Exercise 1.Consider the two period deterministic model having three instruments D01, D02, and D012; a zero coupon bond maturing at t1, a zero coupon bond maturing at t2, and a forward starting zero coupon bond from t1to t2. The non zero cash flows for the bonds are C1 (D01)=1, C2 (D02)=1, C1 (D012)=-D012, and C2 (D012)=1. Given prices X0 (D01)=D01, X0 (D02)=D02, and X0 (D012)=0, find the the value of D012in terms of D01and D02that ensures the model is arbitrage free. (Hint: Find the unique price deflators ?0=1, ?1and ?2.)
Exercise 2.Consider the n period model having instruments Rj, 0=j
Exercise 3.Using the same model as in exercise 2, find the quote, Fijk, at time ti of a forward rate agreement over the interval [tj,tk] having day count fraction djk in terms of D(t,u)=Et [?u]/?t, the price at time t of a zero coupon bond maturing at u. The price of the FRA is zero at ti and has cash flows of -1at time tj and 1+Fijkdjk at time tk.
Exercise 4.Using the notation in exercise 3, show that in an arbitrage free model that receiving the cash flows Fj,j,j+1dj,j+1 at time tj+1for 0=j
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Exercise 1. Consider the two period deterministic model having three instruments D01, D02, and D012; a zero coupon bond maturing at t1, a zero coupon bond maturing at t2, and a forward starting zero coupon bond from t1to t2. The non zero cash flows for the bonds are C1 (D01)=1, C2 (D02)=1, C1 (D012)=-D012, and C2 (D012)=1. Given prices X0 (D01)=D01, X0 (D02)=D02, and X0 (D012)=0, find the the value of D012in terms of D01and D02that ensures the model is arbitrage free. (Hint: Find the unique price deflators ?0=1, ?1and ?2.) Exercise 2. Consider the n period model having instruments Rj, 0=j