Exercise 5.6 Suppose that the short-term rate r (in percentage on a yearly basis) is modeled by a Ornstein-Uhlenbeck process with parameters α = 0.5, β = 2 and σ = 0.9. Suppose also that the market price of risk is q1 = −0.012 and q2 = 0.01

(a) What is the distribution of the process ˜r under the equivalent martingale measure?

(b) What is the long term annual yield of a zero-coupon bond?

(c) Today, the annual yield of a 3-month zero-coupon bond is 4.5%. What is the implied spot rate in percentage?

(d) Give the actual value V0 of a 6-month zero-coupon bond with face value of $100000.

(e) Give the law of r in one month.

(f) Find the value of r such that P r  1 12  ≥ r = 0.01.
(g) Let V be the value in 1 month of a zero-coupon bond expiring in 6 months from now with value $100000. Find the value v such that P(V ≤
v) = 0.01.
(g) Find the 1-month VaR of order 99% for the loss X = V0 − e−r/12V on the investment described in (d). Assume a risk-free rate r of 2%.