We shall use 0.5 as the up-state probability and 0.5 as the down-state probability. These are, of course, the risk neutral probabilities, not the actual probabilities. Finally, recall the following notation:

Consider a three-year cap indexed to the yearly rate displayed in Table 1, and struck at rK = 7.5%. The payments are annual, i.e. = 1, and the notional value of the cap is N = $100. Given the interest rate tree in Table 1 find the value today (i.e. at t = 0) of the cap.

rit1, an up movement in interest rates Tij =) rir1.i+1 a "down" movement in interest rates 0 1 2 t= (years) 0 1 2 6.00% 7.704% 4.673% 9.892% 6.000% 3.639% Table 1: The Risk Neutral Black-Derman-Toy Interest Rate Tree fitted to the zero coupon bonds. Recall that the payoff of a plain vanilla cap as of time T1, T2, ...,T is CF(T;) = A * N * max(rn(T; - A,T;) - TK,0) rit1, an up movement in interest rates Tij =) rir1.i+1 a "down" movement in interest rates 0 1 2 t= (years) 0 1 2 6.00% 7.704% 4.673% 9.892% 6.000% 3.639% Table 1: The Risk Neutral Black-Derman-Toy Interest Rate Tree fitted to the zero coupon bonds. Recall that the payoff of a plain vanilla cap as of time T1, T2, ...,T is CF(T;) = A * N * max(rn(T; - A,T;) - TK,0)