Consider a single-factor affine model.

(a) Use the fact that the short rate r is an affine function of the state variable to show that dr = φ dt −κr dt + 

α + βr dB∗ (18.46)

for constants φ, κ, α, and β.

(b) Assume β > 0 in (18.46). Show that r is a translation of a square-root process—that is, there exist η and Z such that rt = η +Zt , (18.47a)

dZ = φˆ dt − ˆκZdt + ˆσ

√

ZdB∗ (18.47b)

for constants κˆ, θˆ, and σˆ .

(c) The condition φ >ˆ 0 is necessary and sufficient for Zt ≥ 0 for all t in

(18.47b) and hence for the square root to exist. Assuming β > 0, what are the corresponding conditions on the coefficients in (18.46) that guarantee α + βrt ≥ 0 for all t?