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Unformatted text preview: 6.6 BlackScholes formula Many options take place over a time period of one or more months, so it is
natural consider St to be the stock price after t years. We could use a binomial
model in which prices change at the end of each day but it would also be
natural to update prices several times during the day. Let h be the amount of
time measured in years between updates of the stock price. This h will be very
small e.g., 1/365 for daily updates so it is natural to let h ! 0. Knowing what
will happen when we take the limit we will let
p
Snh = S(n 1)h exp(µh +
hXn )
where P (Xn = 1) = P (Xn = 1) = 1/2. This is binomial model with
p
p
u = exp(µh +
h)
d = exp(µh
h) (6.23) Iterating we see that
Snh n
pX
= S0 exp µnh +
h
Xm
m=1 ! (6.24) If we let t = nh the ﬁrst term is just µt. Writing h = t/n the second term
becomes
n
p
1X
t· p
Xm
n m=1
To take the limit as n ! 1, we use the Theorem 6.9. Central Limit Theorem. Let X1 , X2 , . . . be i.i.d. with EXi =
0 and var (Xi ) = 1 Then for all x we have
!...
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 Spring '10
 DURRETT
 The Land

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