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QUESTION

Consider the following discrete time one-period market model. The savings account is $1 at time 0 and $β at time 1. The stock price is given by S0 = 1 and S1 = ξ where ξ is a random variable taking tw

Consider the following discrete time one-period market model. The savings account is $1 at time 0 and $β at time 1. The stock price is given by S0 = 1 and S1 = ξ where ξ is a random variable taking two possible values u and d, each with positive probability. Moreover, assume that 0

(a) Define what is meant by an equivalent martingale measure (EMM). Find, with proof, the EMM of this model. Does this model have arbitrage opportunities? 

(b) Consider a contract which pays D1 = 1/S1 at time 1. Prove that the time 0 price of this contract is given by: D0 = (u + d − β) / (udβ) .

(c) Find the replicating portfolio for this contract.

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