# a4 - Scenario Price of S1 at time one Price of S2 at time...

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Scenario Price of S 1 at time one Price of S 2 at time one ω 1 60/9 40/3 ω 2 60/9 80/9 ω 3 40/9 80/9 Table 1: Prices at time one ActSc 446/846 Winter 2006 Assignment 4 Due Date: April 7,2006, 5.00pm 1. A stock price is currently \$30. Each month for the next two months it is expected to increase by 8% or reduced by 10%. The risk-free interest rate is 5%. Using a two-step tree to calculate the value of a derivative that pays off max [(30 - S T ) , 0] , where S T is the stock’s price two months later. If the derivative is American-style, should it be exercised early? 2. State precisely the first fundamental theorem of asset pricing theorem, for a single-step security mar- ket. 3. Given a single-step security market which is arbitrage-free, prove that the model is complete if and only if there exists only one state-price vector ( second fundamental theorem ). 4. Consider a single-step security market in which N = 2 and M = 2 . The first basic asset, say B , is risk-less asset such that B (0) = 1 , and B (1) = 1 + 1 / 9 . The second basic asset, say S , is risky and S (0) = 5

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