Stochastic

To check the martingale property we need to show that

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Unformatted text preview: previous section we begin with the One period case There are two possible outcomes for the stock called heads (H ) and tails (T ). ⇠⇠ S1 (H ) = S0 u ⇠⇠ ⇠⇠ S 0 XX XX XX S1 (T ) = S0 d We assume that there is an interest rate r, which means that $1 at time 0 is the same as $(1 + r) at time 1. For the model to be sensible, we need 0 < d < 1 + r < u. (6.1) Consider now an option that pays o↵ V1 (H ) or V1 (T ) at time 1. This could be a call option (S1 K )+ , a put (K S1 )+ , or something more exotic, so we will consider the general case. To find the “no arbitrage price” of this option we suppose we have V0 in cash and 0 shares of the stock at time 0, and want to pick these to match the option price exactly: ✓ ◆ 1 1 V0 + 0 S1 (H ) S0 = V1 (H ) (6.2) 1+r 1+r ✓ ◆ 1 1 V0 + 0 S1 (T ) S0 = V1 (T ) (6.3) 1+r 1+r Notice that here we have to discount money at time 1 (i.e., divide it by 1 + r) to make it comparable to dollars at time 0. To find the values of V0 and 0 we define t...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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