# Question

Suppose there are 1-, 2-, and 3-year zero-coupon bonds, with prices given by P1, P2, and P3. The implied forward interest rate from year 1 to 2 is r0(1, 2) = P1/P2 − 1, and from year 2 to 3 is r0(2, 3) = P2/P3 − 1. Denote the rates as r(1) and r(2).

Suppose that you select the 3-year bond as numeraire.

a. Explain why r(2) is a martingale, but r(1) is not.

b. The price ratio P1/P3 is a martingale. What is the interpretation of this ratio?

c. Suppose that P1/P3 and P2/P3 both follow geometric Brownian motion with zero drift, volatilies σ13 and σ23, and correlation ρ dt. What is the process followed by r(1)?

Suppose that you select the 3-year bond as numeraire.

a. Explain why r(2) is a martingale, but r(1) is not.

b. The price ratio P1/P3 is a martingale. What is the interpretation of this ratio?

c. Suppose that P1/P3 and P2/P3 both follow geometric Brownian motion with zero drift, volatilies σ13 and σ23, and correlation ρ dt. What is the process followed by r(1)?

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