Tham khảo tích phân Ito

14.1 Brownian Motion See Fig.. In other words, Brownian motion has absolute volatility 1... Think of t k as the number of shares of the asset acquired at trading datet k and held untilt

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.

14.1 Brownian Motion

(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:

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.,

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

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

= lim

jjjj!0jjjj

TZ

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

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

= 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:

( 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

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 thefollowing 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)

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 motionare 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(

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 untiltrading datet k+1

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

14.7 Properties of the Itˆo integral of an elementary process

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

t t t

We prove the martingale property for the elementary process case

Theorem 7.45 (Martingale Property)

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:

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

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

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

F( s )3

7

IE

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

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 )

1

A2

=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

1n=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

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

nandmare large positive integers Then

1n=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

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

0B(u) dB(u):

We approximate the integrand as shown in Fig 14.5

if (n,1)T n

u < T:

By definition,

Z T

0

B(u) dB(u) = lim

n!1

n,1 X

k =0B

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 +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 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;

(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

T 0

14.10 Quadratic variation of an Itˆo integral

Theorem 10.48 (Quadratic variation of Itˆo integral) Let

This holds even if is not an elementary process The quadratic variation formula says that at eachtime u, the instantaneous absolute volatility of I is  2 ( u ) This is the absolute volatility of theBrownian 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

tk+1 Z

tk

 2 ( u ) du

jjjj!0, , , !

Z t

0  2 ( u ) du:

Itˆo’s Formula

15.1 Itˆo’s formula for one Brownian motion

We want a rule to “differentiate” expressions of the formf ( B ( t )), wheref ( x ) is a differentiablefunction IfB ( t )were also differentiable, then the ordinary chain rule would give

d dtf ( B ( t )) = f0( B ( t )) B0( t ) ;which could be written in differential notation as

df ( B ( t )) = f0

( B ( t )) B0

( t ) dt

= f0( B ( t )) dB ( t )

However,B ( t )is not differentiable, and in particular has nonzero quadratic variation, so the correctformula has an extra term, namely,

0 f0

( B ( u )) dB ( u ) + 1 2Z t

0 f00( B ( u )) du:

Remark 15.1 (Differential vs Integral Forms) The mathematically meaningful form of It ˆo’s

for-mula is It ˆo’s forfor-mula in integral form:

f ( B ( t )),f ( B (0)) =Z t

0 f0( B ( u )) dB ( u ) + 1 2Z t

0 f00( B ( u )) du:

167

This is because we have solid definitions for both integrals appearing on the right-hand side Thefirst,

Z t

0 f0( B ( u )) dB ( u )

is an It ˆo integral, defined in the previous chapter The second,

Z t

0 f0( B ( u )) du;

is a Riemann integral, the type used in freshman calculus.

For paper and pencil computations, the more convenient form of It ˆo’s rule is It ˆo’s formula in

differ-ential form:

df ( B ( t )) = f0

( B ( t )) dB ( t ) + 1 2 f00

( B ( t )) dt:

There is an intuitive meaning but no solid definition for the termsdf ( B ( t )) ;dB ( t )anddtappearing

in this formula This formula becomes mathematically respectable only after we integrate it

15.2 Derivation of Itˆo’s formula

Considerf ( x ) = 1 2 x 2, so that

f0( x ) = x; f00( x ) = 1 :Letx k ;x k+1be numbers Taylor’s formula implies

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

( x k ) + 1 2 ( x k+1,x k ) 2 f00

( x k ) :

In this case, Taylor’s formula to second order is exact becausef is a quadratic function.

In the general case, the above equation is only approximate, and the error is of the order of( x k+1,

x k ) 3 The total error will have limit zero in the last step of the following argument.

FixT > 0and let =ft 0 ;t 1 ;::: ;t ngbe a partition of[0 ;T ] Using Taylor’s formula, we write:

We letjjjj!0to obtain

f ( B ( T )),f ( B (0)) =Z T

0 B ( u ) dB ( u ) + 1 2hBi( T )

| {z }T

15.3 Geometric Brownian motion

Definition 15.1 (Geometric Brownian Motion) Geometric Brownian motion is

S ( t ) = f ( t;B ( t )) :Then

15.4 Quadratic variation of geometric Brownian motion

In the integral form of Geometric Brownian motion,

S ( t ) = S (0)+Z t

0 S ( u ) du +Z t

0 S ( u ) dB ( u ) ;the Riemann integral

15.5 Volatility of Geometric Brownian motion

Fix0  T 1  T 2 Let = ft 0 ;::: ;t ngbe a partition of[ T 1 ;T 2 ] The squared absolute sample

volatility ofSon[ T 1 ;T 2 ]is

1

T 2,T 1

nX ,1 k=0 [ S ( t k+1 ),S ( t k )] 2 '

1

T 2,T 1

T2 Z

T1

 2 S 2 ( u ) du

' 2 S 2 ( T 1 )

As T 2 # T 1, the above approximation becomes exact In other words, the instantaneous relative

volatility ofSis 2 This is usually called simply the volatility ofS

15.6 First derivation of the Black-Scholes formula

Wealth of an investor An investor begins with nonrandom initial wealth X 0 and at each timet,holds
( t )shares of stock Stock is modelled by a geometric Brownian motion:

dS ( t ) = S ( t ) dt + S ( t ) dB ( t ) :

( t ) can be random, but must be adapted The investor finances his investing by borrowing orlending at interest rater.

LetX ( t )denote the wealth of the investor at timet Then

v ( t;S ( t )) :The differential of this value is

To ensure thatX ( t ) = v ( t;S ( t ))for allt, we equate coefficients in their differentials Equating the

dBcoefficients, we obtain the
-hedging rule:

( t ) = v x ( t;S ( t )) :Equating thedtcoefficients, we obtain:

If an investor starts withX 0 = v (0 ;S (0))and uses the hedge
( t ) = v x ( t;S ( t )), then he will have

X ( t ) = v ( t;S ( t ))for allt, and in particular,X ( T ) = g ( S ( T ))

15.7 Mean and variance of the Cox-Ingersoll-Ross process

The Cox-Ingersoll-Ross model for interest rates is

dr ( t ) = a ( b,cr ( t )) dt + q

r ( t ) dB ( t ) ;wherea;b;c;andr (0)are positive constants In integral form, this equation is

IEr ( t ) = r (0) + aZ t

0 ( b,cIEr ( u )) du:

Differentiation yields

d dtIEr ( t ) = a ( b,cIEr ( t )) = ab,acIEr ( t ) ;which implies that

d dt

Variance ofr ( t ) The integral form of the equation derived earlier fordr 2 ( t )is

d dte 2act IEr 2 ( t ) = e 2act

r (0),

b c

15.8 Multidimensional Brownian Motion

Definition 15.2 (d-dimensional Brownian Motion) A d-dimensional Brownian Motion is a

pro-cess

B ( t ) = ( B 1 ( t ) ;::: ;B d ( t ))

with the following properties:

 EachB k ( t )is a one-dimensional Brownian motion;

 Ifi6= j, then the processesB i ( t )andB j ( t )are independent

Associated with ad-dimensional Brownian motion, we have a filtrationfF( t )gsuch that

 For eacht, the random vectorB ( t )isF( t )-measurable;

 For eachtt 1 :::t n, the vector increments

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

are independent of ( t )

15.9 Cross-variations of Brownian motions

Because each componentB iis a one-dimensional Brownian motion, we have the informal equation

var( C  ) = nX ,1

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

nX ,1 k=0 ( t k+1,t k ) =jjjj:T:

Asjjjj!0, we havevar( C  )!0, soC  converges to the constantIEC  = 0

15.10 Multi-dimensional Itˆo formula

To keep the notation as simple as possible, we write the It ˆo formula for two processes driven by a

two-dimensional Brownian motion The formula generalizes to any number of processes driven by

a Brownian motion of any number (not necessarily the same number) of dimensions.

LetXandY be processes of the form

more It ˆo integrals, are called semimartingales The integrands ( u ) ; ( u ) ;and ij ( u )can be anyadapted processes The adaptedness of the integrands guarantees thatXandY are also adapted Indifferential notation, we write

dX = dt +  11 dB 1 +  12 dB 2 ;

dY = dt +  21 dB 1 +  22 dB 2 :Given these two semimartingalesXandY, the quadratic and cross variations are: