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Unformatted text preview: screte time, we can guess that the option price
is its expected value after changing the probabilities to make the stock price a
martingale.
2
Theorem 6.10. Write E ⇤ for expected values when µ = r
/2 in (6.26).
⇤
rT
g (ST ).
The value of a European option g (ST ) is given by E e 200 CHAPTER 6. MATHEMATICAL FINANCE Proof. We prove this by taking limits of the discrete approximation. The risk
neutral probabilities, p⇤ , are given by
h
1 + rh d
.
ud p⇤ =
h (6.28) Using the formulas for u and d in (6.23 and recalling that ex = 1+ x + x2 /2+ · · · ,
p
p2
1
u = 1 + µh +
h + (µh +
h) + . . .
2
p
h + ( 2 /2 + µ)h + . . .
=1+
p
d = 1 + µh
h + ( 2 /2 + µ)h + . . .
so from (6.28) we have
p
h + (r µ
⇤
p
ph ⇡
2h 2 /2)h = 1r
+
2 µ
2 2 (6.29) /2 p h h
h
If X1 , X2 , . . . are i.i.d. with
h
P (X1 = 1) = p⇤
h h
P (X1 = 1) = 1 p⇤
h then the mean and variance are
h
EXi = 2p⇤ =
h (r h
var (Xi ) = 1 2 µ /2) p h h
(EXi )2 ! 1 To apply the central limit theorem we note that
t/h
t/h
pX h
pX h
h
Xm =
h
(Xm
m=1 h
E Xm...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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