4. This question concerns the following binomial model for the 1-year LIBOR rate. The current rate is r(0, 1) = r = 0.0400; in one year, the rate can be ru = 0.0500 or rd = 0.0300. Finally, the current 2-year rate is r(0, 2) = 0.0410. (All rates are with annual compounding.)

a) Consider a European put with maturity T on a zero-coupon bond that matures at T 0 = 2 with a strike of KP = 96 per 100 face. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as this put at time T = 1?

b) What is the price of the put?

c) Consider a caplet maturing at T 0 = 2 that pays to the owner, on date T 0 , interest at rate r(T 0 1, T0 ) net of the fixed rate KC = 0.0420, if the owner elects to receive the payment. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as the value of this caplet at time T = 1 on a notional amount N = $10, 000?

d) What is the value of this contract (on the notional amount N = $1M)?

e) Compute the risk-neutral probabilities consistent with the given tree and zcb prices.

f) What are the possible values of the return between 0 and 1 on the zcb maturing at 2, and what is the expectation of this return under the risk-neutral probability (i.e., the weighted average of the returns using the RNP as weights)? Answer the same questions for the return between 0 and 1 of the put, and compare with the bond. How would you describe the comparison?

g) Verify the answers to parts b) and d) using the risk-neutral probabilities.

Q4. This question concerns the following binomial model for the 1-year LIBOR rate. The current rate is r(0,1)=r=0.0400; in one year, the rate can be ru = 0.0500 or rd = 0.0300. Finally, the current 2-year rate is r(0,2) = 0.0410. (All rates are with annual compounding.) a) Consider a European put with maturity T on a zero-coupon bond that matures at T' = 2 with a strike of Kp = 96 per 100 face. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as this put at time T = 1? b) What is the price of the put? c) Consider a caplet maturing at T' = 2 that pays to the owner, on date T', interest at rate r(T' -1,T") net of the fixed rate Kc = 0.0420, if the owner elects to receive the payment. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as the value of this caplet at time T = 1 on a notional amount N = $10,000? d) What is the value of this contract (on the notional amount N = $1M)? e) Compute the risk-neutral probabilities consistent with the given tree and zob prices. f) What are the possible values of the return between 0 and 1 on the zcb maturing at 2, and what is the expectation of this return under the risk-neutral probability (i.e., the weighted average of the returns using the RNP as weights)? Answer the same questions for the return between 0 and 1 of the put, and compare with the bond. How would you describe the comparison? g) Verify the answers to parts b) and d) using the risk-neutral probabilities. Q4. This question concerns the following binomial model for the 1-year LIBOR rate. The current rate is r(0,1)=r=0.0400; in one year, the rate can be ru = 0.0500 or rd = 0.0300. Finally, the current 2-year rate is r(0,2) = 0.0410. (All rates are with annual compounding.) a) Consider a European put with maturity T on a zero-coupon bond that matures at T' = 2 with a strike of Kp = 96 per 100 face. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as this put at time T = 1? b) What is the price of the put? c) Consider a caplet maturing at T' = 2 that pays to the owner, on date T', interest at rate r(T' -1,T") net of the fixed rate Kc = 0.0420, if the owner elects to receive the payment. What portfolio of the 1-year and 2-year zero-coupon bonds has the same payoff as the value of this caplet at time T = 1 on a notional amount N = $10,000? d) What is the value of this contract (on the notional amount N = $1M)? e) Compute the risk-neutral probabilities consistent with the given tree and zob prices. f) What are the possible values of the return between 0 and 1 on the zcb maturing at 2, and what is the expectation of this return under the risk-neutral probability (i.e., the weighted average of the returns using the RNP as weights)? Answer the same questions for the return between 0 and 1 of the put, and compare with the bond. How would you describe the comparison? g) Verify the answers to parts b) and d) using the risk-neutral probabilities