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The process for the prices of a 5-year maturity zero-coupon bond and of a derivative on the interest rate that matures in three years are decrihed by the following trees. The probablities that an analyst associates 1with going up and down are 60% and 40% at each node of the tree, respectively. (NOTE: These are NOT the risk neutral probabilities.) 5-year zero coupon bond price period==> 20 21 22 23 24 i=5 time(inyears)==> i=0 t=l i=2 t=3 i=4 i=5 zg Z1" 23 z; 23 100 84.33 70.91 100 61.81 87.02 55.55 75.49 100 52. 03 67.07 89.12 63.08 79.05 100 74.84 91.31 86.57 100 95.86 100 Derivative period==:=- i=0 21 22 i=3 time(inyears)==> t=0 t=1 t=2 i=3 154.5 108.72 68.04 103 40.33 49 17 23.92 10 3 4.64 U . Suppose that you hold a portfolio of 10 ve-years zeros and 20 derivatives. How does the portfolio payoff evolve over three years? Construct the tree. . How can you change your position in the derivative in order to make the portfolio riskless between date t = 0 and t = 1? . What is the implied interest-rate tree up to t = 2? . What is the price of a zero-coupon bond that matures at time t = 2