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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