Using the example in Figure 7.7 and the results in Table 7.5

(a) Show the computations for the \(T_{1}\) discount factor curves in each of the \((u, d)\) states starting from the \(T_{2}\) discount factor curves.

(b) Show the computation for the Arrow-Debreu prices of the three \(T_{2}\) states \((u u, u

d, d d)\) from the \(T_{1}\) AD prices.

(c) Calculate the semiannual 1-year CMS (par swap) rate in each of the \((u, d)\) states at \(T_{1}\).

(d) Calculate today's value of a \(\$ 1 \mathrm{M}\) European-style 6 month into 1-year payer swaption with semiannual fixed rate of \(4 \%\) per annum.

Figure 7.7

Table 7.5

Transcribed Image Text:

(State, rem Arrow Debreu Price) P= 1/2, (0,4%, 1.0) To=0 (u, 4.25%, 0.4902) (d, 3.75%, 0.4902) T = 6m (uu, 4.5%, 0.2400) (ud, 4%, 0.4806) (dd, 3.5%, 0.2406) T=1y (uuu, 4.75%, 0.1174) (uud, 4.25%, 0.3529) (dud, 3.75%, 0.3538) (ddd, 3.25%, 0.1182) T3 = 1.5y (uuuu,, 0.0573) (uuud,, 0.2301) (udud,, 0.3464) (ddud,, 0.2318) (dddd,, 0.0582) T = 2y