Exercise 5.2 Suppose that r is the short-term interest rate (expressed in percentage on a yearly basis) and that it is modeled by a Ornstein-Uhlenbeck process with parameters α = 0.8, β = 3.25, σ = 0.9.

(a) Find the stationary distribution of the process.

(b) Define X(t) = 1 36000 r(t/360), t ≥ 0. Interpret X and find its distribution.

(c) Suppose that the market price of risk is q(t, r) = 0.5+0.2r. What is the distribution of ˜r under the equivalent martingale measure? Also find the limiting behavior of the annual yield R(t, T ) on a zero-coupon bond, as T →∞.

(d) Suppose that under the equivalent martingale measure, ˜r is an Ornstein-

Uhlenbeck process with parameters a = 0.6 and b = 1.75. Find the associated market price of risk.