Question 5. (a) Using the notation for the 2-step Black-Derman-Toy interest rate tree from lectures, show that the equations to be solved at the first step are p(1, 2). + p(1.2) = 2e (1.2) and | In ( - In (p(1,2).)) - In ( - In (p(1, 2).))| = 20 (1,2). [2 marks (b) You are given the following term structure of interest rates and volatilities of forward rates: Time to Maturity (Yrs) r(0. T) () 0-Coupon Price* Forward Rate Volatility () 6. 1982 0.9399 G.4030 0.8798 20 6.8721 0.8137 18 17 "Per $1 face value. A 2-step Black-Derman-Toy interest rate tree was constructed from this data, producing the following possible forward rates (6) at the various nodes: r(1.2) = 5.3103. r(1.2). = 7.9221; r(2.3)or = 5.3108, r(2.3).4 - 7.5993: r(2.3).. - 10.8922. (i) Construct the dual tree which contains 0-coupon bond prices at each bode. 12 marks (ii) Verify that these forward rates are compatible with the given term and volatility structure. (It suffices to carry out the verification at just one or two nodes.) 2 marks SEE OVER