The process for the prices of a 5-year maturity zero-coupon bond and of a derivative on the in the interest rate that matures in three years are decribed by the following trees. The probablities that an analyst associates with 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=) i=0 i=1 time(inyears)=) t=0 t=1

53:93 50:51

61:24

i=2 t=2

60:01 65:12 72:66

i=3 t=3

68:85 73:29 76:75 84:05

i=4 i=5 t=4 t=5

100 81:88

1

84:49 86:53 88:65 93:07

100

100

100

100

100

Derivative

period=) i=0 i=1 i=2 i=3 time(inyears) =) t=0 t=1 t=2 t=3

66:06 39:16

23:23

150 105:55

100 47:74

1. Suppose that you hold a portfolio of 10 5-years zeros and 20 derivatives. How does the portfolio payoff evolve over three years? Construct the tree.

2. How can you change your position in the derivative in order to make the portfolio riskless between date t = 0 and t = 1?

3. What is the implied interest-rate tree up to t = 2?

4. What is the price of a zero-coupon bond that matures at time t = 2?

Problem I (6 points) The proces for the prices of a year maturity -coupon bond and of a derivative on the in the interest rate that matures in three years are decribed by the following trees The probabilities that an analystociates with going up and down are 60% 40% at each node of the tree, respectively. (NOTE: These are NOT the risk neutral probabilities) period i 5-year zero coupon bon price -l i -2 -3 -5 30.51 Derivative i period time y o 1. Suppose that you hold a portfolio of 10 five-years eros and 20 derivatives. How does the portfolio p r ove over three years? Construct the tree. 2. How can you change your position in the derivative in order to make the portfolio risk between date and t 1 ? 3. What is the implied interest rate tree up to t 2? 4. What is the price of a zero-coupon bond that matures at timet2! Problem I (6 points) The proces for the prices of a year maturity -coupon bond and of a derivative on the in the interest rate that matures in three years are decribed by the following trees The probabilities that an analystociates with going up and down are 60% 40% at each node of the tree, respectively. (NOTE: These are NOT the risk neutral probabilities) period i 5-year zero coupon bon price -l i -2 -3 -5 30.51 Derivative i period time y o 1. Suppose that you hold a portfolio of 10 five-years eros and 20 derivatives. How does the portfolio p r ove over three years? Construct the tree. 2. How can you change your position in the derivative in order to make the portfolio risk between date and t 1 ? 3. What is the implied interest rate tree up to t 2? 4. What is the price of a zero-coupon bond that matures at timet2