3.4 Quadratic Variation

3.4.1 First-Order Variation

3.4.2 Quadratic Variation

3.4.3 Volatility of Geometric Brownian
Motion
3.4.1 First-Order Variation

We wish to compute the amount of up
and down oscillation undergone by this
function between times 0 and T, with
the down moves adding to rather than
subtracting from the up moves.(計算移
動距離而非位移)
One example
We call this the first-order variation FVT ( f ).
For the function f shown, it is
FVT ( f )  [ f (t1 )  f (0)]  [ f (t2 )  f (t1 )]  [ f (T )  f (t2 )]
t1
t2
T
0
t1
t2
  f (t )dt   ( f (t ))dt   f (t )dt
T
  | f (t ) | dt
0
(3.4.2)
In general, to compute the the first-order variation
of a function up to time T, we first choose a partition
  {t0 , t1 , , tn } of [0,T], which is a set of times
0  t0  t1 
 tn  T
The maximum step size of the partition will be denoted
  max j 0,
, n 1
(t j 1  t j )
We then define
n 1
FVT ( f )  lim  | f (t j 1 )  f (t j ) |
 0
(3.4.3)
j 0
The limit in (3.4.3) is taken as the number n of partition
points goes to infinity and the length of the longest
subinterval t j 1  t j goes to zero.
MVT
Multiplying
f (t j 1 )  f (t j )
t j 1  t j
 f (t *j ) by t j 1  t j ,
we obtain f (t j 1 )  f (t j )  f (t *j )(t j 1  t j ).
n 1
The sum on the right-hand side of
 | f (t
j 0
n 1
may be written as
*

|
f
(
t

j ) | (t j 1  t j ),
j 0
j 1
)  f (t j ) |
which is a R iemann sum for the integral of the function
| f (t *j ) | . Therefor,
FVT ( f )  lim
 0
n 1
 | f (t ) | (t
j 0
*
j
and we have rederived (3.4.2).
T
j 1
 t j )   | f (t ) | dt ,
0
3.4.2 Quadratic Variation
Definition 3.4.1 Let f(t) be a function defined for 0  t  T .
The quadratic variation of f up to time T is
[ f , f ](T )  lim
 0
where   {t0 , t1 ,
n 1
2
[
f
(
t
)

f
(
t
)]
(3.4.5)
 j 1
j
j 0
, tn } and 0  t0  t1 
 tn  T .
Remark 3.4.2. Suppose the function f has a continuous derivative. Then
n 1
n 1
j 0
j 0
2
* 2
2

[
f
(
t
)

f
(
t
)]

|
f
(
t
)
|
(
t

t
)
 j 1
 j j 1 j  
j
and thus [ f , f ](T )  lim [ 
 0
j 0
* 2

|
f
(
t
 j ) | (t j 1  t j )]
j 0
 0
T
n 1
* 2

|
f
(
t
 j ) | (t j 1  t j )
j 0
 lim    | f (t ) |2 dt  0
 0
* 2

|
f
(
t
 j ) | (t j 1  t j )
n 1
 lim   lim
 0
n 1
0
In the last step of this argument, we use the fact that
f (t ) is continuous to ensure that
If

T
0

T
0
| f (t ) |2 dt is finite.
2

| f (t ) | dt is inf inite, then
lim [ 
 0
n 1
* 2

|
f
(
t
 j ) | (t j 1  t j )]
j 0
leads to a 0   situation, which can be anything between
0 and .


Most functions have continuous derivatives,
and hence their quadratic variation are zero.
For this reason, one never consider quadratic
variation in ordinary calculus.
The paths of Brownian motion, on the other
hand, cannot be differentiated with respect
to the time variable.
continuous derivative
[f,f](T)=0
[f,f](T)=0
ex: |t|
not continuous derivative
[f,f](T)  0
ex:布朗運動
Theorem 3.4.3. Let W be a Brownian motion. Then [W,W](T)=T
for all T  0 almost surely.
Proof of Theorem 3.4.3.: Let   {t0 , t1 , , tn } be a partition of
[0,T]. Define the sampled quadratic variation corresponding to
n 1
this partition to be Q   (W (t j 1 )  W (t j )) 2 .
j 0

 0

The sampled quadratic variation is the sum of
independence random variables. Therefore, its mean
and variance are the sums of the means and
variances of these random variables. We have
E[(W (t j 1 )  W (t j )) 2 ]  Var[W (t j 1 )  W (t j )]  t j 1  t j (3.4.6)
which implies
n 1
n 1
j 0
j 0
EQ   E[(W (t j 1 )  W (t j )) 2 ]   (t j 1  t j )  T
Var[(W (t j 1 )  W (t j )) 2 ]
 E[((W (t j 1 )  W (t j ))  (t j 1  t j )) ]
2
2
 E[(W (t j 1 )  W (t j )) 4 ]  2(t j 1  t j ) E[(W (t j 1 )  W (t j )) 2 ]
 (t j 1  t j )
2
 3(t j 1  t j ) 2  2(t j 1  t j )2  (t j 1  t j ) 2
 2(t j 1  t j )
2
(3.4.7)
n 1
n 1
Var (Q )  Var[(W (t j 1 )  W (t j )) ]   2(t j 1  t j ) 2
2
j 0
j 0
n 1
  2  (t j 1  t j )  2  T
j 0
In particular, lim  0Var (Q )=0, and we conclude
that lim  0Q =EQ =T

To unders tan d better the idea behind Theorem3.4.3, we
jT
choose a large value of n and take t j = , j=0,1, ,n.
n
T
Then t j 1  t j =
for all j and
n
2
Y
j 1
(W (t j 1 )  W (t j )) 2  T 
n
Since the random var iables Y1 ,Y2 , Yn are iid, the Law
of Large Numbers implies that

2
Y
n-1 j 1
j=0
the common mean EY j21 as n  .
n
converges to the
we can compute the cross variation of W(t) with t and the
quadratic variation of t with itself, which are
n 1
lim
 0
 (W (t
j 0
j 1
)  W (t j ))(t j 1  t j )  0,
n 1
lim
 0
2
(
t

t
)
 j 1 j  0.
j 0
(3.4.13)
(3.4.12)
n 1
To see that 0 is the lim it in lim
 0
 (W (t
j 0
j 1
)  W (t j ))(t j 1  t j )  0
we observe that
| (W (t j 1 )  W (t j ))(t j 1  t j ) | max | (W (tk 1 )  W (tk )) | (t j 1  t j )
0 k  n 1
and so
n 1
|  (W (t j 1 )  W (t j ))(t j 1  t j ) | max | (W (tk+1 )  W (tk )) | T
j 0
0 k  n 1
Since W is continous, max 0 k  n 1 | (W (tk+1 )  W (tk )) | has lim it
zero as  goes to zero.
n 1
To see that 0 is the limit in lim
 0
 (t
j 0
j 1
 t j )  0,
2
we observe that
n 1
 (t
j 0
n 1
j 1
 t j )  max (tk 1  tk )   (t j 1  t j )    T
2
0 k  n 1
j 0
which obviously has limit zero as   0.
n 1
Just as we capture lim
 0
2
(
W
(
t
)

W
(
t
))
 T by
 j 1
j
j 0
writing dW (t )dW (t )  dt , we capture
n 1
lim
 0
 (W (t
j 0
j 1
)  W (t j ))(t j 1  t j )  0
n 1
lim
 0
2
(
t

t
)
 j 1 j  0
j 0
by writing dW (t )dt  0, dtdt  0.
3.4.3 Volatility of Geometric
Brownian Motion
Let  and  >0 be constants, and define the geometric
Brownian motion
1 2 

S (t )  S (0) exp  W (t )  (   )t 
2


Here we show how to use the quadratic var iation of
Brownian motion to identify the volatility  from a
path of this process.
Let 0  T1 <T2 be given, and suppose we observe the
geometric Brownian motion S(t) for T1  t  T2 .We
may then choose a partition of this interval,
T1  t0  t1  L  tn  T2 , and observe "log returns"
S(t j+1 )
1 2
log
= ((W (t j 1 )  W (t j ))+(   t )(t j 1  t j )
S(t t )
2
over each of the sub int erval [t j ,t j 1 ].
The sum of the squares of the log returns, sometimes called
the realized volatility, is
m-1
S(t j+1 ) 2
( log
)

S(t t )
j=0
m-1
m-1
1
= 2  ((W (t j 1 )  W (t j )) 2 +(   2 ) 2  (t j 1  t j ) 2
2
j=0
j=0
1 2 m 1
+2 (   ) (W (t j 1 )  W (t j ))(t j 1  t j )
2
j 0
(3.4.15)
n 1
By
lim
 0
2
(
W
(
t
)

W
(
t
))
 T2  T1
 j 1
j
j 0
n 1
lim
 0
 (W (t
j 0
j 1
)  W (t j ))(t j 1  t j )  0
n 1
lim
 0
2
(
t

t
)
 j 1 j  0
j 0
We conclude that when the maximum step size  is small,
m-1
1 2 2 m-1
  ((W (t j 1 )  W (t j )) +(   )  (t j 1  t j ) 2
2
j=0
j=0
2
2
1 2 m 1
+2 (   ) (W (t j 1 )  W (t j ))(t j 1  t j )
2
j 0
is approximately equal to  2 (T2 -T1 ), and hence
S(t j+1 ) 2
1 m-1
2
(
log
)



T2 -T1 j=0
S(t t )
(3.4.16)


In theory, we can make this approximation
as accurate as we like by decreasing the step
size. In practice, there is a limit to how small
the step size can be.
On small time intervals, the difference in
prices due to the bid-ask spread can be as
large as the difference due to price
fluctuations during the time interval.

名詞解釋 bid-ask spread
當股票的最高買價大於等於最低賣價的
時候，就會有股票成交；所以當最高買
價小於等於最低賣價的時候不會有股票
成交，且中間就會有一個價差，稱為bidask spread。