Assume dr dY = ( r) ( Y) dt + 0 0 1 0 0 Y dB

Assume

dr dY



=

 κ(θ −r)

γ (φ −Y)



dt +



σ 0 0 η

 1 0 0 √Y

dB∗

1 dB∗

2



(18.50)

for constants κ, θ, σ, γ , φ, and η, where the B∗

i are independent Brownian motions under a risk-neutral probability.

(a) Given the completely affine price of risk specification (18.12b), where X = (r Y) and S(X) =

 1 0 0 √Yt



, show that dr = κ(θˆ −r)dt + σ dB1 for some constant θˆ, where B1 is a Brownian motion under the physical probability. Show that the risk premium of a discount bond depends only on its time to maturity and does not depend on r or Y.

(b) Consider the price of risk specification (18.49), replacing S(X)−1 by

 1 0 0 0

.

Show that the risk premium of a discount bond can depend on r and Y.

Note: This is an essentially affine model. It is an example given by Duffee (2002).