The Itˆo Integral

Chapter 14
The It
ˆ
o Integral
The following chapters deal with Stochastic Differential Equationsin 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:
1.
B 0 = 0;
Technically,
IP f! ; B 0;!=0g=1
,
2.
B t
is a continuous function of
t
,
3. If
0=t
0
t
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

,ofa
function
f t
.
153
154
t
t
1
2
t
f(t)
T
Figure 14.1: Example function
f 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
f
0
t dt +
t
2
Z
t
1
,f
0
t dt +
T
Z
t
2
f
0
t dt:
=
T
Z
0
jf
0
tj dt:

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
n
g
be a partition of
0;T
, i.e.,
0=t
0
t
1
:::  t
n
= T:
The mesh of the partition is defined to be
jjjj = max
k=0;::: ;n,1
t
k+1
, t
k
:
We then define
FV

0;T 
f = lim
jjjj!0
n,1
X
k=0
jf t
k+1
 , f t
k
j:
Suppose
f
is differentiable. Then the Mean Value Theorem implies that in each subinterval
t
k
;t
k+1

,
there is a point
t

k
such that
f t
k+1
 , f t
k
=f

0
t

k
t
k+1
, t
k
:
CHAPTER 14. The ItˆoIntegral
155
Then
n,1
X
k=0
jf t
k+1
 , f t
k
j =
n,1
X
k=0
jf
0
t

k
jt
k+1

, t
k
;
and
FV
0;T 
f = lim
jjjj!0
n,1
X
k=0
jf
0
t

k
jt
k+1
, t
k

=
T
Z
0
jf
0
tj dt:
14.3 Quadratic Variation
Definition 14.2 (Quadratic Variation) Thequadraticvariationof a function

f
on an interval
0;T
is
hf iT = lim
jjjj!0
n,1
X
k=0
jf t
k+1
 , f t
k
j
2
:
Remark 14.1 (Quadratic Variation of Differentiable Functions) If
f
is differentiable, then
hf iT =
0
, because
n,1
X
k=0
jf t
k+1
 , f t
k
j

2
=
n,1
X
k=0
jf
0
t

k
j
2
t
k+1
, t
k

2
jjjj:
n,1
X
k=0
jf
0
t

k
j
2
t

k+1
, t
k

and
hf iT   lim
jjjj!0
jjjj: lim
jjjj!0
n,1
X
k=0
jf
0
t

k
j
2
t
k+1
, t
k

= lim
jjjj!0
jjjj
T
Z
0

jf
0
tj
2
dt
=0:
Theorem 3.44
hB iT =T;
or more precisely,
IP f! 2 ; hB :; !iT = Tg =1:
In particular, the paths of Brownian motion are not differentiable.
156
Proof: (Outline) Let
=ft
0
;t
1
;::: ;t
n
g
be a partition of
0;T
. To simplify notation, set
D
k
=
B t
k+1
 , B t
k


.Definethesample quadratic variation
Q

=
n,1
X
k=0
D
2
k
:
Then
Q

, T =
n,1
X
k=0
D
2
k
, t
k+1
, t
k
:
We want to show that
lim
jjjj!0

Q

, T =0:
Consider an individual summand
D
2
k
, 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
n,1
X
k=0
D

2
k
, t
k+1
, t
k
 = 0:
For
j 6= k
,theterms
D
2
j
, t
j +1
, t
j

and
D
2
k
, t
k+1
, t
k

are independent, so
varQ


, T =
n,1
X
k=0
varD
2
k
, t
k+1
, t
k

=
n,1
X
k=0
IE D
4
k
, 2t
k+1
, t
k
D
2
k
+t
k+1
, t
k


2

=
n,1
X
k=0
3t
k+1
, t
k

2
, 2t
k+1
, t
k

2
+t
k+1
, t
k

2

(if
X
is normal with mean 0 and variance


2
,then
IE X
4
=3
4
)
=2
n,1
X
k=0
t
k+1
, t
k

2
 2jjjj
n,1
X
k=0
t
k+1
, t
k

=2jjjj T:
Thus we have
IE Q


, T =0;
varQ

, T   2jjjj:T :
CHAPTER 14. The ItˆoIntegral
157
As
jjjj!0
,
varQ

, T !0
,so
lim
jjjj!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


2
is 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
n,1
X
k=0
B t
k+1
 , B t
k

2
:
This is approximately equal to
1
T
2
, T
1
hB i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
hB iT =T =
T
Z
0
1dt; 8T  0:
This says that quadratic variation for Brownian motion accumulates at rate 1 at all times along
almost every path.