The Itˆo Integral

The Itˆo IntegralThe following chapters deal with Stochastic Differential Equations in Finance.. Oksendal, Stochastic Differential Equations, Springer-Verlag,1995 2.. Hull, Options, Futu

The Itˆo 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.

(See Fig 13.3.) ;F;P)is given, always in the background, even when not explicitly mentioned

Brownian motion,B ( t;! ) : [0 ;1) !IR, has the following properties:

1 B (0) = 0;Technically,IPf! ; B (0 ;! ) = 0g= 1,

2 B ( t )is a continuous function oft,

3 If0 = t 0t 1 :::t n, then the increments

B ( t 1 ),B ( t 0 ) ; ::: ; B ( t n ),B ( t n,1 )

are independent,normal, and

IE [ B ( t k+1 ),B ( t k )] = 0 ;

IE [ B ( t k+1 ),B ( t k )] 2 = t k+1,t k :

14.2 First Variation

Quadratic variation is a measure of volatility First we will consider first variation,FV ( f ), of a functionf ( t )

153

t

1

2

t

f(t)

T

Figure 14.1: Example functionf ( t ).

For the function pictured in Fig 14.1, the first variation over the interval [0 ;T ]is given by:

FV [0;T] ( f ) = [ f ( t 1 ),f (0)],[ f ( t 2 ),f ( t 1 )] + [ f ( T ),f ( t 2 )]

= t

1 Z

0 f0

( t ) dt + t

2 Z

t1

(,f0

( t )) dt +ZT

t2

f0

( t ) dt:

=ZT

0

jf0

( t )jdt:

Thus, first variation measures the total amount of up and down motion of the path

The general definition of first variation is as follows:

Definition 14.1 (First Variation) Let =ft 0 ;t 1 ;::: ;t ngbe a partition of[0 ;T ], i.e.,

0 = t 0t 1 :::t n = T:

The mesh of the partition is defined to be

jjjj= max k=0;:::;n

,1 ( t k+1,t k ) :

We then define

FV [0;T] ( f ) = lim

jjjj!0

nX ,1 k=0

jf ( t k+1 ),f ( t k )j:

Supposefis differentiable Then the Mean Value Theorem implies that in each subinterval[ t k ;t k+1 ], there is a pointt

ksuch that

f ( t k+1 ),f ( t k ) = f0( t

k )( t k+1,t k ) :

nX ,1 k=0

jf ( t k+1 ),f ( t k )j= nX ,1

k=0

jf0( t

k )j( t k+1,t k ) ;

and

FV [0;T] ( f ) = lim

jjjj!0

nX ,1 k=0

jf0

( t

k )j( t k+1,t k )

=ZT

0

jf0( t )jdt:

14.3 Quadratic Variation

Definition 14.2 (Quadratic Variation) The quadratic variation of a functionfon an interval[0 ;T ]

is

hfi( T ) = lim

jjjj!0

nX ,1 k=0

jf ( t k+1 ),f ( t k )j2 :

Remark 14.1 (Quadratic Variation of Differentiable Functions) Iffis differentiable, thenhfi( T ) =

0, because

nX ,1

k=0

jf ( t k+1 ),f ( t k )j2 = nX ,1

k=0

jf0

( t

k )j2 ( t k+1,t k ) 2

 jjjj: nX ,1

k=0

jf0( t

k )j

2 ( t k+1,t k )

and

hfi( T ) lim

jjjj!0jjjj: lim

jjjj!0

nX ,1 k=0

jf0

( t

k )j2 ( t k+1,t k )

= lim

jjjj!0jjjj

T

Z

0

jf0( t )j

2 dt

= 0 :

Theorem 3.44

hBi( T ) = T;

or more precisely,

IPf!2 hB ( :;! )i( T ) = Tg= 1 :

In particular, the paths of Brownian motion are not differentiable.

Proof: (Outline) Let =ft 0 ;t 1 ;::: ;t ngbe a partition of[0 ;T ] To simplify notation, setD k =

B ( t k+1 ),B ( t k ) Define the sample quadratic variation

Q  = nX ,1 k=0 D 2k :

Then

Q ,T = nX ,1

k=0 [ D 2k,( t k+1,t k )] :

We want to show that

lim

jjjj!0 ( Q ,T ) = 0 :

Consider an individual summand

D 2k,( t k+1,t k ) = [ B ( t k+1 ),B ( t k )] 2,( t k+1,t k ) :

This has expectation 0, so

IE ( Q ,T ) = IE nX ,1

k=0 [ D 2k,( t k+1,t k )] = 0 :

Forj6= k, the terms

D 2j,( t j+1,t j ) and D 2k,( t k+1,t k )

are independent, so

var( Q ,T ) = nX ,1

k=0 var[ D 2k,( t k+1,t k )]

= nX ,1 k=0 IE [ D 4k,2( t k+1,t k ) D 2k + ( t k+1,t k ) 2 ]

= nX ,1 k=0 [3( t k+1,t k ) 2,2( t k+1,t k ) 2 + ( t k+1,t k ) 2 ]

(ifXis normal with mean 0 and variance 2, thenIE ( X 4 ) = 3  4)

= 2 nX ,1 k=0 ( t k+1,t k ) 2

2jjjj

nX ,1 k=0 ( t k+1,t k )

= 2jjjjT:

Thus we have

IE ( Q ,T ) = 0 ;

var( Q  T ) 2  :T:

Asjjjj!0,var( Q ,T )!0, so

lim

jjjj!0 ( Q ,T ) = 0 :

Remark 14.2 (Differential Representation) We know that

IE [( B ( t k+1 ),B ( t k )) 2

,( t k+1,t k )] = 0 :

We showed above that

var[( B ( t k+1 ),B ( t k )) 2

,( t k+1,t k )] = 2( t k+1,t k ) 2 :

When( t k+1,t k )is small,( t k+1,t k ) 2is very small, and we have the approximate equation

( B ( t k+1 ),B ( t k )) 2't k+1,t k ;

which we can write informally as

dB ( t ) dB ( t ) = dt:

14.4 Quadratic Variation as Absolute Volatility

On any time interval[ T 1 ;T 2 ], we can sample the Brownian motion at times

T 1 = t 0 t 1 :::t n = T 2

and compute the squared sample absolute volatility

1

T 2,T 1

nX ,1 k=0 ( B ( t k+1 ),B ( t k )) 2 :

This is approximately equal to

1

T 2,T 1 [hBi( T 2 ), hBi( T 1 )] = T 2,T 1

T 2,T 1 = 1 :

As we increase the number of sample points, this approximation becomes exact In other words,

Brownian motion has absolute volatility 1.

Furthermore, consider the equation

hBi( T ) = T =ZT

0 1 dt; 8T 0 :

This says that quadratic variation for Brownian motion accumulates at rate 1 at all times along

almost every path.

14.5 Construction of the Itˆo Integral

The integrator is Brownian motionB ( t ) ;t  0, with associated filtrationF( t ) ;t  0, and the following properties:

1 st =)every set inF( s )is also inF( t ),

2 B ( t )isF( t )-measurable,8t,

3 Fortt 1 :::t n, the incrementsB ( t 1 ),B ( t ) ;B ( t 2 ),B ( t 1 ) ;::: ;B ( t n ),B ( t n,1 )

are independent ofF( t )

The integrand is ( t ) ;t0, where

1  ( t )isF( t )-measurable8t(i.e.,is adapted)

2 is square-integrable:

IEZT

0  2 ( t ) dt <1; 8T:

We want to define the Itˆo Integral:

I ( t ) =Zt

0  ( u ) dB ( u ) ; t0 :

Remark 14.3 (Integral w.r.t a differentiable function) If f ( t ) is a differentiable function, then

we can define

t

Z

0  ( u ) df ( u ) =Z t

0  ( u ) f0( u ) du:

This won’t work when the integrator is Brownian motion, because the paths of Brownian motion are not differentiable

14.6 Itˆo integral of an elementary integrand

Let =ft 0 ;t 1 ;::: ;t ngbe a partition of[0 ;T ], i.e.,

0 = t 0t 1 :::t n = T:

Assume that  ( t ) is constant on each subinterval[ t k ;t k+1 ] (see Fig 14.2) We call such a an

elementary process.

The functionsB ( t )and ( t k )can be interpreted as follows:

Think ofB ( t )as the price per unit share of an asset at timet

t ) δ(

t )

δ(

δ( ) δ( t = t )

)

t

δ(

0

δ( t )

δ( t )

δ( t )

Figure 14.2: An elementary function.

 Think oft 0 ;t 1 ;::: ;t nas the trading dates for the asset.

 Think of ( t k )as the number of shares of the asset acquired at trading datet k and held until trading datet k+1

Then the It ˆo integralI ( t )can be interpreted as the gain from trading at timet; this gain is given by:

I ( t ) =

8

>

>

>

>

 ( t 0 )[ B ( t ), B ( t 0 )

| {z }

 ( t 0 )[ B ( t 1 ),B ( t 0 )] +  ( t 1 )[ B ( t ),B ( t 1 )] ; t 1 tt 2

 ( t 0 )[ B ( t 1 ),B ( t 0 )] +  ( t 1 )[ B ( t 2 ),B ( t 1 )] +  ( t 2 )[ B ( t ),B ( t 2 )] ; t 2 tt 3 :

In general, ift k tt k+1,

I ( t ) = kX ,1 j=0  ( t j )[ B ( t j+1 ),B ( t j )] +  ( t k )[ B ( t ),B ( t k )] :

14.7 Properties of the Itˆo integral of an elementary process

Adaptedness For eacht; I ( t )isF( t )-measurable

Linearity If

I ( t ) =Zt

0  ( u ) dB ( u ) ; J ( t ) =Zt

0 ( u ) dB ( u )

then

I ( t )J ( t ) =Z t

0 (  ( u ) ( u )) dB ( u )

t t t

t

l+1

.

Figure 14.3: Showingsandtin different partitions.

and

cI ( t ) =Z t

0 c ( u ) dB ( u ) :

Martingale I ( t )is a martingale

We prove the martingale property for the elementary process case

Theorem 7.45 (Martingale Property)

I ( t ) = kX ,1

j=0  ( t j )[ B ( t j+1 ),B ( t j )] +  ( t k )[ B ( t ),B ( t k )] ; t k tt k+1

is a martingale.

Proof: Let0  s  t be given We treat the more difficult case thats and t are in different subintervals, i.e., there are partition pointst ` andt k such thats2[ t ` ;t `+1 ]andt2 [ t k ;t k+1 ](See Fig 14.3)

Write

I ( t ) = `X ,1 j=0  ( t j )[ B ( t j+1 ),B ( t j )] +  ( t ` )[ B ( t `+1 ),B ( t ` )]

+ kX ,1 j=`+1  ( t j )[ B ( t j+1 ),B ( t j )] +  ( t k )[ B ( t ),B ( t k )]

We compute conditional expectations:

IE

2

4

`,1

X

j=0  ( t j )( B ( t j+1 ),B ( t j ))

F( s )

3

5= `X ,1 j=0  ( t j )( B ( t j+1 ),B ( t j )) :

IE

 ( t ` )( B ( t `+1 ),B ( t ` ))

F( s )

=  ( t ` )( IE [ B ( t `+1 )jF( s )],B ( t ` ))

=  ( t ` )[ B ( s ) B ( t ` )]

These first two terms add up toI ( s ) We show that the third and fourth terms are zero.

IE

2

4

k,1

X

j=`+1  ( t j )( B ( t j+1 ),B ( t j ))

F( s )

3

5= kX ,1 j=`+1 IE

IE

 ( t j )( B ( t j+1 ),B ( t j ))

F( t j )

F( s )

= kX ,1 j=`+1 IE

2 6

 ( t j )( IE [ B ( t j+1 )jF( t j )],B ( t j ))

=0

F( s )

3 7

IE

 ( t k )( B ( t ),B ( t k ))

F( s )

= IE 2 6

4 ( t k )( IE [ B ( t )jF( t k )],B ( t k ))

=0

F( s )

3 7 5

Theorem 7.46 (Itˆo Isometry)

IEI 2 ( t ) = IEZ t

0  2 ( u ) du:

Proof: To simplify notation, assumet = t k, so

I ( t ) =Xk

j=0  ( t j )[ B ( t j+1 ),B ( t j )

Dj

]

EachD j has expectation 0, and differentD jare independent

I 2 ( t ) =

0

@

k

X

j=0  ( t j ) D j

1 A

2

=Xk

j=0  2 ( t j ) D 2j + 2X

i<j  ( t i )  ( t j ) D i D j :

Since the cross terms have expectation zero,

IEI 2 ( t ) = Xk

j=0 IE [  2 ( t j ) D 2j ]

=Xk

j=0 IE

 2 ( t j ) IE

( B ( t j+1 ),B ( t j )) 2

F( t j )



=Xk

j=0 IE 2 ( t j )( t j+1,t j )

= IEXk

j=0

tj+1 Z

tj

 2 ( u ) du

= IEZ t

0  2 ( u ) du

0=t0 t1 t 2 t 3 t = T 4

path of

δ4

Figure 14.4: Approximating a general process by an elementary process 4, over[0 ;T ].

14.8 Itˆo integral of a general integrand

FixT > 0 Letbe a process (not necessarily an elementary process) such that

  ( t )isF( t )-measurable,8t2[0 ;T ],

 IER0 T  2 ( t ) dt <1:

Theorem 8.47 There is a sequence of elementary processesf ng

1

n=1such that

lim

n!1IEZ T

0 j n ( t ), ( t )j2 dt = 0 :

Proof: Fig 14.4 shows the main idea.

In the last section we have defined

I n ( T ) =Z T

0  n ( t ) dB ( t )

for everyn We now define

Z T

0  ( t ) dB ( t ) = lim n! 1

Z T

0  n ( t ) dB ( t ) :

The only difficulty with this approach is that we need to make sure the above limit exists Suppose

nandmare large positive integers Then

var( I n ( T ),I m ( T )) = IE Z T

0 [  n ( t ), m ( t )] dB ( t )

!2

(It ˆo Isometry:)= IEZ T

0 [  n ( t ), m ( t )] 2 dt

= IEZ T

0 [j n ( t ), ( t )j+j ( t ), m ( t )j] 2 dt

(( a + b ) 2

2 a 2 + 2 b 2 :)2 IEZ T

0 j n ( t ), ( t )j

2 dt + 2 IEZ T

0 j m ( t ), ( t )j

2 dt;

which is small This guarantees that the sequencefI n ( T )g

1

n=1has a limit

14.9 Properties of the (general) Itˆo integral

I ( t ) = Z t

0  ( u ) dB ( u ) :

Hereis any adapted, square-integrable process

Adaptedness For eacht,I ( t )isF( t )-measurable

Linearity If

I ( t ) =Zt

0  ( u ) dB ( u ) ; J ( t ) =Zt

0 ( u ) dB ( u )

then

I ( t )J ( t ) =Z t

0 (  ( u ) ( u )) dB ( u )

and

cI ( t ) =Z t

0 c ( u ) dB ( u ) :

Martingale. I ( t )is a martingale

Continuity. I ( t )is a continuous function of the upper limit of integrationt

Itˆo Isometry. IEI 2 ( t ) = IER0 t  2 ( u ) du

Example 14.1 () Consider the Itˆo integral

Z T

0 B(u) dB(u):

We approximate the integrand as shown in Fig 14.5

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

Figure 14.5: Approximating the integrandB ( u )with 4, over[0 ;T ].

n(u) =

8

>

>

>

>

B(0) = 0 if 0u < T=n;

B(T=n) if T=nu < 2T=n;

:::

B (n,1)T T



if (n,1)T n

u < T:

By definition,

Z T

0

B(u) dB(u) = lim

n!1

n,1 X

k =0 B

 kT n

  B

 (k + 1)T n

 ,B

 kT n

 :

To simplify notation, we denote

Bk 4

= B

 kT n



;

so

Z T

0

B(u) dB(u) = lim

n!1

n,1 X

k =0

Bk(Bk +1

,Bk):

We compute

1 2 n,1 X

k =0

(Bk +1 ,Bk)2= 1

2 n,1 X

k =0

B2

k +1 , n,1 X

k =0

BkBk +1+1

2 n,1 X

k =0

B2 k

= 1

2B2

n+1 2 n,1 X

j=0

B2 j , n,1 X

k =0

BkBk +1+1

2 n,1 X

k =0

B2

= 1

2B2

n+n,1 X

k =0

B2 k , n,1 X

k =0

BkBk +1

= 1

2B2 n , n,1 X

Bk(Bk +1

,Bk):

n,1 X

k =0

Bk(Bk +1

,Bk) =1

2B2 n , 1 2 n,1 X

k =0

(Bk +1 ,Bk)2;

or equivalently

n,1

X

k =0

B



kT

n

  B

 (k + 1)T n

 ,B

 kT n



= 1

2B2(T),

1 2 n,1 X

k =0

 B

 (k + 1)T n

  k T



2 :

Letn!1and use the definition of quadratic variation to get

Z T

0

B(u) dB(u) = 1

2B2(T),

1

2T:

Remark 14.4 (Reason for the 1 2 T term) Iff is differentiable withf (0) = 0, then

Z T

0 f ( u ) df ( u ) =Z T

0 f ( u ) f0

( u ) du

= 1

2 f 2 ( u )

T 0

= 1 2 f 2 ( T ) :

In contrast, for Brownian motion, we have

Z T

0 B ( u ) dB ( u ) = 1 2 B 2 ( T ),1 2 T:

The extra term 1 2 T comes from the nonzero quadratic variation of Brownian motion It has to be there, because

IEZ T

0 B ( u ) dB ( u ) = 0 (It ˆo integral is a martingale) but

IE 1 2 B 2 ( T ) = 1 2 T:

14.10 Quadratic variation of an Itˆo integral

Theorem 10.48 (Quadratic variation of Itˆo integral) Let

I ( t ) = Z t

0  ( u ) dB ( u ) :

Then

hIi( t ) =Z t

0  2 ( u ) du:

This holds even if is not an elementary process The quadratic variation formula says that at each time u, the instantaneous absolute volatility of I is  2 ( u ) This is the absolute volatility of the Brownian motion scaled by the size of the position (i.e  ( t )) in the Brownian motion Informally,

we can write the quadratic variation formula in differential form as follows:

dI ( t ) dI ( t ) =  2 ( t ) dt:

Compare this with

dB ( t ) dB ( t ) = dt:

Proof: (For an elementary process) Let = ft 0 ;t 1 ;::: ;t ngbe the partition for, i.e., ( t ) =

 ( t k )fort k tt k+1 To simplify notation, assumet = t n We have

hIi( t ) = nX ,1

k=0 [hIi( t k+1 ), hIi( t k )] :

Let us computehIi( t k+1 ), hIi( t k ) Let =fs 0 ;s 1 ;::: ;s mgbe a partition

t k = s 0 s 1 :::s m = t k+1 :

Then

I ( s j+1 ),I ( s j ) = sj

+1 Z

sj  ( t k ) dB ( u )

=  ( t k )[ B ( s j+1 ),B ( s j )] ;

so

hIi( t k+1 ), hIi( t k ) = mX ,1

j=0 [ I ( s j+1 ),I ( s j )] 2

=  2 ( t k ) mX ,1

j=0 [ B ( s j+1 ),B ( s j )] 2

jjjj!0

, , , !

 2 ( t k )( t k+1,t k ) :

It follows that

hIi( t ) = nX ,1

k=0  2 ( t k )( t k+1,t k )

= nX ,1 k=0

tk+1 Z

tk

 2 ( u ) du

jjjj!0

, , , !

Z t

0  2 ( u ) du:

Itˆo Isometry. IEI ( t ) = IER0 t  ( u ) du

Example 14.1 () Consider the Itˆo integral< /b>... guarantees that the sequencefI n ( T )g

1

n=1has a limit

14.9 Properties of the (general) Itˆo integral< /b>... 0 (It ˆo integral is a martingale) but

IE 2 B ( T ) = 2 T:

14.10 Quadratic variation of an Itˆo integral< /b>

Theorem 10.48