This section provides a brief recapitulation of the key results relating to the prices of bonds in the cross section and time series implied by GDTSMs. These results are sub- sequently used in Section 2.3 to construct the model central to this chapter: a GDTSM where risk premia to bonds depend on a linear combination of forward rates.
2.2.1 Affine Bond Pricing
Risk is generated in the economy by n Q-Brownian motions WtQ ≡ (Wt,Q1, . . . , Wt,nQ)T.
The n-dimensional state vector is denotedXt≡(Xt,1, . . . , Xt,n)T and is driven byWtQ.
The short rate of interest,rt, is an affine function of the state vector:
rt=δ0,X+δXTXt. (2.1)
Here δ0,X is a scalar andδX is ann-vector.
Throughout this chapter, the X subscript to parameters denotes that they enter the model in the form given in (2.1) (and below in (2.2) and (2.21)) when the state vector is Xt. This notation is useful later when I make invariant transformations to different state
vectors. For example, ifYtis used as the state vector then I write rt=δ0,Y +δTYYt and,
of course,δ0,Y and δY are related to δ0,X and δX in a way that depends on the relation
between Yt and Xt. I return to this topic in Subsection 2.3.4. A similar approach to
notation is taken with functions and matrices that depend on parameters. GDTSMs impose that the Q evolution of the state vector is described by
dXt=KXQ
θXQ−Xt
dt+ ΣXdWtQ, (2.2)
whereKXQ and ΣX aren×n matrices andθQX is ann-vector.
Duffie and Kan (1996) show that, in this framework, the time t price of the time T maturity zero-coupon bond is given by
Pt(u) =PX(Xt, u)≡eAX(u)−BX(u) TX
t. (2.3)
Here,u≡T−t,AX(u) is a scalar function andBX(u) is ann-valued function. These are
the solutions to the system of Riccati ordinary differential equations (ODEs), provided in Subsection 2.7.1 in the appendix to this chapter. Given (2.3), the bond yield defined by yt(u)≡ − 1 ulnPt(u) (2.4) is given by yt(u) = 1 u(−AX(u) +BX(u) TX t). (2.5)
Finally, letting ft(u) denote the time tinstantaneous forward rate that prevails at time
t+u, it is a standard no-arbitrage result that
ft(u) =− ∂ ∂ulogPX(Xt, u) =− ∂AX(u) ∂u + ∂BX(u)T ∂u Xt. (2.6)
(2.3) provides the time t conditional prices of the cross section of bond maturities. However, we are yet to make a statement about conditional expected returns to bonds in the time series. For this we must specify the market prices of risk.
2.2.2 Expected Returns to Bonds The SDF, denoted by πt, follows the process
dπt
πt
=−rtdt−ΛTtdWtP,
whereWtP ≡(Wt,P1, . . . , Wt,nP)T is ann-vector of P-Brownian motions and Λtis a column
n-vector specifying market prices of risk. Applying Girsanov’s theorem to (2.2) gives the dynamics of the state vector under the P measure as
dXt=KXQ
θQX −Xt
dt+ ΣXΛtdt+ ΣXdWtP. (2.7)
The bond pricing function solves dPX(Xt, u)
PX(Xt, u)
= (rt+et(u))dt+vt(u)TdWtP, (2.8)
whereet(u) is a scalar that denotes the time tinstantaneous risk premium to holding a
bond that matures at time t+u. vt(u) is an n-vector that determines the volatility of
this bond. An application of It¯o’s lemma to (2.3) and using (2.7) reveals that these two quantities are given by
et(u) ≡ −BX(u)TΣXΛt and (2.9)
vt(u)T ≡ −BX(u)TΣX.
The GDTSM is not yet fully specified; (2.9) shows that we must choose a Λtto determine
risk premia. The choice of form for Λtis delicate. It must be sufficiently flexible to ensure
that the model is able to capture the key feature of the data that we wish to expose (that expected excess returns to bonds are linear in forward rates) but also restricted enough so that the model remains econometrically tractable. An essentially affine specification, given in the next subsection, is suitable for this task.
2.2.3 Essentially Affine Market Prices of Risk
When the state vector follows a Gaussian affine process, the essentially affine specifica- tion for the market price of risk vector is
Λt≡λ0,X +λ1,XXt, (2.10)
whereλ0,X andλ1,X are anndimensional column vector andn×nmatrix of parameters
depend on the parameters that governXt. This will become clear below.
The equations for the short rate, the Q-dynamics and the market prices of risk in terms of the state vector, given in (2.1), (2.2) and (2.10) respectively, completely specify the GDTSM in the sense that the statistical properties of bond prices in both the time series and in the cross section of bond maturities are pinned down. The aim is to make risk premia depend on a linear combination of forward rates. By inspecting (2.10) and the equation for forward rates in (2.6) it is immediately apparent that, with appropriate choices of λ0,X andλ1,X, it is possible to ensure this aim. The next step is to formalise
this observation.