for m>0,and K is either upper or lower triangular for m=0, = 6x1 ∑=I, 0N-m)×1 0m×1 B= Imxm B股N-m 1(-m)x1 O(N-m)xm O(N-m)x(N-m) with the following parametric restrictions imposed: d≥0，m+1≤i≤N, K,0三∑K日，>0,1≤i≤m,K≤0,1≤j≤m,j≠i, j=1 Θ：≥0,1≤i≤m,Bi≥0,1≤i≤m,m+1≤j≤N. Then the sub-family Am(N)of affine term structure models is obtained by inclusion of all models that are invariant transformations of this canonical model or nested special cases of such transformed models.For the case of N risk factors,this gives N+1 non-nested sub-families of admissible affine mod- els.?Members of the families Am(N)include,among others,Vasicek [1977, Langetieg [1980],Cox,Ingersoll,and Ross [1985],Longstaff and Schwartz (1992],Chen and Scott [1993],Pearson and Sun [1994],Duffie and Singleton 1997,Balduzzi,Das,Foresi,and Sundaram 1996,Balduzzi,Das,and Foresi 1998,Duffie and Liu [2001],and Collin-Dufresne and Goldstein [2001al. Quadratic Gaussian Models If G(Yi,t;T)is quadratic in Yi,i.e., G(Yi,t;T)=A(T-t)+B(T-t)Yi+YC(T-t)Yi, then it must be the case that rt =a+B'Y:+YYi,where a =A'(0), B=B'(0),and y=C(0).Without loss of generality,we can assume that C(r)is symmetric.Thus y must also be symmetric and (14)becomes A+YB+Y:CY:=(a+YB+YyYi)+u(Yi,t)'[B(T)+2CY] (18) +Trace[a(,yCa,t训-号B+20Ya,a,y[B+20Y- 7Although the classification scheme was originally used by Dai and Singleton [2000] to characterize the state dynamics under the actual measure,it is equally applicable to the state dynamics under the Q-measure.Note that not all admissible affine DTSMs are subsumed by this classification scheme (i.e.,not all admissible models are invariant transformations of a canonical model). 15
for m > 0, and K is either upper or lower triangular for m = 0, Θ = ΘB m×1 0(N−m)×1 , Σ = I, α = 0m×1 1(N−m)×1 , B = Im×m BBD m×(N−m) 0(N−m)×m 0(N−m)×(N−m) , with the following parametric restrictions imposed: δi ≥ 0, m + 1 ≤ i ≤ N, KiΘ ≡ Xm j=1 KijΘj > 0, 1 ≤ i ≤ m, Kij ≤ 0, 1 ≤ j ≤ m, j 6= i, Θi ≥ 0, 1 ≤ i ≤ m, Bij ≥ 0, 1 ≤ i ≤ m, m + 1 ≤ j ≤ N. Then the sub-family Am(N) of affine term structure models is obtained by inclusion of all models that are invariant transformations of this canonical model or nested special cases of such transformed models. For the case of N risk factors, this gives N +1 non-nested sub-families of admissible affine models.7 Members of the families Am(N) include, among others, Vasicek [1977], Langetieg [1980], Cox, Ingersoll, and Ross [1985], Longstaff and Schwartz [1992], Chen and Scott [1993], Pearson and Sun [1994], Duffie and Singleton [1997], Balduzzi, Das, Foresi, and Sundaram [1996], Balduzzi, Das, and Foresi [1998], Duffie and Liu [2001], and Collin-Dufresne and Goldstein [2001a]. Quadratic Gaussian Models If G(Yt, t; T) is quadratic in Yt, i.e., G(Yt, t; T) = A(T − t) + B(T − t) 0 Yt + Y 0 t C(T − t)Yt, then it must be the case that rt = α + β0 Yt + Y 0 t γYt, where α = A0 (0), β = B0 (0), and γ = C0 (0). Without loss of generality, we can assume that C(τ ) is symmetric. Thus γ must also be symmetric and (14) becomes A˙ + Y 0 t B˙ + Y 0 t CY˙ t = (α + Y 0 t β + Y 0 t γYt) + µ(Yt, t) 0 [B(τ )+2CYt] (18) +Trace [σ(Yt, t) 0 Cσ(Yt, t)] − 1 2 [B + 2CYt] 0 σ(Yt, t)σ(Yt, t) 0 [B + 2CYt] . 7Although the classification scheme was originally used by Dai and Singleton [2000] to characterize the state dynamics under the actual measure, it is equally applicable to the state dynamics under the Q-measure. Note that not all admissible affine DTSMs are subsumed by this classification scheme (i.e., not all admissible models are invariant transformations of a canonical model). 15
In order for (18)to hold for any Yi,it is sufficient that 1.u(Yi,t)=a+byt,where the N x 1 vector a and N x N matrix b are constants; 2.o(Y,t)=o,where o is a N x N constant matrix; 3.A(T),B(T),and C(r)satisfy the following ODEs: A=a+dB-Bad'B+Tcelo'Col, (19) B =B+B-2C'aa'B+2C'a, (20) C =+[6C+C'b]-2C'ao'C. (21)】 For suitable choices of (a,B,y;a,b;o),the Ricatti equations (19)-(21)admit a unique solution (A(T),B(T),C(T))under the initial conditions A(0)=0, B(0)=ONx1,and C(0)=ONxN.It is easy to verify that the solution has the property that A'(0)=a,B(0)=B,and C(0)=y. The canonical representation of the QG models is simpler than in the case of affine models,because shocks to Y are homoskedastic.To derive their canonical model,Ahn,Dittmar,and Gallant [2002]normalize the diagonal elements of y to unity,set B=0,have K (the mean reversion matrix for Y) being lower triangular,and have diagonal.They show that the QG models in Longstaff [1989],Constantinides [1992],and Lu [1999]are restricted special cases of their most flexible canonical model. 4 DTSMs with Jump Diffusions Suppose that rt =r(Yi,t)is a function of a jump-diffusion process Y with risk-neutral dynamics dY=μ(Y,t)dt+o(Y,t)dW+△YdZ, (22) where Zt is a Poisson counter with risk-neutral intensity At,and the jump size AYi is drawn from a risk-neutral distribution v(r)=v(x;Yi,t). No arbitrage implies that the zero-coupon bond price D(t,T)satisfies [品+A]p)-r心pen=0 (23) 16
In order for (18) to hold for any Yt, it is sufficient that 1. µ(Yt, t) = a + bYt, where the N × 1 vector a and N × N matrix b are constants; 2. σ(Yt, t) = σ, where σ is a N × N constant matrix; 3. A(τ ), B(τ ), and C(τ ) satisfy the following ODEs: A˙ = α + a0 B − 1 2 B0 σσ0 B + Trace [σ0 Cσ] , (19) B˙ = β + b 0 B − 2C0 σσ0 B + 2C0 a, (20) C˙ = γ + [b 0 C + C0 b] − 2C0 σσ0 C. (21) For suitable choices of (α, β, γ; a, b; σ), the Ricatti equations (19)–(21) admit a unique solution (A(τ ), B(τ ), C(τ )) under the initial conditions A(0) = 0, B(0) = 0N×1, and C(0) = 0N×N . It is easy to verify that the solution has the property that A0 (0) = α, B0 (0) = β, and C0 (0) = γ. The canonical representation of the QG models is simpler than in the case of affine models, because shocks to Y are homoskedastic. To derive their canonical model, Ahn, Dittmar, and Gallant [2002] normalize the diagonal elements of γ to unity, set β = 0, have K (the mean reversion matrix for Y ) being lower triangular, and have Σ diagonal. They show that the QG models in Longstaff [1989], Constantinides [1992], and Lu [1999] are restricted special cases of their most flexible canonical model. 4 DTSMs with Jump Diffusions Suppose that rt = r(Yt, t) is a function of a jump-diffusion process Y with risk-neutral dynamics dYt = µ(Yt, t) dt + σ(Yt, t) dWQ t + ∆Yt dZt, (22) where Zt is a Poisson counter with risk-neutral intensity λt, and the jump size ∆Yt is drawn from a risk-neutral distribution νt(x) ≡ ν(x; Yt, t). No arbitrage implies that the zero-coupon bond price D(t, T) satisfies ∂ ∂t + A D(t, T) − r(Y, t) D(t, T)=0, (23) 16