Consider the Vasicek model with time-dependent parameters: drt = (t) (t) rt dt + (t)dB t , (18.51) where B is

Consider the Vasicek model with time-dependent parameters:

drt = κ(t)



θ (t) −rt



dt + σ (t)dB∗

t , (18.51)

where B∗ is a Brownian motion under a risk-neutral probability. Define rˆt = exp

−

t 0

κ(s)ds



r0 +

t 0

exp

−

t u

κ(s)ds



σ (u)dB∗

u (18.52a)

g(t) =

t 0

exp

−

t u

κ(s)ds



κ(u)θ (u)du. (18.52b)

(a) Show that rˆ defined in (18.52a) satisfies drˆt = −κ(t)rˆt dt +σ (t)dB∗

t .

(b) Define rt = ˆrt +g(t). Show that r satisfies (18.51).

(c) Given any functions κ(·) and σ (·), explain how to choose θ (·) to fit the current yield curve.