Exercise 24.2.10 Suppose that the effective annual interest rate follows dre re = (t) dt +(t)dW. Prove

Exercise 24.2.10 Suppose that the effective annual interest rate follows dre re

= μ(t) dt +σ(t)dW.

Prove that drc(t)

1−e−rc(t)

=



μ(t)− 1 2



1−e−rc(t) 

σ(t)2



dt +σ(t)dW.

(The continuously compounded rate is approximately lognormally distributed when rc(t) = o(dt) as 1−e−rc(t) ≈ rc(t)+o(dt2) and converges to a normal distribution when rc(t)→∞.)