Stochastic

# 4 68 2 06 3216 4 0 t 096 notice that vn ah

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online