Suppose that you estimated the risk neutral tree for interest rates in Table 11.23, where there is equal risk neutral probability to move up or down the tree. Assume also for simplicity that each interval of time represents 1 year, that is, ( = 1.
(a) Compute the price of a 1-, 2-, and 3-year zero coupon bond.
(b) Compute the swap rate c (3) for a plain vanilla swap with annual cash flows and maturing on i = 3. Recall that cash flows are given by
CFi,j (i + 1) = N ( (r1 (i, j) - c(3))
Where, r1 (i, j) = (eri,j ( 1 - 1) is the annually compounded rate that corresponds to ri,j.
(c) Consider an option with maturity i = 1 with the following payoff

Where r1(1,j) is the annually compounded rate at time i = 1, and Z1 (3) is the zero coupon at time i = 1 that pays 1 at time i = 3.
i. What is the value of this option?

ii. If you sell this option, how can you hedge it? Write down the hedging strategy and confirm its performance at i = 1. Be precise in the description of the steps.
(d) Procter & Gamble Leveraged Swap: In November 1993, Procter & Gamble (P&G) entered a swap with Bankers Trust (BT) where BT would pay P&G a fixed rate rÌ…, and P&G would pay BT a floating rate plus a spread. The spread was going to be equal to 0 at time of initiation, and would be set at time i = 1 equal to the value s1 in Equation 11.43. The spread remains constant thereafter. To provide an example, suppose that the interest rate increases at i = 1 to r1,u and decreases afterwards to r2, ud. The spread is set to S1,u at time i = l, implying that P&G has to pay at time i = 1 simply r0 ( N, at time i = 2 the cash flow (r1(l, u) + Si,u) ( N, and at time i = 3 the cash flow (r1(2,ud) + S1, u) ( N, where N = 100 is the notional. (Remember that in swaps, the floating rate at time i determines the cash flows at time i + 1).
i. Assume the maturity of the levereged swap is 3 years. What is the value of the swap for P&G if rÌ… = c (3), where c (3) is the swap rate determined in Part (b)?
ii. Given your answer to part i, the value off that makes the swap value equal 0 at i = 0 is higher or lower than c (3)? Provide an brief intuition.
iii. Using a spreadsheet, compute the value of rÌ… (this can be done by using solver in Microsoft Excel, or simply by trial and error)?

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81 = max (0.98 × (ru) .06 i-0 i=1 i=2 2.0.08 r2.ud = 0.05 r2.du = 0.04 r1.d = 0.03 T2.dd = 0.02