . . . . . . . . . . . . . . 29

3.4.1 Mixed fBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 Market with transaction costs . . . . . . . . . . . . . . . . . . 30

3.4.3 Non-continuous trading . . . . . . . . . . . . . . . . . . . . . 313.5 Pricing by risk preferences . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5.1 Price of an European option . . . . . . . . . . . . . . . . . . . 323.5.2 Role of the Hurst parameter . . . . . . . . . . . . . . . . . . . 35

4 Application to real market data 394.1 Estimating the Hurst parameter . . . . . . . . . . . . . . . . . . . . 394.2 Estimating volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Market data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Bibliography 45

i


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ii


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Introduction

Since the introduction of the classical Black-Scholes model for pricing financialderivatives its assumptions have been commonly criticised. During the study ofstatistical properties of the financial data, several stylised factshave been recog-nised to be common to a wide variety of markets, i.a.: long-range dependence, heavytails, skewness (gain/loss asymmetry), jumps, volatility clustering.

All of the above is in conflict with the BlackScholes model. To overcome the

first point, it has been proposed to replace the Brownian motion by the fractionalBrownian motion. We shall focus on this area of research in our thesis.

Fractional Brownian Motion (fBM) is a stochastic process introduced by Kol-mogorov [Kol] in 1940 for the turbulence modelling. In 1968 Mandelbrot and vanNess [MvN] gave a representation theorem for Kolmogorovs process, and introducedthe name of fractional Brownian Moon.

Fractional Brownian motion is a continuous, zero-mean process with stationaryincrements and variance

E(BHt ) = t2H

The Hurst parameter H > 0.5 determines the statistical long-range dependence.

For H = 0.5 we obtain the standard Brownian Motion. It has been shown that forfinancial data H ( 12 , 1).After some initial enthusiasm, the idea of involving fBM into quantitative finance

was abandoned for many years because, as the process had occurred not to be asemimartingale, and no integration theory was known. Then Hu and Oksendal [HO]developed new methods of stochastic calculus and the hope arised again.

Because of the non-semimartingale-property fBM pricing models admit arbitragepossibilities with continuous trading. Therefore in order to exclude arbitrage severalclasses of admissible portfolios have been suggested. Such a class must be: smallenough to eliminate arbitrage possibilities, large enough to contain hedging strategiesfor relevant options, economically meaningful.

Recently Rostek and Schobel [RS] gave explicit formulae for the price of an Euro-pean option in the model in which arbitrage is excluded by imposing some minimalamount of time between two consecutive transactions. Preference based approach isused to derive formulae.

The goals of this thesis are:

Introduce the fractional Brownian Motion and study its basic properties im-portant from the modelling point of view (Chapter 1)

Study the pathwise (Chaper 1, 1) and Wick-Ito (Chaper 1, 2) integration

theories with respect to fBM

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iv

Study conditional distribution of fBM (Chapter 2, 3)

Give a survey on the history of fBM application to financial modelling (Chapter3, 1-4)

Present the Rostek-Schobel model (Chapter 3, 5)

Discuss the issue of parameter estimation for the model and perform empiricalstudies of the formulae derived (Chapter 4)

Podzi ekowania

Dzi ekuj e Panu doktorowi hab. Piotrowi Kobakowi za wskazanie tematu zastosowanulamkowego procesu Wienera do modelowania rynkow finansowych oraz inspiracj edo napisania niniejszej pracy, jak rowniez za opiek

e nad prac a, czas poswi econy na

dyskusje prezentowanych rezultatow oraz liczne uwagi i poprawki.Sluchaczom Seminarium Zakladu Matematyki Finansowej UJ, w szczegolnosci

Panu prof. Armenowi Edigarianowi, za wysluchanie moich referatow oraz wskazanieniedocigni.

Rodzicom dziekuj

e za wsparcie, ktore otrzymalem w calym okresie studiow.

Prac e magistersk a dedykuj e Madzi.


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Chapter 1

Fractional Brownian Motion

In this chapter we shall introduce the self-similarity property and define the frac-tional Brownian Motion (fBM). Next we shall study other basic properties of thefBM such as: long-range dependence, Holder continuity, path differentiability, etc.

The fractional Brownian Motion was first introduced by Kolmogorov in 1940[Kol], and then Maldelbrot and Van Ness [MvN] discovered many of its properties.

1.1 Definition and existence

Definition 1.1.1. A stochastic process X : T R is Gaussian if for anypositive integer n, and any collection t1, . . . , tn T the distribution of the randomvector (Xt1, . . . , X tn) is normal.

Definition 1.1.2. A stochastic process X : [0, +) R is called H-self-similarwith H

(0, 1) if

a > 0, t 0 Xat aHXt (1.1)where denotes the equality of the finite-dimensional distributions.Definition 1.1.3. We say that a process X : [0, +) R has stationaryincrements if

s, t 0 Xt+s Xt Xs X0 (1.2)Directly from the above definitions one can easily obtain:

Lemma 1.1.1. [EM, Prop. 2.2.] A stochastic process X : [0, +) R whichis H-self-similar, with stationary increments and finite variance has the followingproperties:

1. X0 = 0 a.s.

2. t 0 E(Xt) = 03. t 0 E(X2t ) = t2H2, where 2 = E(X21)

4. s, t 0 Cov(Xt, Xs) = 22 (t2H + s2H |t s|2H)Proof. 1. X0 = Xa0 aHX0 for each a > 0.

2. By self-similarity we have E(X2t) = 2HE(Xt), on the other hand by stationar-

ityE

(X2t) =E

(X2t Xt) +E

(Xt) = 2E

(Xt).

1


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2 CHAPTER 1. FRACTIONAL BROWNIAN MOTION

3. E(X2t ) = E(tHX1)

2 = t2HE(X1)

4.

Cov(Xt, Xs) = E(XtXs) =1

2(EX2t + EX

2s E(Xt Xs)2)

= 2

2(t2H + s2H |t s|2H)

Our next goal is to show that H-self-similar processes with stationary incrementsin fact exist. To achieve this we shall need the following proposition:

Proposition 1.1.2. [Taq, Prop. 2.2] Let H (0, 1). The functionRH(s, t) = t

2H + s2H |t s|2H (1.3)is nonnegative-definite.

Proof. Let t1, . . . , tn 0, 1, . . . , n R. We ask ifn

i,j=1

RH(ti, tj)ij 0

Set t0 = 0 and 0 = n

i=1 i. Becausen

i=0 i = 0 we get

ni,j=1

RH(ti, tj)ij =n

i,j=1

(t2Hi + t2Hj |ti tj |2H)ij =

ni,j=0

|ti tj |2Hij

For any > 0 we have

ni,j=0

e|titj|2H

ij =n

i,j=0

(e|titj |2H 1)ij =

ni,j=0

|ti tj |2Hij + o()

Now, as 0, it is enough to show thatn

i,j=0

e|titj |2H

ij 0

but this follows from the fact that the function e|t|2H

is a characteristic function(cf. [JS, Chapter 9.2]).

Theorem 1.1.3. [Shi2, Theorem II.9.1] (Kolmogorovs consistency theorem)For all k N, t1, . . . , tk T let Qt1,...,tk be probability measures onRnk such that

for all Borel sets Fi and all permutations on {1, . . . , k}

Q(t1),...,(tk)(F1 Fk) = Qt1,...,tk(F1(1) F1(k)) (1.4)and

Qt1,...,tk(F1 Fk) = Qt1,...,tk,tk+1(F1 Fk R) (1.5)Then there exists a probability space (,F,P) and a stochastic process X : Rnsuch that

Qt1,...,tk(F1 Fk) = P(Xt1 F1, . . . , X tk Fk) (1.6)

for all ti T

, k N

and all Borel sets Fi.


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1.1. DEFINITION AND EXISTENCE 3

Now for each collection t1, . . . , tk 0 by proposition 1.1.2 we know that there ex-ists a Gaussian vector (Z1, . . . , Z k) with the covariance matrix [

2RH(ti, tj)]i,j=1,...,kwhich induces a probability measure Qt1,...,tk . One can easily verify that a fam-ily {Qt1,...,tk : t1, . . . , tk 0} is consistent. Consequently by theorem 1.1.3 thereexists a probability distribution and a stochastic process that Qt1,...,tk are its finite-

dimensional distributions.

Definition 1.1.4. Let H (0, 1). A stochastic process BH : [0, +) Rwhich is Gaussian, H-self-similar, has stationary increments and 2 = 1 is called the

fractional Brownian Motion.

Figure 1.1: Sample path of the fBM for H = 0.3




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The index H is called Hurst exponent after the British hydrologist H. E. Hurst.Note that for H = 12 we get the standard Brownian Motion, which we shall furtherdenote by Wt.

1.2 Long range dependence

Long-range dependence is a phenomenon that may arise in the analysis of time seriesdata. It relates to the rate of decay of statistical dependence. More formally:

Definition 1.2.1. A stationary random sequence {Xn}nN exhibits long-range de-pendence if the autocovariance function (n) = EX0Xn fulfils

n=1

(n) = . (1.7)

Definition 1.2.2. The sequence

Xn = BHn+1 B

Hn (1.8)

is called the fractional Gaussian noise (fGn) with Hurst index H.

Figure 1.4: Autocovariance function of fBM for the case of persistence.

For H = 12 we know that the increments of the Brownian Motion are independent,therefore uncorrelated. On the other hand, when H = 12 the increments are notindependent, more precisely we have

(n) =1

2((n + 1)2H + (n

1)2H

2n2H)

H(2H

1)n2H2 (1.9)


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1.3. SAMPLE PATH PROPERTIES 5

as n . In particular, when H > 12 the increments of the fractional BrownianMotion are positively correlated and exhibit the long-range dependence. In thecase when H < 12 increments are negatively correlated and have the short-rangedependence property (i.e.

n=1 (n) < ).

For us the interesting case is when H (12 , 1) because, as some research showed,for financial data H 0.6.

1.3 Sample path properties

Let us begin with recalling the Kolmogorovs criterion

Definition 1.3.1. Let X, Y : T R be stochastic processes on (,F,P). Wesay that X is a version of Y if for all t T

P(Xt = Yt) = 1 a.s.

Theorem 1.3.1. [Pro][Cor. VII.1](Kolmogorovs criterion) Suppose the processX : [0, +

)

R satisfies the following condition: For all T > 0 there exist

, , C > 0 such that

0 t, s T E[|Xt Xs|] C|t s|1+ (1.10)Then there exists a version of X which is Holder continuous of order [0, ) a.s.Proposition 1.3.2. [S, Prop. 3.2] The fractional Brownian Motion admits a ver-sion with almost all sample paths Holder continuous of order strictly less than H.

Proof. For BH we have:

E|BHt BHs | = E|BH1 ||t s|H

It is enough now to apply the Kolmogorovs criterion 1.3.1.

Proposition 1.3.3. [MvN, Prop. 4.2.] The sample paths of the fractional BrownianMotion are nowhere differentiable a.s.

Proof. By self-similarity:

BHt BHt0t t0 (t t0)

H1BH1

Let us consider the event

A(t) =

sup

t0st

BHs BHt0s t0

> d

.

For any sequence tn decreasing to t0, we have

A(tn) A(tn+1)and

A(tn) BHtn BHt0tn t0

> d

Finally

P

BHtn BHt0tn t0

> d

= P({|BH1 | > (tn t0)1Hd}) 1, as n


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1.4 p-variation and quadratic variation

Another criterion of the regularity of trajectories is the p-variation. We follow mostly[S]. Consider partitions = {0 = t0 < t1 < .. . < tn = T} of the interval [0, T]. For

p 1 and f : [0, T] R we denote:

vp(f; ) =n

k=1

|f(tk) f(tk1)|p (1.11)

Definition 1.4.1. We say that f has finite p-variation if the limit

v0p(f) = lim||0

vp(f; ) (1.12)

exists and is finite.1

We say that f has bounded p-variation if

vp(f) = sup

vp(f; ) < (1.13)

By Wp([0, T]) we denote the Banach space of all functions with bounded p-variation equipped with the norm

||f||[p] = ||f||(p) + ||f||

where||f||(p) = vp(f)

1

p and ||f|| = supt[0,T]

|f(t)|

For more details on the notion of p-variation see [DN].

Definition 1.4.2. Let {n} be a sequence of partitions of [0, T] such that |n| 0.For a stochastic process Xt by the quadratic variation along the sequence {n} wemean

[X, X]T = limn

nk=1

(Xtk Xtk1)2

if the limit exists (in probability).

Further we shall use the following notions and theorems from ergodic theory:

Definition 1.4.3. [Shi, Def. V.3.1] A set A F is invariant with respect to thesequence {n} of random variables if there is a Borel set set B B(R) such thatfor n 1

A = { : (n, n+1, . . .)

B}

Definition 1.4.4. [Shi, Def. V.3.2] A stationary sequence {n} is ergodic if themeasure of every invariant set is either 0 or 1.

Theorem 1.4.1. [Shi, Theorem V.3.3](Birkhofs Ergodic Theorem) Let{n}be a stationary (strict sense), ergodic random sequence withE|1| < . Then

limn

1

n

nk=1

k() = E(1)

P-a.s. and in L1.1Hee lim

||0

means the common limit for all sequences of partitions.


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1.4. P-VARIATION AND QUADRATIC VARIATION 7

Lemma 1.4.2. (cf. [Shi, Theorem V.3]) Let{n} be a Gaussian stationary sequencewithEn = 0, and autocovariance function (n) 0. Then{n} is ergodic.

Basing on the ergodic theory Rogers [Rog] obtained the following:

Lemma 1.4.3. [Rog] Let n = {j2n : j = 0, . . . , 2

n} be a partition of [0, 1]. Then

limn

vp(BH; n) =

, if pH < 1E|BH1 |

p, if pH = 10, if pH > 1

(1.14)

in probability.

Proof. Set Yn := (2n)pH1

2nk=1 |B

Htk

BHtk1|p. By self similarity:

Yn (2n)pH12nk=1

|tk tk1|pH|BHk BHk1|p = 2n2nk=1

|BHk BHk1|p

The sequence of one step increments of BHk BHk1 is stationary, centred Gaussianand with covariance function (n) which tends to 0. Therefore it is ergodic.

By Birkhofs Ergodic Theorem:

2n2nk=1

|BHk BHk1|p E|BH1 |p =: Cp

a.s. and in L1.Therefore Yn

P Cp. Because vp(BH; n) = (2n)pH1Yn the proof is finished.

Proposition 1.4.4. [S, Prop. 3.8]

1. For pH > 1 v0p(BH) = 0 a.s.

2. For pH < 1 vp(BH) = + and v0p(BH) does not exists.

Proof. 1. Let be a partition of [0, 1]. By proposition 1.3.2:

nk=1

|BHtk BHtk1|p Cpn

k=1

|tk tk1|p Cp||p1n

k=1

|tk tk1| a.s.

for < H and p > 1. Now it is enough to let ||

0.

2. Let n be a sequence of partitions like in lemma 1.4.3. By Riesz theorem wecan take its subsequence n such that v 1

H(BH, n) E|BH1 |

1

H a.s. There-

fore vp(BH, n) + a.s., so v0p(BH) cannot exist. This also proves that

vp(BH) = + a.s.

Corollary 1.4.5. [Azm2, Theorem 2.3] For the fBM we have [BH, BH]T = 0 ifH > 12 and [B

H, BH]T does not exist if H 1 and have no common discontinuities. Then the Riemann-Stieltjes

integral exists.

We may apply this theorem to the fBM and obtain:

Proposition 2.1.2. [S, Theorem 6.2] Let f : [0, T] R be a stochastic processwith sample paths inWq a.s. with q ) is a Hilbert space.

Definition 2.2.2. For a deterministic function f L2([0, T]) we define its Wickintegral in an ordinary way. Let n = {0 = t0 < . . . < tn = T} be a sequence ofpartitions of [0, T] such that |n|

0, and fn be the step functions approximating

f:

fn(t) =i

ani 1l[ti,ti+1)(t)(L2) f(t)

Then we put T0

fn(t)dBHt :=

i

ani (BHti+1

BHti ) (2.7)

and T0

f(t)dBHt(L2):= lim

n

T0

fn(t)dBHt (2.8)

1Note that = C0(R+,R), a space of continuous functions such that (0) = 0.


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2.2. WICK-ITO INTEGRATION 13

Lemma 2.2.1. [DHPD, Lemma 2.1] If f, g L2([0, T]) thenT0 f d

BH,T0 gd

BH

are well defined Gaussian random variables. Moreover:

1. E(

T0 f d

BH) = 0

2. E(T0 f dBHT0 gdBH) = f, g3. E(

T0 f d

BH)2 = ||f||2 (Wick-Ito isometry)

Proof. Step 1. Let f, g be simple functions, that is: f =n

i=1 ai1l[ti,ti+1), g =ni=1 bi1l[ti,ti+1).

E(

T

0 f dBH) =

ni=1 aiE(B

Hti+1

BHti ) =

ni=1 ai[E(B

Hti+1

) E(BHti )] = 0

E

T0

f dBHT0

gdBH

=n

k,j=1

akbjE[(BHtk+1

BHtk )(BHtj+1 BHtj )]

=n

k,j=1

akbj [E(BHtk+1

BHtj+1) + E(BHtk

BHtj ) E(BHtk+1BHtj ) E(BHtk BHtj+1)]

=1

2

nk,j=1

akbj [|tk tj+1|2H + |tk+1 tj |2H |tk tj |2H |tk+1 tj+1|2H]

=

nk,j=1

akbj tk+1tk

tj+1tj

|s t|2H2

dsdt

=

T0

T0

f(s)g(t)(s, t)dsdt = f, g

Step 2. Let {fn}nN be a sequence of step functions such that fn f in L2. ByStep 1. we have:

E(

T0

(fn fm)dBH)2 = ||fn fm||2 0

so we can proceed with a passage to the limit.

Step 3. Let {fn}nN,{gn}nN be sequences of step functions such that fn f,gn f in L2. Using again Step 2.:

E(

T0

(fn gn)dBH)2 = ||fn gn|| ||fn f|| + ||gn f|| 0

so the limit does not depend on the choice of the approximating sequence.


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14 CHAPTER 2. INTEGRATION WITH RESPECT TO FBM

2.2.2 Wick exponential

Our goal is to generalise the concept of integration to random variables ofLp(,F,P).In the following we shall show how such variables can be approximated by the linearcombinations of the, so called, Wick exponentials:

Definition 2.2.3. By the Wick exponential we mean : L2([0, T]) L2(,F,P)defined by:

(f) = exp

T0

f(t)dBHt 1

2||f||2

(2.9)

Note that such an exponential is a random variable. By Ewe denote the linear spanof the exponentials, that is:

E= span{(f) : f L2([0, T])} (2.10)

Lemma 2.2.2. (f)(g) = (f + g)ef,g

Proof. We compute:

(f)(g) = exp

T0

f(t)dBHt ) 1

2

T0

T0

f(s)f(t)(s, t)dsdt

exp

T0

gf(t)dBHt ) 1

2

T0

T0

g(s)g(t)(s, t)dsdt

= (f + g)exp

T0

T0

f(s)g(t)(s, t)dsdt

= (f + g)ef,g

The next two propositions reduce many verifications for variables in L2

(,F,P)to verifications of exponentials in E:

Proposition 2.2.3. [DHPD, Theorem 3.1] E is dense in Lp(,F,P) for p 1.Proof. Step 1. F : R is a polynomial of the fBM if

F = p(BHt1 , . . . , BHtn

)

where p is a polynomial, and 0 t1 . . . tn. By the Stone-Weierstrasstheorem the set of all polynomials of the fBM is dense in Lp.

Step 2. By lemma 2.2.2 the product of elements of E is still in E, therefore it isenough to show that BHt can be approximated by elements of E.

Indeed, for > 0 we define a function f = 1l[0,t], f L2. The Wickexponential (f) = Ce

BHt for some C > 0. We also put

F =(f) C

C=

eBHt 1

F E and F BHt in Lp if 0.

Lemma 2.2.4. Wick exponential moments share the following properties:


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1. E((f)) = 1

2. E((f)(g)) = ef,g

3. E((f)2) = e||f||2

Proof. 1. By lemma 2.2.1 T0 f(t)dBHt is normally distributed with zero meanand variance ||f||2. Therefore exp

T0 f(t)d

BHt

) is log-normally distributed

with mean exp( 12 ||f||2).

E((f)) = E

exp(

T0

f(t)dBHt )

exp

1

2||f||2

= exp

1

2||f||2

exp

1

2||f||2

= 1

2. By lemma 2.2.2:

E((f)(g)) = ef,g

Proposition 2.2.5. [DHPD, Theorem 3.2] If f1, . . . , f n L2([0, T]) such that||fi fj || = 0 fori = j then(f1), . . . , (fn) are linearly independent inL2(,F,P).Proof. Let 1, . . . , n R be such that

||1(f1) + + n(fn)||L2 = 0

For every g L2 we have

1(f1) + + n(fn), (g)L2 = E[(1(f1) + + n(fn))(g)] = 0

By lemma 2.2.4 this means that

1ef1,g + + ne

fn,g = 0

Replacing g by g we obtain

1ef1,g + + ne

fn,g = 0

Expanding in the powers of and comparing coefficients we obtain a system of linear

equations

1 + + n = 01f1, g + + nfn, g = 0

1f1, gn1 + + nfn, gn1 = 0By the Vandermonde formula the determinant of the above system is

i


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Ds

T0

f(u)dBHu =

T0

f(u)g(s)(u, s)du = (f)(s) (2.18)

Dg(f) = (f) T

0 T

0

f(s)g(t)(s, t)dsdt = (f)

f, g

(2.19)

Ds (f) = (f)

T0

f(u)g(s)(u, s)du = (f)(f)(s) (2.20)

where f, g L2([0, T]).

Proof. Let

F() =

T0

f(s)dBHs () =

T0

f(s)d(s)

We have

F( + g) = T0

f(s)(d(s) + dg(s))

=

T0

f(s)dBHs () +

T0

T0

f(s)f(t)(s, t)dsdt

=

T0

f(s)dBHs () + f, g

Finally

Dgf() = lim0

F() F( + g)

= f, g

2.2.4 Wick product

In this subsection we define the Wick product . We shall need it to extend thetheory of stochastic calculus to the fractional Brownian Motion case.

Definition 2.2.7. By the Wick product for Wick exponentials we mean:

(f) (g) = (f + g) (2.21)

Because for distinct f1, . . . , f n their exponentials are linearly independent, theabove definition can be extended to define the Wick product of two randomvariables , E and further to any random variables of Lp(,F,P).

Note that the Wick product of two random variables cannot be interpreted in apathwise sense. This means that when one knows only the values (), () he willbe not able to compute ( )().

Lemma 2.2.8. Letf, g L2([0, T]), then:

1. E((f) (g)) = E((f))E((g))

2. E((f)

(g))2 = e||f+g||2


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Proof.E((f) (g)) = E((f + g)) = 1 = E((f))E((g))

E((f) (g))2 = E((f + g))2 = e||f+g||2

Proposition 2.2.9. [DHPD, Proposition 3.4] Letg L2([0, T]), , Dg L2(,F,P)then

T0

g(s)dBHs =

T0

g(s)dBHs Dg (2.22)

Proof. By the definition of a Wick product we have

(f) (g) = (f + g) RDifferentiating with respect to and evaluating at = 0 we obtain

(f) T0

g(s)dBHs = (f) T0

g(s)dBHs T0

T0

f(s)g(t)(s, t)dsdt= (f)

T0

g(s)dBHs (f)f, g

= (f)

T0

g(s)dBHs Dg(f)

By linearity and proposition 2.2.3 the proof is finished.

Proposition 2.2.10. [DHPD, Proposition 3.5] Let f, g L2([0, T]), E then:

1. E(f) T0 g(t)dBHt 2

= exp(||f||2) (f, g)2 + ||g||22. E

T0 g(t)dBHt 2 = E(Dg)2 + 2||g||2

Proof. By lemmas 2.2.4, 2.2.8 for , R we haveE[((f) (g))((h) (g))] = ef+g,h+g

Taking partial derivatives 2

to both sides and evaluating at (, ) = (0, 0) we

get

E

(f)

T

0g(s)dBHs

(h)

T

0g(s)dBHs = ef,h(f, gh, g + g, g)

= E(Dg(f)Dg(h) + (f)(h)g, g)By bilinearity we finish the proof.

Corollary 2.2.11. [DHPD, Corollary 3.6] Let g, h L2([0, T]), , E then:

E

T0

g(t)dBHt

T0

h(t)dBHt

= E[DhDg + g, h] (2.23)

Proof. The formula may be obtained by applying the polarisation technique.


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26/53


By corollary 2.2.11:

i,j = E

tj+1tj

Ds Xtids

ti+1ti

Dt Xtjdt + XtiXtj

ti+1ti

tj+1tj

(u, v)dudv

therefore

E(S2) =n1i,j=0

E

tj+1tj

Ds Xtids

ti+1ti

Dt Xtjdt + XtiXtj

ti+1ti

tj+1tj

(u, v)dudv

Condition W5 guarantees that Sn is a Cauchy sequence in L2(,F,P) when

|n| 0.

Corollary 2.2.13. [DHPD, Theorem 3.7] For any X L(0, T) :

1. E(T0 Xsd

BHs ) = 0

2. E(T0 Xsd

BHs )2 = E

(T0 D

s Xsds)

2 + ||X||2

(Wick-Ito isometry)

Proposition 2.2.14. [DHPD, Theorem 3.7] Let X, Y L(0, T).

1.T0 (aXs + bYs)d

BHs = aT0 Xsd

BHs + bT0 Ysd

BHs a.s.

2. If E[ sups[0,T]

|Xs|]2 < +, sup

s,t[0,T]E[|Ds Xt|

2] < + then

t0 Xsd

BHs has a

continuous version.

Proof. We denote Zt =t0 Xsd

BHs . Applying Wick-Ito isometry we have:

E(|Zt Zs|2) E|

ts

XudBHu |

2

E

ts

ts

|DuXv|2dudv +

ts

ts

XuXv(u, v)dudv

C(t s)2 + D(t s)2H

By the Kolmogorovs criterion 1.3.1 the proof is finished.

2.2.6 Wick-Ito formula

Further we get Ito formula for the new type of integration:

Proposition 2.2.15. [DHPD, Theorem 4.1] Let F C2(R) with bounded secondderivative, then

F(BHt ) F(BHs ) = t

s

F(BHu )dBHu + H

t

s

F(BHu )u2H1du (2.26)


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Proof. Let = {0 = t0, . . . , tn = T} be a partition of [0, T]. By Taylors formula

F(BHT ) F(BH0 ) =n1i=0

[F(BHti+1) F(BHti )]

=

n1

i=0

F

(BH

ti )[BH

ti+1 BH

ti ] +

1

2

n1

i=0

F

(i)[BH

ti+1 BH

ti ]2

=n1i=0

F(BHti ) [BHti+1 BHti ] +n1i=0

ti+1ti

Ds F(BHti )ds

+1

2

n1i=0

F(i)[BHti+1

BHti ]2 = I1 + I2 + I3

where i (BHti , BHti+1).I1

T

0 F(BHu )d

BHu in L2.

By the chain rule, for s

[ti, ti+1)

Ds F(BHti ) = F

(BHti )Ds B

Hti

= F(BHti )ti0

(u, s)du

= HF(BHti )(s2H1 (s ti)2H1)

Therefore I2 HT0 u

2H1F(BHu )dBHu .

Finally I3 0 because H > 12 and F is bounded.The following proposition shows how the -derivative of a stochastic Wick-Ito

integral is computed.

Proposition 2.2.16. [DHPD, Theorem 4.2] LetX L(0, T) and sups[0,T]

E[|Ds Xs|2] < + ,

and let

Yt =

t0

XudBHu

then for s, t [0, T]

Ds Yt =

t0

Ds XudBHu +

t0

Xu(s, u)du a.s. (2.27)

Finally we have he general Wick-Ito formula:

Theorem 2.2.17. [DHPD, Theorem 4.3] LetX L(0, T) and letYt =t0 Xud

BHu .Assume that there exists > 1 H such that

E|Xu Xv|2

C|u v|2

where |u v| for some > 0 andlim

0u,vt;|uv|0E|Du(Xu Xv)|2 = 0

Let F C1,2([0, T] R;R) be a function with bounded derivatives. Moreover letE[

T0 |XsD

s Ys|] < + and F(s, Ys)Xs L(0, T). Then

F(t, Yt) = F(0, 0) +

t0

F

s(s, Ys)ds +

t0

F

x(s, Ys)Xsd

BHs

+

t

0

2F

x2(s, Ys)XsD

s Ysds a.s. (2.28)


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The following is a version of the Wick-Ito formula useful in applications.

Theorem 2.2.18. [DHPD, Theorem 4.5] Let X L(0, T) satisfy the conditionsof the Wick-Ito theorem, Y be such thatE[sups[0,T] |Ys|] < + and let

Zt = a + t

0 Yudu + t

0 Xud

BH

u , a RLetF C1,2([0, T] R;R) be a function with bounded derivatives and F(s, Ys)Xs L(0, T). Then

F(t, Zt) = F(0, a) +

t0

F

s(s, Zs)ds +

t0

F

x(s, Zs)Ysds

+

t0

F

x(s, Zs)Xsd

BHs +

t0

2F

x2(s, Ys)XsD

s Zsds a.s.(2.29)

We omit the proofs as they are rather long and technical.

2.3 Conditional distribution of fBM

In this section we consider a two-sided fractional Brownian Motion, that is a stochas-tic process BH : R R sharing the same properties as in the previous case. Weare interested in the conditional distribution ofBHT,t := E[BHT |FHt ]where T > t and FHt = (B

Hs : s t). We follow [RS].

Gripenberg and Norros [GN] gave the following prediction formula:

Proposition 2.3.1. [GN][Theorem 3.1] Let H (1

2 , 1). For each T > t > 0, theconditional expectation of BHT can be represented by:

BHT,t = BHt + t

g(T t, s t)dBHs (2.30)

where

g(v, w) =sin((H 12))

(w)H+ 12

v0

xH1

2

x w dx

=sin((H 12))

1

H

1

2 w

v H+ 1

2 vvw

H 12

,3

2 H

and denotes the incomplete Beta function.

They also proved a technical lemma:

Lemma 2.3.2. [GN][Corollary 3.2]0

0

g(T t, x)g(T t, y)(x, y)dxdy = (T t)2H(1 H) (2.31)

where

H =sin((H 12))( 32 H)2

(H

12)(2

2H)


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2.3. CONDITIONAL DISTRIBUTION OF FBM 23

We omit the proofs of the above results as they are rather technical and relymostly on computations of integrals.

Let [1]t denotes the equivalence class

[1]t = { : BHs () = BHs (1), s (, t]}

The following theorem characterises the conditional distribution of BHT withinits equivalence class:

Proposition 2.3.3. [RS][Theorem 3.2] Let1 be a representative of an equivalenceclass [1]t. The conditional distribution of B

HT based on the observation [1]t is

N(BHt + T,t ,2T,t), where:T,t := E[BHT |FHt ](1) = t

g(T t, s t)dBHs (1) (2.32)

2T,t := V ar[B

HT |F

Ht ](1) = E[(B

HT

BHT,t)

2|FHt ](1) = H(T t)2H(2.33)Proof. The normality of the conditional distribution follows from the Gaussian char-acter of the fBM. By the definition of the conditional expectation:

[1]t

BHT dP =

[1]t

BHT dP = BHT (1)P([1]t)because BHT is constant on [1]t. Therefore

BHT (1) = [1]t

BHT dP

P([1]t)=

[1]t

BHT dP

Respectively for the conditional variance:

2T,t = [1]t

(BHT BHT )2dP = E (BHT BHT,t)2|FHt (1)BHT is the orthogonal projection of BHT on span{BHs : s t}, so BHT BHT,t is or-

thogonal to span{BHs : s t}, therefore independent. As a conclusion E

(BHT BHT,t)2|FHt is deterministic.

2T,t = E

(BHT

BHt )

2|FHt

= E

E

((BHT

BHt )

T,t)

2|FHt

= EE (BHT BHt )2|FHt 2E (BHT BHt )T,t|FHt + E (T,t)2|FHt = E(BHT BHt )2 2E(T,t)2 + E(T,t)2= E(BHT BHt )2 E(T,t)2 = (T t)2H E(T,t)2

E(T,t)2 = Et

g(T t, s t)dBHs2

=

t

t

tg(T t, v t)tg(T t, w t)(v, w)dvdw

=

0

0

g(T

t,

x)g(T

t,

y)(x, y)dxdy = (T

t)2H(1

H)


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Finally: 2T,t = (T t)2H(1 (1 H)) = H(T t)2HGripenberg and Norros [GN] considered also prediction based on a partial knowl-

edge about the past, that is focused on the distribution of BHT conditional on(BHs : s [t a, t]. In this setting they obtained coefficients H,a, and 2T,t,aand showed that if the observation interval becomes as large as the interval to bepredicted, that is a (T t) then H,a H and 2T,t,a 2T,t. So concerning thevariance, a limited historical observation is justified.

Rostek [Ros] concludes that it is more comfortable to consider the theoreticalcase of unlimited history.

2.3.1 Conditional Wick-Ito formula

Following the proof of the Wick-Ito formula one can obtain its conditional version

for the process BHT :Proposition 2.3.4. [RS][Theorem 4.1] Let F C1,2([0, T] R;R) be a functionwith bounded derivatives, then under some regularity conditions:

F(t, BHt ) = F(0, 0) + t0

F

s(s, BHs )ds + t

0

F

x(s, BHs )BHs d BHs

+HH

t0

s2H12F

x2(s, BHs )(BHs )2ds a.s. (2.34)


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Chapter 3

fBM as a model in finance

3.1 Basics of financial markets modelling

In recent years many authors have suggested to replace the classical Brownian motion

as driving process in the modelling of stock prices with the fractional one. Let usstart with recalling some basic notions from the theory of arbitrage pricing. A goodmonography on this subject is [DS].

Definition 3.1.1. By a market model we mean a 7-tuple:

M = (,F, {Ft}t[0,T],P,B,S,A)

where

(,F, {Ft}t[0,T],P) is a probability space with filtration

B = (Bt)t[0,T] is a deterministicR

-valued function modelling the risk freebank account

S = (St)t[0,T] is an {Ft}-adapted R-valued stochastic process modelling dis-counted stock prices

A is a set of admissible trading strategies.

The interpretation is the following. Our market consists of two tradable assets:a discounted bank account B (usually Bt = e

rt, r > 0) and the stock S which pricesare measured in the same units as B. (Therefore B is sometimes called a numeraire.)

Definition 3.1.2. By a portfolio (or strategy) we mean a pair of {Ft}-adapted

processes = (t, t)t[0,T].

Again we look at t as the amount of money stored (or borrowed) on the bankaccount at time t, and at t as the number of stock shares possessed at time t.

With a portfolio we associate the wealth process given by:

Vt() = tBt + tSt (3.1)

Following [BSV2] we introduce several definitions of arbitrage:

Definition 3.1.3. A portfolio is an arbitrage opportunity if

1. V0() = 0

25


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26 CHAPTER 3. FBM AS A MODEL IN FINANCE

2. VT() 0 P-a.s., and3. P(VT() > 0) > 0.

Definition 3.1.4. A portfolio is a strong arbitrage if

1. V0() = 0

2. c > 0 such that VT() c P-a.s.Definition 3.1.5. A sequence of portfolios {n}nN is an approximate arbitrage if

1. V0(n) = 0 for all n

2. VT = limnVT(n) exists in probability

3. VT 0 P-a.s., and4. P(VT > 0) > 0.

Definition 3.1.6. A sequence of portfolios {n}nN is a strong approximate arbi-trage if it is approximate arbitrage and c > 0 such that VT c P-a.s.Definition 3.1.7. A sequence of portfolios {n}nN is free lunch with vanishing riskif it is approximate arbitrage and

limn

esssup|VT(n)()1l{VT(n)


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3.2. PATHWISE FRACTIONAL BLACK-SCHOLES MODEL 27

Note that the meaning of the above definitions depends on the integration theoryused.

[BSV2] introduce the following:

Definition 3.1.9. We call a self-financing portfolio nds-admissible(no doublingstrategies) when there exists a constant M

0 such that Vt()

M P-a.s. for

all t [0, T] This class of portfolios we denote by Ands.Definition 3.1.10. We call a portfolio simple when there exists a finite numberof stopping times 0 0 . . . n T such that is constant on (k, k+1]. Inother words

t =n1k=0

k (k,k+1](t)

where k s are Fk -measurable. The class of simple and self-financing portfolios wedenote by Asi.

Definition 3.1.11. We call a portfolio -simplewhen it is simple, and k+1

k >

for all k. The class of -simple and self-financing portfolios we denote by A,si.

Definition 3.1.12. We call a portfolio almost simplewhen there exists a sequence{k}kN of stopping times 0 0 . . . n T for all n, such that P({k N k =T}) = 1 and is constant on (k, k+1]. In other words

t =N1k=0

k (k,k+1](t)

where k s are Fk-measurable and N is an a.s. N-valued random variable. Theclass of almost simple and self-financing portfolios we denote by Aas.

In this chapter we are interested in modelling the stock dynamics with fBM. Weassume H (12 , 1).

3.2 Pathwise fractional Black-Scholes model

The classical Black-Scholes model of the stock market is described by the stochasticdifferential equations:

dBt = rBtdt (3.3)

dSt

= Stdt + S

tdW

t(3.4)

The first try to involve the fractional Brownian motion in the modelling of themarket was simply to replace the process W with BH in the above equations, and tointerpret the integrals in the pathwise way, which is natural from a modelling pointof view. In this case our equations become:

dBt = rBtdt (3.5)

dSt = Stdt + StdBHt (3.6)

By the proposition 2.1.3 we get the solution to the above equation:

S(t) = s0e

t+BHt

(3.7)


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Unfortunately this model admits arbitrage opportunities. Probably the first toprove it was Rogers [Rog]. Here we show a simple example by Shiryaev [Shi]:

Let t = 1 e2BHt , t = 2(eBHt 1). For simplicity we assume = r and = s0 = 1. For this portfolio we have:

Vt() = tBt + tSt = (1 e2BHt

)ert

+ 2(eBHt

1)(eBHt +rt

)= ert(eB

Ht 1)2 > 0

Now using proposition 2.1.3 we obtain:

dVt() = rert(eB

Ht 1)2dt + 2ert(eBHt 1)eBHt dBHt

= (1 e2BHt )rertdt + 2(eBHt 1)eBHt rertdt + 2ert(eBHt 1)eBHt dBHt= tdBt + tSt(rdt + dB

Ht ) = tdBt + tdSt

hence the portfolio is self-financing.Note that the existence of arbitrage in this pathwise model is a direct consequence

of the following theorem by Delbaen and Schachermeyer:

Theorem 3.2.1. [DS][Theorem 9.7.2] Let S be an adapted cadlag process. If S islocally bounded and satisfies no free lunch with vanishing risk in the classAsiAnds,then S is a semimartingale.

The theorem says that if there is no arbitrage using simple portfolios (withpathwise products) then the price process is a semimartingale. However, since ourprocess is not a semimartingale, an arbitrage must exist. In view of this result thepathwise fBS model is not suitable in finance.

3.3 Wick-fractional Black-Scholes modelUnder Wick-Ito integrals our stock market model becomes:

dBt = rBtdt (3.8)

dSt = Stdt + StdBHt (3.9)

Thanks to the proposition 2.2.15 we get the solution to the above stochasticequation:

St = s0eBHt +t

2

2t2H (3.10)

Unfortunately, in this model also arbitrage strategies can be constructed. Cherid-

ito proved the following:

Proposition 3.3.1. [Che2, Theorems 3.1, 3.2 and 4.3] In the model given by 3.9there is:

1. free lunch with vanishing risk in the classAsi Ands

2. strong arbitrage in the classAas Ands

3. no arbitrage in the classA,si for every > 0.

To exclude arbitrage, but still keeping to continuous trading, Hu and Oksendal

[HO] suggested to adjust the definition of a self-financing portfolio:


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3.4. OTHER APPROACHES TO EXCLUDE ARBITRAGE 29

Definition 3.3.1. We say that the portfolio = (t, t) is Wick-self-financing iffor all t [0, T]

Vt() = V0() +

t0

sdBs +

t0

sdSs

and similarly for the wealth process:

Vt() = tBt + t StHu and Oksendal [HO] showed that the Wick-fractional Black-Scholes model is

free of arbitrage with the class of Wick-self-financing portfolios and complete.Finally, using their results Necula [Nec] proved the following version of a Black-

Scholes formula:

Theorem 3.3.2. [Nec][Theorem 4.2] The price of an European call option withstrike K and maturity T at time t [0, T] is given by:

C(t, St) = StN(d+) Ker(Tt)N(d) (3.11)where

d =ln St

K+ r(T t) 22 (T2H t2H)

T2H t2H (3.12)

There is a lot of controversy about this model. Although the result by Hu,Oksendal and Necula is mathematically a good analogue of the no-arbitrage resultfor classical Black-Scholes model, the usage of the Wick products does not admit anyeconomic interpretation. In particular, the proof of the no-arbitrage result for Wick-self-financing portfolios does not assume that the portfolio is adapted to the filtrationgenerated by the stock price process. So, even if the investor has full informationabout the future stock prices, he cannot generate arbitrage in this model, whichwould be obviously possible in the real world economy.

The second surprising feature appears when one takes a deeper look a the Wick-fBS formula itself. If we have t1 t2 t T and two options with the samematurity T written in date t1 and t2 respectively the Wick-fractional Black-Scholesprices (at time t) for these two options will be different. Suppose one option ischeaper than the other. Now we can buy a (C2 C1)er(Tt) ).

This means one can do arbitrage in a non-arbitrage market!Oksendal [Oks] introduced the concept of a market observer to justify the defi-

nition of the Wick self-financing portfolios. All formulae containing Wick productsare interpreted as abstract quantities that become real prices by an observation.

A good summary was given by C. Necula: Nowadays the Wick-fractional BSmarket based on Wick integrals is considered a beautiful mathematical construct

but with limited applicability in finance.1

3.4 Other approaches to exclude arbitrage

3.4.1 Mixed fBM

Cheridito [Che1] proposed to modify the stochastic process of the stock price inorder to get a semimartingale. The so called mixed fractional Brownian Motion isa linear combination of fBM and a classical Brownian Motion. We take

Zt = BHt + Wt (3.13)

1Ciprian Necula, May 2010, private communication


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where BH and W are independent. For H ( 34 , 1) and > 0 by [Che1, Theorem1.7]

BHt + Wt WtWe observe that

Cov(Wt + BH

t , Ws + BH

s ) = 2

min(s, t) + Cov(BH

t , BH

s )

In the mixed model the stock price process is

St = s0et+(BHt +Wt)

1

22t (3.14)

The model is arbitrage-free in the class Ands and complete. For each > 0 onecan price assets with respect to the unique martingale measure Q, and get (at timet = 0):

C() = EQ [max(s0eT+(BHt +Wt) erTK, 0)] = BS(0, s0, ) (3.15)

where BS(0, s0, ) denotes the classical Black-Scholes price with initial price s0,

and volatility .

3.4.2 Market with transaction costs

Guasoni et. al. [GRS2] introduced transaction costs into the fBS model and showedthat in this context arbitrage disappears.

By C+s0([0, T]) we denote the space of continuous processes with positive trajec-tories starting from s0.

For the markets with transaction costs the following notion of consistent pricesystems corresponds to equivalent martingale measures in the frictionless case:

Definition 3.4.1. Let S

C+s0([0, T]). By an -consistent price system we mean a

pair (S,Q) such that Q P, S is a Q-martingale such that S0 = S0 and for t [0, T](1 )St St (1 + )St, a.s. (3.16)

In this subsection a trading portfolio = (, ) is a predictable, finite-variationR2-valued process such that 0 = T = 0 (a strategy must begin and end with acash position only). By the total variation of we mean the process:

Vt() = sup0t0...tn=t

ni=1

|ti ti1| (3.17)

The wealth process is:

V() =T0

sdBs +T0

sdSs T0

SsdV()s

Vt () = V(1l(0,t))

Note that the integrals are a.s. well-defined pathwise, in the Riemann-Stieltjes sense,since V() is a.s. finite.

Proportional transaction exclude the possibility of continuous trading. Duringthe interval [t, t + t] the value of the position in stock (without considering trans-action costs) changes by tdSt, while the transaction cost StV()t is charged to thecash account. The term 1l(0,t) in the definition of V

t () accounts for the liquidation

cost.


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3.4. OTHER APPROACHES TO EXCLUDE ARBITRAGE 31

Definition 3.4.2. We say that is admissible for -transaction costs if there existsM 0 such that for all t [0, T] Vt () M a.s.. We denote this class ofportfolios by A,nds.

Theorem 3.4.1. [GRS2, Theorem 1.11](The fundamental theorem of asset

pricing with -transaction costs) Let S C+

s0([0, T]). The following conditionsare equivalent:

1. For each > 0 there exists an -consistent price system.

2. For each > 0 there is no arbitrage in the classA,nds.

Definition 3.4.3. A continuous process S : [0, T] R+ satisfies the conditionalfull support (CFS) condition if for all t [0, T)

supp P(S|[t,T]|Ft) = C+St

([t, T])

The following shows that the Wick-fractional BS model does not have arbitrageunder transaction costs in the class A,nds.

Proposition 3.4.2. [GRS, Theorem 1.2] Conditional full support implies the exis-tence of an -consistent price system for every > 0

Lemma 3.4.3. [C] Fractional Brownian Motion has CFS.

The problem of option price remains unsolved in this setting.

3.4.3 Non-continuous trading

Cheridito [Che2] arguments that in reality the seller of an option can only carryout finitely many transactions to hedge the option. Moreover he cannot buy andsell within nanoseconds. To find a reasonable option price, one should postulate theexistence of an arbitrary small minimal amount of time > 0 that must lie betweentwo consecutive transactions, that restricts trading strategies to the class >0A,si.As we mentioned before (cf. Prop. 3.3.1) there is no arbitrage in this class.

Having given up on continuous trading dynamical hedging and replication meth-ods are no longer available. In a complete market no arbitrage ensures the existenceof unique equivalent martingale measure. In this setting the measure is not unique.However, we may impose another economic equilibrium condition to find the best

pricing measure. We shall discuss this approach in the next section.Bender, Sottinen adn Valkeila [BSV2] noticed that the class >0A,si is some-

what to much restrictive. To remedy this problem they proposed the following moregeneral class of admissible portfolios.

Definition 3.4.4. A sequence of stopping times 0 0 . . . n = T satisfies thedelay property if for all k there exist:

1. an Fk -measurable open set Uk C+Sk ([k, T]) and

2. an Fk -measurable a.s. positive random variable k such that k+1 k kin the set Uk {k+1 > k}.


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We call a portfolio delay-simple if it is of the form

t =n1k=0

k (k,k+1](t)

where k s are Fk -measurable. The class of delay-simple and self-financing port-folios we denote by Adesi.

Note that >0A,si Adesi.Basing on the CFS property the authors obtained the following:

Proposition 3.4.4. [BSV2, Theorem 4.2] In the Wick-fractional BS model there isno arbitrage in the classAdesi.

3.5 Pricing by risk preferences

In this section our market model is the Wick-fractional market model with the classof delay-simple portfolios. We keep the notation from section 2.3.

3.5.1 Price of an European option

Here we study pricing approach suggested by Rostek and Schobel [RS], which theycall preference based equilibrium pricing. Their idea is to focus on a two-timeapproach. They postulate some equilibrium condition, namely that investor shouldbe indifferent between buying the stock and holding the amount St of the risklessasset. Formally we demand:

E[er(Tt)

ST|FHt ] = St (3.18)

This is less than being a martingale.2 However, this condition is not achieved bychanging the measure like in the standard Black-Scholes model, but by adjusting thedrift rate of the stock process. Economic argumentation given by the authors is thatin a world where all investors are risk-neutral (but possessing and using informationabout the past), the basic asset cannot be of an arbitrary shape, but has to be inequilibrium itself.

The price of an European call option in this model is defined to be:

C(T , t , S t,K,r, , H) = er(Tt)E[max(ST K, 0)|FHt ] (3.19)

Lemma 3.5.1. [RS, Section 4] Let

St = s0e BHt +t22 t2Hbe the conditional stock process. ln(ST) is normally distributed with moments

m = ln St + r(T t) 12

H2(T t)2H (3.20)

v = H2(T t)2H (3.21)

2 Recently, the same authors have shown that this condition may be obtained by maximisinginvestors expected utility, cf. [RS2].


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3.5. PRICING BY RISK PREFERENCES 33

Proof. Applying the fractional Wick-Ito formula to F(t, BHt ) = ln St we haveln(ST) = ln(St) + (T t) 1

2H

2(T t)2H + (BHT BHt )Therefore ln(ST) N(m, v) (on the space ([1]t, ([1]t),P)) where

m = ln St + (T t) 12

H2(T t)2H + T,t

v = H2(T t)2H

and ST logN(M, V) withM = exp(m +

1

2v) = Ste

(Tt)+ T,t

V = exp(2m + 2v) exp(2m + v) = S2t e2(Tt)(eH2(Tt)2H 1)

The mean of the conditional process St must equal the conditional mean of theprocess St, that is:E(ST) = E(ST|FHt )

therefore

Ste(Tt)+ T,t = Ste

r(Tt)

where is the adjusted drift rate. Finally we get:

(T t) = r(T t) T,t (3.22)Combining the last equality with previous results we obtain the conditional mo-

ments of ln(ST) to bem = ln St + r(T t) 1

2H

2(T t)2H

v = H2(T t)2H

Lemma 3.5.2. LetX N(m, v), K > 0. Then

E(eX1l{XK}) = em+v

2 N

K m v

v (3.23)

Proof. Let Q be a measure such that X N(m + v, v) under Q. The followingfunctions

f(x) =12v

exp

(x m)

2

2v

g(x) =

12v

exp

(x m v)

2

2v

are densities of X under measures P and Q respectively. Put

(x) = f(x)g1(x) = exp(x m v2 )


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For each h measurable and such that E|h(X)| < + we have:

EP(h(X)) =

h(X)dP =

R

h(x)f(x)dx =

R

h(x)(x)g(x)dx

=

h(X)(X)dQ = EQ(h(X)(X))

Therefore

E(eX1l{XK}) = EQ((X)eX1l{XK}) = EQ(e

X+m+v2 eX1l{XK})

= EQ(em+v

2 1l{XK}) = em+v

2Q(1l{XK})

But

Q(1l{XK}) = Q

X m v

v K m v

v

= N

K m v

v

Combining above equalities we finish the proof.

Proposition 3.5.3. [RS, Theorem 4.2] The price of an European call option withstrike K and maturity T at time t [0, T] valued by a risk-neutral investor is givenby the following formula:

C(T , t , S t,K,r, , H) = StN(dH+ ) Ker(Tt)N(dH ) (3.24)

where

dH =ln(St

K) + r(T t) 12H2(T t)2H

H(T t)H (3.25)

Proof. We start with exploiting the definition:

C(T , t , S t,K,r, , H) = E

er(Tt)(ST K)1l{STK}|FHt

= E

er(Tt)ST1l{ STK} + Eer(Tt)K1l{ STK}= I1 + I2

Using previous lemmas:

I1 = er(Tt)E

ST1l{ STK} = er(Tt)Eeln ST1l{ln STlnK}= er(Tt)elnSt+r(Tt)N

ln K ln St r(T t) 12H2(T t)2H

H(T t)H = StN(d

H+ )

Finally

I2 = er(Tt)KE

1l{ STK}

= er(Tt)KP

1l{ln STlnK}

= er(Tt)KN

ln K ln St r(T t) +

12H

2(T t)2HH(T t)H

= Ker(Tt)N(dH )


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Note that for the case H = 12 one obtains standard Black-Scholes pricing formula.

Corollary 3.5.4. The price of a fractional European put option is

P(T , t , S t,K,r, , H) = Ker(Tt)

(dH ) St(dH+ ) (3.26)Corollary 3.5.5. (Put-Call parity)

C(T , t , S t,K,r, , H) P(T , t , S t,K,r, , H) = St Ker(Tt) (3.27)

3.5.2 Role of the Hurst parameter

The conditional variance of the fBM consists of two factors. The first H dependsonly on the Hurst parameter. The following figure shows the shape of this factor:

Figure 3.1: Shape of the factor H for H [0.5, 1].

We see that for H = 12 H reaches its maximum and as H 1, H 0, whichmeans that in case of total persistence the uncertainty vanishes.

The second factor is (T t)2H and for H ( 12 , 1) the relation between time tomaturity and the variance has convex shape.

Now let us take a look at the values of option for different Hurst parameters.From the following diagram we see that when increasing H the price is decreasing.

To examine this effect in details we have to calculate a partial derivative withrespect to H.

Proposition 3.5.6. [RS, Theorem 4.3] The partial derivative of the call price Cwith respect to the Hurst parameter H is given by

C

H= Stf(d

H+ )

H(Tt)H

(1 H) (H+ 12) + 2ln 2 + 2ln(T t)

2

(3.28)

wheref is a density of standard normal distribution, and is the digamma function.3

3The digamma function is defined as the logarithmic derivative of the gamma function:

(x) =d

dxln(x) =

(x)

(x)


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Figure 3.2: Shape of the factor (T t)2H.

The following are immediate consequences of the properties of natural logarithm:

Proposition 3.5.7. [RS, Theorem 4.3] The partial derivative of the call price Cwith respect to the Hurst parameter H has the following properties:

1. For := T t 1 it holdsC

H(H) < 0, H ( 1

2, 1)

2. For := T t > 1 there exists H (12 , 1) such thatC

H( H) = 0

C

H(H) > 0, H ( 1

2, H)

C

H(H) < 0, H ( H, 1)

We omit the technical proofs.

The next figure shows these relations between the Hurst parameter and the callprice in a graphical way:


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Figure 3.3: Price of the fractional European call option with varying Hurst parameterH (chosen parameters: r = 0, 02, K = 100, = 0, 2, T t = 0, 25).


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Figure 3.4: Maturity effect on the relation between the price of the fractional Eu-ropean call and Hurst parameter H for maturity = 0, 25, = 0, 75 and = 2

(chosen parameters: r = 0, 02, S = 100, K = 100, = 0, 2)


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Chapter 4

Application to real market data

In this chapter we perform empirical analysis of the formula derived in section 3.4for the intraday data from Warsaw Stock Exchange in Poland with comparison tothe standard Black-Scholes one.

4.1 Estimating the Hurst parameter

The scaling character of H-self-similarity suggests that the self-affine random func-tion BH(t) is proportional to |t|H. Several methods of estimating H make useof this property.

The simplest, the oldest and the most popular is R/S analysis. This method wasintroduced by Hurst. Given a physical time series of length n ({Xi : i {1, . . . , n}}we divide it into k non-overlapping blocks. Next we compute two numbers

R(ti, d) = max{0, W(ti, 1), . . . , W (ti, d)} min{0, W(ti, 1), . . . , W (ti, d)} (4.1)where ti = n/k(i 1) + 1 are the starting points of the blocks which satisfy(ti 1) + d n and

W(ti, j) =

jl=1

Xti+l1 j

d

dl=1

Xti+l1, j = 1, . . . , d

and S2((ti, d)) is the sample variance of Xti, . . . , X ti+d1. For each value of d weobtain a number of R/S samples. We compute these samples for logarithmicallyspaced values of d. For the fractional Gaussian noise the ratio R/S follows

E[R(d)/S(d)] CdH

Hence a plot of log R(d)/S(d) versus log d should have a constant slope. Finallythe slope of the regression line for the R/S samples is an estimate for the Hurstparameter. More on the methods of estimation the Hurst parameter can be foundin [Rose] or [Cle].

4.2 Estimating volatility

Here we present the procedure of volatility estimation introduced by Cajueiro andBarbachan [CB]. Let

Xt = ln(St

S0) = t

1

2

2t2H + BHt

39


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40 CHAPTER 4. APPLICATION TO REAL MARKET DATA

Let us assume that in order to estimate volatility () we make observations of therealisation of this process at times tk =

kTn

for k = 0, 1, . . . , n. We denote

Zk = Xk+1n

T Xk

nT

Therefore Zk = (BHk+1n

T BHk

nT

) + n

12

2(t2Hk+1 t2Hk )

Dividing both sides by TH we obtain:

ZkTH

=

nHk +

THn 1

2TH2(t2Hk+1 t2Hk )

where k =nH

TH(BHk+1

nT

BHknT

). By self-similarity

(0, . . . , n1)

BH1 BH0 , . . . , BHn BHn1

We see that E(k) = 0 and E(2k) = 1. We refer to [KMV] for the proof that one canuse sample variance to estimate :

2n =n

n 1 (y2n yn2)

where yk =nHZkTH

and yn =y0++yn1

n.

The above procedure gives different values of estimated volatility. Let s be thesample variance of observed Zks. By 2H we shall denote the estimated variance withassumed Hurst parameter H. Therefore, when we use daily data for the purpose ofestimating annual volatility we have for the classical Black-Scholes model:

212 = 250sand for the fractional one 2H = 250HsAs we shall see in the next section for the stock market H 0.63 which means 2H istwo times bigger than 21

2

. This property affects the option price obtained from the

proper formulae. The next figure shows the relation between FBS prices for severalvalues of H, but for volatility estimated basing on the same sample data.

4.3 Market data

All the calculations presented here are based on the WIG201 option intradaypricesfrom the period 01/01/2005 - 31/12/2009.2 We eliminated all transactions madebefore 120 days to maturity, because of small liquidity. As a result, taking in ouropinion the most reliable data we obtained the set of 152 000 transactions.

First we observed that by performing R/S analysis for different subperiods weobtain very similar values of the estimated Hurst parameter, therefore since the

1The WIG20 is the index of the twenty largest companies on the Warsaw Stock Exchange.2To be precise, because of the data availability, we took all the options series with the follow-

ing maturity dates: 18/03/2005, 17/06/2005, 16/09/2005, 16/12/2005, 17/03/2006, 16/06/2006,15/09/2006, 15/12/2006, 16/03/2007, 21/03/2008, 20/06/2008, 19/09/2008, 19/12/2008,20/03/2009, 19/06/2009, 18/09/2009, 18/12/2009.


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4.3. MARKET DATA 41

Figure 4.1: Price of the fractional European call option with varying Hurst parameterH (chosen parameters: r = 0, 02, K = 100,

212

= 0, 2, T t = 0, 25).

impact of the slight difference in that parameter on the option price is negligiblewe decided to use the value estimated for the whole sample period which is H =0.6378215. For this value of the Hurst parameter we calculated the H coefficientto be H = 0.9316925.

We estimated historical volatility for the two-months periods. For the risklessrate we took the yield of 52-week Polish Treasury notes.

To be able to price index options in both standard BS and Rosteks fractionalBS models one needs to be able to estimate the dividend yield. Because performingthis task would be very time consuming or even impossible we instead used formulaefor the price of the option on futures (Blacks formula). Futures contract follow the

behaviour of the index itself and at the same time are sensible to the payment ofdividends. The price of an option on the stock index is equivalent to the price ofthe option on futures.

In the standard Black-Scholes model the Black formula is:

C(T , t , F t,K,r, ) = er(Tt)(FtN(d+) KN(d)) (4.2)

where

d =ln(Ft

K) 12

2(T t)(T t) (4.3)

and in Rosteks fractional model:


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C(T , t , F t,K,r, , H) = er(Tt)(FtN(d

H+ ) KN(dH )) (4.4)

where

dH =ln(Ft

K) 12H

2(T t)2H

H

(T

t)H(4.5)

Here Ft denotes the price of the futures contract (written on the same asset asthe option) with maturity T at time t.

For our calculations we also took intraday data for respective futures contractlisten on the WSE.

To estimate the forecasting power of the formulae obtained we perform similaranalysis to the one that can be found in [MW]. For each transaction we computeBS, and FBS price and then compare them in the means of average mean squareerror.

AMSE=

1

n

n

i=1

Cmarketi CcalciCmarketi 2

(4.6)

where Cmarketi denotes the market option price for i-th sample, and Ccalci is the price

from BS or FBS formula. Obviously the smaller the AMSE is, the better the pricingformula is.

The following table shows the AMSEs of BS and FBS calculated for the fullsample set.

BS FBS

AMSE 0.6596 10.44

Table 4.1: AMSEs of BS and FBS

Judging only by the numbers above we find that BS gives better approximationof the market price in average. Performing further investigation, we shall see howvolatility, time to maturity and moneyness can influence the comparison result.

For an option we define the moneyness in the following way:

m =StK

1 (4.7)

where as usual St stands for the stock price at time t and K is the strike price.

We divided our sample set into separable groups to check for which values ofm which formula performs better. Therefore we call an option deep out-of-the-money if m < 10%, out-of-the-money for m [10%, 2%), at-the-money form [2%, 2%), in-the-money for m [2%, 10%) and deep in-the-money whenm 10%. The following table shows the AMSEs for the introduced option classes:

deep OTM OTM ATM ITM deep ITM

BS 0.0027 0.0076 3.0124 0.1482 0.3400FBS 0.0119 0.0483 2.2930 1.2915 26.2758

Table 4.2: AMSEs of BS and FBS depending on the moneyness


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4.3. MARKET DATA 43

From the above we see that for most classes of options BS approximates theirprices better, but FBS is much better then BS for at-the-money options (most liquid,and most traded). We may also notice that the average error is very high for deep-OTM option.

Next we divided our sample set according to the time to maturity. We say an

option is short-term if T t < 45 days, middle-term for T t [45 days, 90 days)and long-term when T t 90 days. The following table shows the AMSEs for theintroduced time-to-maturity classes:

short middle long

BS 1.0604 0.1681 0.1990FBS 24.4137 8.2451 9.4993

Table 4.3: AMSEs of BS and FBS depending on the time to maturity

From the above we see that BS formula outperforms FBS in all time-to-maturity

classes.Finally we examine how the two pricing formulae behave in dependence on the

volatility. We introduce three volatility intervals: low volatility when 12

< 20%,

medium volatility for 12

[20%, 35%) and high volatility if 12

35%. The followingtable shows the AMSEs for the introduced volatility classes:

small medium high

BS 0.1534 0.1911 2.0724FBS 2.4060 12.8238 10.4624

Table 4.4: AMSEs of BS and FBS depending on volatility

In conclusion, we see that although the average error of the whole sample set islower for the BS formula, examination in more details showed that FBS may outper-form the standard model under certain circumstances. This is not as astonishing,because Meng and Wang [MW] earlier obtained similar results (yet for the differ-ent formula) and concluded that because most investors and market makers tradeaccording to the prices obtained from the classical Black-Scholes formula the othermodels can outperform BS only when the market mechanism is given a full scope.

The result shown here proves that the fractional Black-Scholes formula for optionprice, more accurate from the theoretical point of view, may be more consistent

with the market price then the classical Black-Scholes formula, even though mostmarket participants trade according to the standard model, but only in certaincircumstances. However, we accentuate that in order to investigate this phenomenonin more details data from more liquid stock or FX market should be taken intoconsideration.

Also further modifications to the model presented here may be considered, forinstance introducing stochastic volatility process, stochastic interest rates or random

jumps, that would increase the exactness of option pricing. In our opinion Rosteksmodel opened a new, probably more fecund than the previous ones, way of involvingthe fractional Brownian Motion into stochastic finance.


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