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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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