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Unformatted text preview: but it is no more di cult to consider path dependent
options, so we derive the basic formulas in that generality.
Given a sequence a of heads and tails of length n let gn (a) be the value if
we exercise at time n. Our ﬁrst goal is to compute the value function Vn (a) 195 6.5. AMERICAN OPTIONS for the N period problem. To simplify some statements, we will suppose with
essentially no loss of generality that gN (a)
0, so VN (a) = gN (a). To work
backwards in time note that at time n we can exercise the option or let the
game proceed for one more step. Since we will stop or continue depending on
which choice gives the better payo↵:
⇢
1
⇤
[p⇤ (a)Vn+1 (aH ) + qn (a)Vn+1 (aT )]
(6.20)
Vn (a) = max gn (a),
1+r n
⇤
where p⇤ (a) and qn (a) = 1 p⇤ (a) are the riskneutral probabilities which make
n
n
the underlying stock a martingale. 64, 0
32, 0
0, 0* A 16, 0.96
0, 0.96* 8, 2.874 A A A 2, 2.784*A
A A 4, 6
6*,4.16 A 16, 0 A
A 4, 6 A
A 1, 9 8, 2.4
2, 2.4* A A A A
A 2, 8
8*, 6 A Example 6.7. For a concrete example, suppose as we did in Example 6.3 that
the stock price follows the binomial model with S0 = 8, u = 2, d = 1/2, r = 1/...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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