Exercise 5.5 Consider the CIR model for the short-term interest rate r.

(a) Interpret the so-called mean-reverting property.

(b) Show that limt→∞ E{r(t)} = β.

(c) Prove that limt→∞ V ar{r(t)} = βσ2 2α .

(d) Show that r(t) converges in law as t → ∞ and find the limiting distribution.

(e) For any x ∈ R and u ≥ 0, find E eu˜r(T)I {˜r(T ) ≤ x} e

−

T t ˜r(s)ds

˜r(t) = r

.

The last expectation can be useful for approximating the value of a Bermudan call or put option on a bond.