Assume there is an SDF process M with dM M = r dt S(Xt) +S(Xt) 1Xt

Assume there is an SDF process M with dM M = −r dt −

S(Xt)λ +S(Xt)

−1ΛXt



dB+

dε

ε , (18.49)

where B is a vector of independent Brownian motions under the physical probability, ε is a local martingale uncorrelated with B, S(X) is a diagonal matrix the squared elements of which are affine functions of X, S(X)−1 denotes the inverse of S(X), λ is a constant vector, and Λ is a constant matrix. Assume MR is a martingale, so there is a risk-neutral probability corresponding to M. (Warning:

This assumption is not valid in general. See the end-of-chapter notes.)

(a) Assume r = δ0 +δ

X and dX = (φ + KX)dt + σS(X)dB∗, where B∗ is a vector of independent Brownian motions under the risk-neutral probability, δ0 is a constant, δ and φ are constant vectors, and K and σ

are constant matrices. Show that dX = (φˆ + KXˆ )dt +σS(X)dB for a constant vector φˆ and constant matrix Kˆ .

(b) Using the fact that bond prices are exponential-affine, calculate

−

dP P

dM M



to show that the risk premium of a discount bond is affine in X.

(c) Consider the Vasicek model with the price of risk specification (18.49).
Show that, in contrast to the completely affine model considered in Exercise 18.5, the risk premium of a discount bond can depend on the short rate.