Problem 7 6marks the initial value s of the index is

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Problem 7: [6marks] 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 18 22 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 . 8 S T S 0 parenrightbigg at maturity, where S T is the value of the index at time T and k > 0 and G > 0 . G is a minimum guaranteed rate. (a)[ 2pts] Find the price of this product (as a function of G ) by giving explicitly the strategy that replicates the payoff. (b)[ 1pt] If G = 1% , 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. Is there a value of G that gives a fairly priced index linked contract? Consider now that: 1000 max parenleftbigg 1 + G , 0 . 95 S T S 0 parenrightbigg (c)[ 2pt] Find the price of this product (as a function of G ) by giving explicitly the strategy that replicates the payoff. (d) [ 1pt] Which value of G gives a fairly priced index linked contract?
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