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Unformatted text preview: or the European version. It certainly cannot be strictly
less since one possibility in the American option is to continue each time and
this turns it into a European option.
For some of the theoretical results it is useful to notice that
✓
◆
g⌧
⇤
V0 = max E
⌧
(1 + r)⌧ (6.21) where the maximum is taken over all stopping times ⌧ with 0 ⌧ N . In
Example 6.7 ⌧ (T ) = 1 and ⌧ (H ) = 3, i.e., if the stock goes down on the ﬁrst
step we stop. Otherwise we continue until the end.
1
41
V0 = · 6 · + · 6 ·
2
58 ✓ ◆3
4
= 2.4 + 0.384
5 Proof. The key to prove the stronger statement
⇤
Vn (a) = max En (g⌧ /(1 + r)⌧
⌧ n n ) ⇤
where En is the conditional expectation given the events that have occurred up
to time n. Let Wn (a) denote the righthand side. If we condition on the ﬁrst n
outcomes being a then P (⌧ = n) is 1 or 0. In the ﬁrst case we get gn (a). In the
⇤
second case Wn (a) = [p⇤ (a)Wn+1 (aH ) + qn (a)Wn+1 (aT )]/(1 + r), so Wn and
n
Vn satisfy the same r...
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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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