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A3 - STAT/ACTSC 446/846 Assignment#3(due March 13th 2009...

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STAT/ACTSC 446/846 Assignment #3 (due March 13th, 2009) Over 50 marks Note: Recall that, when handing in your assignment, you are requested to use a cover page showing only your UWID number and your section (846 students: also write “846” on the cover page) and to write your full name on the next page. Assignments must be handed in during the TAs office hours in MC 6095 before or on the due date (see the Calendar on UW-ACE for details). This assignment looks long but answers are often shorter than the question! Problem 1: [4marks] The initial value S 0 of the index is 20. The index is modeled in a one period framework with time step T=1 year: ω 1 ω 2 S 1 19 25 The risk-free rate r (annual rate, continuously compounded) is 5%. A company sells index-linked contracts with a maturity T = 1 year for which an initial investment of \$1000 yields a payoff given by 1000 max parenleftbigg 1 + G , 0 . 9 S T S 0 parenrightbigg at maturity, where S T is the value of the index at time T and k > 0. G is a minimum guaranteed rate. (a)[ 2pts] Calculate the risk neutral probability. And find the price by calculating the expectation of the discounted payoff under the risk neutral probability. (b)[ 1pt] If G = 0, is the contract priced fairly (that is, is the initial investment of \$1000 equal to the no-arbitrage price of this derivative contract)? Justify your answer. (c)[ 1pt] Which value of G gives a fairly priced index linked contract? Problem 2: [6marks] Consider a market model with T = 2, i.e. t = 0 , 1 , 2, and riskless interest rate r = 1 4 ( effective interest rate per period, but you will also get full credit if you considered a continuously compounded interest rate since the question was ambiguous. ), and one risky asset whose prices are described by the following diagram: S 2 = 12 ω 1 S 1 = 8 3 4 d57 d115 d115 d115 d115 d115 d115 d115 d115 d115 d115 1 4 d37 d75 d75 d75 d75 d75 d75 d75 d75 d75 d75 S 2 = 6 ω 2 S 0 = 5 2 5 d67 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 d7 3 5 d27 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 d55 S 2 = 6 ω 3 S 1 = 4 2 7 d57 d115 d115 d115 d115 d115 d115 d115 d115 d115 d115 5 7 d37 d75 d75 d75 d75 d75 d75 d75 d75 d75 d75 S 2 = 3 ω 4 (Do not use decimal approximations in calculations.) 1. (2pt) Compute a hedge for the European put option with the exercise price equal to 5.

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