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Question: consider the following discrete time twoperiod market model the savings...

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  1. Consider the following discrete time two-period market model. The savings account is given by βt = β^t for t = 0,1,2. The stock price is given by S0 = 1,S1 = ξ1 and S2 = ξ1ξ2 where ξ1 and ξ2 are random variables, each taking two possible values u and d with positive probabilities. Moreover, assume that d < β < u.

    Consider a contract which pays H2 = S1S2 at time 2.

    1. (a)  Assume that the model has an equivalent martingale measure (EMM).

      Find H1, the time 1 price of the contract.

    2. (b)  Hence or otherwise, prove that the time 0 price of this contract is given by

H0 = u + d − ud/β

 

(c) Find the replicating portfolio for this contract. 

(d) Without the assumption from part (a), this model does not necessarily have an EMM! Find an example where this is the case. Using the stock and the savings account, construct an arbitrage strategy for this case.

 

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