23
Chapter 14 The Itˆ o Integral The following chapters deal with Stochastic Differential Equations in Finance. References: 1. B. Oksendal, Stochastic Differential Equations , Springer-Verlag,1995 2. J. Hull, Options, Futures and other Derivative Securities, Prentice Hall, 1993. 14.1 Brownian Motion (See Fig. 13.3.) is given, always in the background, even when not explicitly mentioned. Brownian motion, , has the following properties: 1. Technically, , 2. is a continuous function of , 3. If , then the increments are independent,normal, and 14.2 First Variation Quadratic variation is a measure of volatility. First we will consider first variation, , of a function . 153

The Itoˆ Integral - Russell Davidsonrussell-davidson.arts.mcgill.ca/e761/Ito.pdf · 158 14.5 Construction of the Itoˆ Integral The integrator is Brownian motion . , with associated

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Chapter 14

The It o Integral

Thefollowing chaptersdealwith StochasticDifferentialEquationsin Finance. References:

1. B. Oksendal,StochasticDifferentialEquations, Springer-Verlag,1995

2. J.Hull, Options,FuturesandotherDerivativeSecurities,PrenticeHall, 1993.

14.1 Brownian Motion

(SeeFig. 13.3.)�����������

is given,alwaysin thebackground,evenwhennot explicitly mentioned.Brownian motion, � �� �������������������� ��! " , hasthefollowing properties:

1. � ���#�$%�'& Technically,! (*)+�,& � �-�'�.�/�$%�102$43

,

2. � �� . is a continuousfunctionof ,

3. If�5$6 87,9� �:;9=<><><19� 8?

, thentheincrements

� �� : �@ � �A 87BC��<B<C<D� � �� E?��@ � �A 8?#F :

areindependent,normal,and

! GH� � �A JILK : M@ � �A JIN-OP$=�'�! G�� � �A EILK : Q@ � �A EINEOAR�$S .I�K : @� .IT<

14.2 First Variation

Quadraticvariationis a measureof volatility. First we will considerfirst variation, U,V ��WP , of afunction

W��� ..

153

154

t

t

1

2t

f(t)

T

Figure14.1:ExamplefunctionW��� .

.

For thefunctionpicturedin Fig. 14.1,thefirst variationover theinterval��������O

is givenby:

U,V�� 7�� ��� ��WD�$4� W/�A :�@ W����#EO @ � W/�A R �@ W��� :�EO � W���� /@ W��� R EO$

�� �7W��A�� .�� � ����

�� �J@ W��-�A E��� �

��� �W����� J��# L<

$��7�� W��A�A J � � �<

Thus,first variationmeasuresthetotalamountof upanddown motionof thepath.

Thegeneraldefinitionof first variationis asfollows:

Definition 14.1(First Variation) Let � $ )B 7 � : �C<B<C<��� ? 0 bea partition of��������O

, i.e.,

��$S 7 9� : 9=<><><19 ? $��2<Themeshof thepartitionis definedto be

��� � ��� $ "!$#I&% 7��(' ' '�� ?#F : �� I K : @� I +<We thendefine

U,V � 7)� ��� ��WD�$ *,+� -(- ./-(- � 7? F :0I1% 7 � W/�A EI K : M@ W/�A JIN � <

SupposeW

isdifferentiable.ThentheMeanValueTheoremimpliesthatin eachsubinterval� JI � JI K : O

,thereis apoint

32Isuchthat

W��� .I K :+�@ W��� JI �$=W � �� 2I C�� .I K :�@ .IN <

CHAPTER14. TheIto Integral 155

Then ?#F :0I&% 7 � W/�A JILK : M@ W/�A EIN � $

? F :0I&% 7 � W � �� 2I � �� .I�K : @ EIN+�

and

U,V � 7)� ��� ��WD�$ * +� - - ./- - � 7?#F :0I&% 7 � W � �� 2I � �� JILK : @ JIN

$��7 � W��-�A J � � �<

14.3 Quadratic Variation

Definition 14.2(Quadratic Variation) Thequadraticvariationof afunctionW

onaninterval����� ��O

is� W�� � � $ * + -(- . -(- � 7

? F :0I1% 7 � W �A EI K :+�@ W �� JIN � R <

Remark 14.1(Quadratic Variation of DifferentiableFunctions) IfW

isdifferentiable,then�EW��B��� Q$

�, because ? F :0

I1% 7 � W �� JI K :��@ W �� JIN � R $? F :0I&% 7 � W�� �A 2I � R �A EI K :�@� JIN R

9 ��� � ��� <? F :0I1% 7 � W � �A 2I � R �A EI K :�@� JIN

and

�8W��>��� 9 * +� - - ./- - � 7 ��� � ��� < * +� - - ./- - � 7? F :0I&% 7 � W � �� 2I � R �� JI K :�@ JIN

$ * +� - - ./- - � 7 ��� � �����7 � W � �� J � R �

$=�'<

Theorem 3.44� � �>���2�$�� �

or moreprecisely, ! ( ) ��� ��& � � �.< ���/��C���; $ �50;$43 <In particular, thepathsof Brownianmotionarenot differentiable.

156

Proof: (Outline)Let � $ )B 87#� �: �><><><�� -? 0bea partitionof

����� � O. To simplify notation,set � I*$

� �A ILK : �@ � �A I . Definethesamplequadratic variation

� . $?#F :0I1% 7 � RI <

Then� . @ �S$ ? F :0

I1% 7 � � RI @ �� .I�K : @� .I EOJ<We wantto show that *,+� -(- ./-(- � 7 � � . @ � $6��<Consideranindividualsummand

� RI @ �� JI K :�@ JIN�$ � � �� .I K :LQ@ � �� .IBJO R @S�� JI K :�@ .IBC<

Thishasexpectation0, so

! G � � . @ � $%! G ? F :0I&% 7 � � RI @ �� .I�K : @ .I>EO $%�'<

For ���$�� , theterms� R� @ �� � K : @� � and � RI @ �� I K : @ I

areindependent,so

� !>� � . @ �2�$ ? F :0I1%�7 � !+� � RI @ �A EI K :�@ .I>EO

$? F :0I1%�7 ! G�� ���I @� ��� JI K :�@� .IB � RI 6�� .I K :�@� JIN R O

$? F :0I1%�7 �����A I K : @� I R @� �� I K : @ I R �A I K : @ I R O

(if � is normalwith mean0 andvariance� R , then! G � ��� �$�� ��� )

$� ? F :0I1%�7 �� I K : @� I R

9� � � � � �? F :0I�% 7 �� JI K : @ JIC

$ � � � � � �;<Thuswe have

! GH� � . @ �2�$%�'�� !�C� � . @ �2�9� ��� � ��� <(�;<

CHAPTER14. TheIto Integral 157

As ��� � ��� � � , � !B� � . @ �2����, so *,+� -(- ./-(- � 7 � � . @ �2�$6��<

Remark 14.2(Differential Representation) We know that! G���� � �� .I K : �@ � �� JIN� R @ �� JI K : @� JI EO $%��<

We showedabove that

� !C� � � �A JI K : M@ � �� .I . R @ �� JI K : @� .IBJOD$� �� .I K : @ .I> R <When

�� I�K : @ I is small,

�A ILK : @ I Ris verysmall,andwehave theapproximateequation

� � �� JI K : Q@ � �� JI R�� EI K : @� JIT�

whichwe canwrite informally as � � �� .�� � �� .�$ �# L<

14.4 Quadratic Variation asAbsoluteVolatility

Onany time interval� � : ��� R O , we cansampletheBrownianmotionat times� : $ E7*9� : 96<><B< 9 8? $�� R

andcomputethesquaredsampleabsolutevolatility

3� R @ � :? F :0I&% 7 � � �A JILK : �@ � �A JIN� R <

This is approximatelyequalto3� R @ � : � � � �>��� R M@ � � �C� � : JO�$ � R @ � :� R @ � : $ 3 <

As we increasethe numberof samplepoints,this approximationbecomesexact. In otherwords,Brownianmotionhasabsolutevolatility 1.

Furthermore,considertheequation

� � �>��� �$�� $��73 � L� ����� ��<

This saysthat quadraticvariation for Brownian motion accumulatesat rate 1 at all timesalongalmosteverypath.

158

14.5 Construction of the It o Integral

The integrator is Brownian motion � �� . � � �, with associatedfiltration

� �A J+�� � �, andthe

following properties:

1. � 9� .$�� everysetin� � � is alsoin

� �A J,

2. � �� . is� �� J

-measurable,��

,

3. For 9� : 9=<><><19 8?

, theincrements� �A : M@ � �A E+� � �� R �@ � �� : +�C<B<C<�� � �� 8?1�@ � �� 8? F : areindependentof

� �� ..

The integrand is� �A J � � �

, where

1.� �A J

is� �� .

-measurable�

(i.e.,�

is adapted)

2.�

is square-integrable:

! G��7� R �� .�� �� �S� ���2<

We wantto definetheIt o Integral:

!D�� .�$��7

� ��� �� � ��� � �� �'<

Remark 14.3(Integral w.r.t. a differentiable function) IfW��� .

is a differentiablefunction, thenwecandefine ��

7� ��� ��8W������$ � �

7 � ��� �W��A��� ��� <This won’t work whenthe integratoris Brownianmotion, becausethe pathsof Brownianmotionarenotdifferentiable.

14.6 It o integral of an elementaryintegrand

Let � $ )C 7 � : �><><><'� ? 0 beapartitionof���'����O

, i.e.,

��$S E7*9� :29=<><><19 E? $��2<Assumethat

� �� .is constanton eachsubinterval

� .I1� JILK : O(seeFig. 14.2). We call sucha

�an

elementaryprocess.

Thefunctions� �� J and� �A I

canbeinterpretedasfollows:

Think of � �� J asthepriceperunit shareof anassetat time .

CHAPTER14. TheIto Integral 159

t )δ(

t )δ(δ( ) δ( t )=t

)tδ(

0=t0t t t = T2 3 4t1

0

δ( t )= 1

δ( t )= 2

δ( t )= 3

Figure14.2:Anelementaryfunction�.

Think of E7#� :+�C<B<>< � 8?

asthetradingdatesfor theasset. Think of

� �A JI>asthenumberof sharesof theassetacquiredat tradingdate

JIandhelduntil

tradingdate .I K :

.

ThentheIto integral! �� .

canbeinterpretedasthegainfromtradingat time ; thisgainis givenby:

! �A J�$������ �����

� �� 87> � � �� .�@ � �� 87>� ��� �%�� 7� �% 7 OJ� � 9� �9 :

� �� 87> � � �� : �@ � �� E7>EO � �� : +� � �A J�@ � �� : EOJ� : 9� �9� R� �� 7 � � �� : �@ � �� 7 EO � �� : � � �A R �@ � �� : EO � �A R � � �A EM@ � �A R EO � R 9 9 �� <

In general,if EI 9 9 JI K :

,

! �� .�$I+F :0� % 7 � �� � C� � �� � K : M@ � �� � EO� � �A JICC� � �� .�@ � �� JINEOJ<

14.7 Propertiesof the It o integral of an elementaryprocess

AdaptednessFor each �N! �� .

is���� .

-measurable.

Linearity If

!D�� J�$��7� ��� �� � ����+� ���A E�$ ��

7�� ��� /� � ��� then ! �A J������� J $ � �

7 � � ��� �� � �����/� � ���

160

t tttl+1l k k+1

s t

. . . . .

Figure14.3:Showing� and

in differentpartitions.

and �!D�� J $ � �

7�� ��� �� � ��� <

Martingale!D�� J

is a martingale.

We prove themartingalepropertyfor theelementaryprocesscase.

Theorem 7.45(Martingale Property)

!D�� J�$I F :0� % 7 � �� � +� � �� � K : �@ � �� � EO � �� .IBC� � �� .�@ � �A JINEOE� JI 9� �9 EILK :

is a martingale.

Proof: Let�S9 � 9

be given. We treat the moredifficult casethat � and

are in differentsubintervals,i.e., therearepartitionpoints

��and

Isuchthat �

��� ��+�� ��-K : Oand

� � I � I K : O(See

Fig. 14.3).

Write

!P�� . $�JF :0� % 7 � �� � +� � �� � K : �@ � �� � JO � �� ��� � � �� ��8K : �@ � �A ���EO I+F :0� % � K : � �� � +� � �� � K : �@ � �A � 8O � �A EIN+� � �� .�@ � �� .I EO

We computeconditionalexpectations:

! G�� �JF :0� %�7 � �� � B� � �� � K : Q@ � �� � ����� � � � � $

�JF :0� % 7 � �� � >� � �� � K : �@ � �A � <

! G�� � �� � C� � �� � K : �@ � �� � E ���� � � � 5$ � �A � ��! GH� � �� � K : � � � � -O @ � �� � �$ � �A � � � � � �@ � �� � EO

CHAPTER14. TheIto Integral 161

Thesefirst two termsaddup to! � � . Weshow thatthethird andfourth termsarezero.

! G�� I+F :0� % � K : � �A � >� � �� � K : �@ � �� � . ���� � � � � $

I+F :0� % � K : ! G � ! G � � �� � B� � �� � K : Q@ � �� � ���� � �� � ���� � � �

$I F :0� % �8K : ! G

���� �A � Q� ! G � � �� � K : � � �� � EO @ � �� � �� ��� �% 7 ���� � � � ���

! G � � �� .I >� � �� .�@ � �A JIN ���� � � � $=! G���� �� .IC ��! G�� � �� J � � �� JIB8O @ � �A JIN�� ��� �%�7 ���� � � � � �

Theorem 7.46(It o Isometry)

! G !TR �A J $=! G � �7 � R � ���� � <

Proof: To simplify notation,assume �$S JI

, so

! �A J�$I0� %�7 � �� � C� � �� � K : �@ � �� � � � � ���� O

Each� � hasexpectation0, anddifferent � � areindependent.

! R �A J�$�� I0� % 7 � �A � � ��

R

$I0� % 7 � R �� � � R� 0 � � � � ��

� � �� � �

�� � <

Sincethecrosstermshave expectationzero,

! G !1RB�A J�$I0� % 7 ! G�� � R �� � � R� O

$I0� % 7 ! G � � R �� � ! G � � � �� � K :��@ � �� � . R ���� � �� �

$I0� % 7 ! G � RN�� � >�� � K : @� �

$=! GI0� % 7

� ��� �� � � R ��� �� �

$=! G � �7 � R#��� ���

162

0=t0t t t = T2 3 4t1

path of path of

δδ4

Figure14.4:Approximatinga general processbyanelementaryprocess�� , over

��������O.

14.8 It o integral of a generalintegrand

Fix�����

. Let�

bea process(not necessarilyanelementaryprocess)suchthat

� �A Jis� �� .

-measurable,�� � � ��� ��O

,

! G�� �7 � R �� .�� �� �S<Theorem 8.47 There is a sequenceof elementaryprocesses

) � ?�0��? % :such that

* +� ? � � ! G � �7 � � ? �A J�@ � �A J � R �# �$=��<

Proof: Fig. 14.4showsthemainidea.

In thelastsectionwehave defined

!�? ���2�$ � �7 � ?P�� .�� � �A E

for every � . Wenow define

� �7 � �A J�� � �� .�$ *,+� ? � � � �

7 � ? �� .�� � �� . <

CHAPTER14. TheIto Integral 163

Theonly difficulty with thisapproachis thatwe needto makesuretheabove limit exists. Suppose� and � arelargepositiveintegers.Then

� ! � ! ? � � Q@ ! � � � $=! G � � �7 � � ? �A EM@ � � �� .EO�� � �� .�� R

(Ito Isometry:)$ ! G � �

7 � � ? �� JM@ � � �� .EO R � $ ! G � �

7 � � � ? �A EM@ � �� J � � � �A EQ@ � � �� J � O R �# ���� �� R 9� � R � � R � �9� #! G � �

7 � � ?D�� .�@ � �� J � R � �� ! G� �7 � � � �� J�@ � �� . � R � ��

which is small.Thisguaranteesthatthesequence) !�? � � L0 �? % :

hasa limit.

14.9 Propertiesof the (general)It o integral

!D�� J�$ � �7 � ������ � � ��+<

Here�

is any adapted,square-integrableprocess.

Adaptedness.For each ,! �A J

is� �� .

-measurable.

Linearity . If

!D�� J�$��7� ��� �� � ����+� ���A E�$ ��

7 � ��� /� � ��� then ! �A J������� J $ � �

7 � � ��� �� � �����/� � ��� and �

!D�� J�$ � �7�� ��� �� � ��� +<

Martingale.!D�� .

is amartingale.

Continuity.!D�� .

is acontinuousfunctionof theupperlimit of integration .

It o Isometry.! G ! R �� .�$6! G � �7 � R � ���� �

.

Example14.1() ConsidertheIto integral ���� ��������� ��������We approximatetheintegrandasshown in Fig. 14.5

164

T/4 2T/4 3T/4 T

Figure14.5:Approximatingtheintegrand � ��� with�� , over

� ��� ��O.

��� � ����������� ����� � � ��� �

if��� �� ��� ��

� ��� � if��� �� ����� ��� ����� � ��� ������� � if� � �!�"� � � ��� �

By definition,

� � � � ��� �� � ���$#&%&'�)(+* ���!�0,.- � 0/21 43 � 5/ � 17698 � 3+: 5/;1 <3 �To simplify notation,wedenote

,>=� ?/�1 3A@so � � ��� ��� �� � ���B#&%&'�C(+* � �!�0,D- � , �� ,.E � : , � �We compute �F ���!�0,.- � � ,DE � : , � F � �F ���!�0,D- � F,DE � :

� �!�0,D- � , ,.E � 6 �F � �!�0,D- � F,� �F F� 6 �F �G�H�0IJ- � FI :

�����0,D- � , ,.E � 6 �F �����0,D- � F,� �F F� 6

� �!�0,D- � F, :�G�!�0,D- � , ,DE �

� �F F� :� �!�0,D- � , �� ,.E � : , � �

CHAPTER14. TheIto Integral 165

Therefore, �G�!�0,D- � , � ,DE � : , ��� �F F� : �F �G�!�0,.- � � ,DE � : , � F @or equivalently�����0,.- � 5/21 3 � ?/ � 1 6<8 � 34: /�1 3 � �F F � � : �F � �!�0,D- � � 5/ � 1 698 � 3 / 1 3 F �Let

(� andusethedefinitionof quadraticvariationto get� � ���� ��� ���� � � �F F � � : �F �

Remark 14.4(Reasonfor the:R � term) If

Wis differentiablewith

W����#/$6�, then

� �7 W���� ��8W���� M$ � �

7 W/� ��W��-��� �� �$ :R W R ���� ����

�7

$ :R W R ���2+<

In contrast,for Brownianmotion,we have� �7 � ��� � � �����$ :

R � R ���;�@ :R � <The extra term

:R � comesfrom thenonzeroquadraticvariationof Brownian motion. It hasto be

there,because ! G � �7 � ������ � ��� $6� (Ito integral is a martingale)

but ! G :R � R ���2Q$ :R �;<

14.10 Quadratic variation of an It o integral

Theorem 10.48(Quadratic variation of It o integral) Let

!D�� J�$ � �7 � ������ � � ��+<

Then� ! �C�� .�$ � �

7 � RN��� �� � <

166

Thisholdsevenif�

is notanelementaryprocess.Thequadraticvariationformulasaysthatat eachtime

�, the instantaneousabsolutevolatility of

!is

� R ��� . This is the absolutevolatility of the

Brownianmotionscaledby thesizeof theposition(i.e.� �� J

) in theBrownianmotion. Informally,wecanwrite thequadraticvariationformulain differentialform asfollows:�T! �� .�� !D�� . $ � R �� .��# L<Comparethiswith � � �� .�� � �� .�$ �# L<Proof: (For anelementaryprocess

�). Let � $ )> -7 �� : �><><><��� 8?'0

bethepartitionfor�, i.e.,

� �� . $� �� JIB

for JI*9 �9� .I�K :

. To simplify notation,assume /$6 8?

. We have

�8! �>�� E�$? F :0I1%�7 � �8! �C�� .I K : Q@ �8! �>�� .I EO1<

Let uscompute�8! �C�� JI K :�M@ �8! �B�� JI>

. Let� $ ) � 7B� � :>�><><><'� � � 0 bea partition

JI2$ � 7*9 � : 9=<><>< 9 � � $ JI K : <Then

!D� � � K : M@ ! � � � �$� ��� �� � � �� JI>�� � ����

$ � �� .IC � � � � � K :+�@ � � � � EO �

so

�E! �C�A JILK : �@ �8! �B�� JI �$ � F :0� % 7 � !D� � � K : @ !D� � � EO R

$ � R �� .I> � F :0� % 7 � � � � � K : �@ � � � � JO R

��� � ��� ���@ @�@�@'@T� � R �� JI>>�� .I�K : @ .I +<

It followsthat

�8! �B�� J�$? F :0I&% 7 � R �A EIN>�� .I K : @ EIN

$?#F :0I&% 7

��� � �� �

� R#��� �� �

� � � � � � �@8@�@'@'@ @ �� �7 � RB� ���� � <

Chapter 15

It o’s Formula

15.1 It o’s formula for oneBrownian motion

We wanta rule to “dif ferentiate”expressionsof the formW�� � �� J� , where

W/����is a differentiable

function.If � �� . werealsodifferentiable,thentheordinarychain rule wouldgive�� W�� � �� J�M$=W � � � �A J � � �� J+�whichcouldbewritten in differentialnotationas

�EW/� � �� .�$=W � � � �� J � � �A E�� $=W��8� � �� J�� � �� JHowever, � �� . is notdifferentiable,andin particularhasnonzeroquadraticvariation,sothecorrectformulahasanextra term,namely,

�8W�� � �A J $6W��-� � �� .�� � �A E :R W�� �-� � �� .� �# ��� ���

� �� � � �� � <

This is Ito’s formulain differential form. Integratingthis,weobtainIto’s formulain integral form:

W�� � �A J�@ W�� � ���#� ��� �� � 7�

$ � �7 W��8� � ��� �� � ��� :

R� �7 W�� �8� � ������ � <

Remark 15.1(Differential vs. Integral Forms) Themathematicallymeaningfulform of Ito’sfor-mulais Ito’s formulain integral form:

W�� � �A J�@ W�� � ���#�$� �7 W��8� � ��� �� � ��� :

R� �7 W�� �8� � ������ � <

167

168

This is becausewe have solid definitionsfor both integralsappearingon the right-handside. Thefirst, � �

7 W � � � � ���� � ��� is anIto integral, definedin thepreviouschapter. Thesecond,

� �7 W�� �8� � � ���� � �

is aRiemannintegral, thetypeusedin freshmancalculus.

For paperandpencilcomputations,themoreconvenientform of Ito’s rule is Ito’s formulain differ-ential form: �8W�� � �� J $6W � � � �� .�� � �� . :

R W � � � � �� .��� �<Thereis anintuitivemeaningbut nosoliddefinitionfor theterms

�8W�� � �� J�+��� � �� . and�

appearingin this formula.This formulabecomesmathematicallyrespectableonly afterwe integrateit.

15.2 Derivation of It o’s formula

ConsiderW � �� $ :

R � R , sothat W��8� � �$ � � W�� �8� ���$ 3#<

Let��I � � I K :

benumbers.Taylor’s formulaimplies

W�� ��ILK : �@ W�� ��IN�$ ����ILK : @ � IN�W��A� � IC� :R ����ILK : @ ��IB�R�W��(�8� � IC+<

In thiscase,Taylor’s formulato secondorderis exactbecauseW

is aquadratic function.

In thegeneralcase,theaboveequationis only approximate,andtheerroris of theorderof� ��ILK : @

� I �. Thetotalerrorwill have limit zeroin thelaststepof thefollowing argument.

Fix�����

andlet � $ )> E7#� : �><><C< �� E?�0 bea partitionof� �'����O

. UsingTaylor’s formula,wewrite:

W � � � � �@ W � � � � �$ :R � R � � @ :

R � R � � $? F :0I&% 7 ��W�� � �� JI K : �/@ W/� � �� .IB�EO

$?#F :0I&% 7 � � �� JILK : �@ � �� JINEOBW��8� � �� JI �

:R? F :0I�% 7 � � �� JILK : �@ � �� JINEO R W��(�A� � �� .IB�

$?#F :0I&% 7 � �� JIN � � �� JILK : �@ � �� .I JO

:R?#F :0I&% 7 � � �� .I�K : M@ � �� .I JO R <

CHAPTER15. Ito’sFormula 169

We let ��� � ��� � � to obtain

W�� � ���2��@ W�� � ���#�$ � �7 � ������ � ����/ :

R � � �C� �2� � � ��$ � �

7 W��8� � ��� ��� � � ��� :R� �7 W��(�A� � ������ � � �:

�� <

This is Ito’s formulain integral form for thespecialcaseW�� �� $ :

R � R <

15.3 GeometricBrownian motion

Definition 15.1(GeometricBrownian Motion) GeometricBrownianmotionis� �� J�$ � ��� ��)#���� � � �� ./ ��� @ :R � R � *�

where�

and � � � areconstant.

DefineW/�A �� � �$ � �-� ��)#���� � � � � @ :

R � R � � �so

� �� .�$=W��� �� � �� .+<Then

W � $ � � @ :R � R � W ��W��,$ � W ��W����*$ � R W�<

Accordingto Ito’s formula,� � �� JQ$ �-W��� L� � �� J$=W � �# W � � � :R W ��� � � � �� ��� �

� �$ � � @ :R � R �W � � � W � � :

R � R W �# $ � � �� .�� � � � �� J�� � �� .

Thus,GeometricBrownianmotionin differential form is� � �� JQ$ � � �� .�� � � � �� J�� � �� .+�andGeometricBrownianmotionin integral form is

� �� .�$ � �-� � � �7 � � � ���� � � �

7 � � ��� /� � � �� <

170

15.4 Quadratic variation of geometricBrownian motion

In theintegral form of GeometricBrownianmotion,

� �� J $ � � � � � �7 � � ��� �� � � �

7 � � � ���� � ��� �theRiemannintegral

U �� .�$� �7 � � ��� ���

is differentiablewith U � �A JM$ � � �� J. This termhaszeroquadraticvariation.TheIto integral

� �� .�$ � �7 � � ��� /� � � ��

is notdifferentiable.It hasquadraticvariation

� � � �A J $ � �7 � R � R ������ � <

Thusthequadraticvariationof�

is givenby thequadraticvariationof�

. In differentialnotation,wewrite � � �� .�� � �� J $ � � � �A J�� � � � �� J�� � �� J� R $ � R � R �A J��

15.5 Volatility of GeometricBrownian motion

Fix��9 � : 9 � R . Let � $ )> 87#�B<C<B<�� E?'0

bea partitionof�(� : ��� R O . Thesquaredabsolutesample

volatility of�

on� � : ��� R O is

3� R @ �Q:? F :0I&% 7 � � �� JI K :��@ � �� .I>EO R � 3� R @ �Q:

� ��� �

R � R � ���� �� � R � R � � :

As� R�� � :

, the above approximationbecomesexact. In otherwords,the instantaneousrelativevolatility of

�is � R . This is usuallycalledsimply thevolatility of

�.

15.6 First derivation of the Black-Scholesformula

Wealth of an investor. An investorbeginswith nonrandominitial wealth � 7 andat eachtime ,

holds ��� .

sharesof stock.Stockis modelledby a geometricBrownianmotion:

� � �� JQ$ � � �A J�� � � � �� J�� � �� J <

CHAPTER15. Ito’sFormula 171

��� .

can be random,but mustbe adapted.The investorfinanceshis investingby borrowing orlendingat interestrate � .Let � �� J denotethewealthof theinvestorat time

. Then� � �� JQ$ � �A J�� � �� .� � � � �� . @ � �� . � �� .EO �# $

��A J � � � �� . � � � � �� J�� � �A EEO � � � �� J�@ � �� J � �� JJO)� $ � � �� J � �

�� J � �� J � � @ � � ��� �Riskpremium

� ��A E � �� J � � � �� J+<

Valueof an option. ConsideranEuropeanoptionwhichpays� � � ���2 attime�

. Let � �� �� � denotethe valueof this option at time

if the stockprice is

� �� . $ �. In otherwords,the valueof the

optionateachtime � � ��� ��O

is� �� �� � �� .�+<

Thedifferentialof thisvalueis� � �� L� � �A J $ � � � � � �$� � :R � ��� � � � �

$ � � � � � � � � � � � � � � O :R � ��� � R � R � $�� � � � � � � :

R � R � R � ����� �# � � � � � �A hedgingportfolio startswith someinitial wealth � 7 andinvestssothat thewealth � �A E at eachtime tracks � �A �� � �� J� . We saw abovethat� � �� JQ$4� � �

�� � @ � � O � � � � � � � <

To ensurethat � �� J�$ � �� L� � �A J for all , weequatecoefficientsin theirdifferentials.Equatingthe� � coefficients,weobtainthe � -hedgingrule:

��� J $ � � �� �� � �� . <

Equatingthe�#

coefficients,weobtain:

� � � � � � :R � R � R � ��� $ � �

�� � @ � � <

But wehaveset �$ � � , andweareseekingto cause� to agreewith � . Makingthesesubstitutions,

weobtain� � � � � � :

R � R � R � ��� $ � � � � � � @ � � �(where � $ � �A �� � �� E� and

� $ � �� .) whichsimplifiesto

� � � � � � :R � R � R � ��� $ � � <

In conclusion,weshouldlet � bethesolutionto theBlack-Scholespartial differentialequation

� � �A � � � � � � � �A � � :R � R � R � ��� �� � �� $ � � �� � ��

satisfyingtheterminalcondition� ���2� � $ � � � +<

If aninvestorstartswith � 7 $ � �-��� � ���# andusesthehedge��� .�$ � � �� �� � �� . , thenhewill have

� �� J�$ � �� L� � �� J for all , andin particular, � ���2Q$ � � � ��� � .

172

15.7 Mean and varianceof the Cox-Ingersoll-Rossprocess

TheCox-Ingersoll-Rossmodelfor interestratesis

� � �� .�$ � � ��@ �� �A J � � ��� � �� J�� � �A E+�

where��� � � � � � and � ��� arepositiveconstants.In integral form, thisequationis

� �� E�$ � �-�#� � � �7 ��� @ �

� ��� � � � � �7 � � � ���� � ��� <

We applyIto’s formulato compute� � R �� . . This is

� W � � �� J� , whereW ���� $ � R

. We obtain� � R �A J�$��EW/� � �A J�$6W��8� � �� J��� � �� .� :R W�� ��� � �A E��� � �� .�� � �� .

$ � �� . � � ��� @ �� �� . �# ��� � �A J�� � �� . � � � ��@ �

� �� .��# ��� � �� J�� � �� . R$� ��� � �� J�� @� �

�� R �� J�� � � ���� �� J�� � �� . � R � �� .��#

$ � ���� � R � �� .��# Q@ ��� R �� J��# � ��� � �� J�� � �A E

The meanof � �A J . Theintegral form of theCIR equationis

� �� E�$ � �-�#� � � �7 ��� @ �

� ��� � � � � �7 � � � ���� � ��� <

Takingexpectationsandrememberingthattheexpectationof anIto integral is zero,we obtain

! G � �� . $ � ��� � � � �7 ��� @ �

! G � ��� � � � <Differentiationyields �� ! G � �� J�$ �P����@ �

! G � �� .� $ ����@ � � ! G � �� J+�which impliesthat �� ����� � ! G � �� . � $���� � � � � ! G � �A J� �� ! G � �� . $ ���� � ��� <Integrationyields � �� � ! G � �� JQ@ � ��� $ ��� � �

7 � ���� � � $ �� ��� �� � @ 3>+<We solvefor

! G � �� J : ! G � �� J�$�� � F �� � / � ���#/@ �� 3 <

If � ��� Q$�� , then! G � �� JQ$�� for every

. If � ��� �$�� , then � �� . exhibits meanreversion:

* +� � � � ! G � �� J�$�� <

CHAPTER15. Ito’sFormula 173

Varianceof � �� . . Theintegral form of theequationderivedearlierfor� � R �� . is

� R �� . $ � R � � =� � �/ � R � �7 � ��� �� � @� �

� � �7 � R ��� �� � � � � �

7 � �� ������ � ��� <Takingexpectations,weobtain

! G � R �� . $ � R � � %� ��� � R � �7 ! G � ��� �� @� �

� � �7 ! G � R � ���� � <

Differentiationyields �� ! G � R �� J $ � ���/ � R ! G � �� J @ ��! G � R �� J+�

which impliesthat �� � R �� � ! G � R �� J�$ � R �� � � � � ! G � R �A J� �� ! G � R �� J $�� R �� � � ���/ � R ! G � �� .+<

Using the formula alreadyderived for! G � �� J andintegratingthe last equation,after considerable

algebraweobtain

! G � R �A J�$� � R � � R � R�

R / � ��� Q@ �� 3 � � R� � �� � � F �� � / � ��� @ �� 3 R � R� � � F R �� � � R� � / �

� @ � ���# 3 � F R �� � <

� ! � �A J $=! G � R �� J�@S� ! G � �� J� R$ � � R � � R / � ��� Q@ �� 3 � R� � � F �� � � R� � / �

� @ � ���# 3 � F R �� � <

15.8 Multidimensional Brownian Motion

Definition 15.2(�-dimensionalBrownian Motion) A

�-dimensionalBrownianMotion is a pro-

cess� �� .�$ � � : �� . �><B<C<'� � � �� .

with thefollowing properties:

Each� I��� E is a one-dimensionalBrownianmotion; If

� �$ � , thentheprocesses���A J

and � � �� J areindependent.

Associatedwith a�-dimensionalBrownianmotion,wehave afiltration

) � �A E 0suchthat

For each , therandomvector � �� J is

� �A E-measurable;

For each 9 : 96<B<><19 ?

, thevectorincrements

� �� �: Q@ � �� J+�B<C<B<�� � �� 8?T�@ � �� 8? F : areindependentof

� �� ..

174

15.9 Cross-variations of Brownian motions

Becauseeachcomponent��is aone-dimensionalBrownianmotion,wehavetheinformalequation� �

��� .�� �

��� J�$ �# L<

However, wehave:

Theorem 9.49 If� �$ � , � �

��� .�� � � �� J�$=�

Proof: Let � $ )B 87#�><B<C< � 8?�0bea partitionof

���'����O. For

� �$ � , definethesamplecrossvariationof �

�and � � on

� �'����Oto be

� . $? F :0I1% 7 � �

��� JI K :��@ �

��� JIB8O1� � � �� .I K :��@ � � �� JINEO#<

Theincrementsappearingon theright-handsideof theabove equationareall independentof oneanotherandall have meanzero.Therefore,

! G � . $6��<We compute� ! � � . . First notethat

� R. $? F :0I&% 7 � � �

�� .I�K : Q@ ���� JI R � � � �� JILK : �@ � � �� .IN R

? F :0� � I � � ��A � K : �@ � �

�� � JON� � � �� � K : �@ � � �� � JO <>� � ��� .I�K : M@ �

��� .ICEO1� � � �A JILK : �@ � � �� .I>8O

All the incrementsappearingin the sumof crosstermsare independentof oneanotherandhavemeanzero.Therefore,

� ! C� � . �$%! G � R.$%! G

? F :0I1% 7 � �

��� JI K :��@ �

��� JINEO R � � � �A EI K : �@ � � �� JIBEO R <

But� �

��� .I K : Q@ �

��� EINEO R

and� � � �� .I K :LM@ � � �� JI>EO R areindependentof oneanother, andeachhas

expectation�� .I�K : @� .I

. It followsthat

� !>� � . �$? F :0I1% 7 �A JI K :�@� JI R 9 � � � � �

? F :0I1% 7 �� JI K :�@ .IB�$ ��� � ��� < �;<

As ��� � ��� � � , we have � !B� � . ��� , so� . convergesto theconstant

! G � . $=� .

CHAPTER15. Ito’sFormula 175

15.10 Multi-dimensional It o formula

To keepthenotationassimpleaspossible,we write the Ito formulafor two processesdrivenby atwo-dimensionalBrownianmotion. Theformulageneralizesto anynumberof processesdrivenbya Brownianmotionof anynumber(notnecessarilythesamenumber)of dimensions.

Let � and � beprocessesof theform

� �� J�$ � ��� � � �7�� ��� �� � � �

7 � :�: ������ � : � �� � �7 � : R ��� �� � R � ��+�

� �� .�$ � �-�#� � �7�� ������ � � �

7 � R : ��� �� � : ��� � � �7 � RR ��� �� � R ��� +<

Suchprocesses,consistingof a nonrandominitial condition,plusa Riemannintegral, plusoneormoreIto integrals,arecalledsemimartingales. The integrands� ���� � � ��� � and

��� ��� canbeany

adaptedprocesses.Theadaptednessof theintegrandsguaranteesthat � and � arealsoadapted.Indifferentialnotation,wewrite � � $ � � � � :�: � � :/ � : R � � R �� � $ � �# � � R : � � : � R�R � � R <Giventhesetwo semimartingales� and � , thequadraticandcrossvariationsare:

� � � � $ � � � � � :: � � : � : R � � R R �$ �NR:�: � � : � � :� ��� �� �

� :: � : R � � : � � R� � � �7 �NR: R � � R � � R� ��� �

� �$ � � R:�: � R: R R � �� � � � $ � � � � � R : � � : � RR � � R R$ � � RR : � RRR R � ��� � � � $ � � � � � :: � � : � : R � � R >� � � � R : � � : � RR � � R $ � � :�: � R : � : R � RR �� Let

W��� L� � ���1bea functionof threevariables,andlet � �A E and � �� . besemimartingales.Thenwe

have thecorrespondingIto formula:�8W��� L� � ���1 $ W � � W � � � W�� � � :R ��W ��� � � � � W � � � � � � �W�� � � � � O <

In integral form, with � and � asdecribedearlierandwith all thevariablesfilled in, thisequationis W��� �� � �� J+� � �A JQ@ W������ � ���#+� � ���

$ � �7 � W � � W � � W�� :

R � � R:: � R: R W ��� � � :: � R : � : R � RR W � � :R � � RR : � RRR �W���� O � �

� �7 � � :: W � � R : W � O � � : � �

7 � � : R W � � RR W � O � � R �where

W $=W���� � � ����+� � ��� , for� � � � ) 3#� T0 , �

�� $ �

�� ���� , and �

�$ �

�� ��

.