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Unformatted text preview: d the history a.
Taking p⇤ (a)(6.8) + (1 p⇤ (a))(6.9) and using (6.10) we have
n
n
Vn (a) = 1
[p⇤ (a)Vn+1 (aH ) + (1
1+r n p⇤ (a))Vn+1 (aT )]
n (6.12) i.e., the value is the discounted expected value under the risk neutral probabilities. Taking the di↵erence (6.8) (6.9) we have
◆
✓
1
1
(a)
(Sn+1 (aH ) Sn+1 (aT )) =
(Vn+1 (aH ) Vn+1 (aT ))
n
1+r
1+r
which implies that
n (a) = Vn+1 (aH )
Sn+1 (aH ) Vn+1 (aT )
Sn+1 (aT ) (6.13) In words, n (a) is the ratio of the change in price of the option to the change
in price of the stock. Thus for a call or put  n (a) 1.
The option prices we have deﬁned were motivated by the idea that by trading
in the stock we could replicate the option exactly and hence they are the only
price consistent with the absence of arbitrage. We will now go through the
algebra needed to demonstrate this for the general n period model. Suppose
we start with W0 dollars and hold n (a) shares of stock between time n and
n + 1 when the otucome of the ﬁrst n events is a. If we invest the...
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 Spring '10
 DURRETT
 The Land

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