Question 5. (a) Using the notation for the 2-star: Black-DermanToy interest rate tree from lectures, show that the equations to be solved at the rst step are M13)\" +P[1.2}e = 23-11313] and Iln(_ln(p(1'2)\")l'1\"(1I1(P(1:2}e})|= Zone). [2 marks] (h) You are given the following tern: structure of interest rates and 1.urolatilities of forward rates: "Time to It'Iaturityr {Yrs Forward Bate Volatility (9:3) *Per $1 face value A 2-step Blask-Derman-Tu}.r interest rate tree was constructed from this data, producing the following possible forward rates {1%) at the various nodes: r(12)d = 5.3103, rug)\" = 19221; es)\" = 5.310s, :12, and = 7.5993; 112,3)..1m = losses. (1) Construct the dual tree which contains [Ir-coupon bond prices at each node [2 marks] (ii) Verify that these forward rates are compatible with the given term and volatility structure. [It sufces to carry out the verication at just one or two nodes} l2 mm]