Springer Finance

106 3 Brownian Motion

and so

Since W is continuous, maxo9�n-l !W(tk+l)-W(k)! has limit zero as 111111 ,

the length of the longest subinterval, goes to zero. To see that 0 is the limit

in (3.4.1 3 ), we observe that

which obviously has limit zero as 111111 --+ 0.

Just as we capture (3.4. 11 ) by writing (3.4. 10 ), we capture (3.4. (^1) 2) and
(3.4.1 3) by writing
dW(t) dt = 0, dt dt = 0.

3.4.3 Volatility of Geometric Brownian Motion

(3.4.14)

0

Let a and a > 0 be constants, and define the geometric Brownian motion

S(t) = S(O) exp { aW(t) + (a-�a^2 ) t}.

This is the asset-price model used in the Black-Scholes-Merton option-pricing
formula. Here we show how to use the quadratic variation of Brownian motion
to identify the volatility a from a path of this process.
Let 0 � T 1 < T 2 be given, and suppose we observe the geometric Brownian
motion S(t) for T 1 � t � T 2. We may then choose a partition of this interval,
T1 = to< t 2 < · · · < tm = T 2 , and observe "log returns"

S(ti+I) (

(

)

(

(^1 2)
log )
S(tj)
=a W(tj+I)-W ti) + a- 2 a (ti+l-ti)
over each of the subintervals [tj, ti +Il· The sum of the squares of the log
returns, sometimes called the realized volatility, is
E (log
S�t��I)
)
2
j=O ( J)
= a^2 �1 (W(ti+I)-W(ti))^2 + (a-�a^2 )
2
� (ti+l -ti)^2
+2a (a-�a^2 ) Y: (W(tj+I) -W(tj))(ti + 1 - ti )· (3.4.^15 )
]= 0