Assume the short rate is an Ornstein-Uhlenbeck process under the physical probability, that is, dr = ˆκ(θˆ − r)dt +σ dB, for constants κˆ, θˆ, and σ, where B is a Brownian motion under the physical probability. Assume there is an SDF process M with dM M = −r dt −λdB+

dε

ε , where λ is a constant and ε is a local martingale uncorrelated with B.

(a) Show that the short rate is an Ornstein-Uhlenbeck process under the risk-neutral probability corresponding to M (that is, the Vasicek model holds).

(b) Calculate the risk premium of a discount bond.