3.4 quadratic variation. 3.4.1 first-order variation 3.4.2 quadratic variation 3.4.3 volatility of...
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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.
One example
1 2
1 2
1 2 1 2
0
We call this the first-order variation ( ).
For the function f shown, it is
( ) [ ( ) (0)] [ ( ) ( )] [ ( ) ( )]
( ) ( ( )) ( )
| ( ) |
T
T
t t T
t t
FV f
FV f f t f f t f t f T f t
f t dt f t dt f t dt
f t dt
0 (3.4.2)
T
2 1 1 2
1 2
The middle term
[ ( ) ( )] ( ) ( )
is included in a way that guarantees that the magnitude
of the down move of the function f(t) between t and t
is added to rather than subtracted
f t f t f t f t
from the tatal.
0 1
0 1
In general, to compute the the first-order variation
of a function up to time T, we first choose a partition
{ , , , } of [0,T], which is a set of times
0
The maximum s
n
n
t t t
t t t T
0, , 1 1
tep size of the partition will be denoted
max ( )j n j jt t
1
10
0
We then define
( ) lim | ( ) ( ) | (3.4.3)
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
n
T j jj
FV f f t f t
t
1 goes to zero. j jt
Our first task is to verify that the definition (3.4.3) is consistent with the formula (3.4.2) for the function shown in Figure 3.4.1.
To do this, we use the Mean value Theorem, which applies to any function whose derivative is defined everywhere.
( )f t( )f t
*1
1 *
1
1
The Mean value Theorem says that in each subinterval
[ , ] there is a point such that
( ) ( ) ( ) (3.4.4)
In other words, someehere between and ,
j j j
j jj
j j
j j
t t t
f t f tf t
t t
t t
+1 +1
the
tan gent line is parallel to the chord connecting the
points ( ,f( )) and ( ,f( )).j j j jt t t t
1 *1
1
*1 1
1
10
1*
10
( ) ( )Multiplying ( ) by ,
we obtain ( ) ( ) ( )( ).
The sum on the right-hand side of | ( ) ( ) |
may be written as | ( ) | ( ),
j jj j j
j j
j j j j j
n
j jj
n
j j jj
f t f tf t t t
t t
f t f t f t t t
f t f t
f t t t
*
1*
1 000
which is a R iemann sum for the integral of the function
| ( ) | . Therefor,
( ) lim | ( ) | ( ) | ( ) | ,
and we have rederived (3.4.2).
j
n T
T j j jj
f t
FV f f t t t f t dt
3.4.2 Quadratic Variation
12
10
0
0 1 0 1
Definition 3.4.1 Let f(t) be a function defined for 0 .
The quadratic variation of f up to time T is
[ , ]( ) lim [ ( ) ( )] (3.4.5)
where { , , , } and 0 .
n
j jj
n n
t T
f f T f t f t
t t t t t t T
1 1 12 * 2 2 * 2
1 1 10 0 0
* 21
00
Remark 3.4.2. Suppose the function f has a continuous derivative. Then
[ ( ) ( )] | ( ) | ( ) | ( ) | ( )
and thus [ , ]( ) lim [ | ( ) | ( )
n n n
j j j j j j j jj j j
n
j j jj
f t f t f t t t f t t t
f f T f t t t
1
1* 2
10 0
0
2
00
]
lim lim | ( ) | ( )
lim | ( ) | 0
n
j j jj
T
f t t t
f t dt
2
0
2
0
1* 2
10
0
In the last step of this argument, we use the fact that
( ) is continuous to ensure that | ( ) | is finite.
If | ( ) | is inf inite, then
lim [ | ( ) | ( )]
leads
T
T
n
j j jj
f t f t dt
f t dt
f t t t
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
not continuous derivative
[f,f](T)=0
[f,f](T)=0ex: |t|
[f,f](T) 0ex: 布朗運動
0 1
Theorem 3.4.3. Let W be a Brownian motion. Then [W,W](T)=T
for all 0 almost surely.
Proof of Theorem 3.4.3.: Let { , , , } be a partition of
[0,T]. Define the sampled quadratic variation correspn
T
t t t
12
10
onding to
this partition to be ( ( ) ( )) .n
j jj
Q W t W t
We must show that this sampled quadratic variation, which is a random variable converges to T as .
We shall show that it has expected value T, and its variance converges to zero. Hence, it converges to its expected value T, regardless of the path along which we are doing the computation.
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
21 1 1
1 12
1 10 0
[( ( ) ( )) ] [ ( ) ( )] (3.4.6)
which implies
[( ( ) ( )) ] ( )
j j j j j j
n n
j j j jj j
E W t W t Var W t W t t t
EQ E W t W t t t T
21
2 21 1
4 21 1 1
21
2 2 21 1 1
21
[( ( ) ( )) ]
[(( ( ) ( )) ( )) ]
[( ( ) ( )) ] 2( ) [( ( ) ( )) ]
( )
3( ) 2( ) ( )
2( ) (3.4.7)
j j
j j j j
j j j j j j
j j
j j j j j j
j j
Var W t W t
E W t W t t t
E W t W t t t E W t W t
t t
t t t t t t
t t
1 12 2
1 10 0
1
10
0
0
( ) [( ( ) ( )) ] 2( )
2 ( ) 2
In particular, lim ( )=0, and we conclude
that lim =E =T
n n
j j j jj j
n
j jj
Var Q Var W t W t t t
t t T
Var Q
Q Q
Remark 3.4.4. In the proof above, we derived (3.4.6) and (3.4.7):
21 1
2 21 1
[( ( ) ( )) ]
and
[( ( ) ( )) ] 2( )
j j j j
j j j j
E W t W t t t
Var W t W t t t
21 1
21
1
It is tempting to arg ue that when is small, ( )
is very small,and therefore ( ( ) ( )) , although
random, is with high probability near its mean .
We could therefore claim tha
j j j j
j j
j j
t t t t
W t W t
t t
21 1
t
( ( ) ( )) (3.4.8)j j j jW t W t t t
21
1
21 1
A better way to try to capture what we think is going
on is to write
( ( ) ( )) 1 (3.4.9)
instead of ( ( ) ( )) (3.4.8)
j j
j j
j j j j
W t W t
t t
W t W t t t
21
1
1
11
1
( ( ) ( ))However, 1
is in fact not near 1, regardless of how small we make
. It is the square of standard normal random variable
( ) ( )
and its d
j j
j j
j j
j jj
j j
W t W t
t t
t t
W t W tY
t t
1
istribution is the same, no matter how small we
make .j jt t
j
1
212
1
To unders tan d better the idea behind Theorem3.4.3, we
jTchoose a large value of n and take t = , j=0,1, ,n.
nT
Then = for all j and n
( ( ) ( ))
Since the random var iab
j j
jj j
t t
YW t W t T
n
1 2 n
2n-1 1
j=0
21
les Y ,Y , Y are iid, the Law
of Large Numbers implies that converges to the
the common mean E as n .
j
j
Y
n
Y
n-1 21j=0
21
1
This mean is 1, and hence ( ( ) ( )) converges
to T. Each of the terms ( ( ) ( )) in the sum can be
Tquite different from its mean = , but when we sum
nmany terms like this, th
j j
j j
j j
W t W t
W t W t
t t
e differences average out to zero.
We write informally
( ) ( )dW t dW t dt
21 1
21
1
but this should not be int erpreted to either (3.4.8) or (3.4.9).
( ( ) ( )) (3.4.8)
( ( ) ( )) 1 (3.4.9)
It is only when we sum both sides of (3.4.9) and
j j j j
j j
j j
W t W t t t
W t W t
t t
call upon
the Law of Large Numbers to cancel errors that we get a
correct statement.
The statement is that on an interval [0,T], Brownian motion accumulates T units of quadratic variation.
Brownian motion accumulates quadratic variation at rate one per unit time.
In particular, the dt on the right-hand side of
is multiplied by an understood 1.
( ) ( )dW t dW t dt
0 1
0 1
12
10
0
Remark 3.4.5. Let { , , , } be a partition of [0,T]
(i.e.,0 ). In addition to computing the
quadratic variation of Brownian motion
lim ( ( ) ( )) , (3
n
n
n
j jj
t t t
t t t T
W t W t T
.4.11)
1
1 10
0
12
10
0
we can compute the cross variation of W(t) with t and the
quadratic variation of t with itself, which are
lim ( ( ) ( ))( ) 0, (3.4.12)
lim ( ) 0.
n
j j j jj
n
j jj
W t W t t t
t t
(3.4.13)
1
1 10
0
1 1 k 1 k 10 1
1
1 10 1
0
To see that 0 is the limit in lim ( ( ) ( ))( ) 0
we observe that
| ( ( ) ( ))( ) | max | ( ( ) ( )) | ( )
and so
| ( ( ) ( ))( ) | max | (
n
j j j jj
j j j j j jk n
n
j j j jk n
j
W t W t t t
W t W t t t W t W t t t
W t W t t t
k+1 k
0 1 k+1 k
( ) ( )) |
Since W is continous, max | ( ( ) ( )) | has limit
zero as goes to zero.k n
W t W t T
W t W t
12
10
0
1 12
1 k 1 k 10 1
0 0
To see that 0 is the limit in lim ( ) 0,
we observe that
( ) max ( ) ( )
which obviously has limit zero as 0.
n
j jj
n n
j j j jk n
j j
t t
t t t t t t T
12
10
0
1
1 10
0
12
10
0
Just as we capture lim ( ( ) ( )) by
writing ( ) ( ) , we capture
lim ( ( ) ( ))( ) 0
lim ( ) 0
by writing ( ) 0, 0.
n
j jj
n
j j j jj
n
j jj
W t W t T
dW t dW t dt
W t W t t t
t t
dW t dt dtdt
3.4.3 Volatility of Geometric Brownian Motion
2
Let and >0 be constants, and define the geometric
Brownian motion
1 ( ) (0)exp ( ) ( )
2
Here we show how to use the quadratic var iation of
Brownian motion to identify the volati
S t S W t t
lity from a
path of this process.
1 2
1 2
1 0 1 2
j+1
t
Let 0 T <T be given, and suppose we observe the
geometric Brownian motion S(t) for T t T .We
may then choose a partition of this interval,
T , and observe "log returns"
S(t )log =
S(t )
nt t t T
L
21 1
1
1(( ( ) ( ))+( )( )
2
over each of the subint erval [ , ].
j j j j
j j
W t W t t t t
t t
m-1j+1 2
j=0 t
m-1 m-12 2 2 2 2
1 1j=0 j=0
21 1
The sum of the squares of the log returns, sometimes called
the realized volatility, is
S(t )( log )
S(t )
1= (( ( ) ( )) +( ) ( )
2
1 +2 ( ) ( ( ) ( ))(
2
j j j j
j j j
W t W t t t
W t W t t
m 1
0
) (3.4.15)jj
t
12
1 2 100
1
1 100
12
100
21
By lim ( ( ) ( ))
lim ( ( ) ( ))( ) 0
lim ( ) 0
We conclude that when the maximum step size is small,
(( (
n
j jj
n
j j j jj
n
j jj
j
W t W t T T
W t W t t t
t t
W t
m-1 m-12 2 2 2
1j=0 j=0
m 12
1 10
22 1
m-1j+1 2 2
j=02 1 t
1) ( )) +( ) ( )
2
1 +2 ( ) ( ( ) ( ))( )
2
is approximately equal to (T -T ), and hence
S(t )1 ( log ) (3.4.16)
T -T S(t )
j j j
j j j jj
W t t t
W t W t t t
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bid-ask spread 。
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