Constantinides (1992) develops the so-called SAINTS model of the nominal term structure of interest rates by specifying exogenously the nominal state-price deflator ζ˜.

In a slightly simplified version, his assumption is that

ζ˜

t = ke−gt+(Xt−α)2

, where k, g, and α are constants, and X = (Xt) follows the Ornstein–Uhlenbeck process dXt = −κXt dt + σ dzt, where κ and σ are positive constants with σ 2 < κ and z = (zt) is a standard onedimensional Brownian motion.

(a) Derive the dynamics of the nominal state-price deflator. Express the nominal shortterm interest rate, r˜t, and the nominal market price of risk, λ˜t, in terms of the variable Xt.

(b) Find the dynamics of the nominal short rate.

(c) Find parameter constraints that ensure that the short rate stays positive. Hint: The short rate is a quadratic function of X. Find the minimum value of this function.

(d) What is the distribution of XT given Xt?

(e) Let Y be a normally distributed random variable with mean μ and variance v2. Show that E



e

−γ Y2 

= (1 + 2γ v2)

−1/2 exp!

− γ μ2 1 + 2γ v2

"

.

(f) Use the results of the two previous questions to derive the time t price of a nominal zero-coupon bond with maturity T, that is B˜ T t . It will be an exponential-quadratic function of Xt. What is the yield on this bond?

(g) Find the percentage volatility σ T t of the price of the zero-coupon bond maturing at T.

(h) The instantaneous expected excess rate of return on the zero-coupon bond maturing at T is often called the term premium for maturity T. Explain why the term premium is given by σ T t λ˜t and show that the term premium can be written as 4σ 2α2 (1 − F(T − t))
Xt α − 1 
Xt α − 1 − F(T − t)eκ(T−t)
1 − F(T − t)

, where F(τ ) = 1 σ 2 κ +
(
1 − σ 2 κ
)
e2κτ .
For which values of Xt will the term premium for maturity T be positive/negative?
For a given state Xt, is it possible that the term premium is positive for some maturities and negative for others?