KO>o2/2 (the so called "Feller condition").Further,the distribution of rt conditional on rt-1 is known to be a generalized Bessel process (Eom[1998). The solution to (10)has r >0 for all n E[0,1),including n =0,so long as the Feller condition is satisfied.However,we are not aware of closed-form solutions for D(t,T)in this model outside of the case of 7=.5. Square-root model (Cox,Ingersoll,and Ross [1985])For this special case of(10)with n=.5 the discount function is given by D(t,T)=A(T)e-B(T)",Tt,T=(T-t), where,with y=VK2 +202, 72x0/a2 A(r)= 2ye(x+7)7/2 2(er-1) (k+y)(er-1)+2yJ ,B(T)=K+)(er-1)+2列 The Green's function for the CIR model is given by G(rt,t;r,T)=D(t,T)2cx2(2cr;v,o), where x2(,v,o)is the non-central chi-square density with the degrees of freedom v and the parameter of noncentrality o,defined by c= K+y-(k-y）e-Y(T-) 4K0 o2[1-e-T--,V= 02, 8y2e-7T-t)rt o2(1-e-T-)[2y+(k-y)(1-e-T-0)】 Log-normal model (Black,Derman,and Toy 1990).As n-1 in (10),the process for r converges to that of a log-normal process. Though widely used in the financial industry (often in this one-factor formulation with time-dependent parameters),we are not aware of closed-form solutions for discount curves in this model. .Three-halves model (Cox,Ingersoll,and Ross [1980).r follows the process dr(t)k(0-r(t))r(t)dt+or(t)'5dWR(t). (11) 10
κθ > σ2/2 (the so called “Feller condition”). Further, the distribution of rt conditional on rt−1 is known to be a generalized Bessel process (Eom [1998]). The solution to (10) has r > 0 for all η ∈ [0, 1), including η = 0, so long as the Feller condition is satisfied. However, we are not aware of closed-form solutions for D(t, T) in this model outside of the case of η = .5. • Square-root model (Cox, Ingersoll, and Ross [1985]) For this special case of (10) with η = .5 the discount function is given by D(t, T) = A(τ ) e−B(τ) rt , T ≥ t, τ ≡ (T − t), where, with γ ≡ √κ2 + 2σ2, A(τ ) = 2γe(κ+γ)τ /2 (κ + γ) (eγτ − 1) + 2γ 2κθ/σ2 , B(τ ) = 2 (eγτ − 1) (κ + γ) (eγτ − 1) + 2γ . The Green’s function for the CIR model is given by G(rt, t; r, T) = D(t, T) 2c χ2 (2c r; ν, φ), where χ2(·, ν, φ) is the non-central chi-square density with the degrees of freedom ν and the parameter of noncentrality φ, defined by c = κ + γ − (κ − γ) e−γ (T −t) σ2 [1 − e−γ (T −t)] , ν = 4κθ σ2 , φ = 8 γ2 e−γ(T −t) rt σ2 (1 − e−γ(T −t)) [2γ + (κ − γ) (1 − e−γ(T −t))]. • Log-normal model (Black, Derman, and Toy [1990]). As η → 1 in (10), the process for r converges to that of a log-normal process. Though widely used in the financial industry (often in this one-factor formulation with time-dependent parameters), we are not aware of closed-form solutions for discount curves in this model. • Three-halves model (Cox, Ingersoll, and Ross [1980]). r follows the process dr(t) = κ(θ − r(t))r(t) dt + σr(t) 1.5 dWQ(t). (11) 10
This process is stationary and zero is entrance if K and o are greater than 0.D(t,T)is given by (see Ahn and Gao 1999]) Dk,=IT32M,日，-e)z0, T() where I()is the "gamma"function,M()is a confluent hypergeometric function (computed through a series expansion),() .2h and [VGo-aP+2g-(52-小，月-2[-a+1+. 1 y=1 Gaussian Model (Vasicek [1977).r follows the diffusion with linear drift (-r(t))dt and constant diffusion coefficient o.In this case,the discount function is given by D(t,T)= eA(T-t-Bt,T),T≥t, where,A(r) 2 6-aol+8G,0)-1 The Green's function for this model is given by (see Jamshidian [1989): r-6①2 2(t,1) G（rt,t:t,T)=D(t,T) V27v(t,T where f(化，T)=eT-r+(1-e-*T-)0-茶(1-eT-)2 is the instantaneous forward rate and (t,T)(1-e()is the con- ditional volatility of the spot short rate. An alternative means of constructing tractable one-factor models is to maintain a simpler representation of the state Y and to let rt=g(Yi,t), for some nonlinear function g.For example,the one-factor Quadratic Gaus- sian (QG)model (see Beaglehole and Tenney [1991])is obtained by letting g(Yi,t)=a+BY+Y2 and Yi following a Gaussian diffusion.The discount function in this model is given by D化，T)=eA()-B(r)-C)竖. 11
This process is stationary and zero is entrance if κ and σ are greater than 0. D(t, T) is given by (see Ahn and Gao [1999]) D(t, T) = Γ(β − γ) Γ(β) M(γ, β, −x(t)) x(t) γ , where Γ(·) is the “gamma” function, M(·) is a confluent hypergeometric function (computed through a series expansion), x(t) = −2b σ2(eb(T−t)−1)r(t) , and γ = 1 σ2 hp(.5σ2 − a)2 + 2σ2 − (.5σ2 − a) i , β = 2 σ2 −a + (1 + γ)σ2 . • Gaussian Model (Vasicek [1977]). r follows the diffusion with linear drift κ(θ −r(t))dt and constant diffusion coefficient σ. In this case, the discount function is given by D(t, T) = e−A(T −t)−B(t,T) rt , T ≥ t, where, A(τ ) = δ − σ2 2κ2 [τ − B(τ )] + σ2 4κ B(τ ) 2 , B(τ ) = 1 − κτ κ . The Green’s function for this model is given by (see Jamshidian [1989]): G(rt, t;t, T) = D(t, T) e − (r−f(t,T ))2 2v(t,T ) p2πv(t, T) , where f(t, T) = e−κ(T −t) r + (1 − e−κ(T −t) )θ − σ2 2κ2 1 − e−κ(T −t) 2 is the instantaneous forward rate and v(t, T) = σ2 2κ 1 − e−2κ(T −t) is the conditional volatility of the spot short rate. An alternative means of constructing tractable one-factor models is to maintain a simpler representation of the state Y and to let rt = g(Yt, t), for some nonlinear function g. For example, the one-factor Quadratic Gaussian (QG) model (see Beaglehole and Tenney [1991]) is obtained by letting g(Yt, t) = α + βYt + γY 2 t and Yt following a Gaussian diffusion. The discount function in this model is given by D(t, T) = e−A(τ)−B(τ) Yt−C(τ) Y 2 t , 11
where B(r)=C(r) 出+引 x(e2rT-1) and C(T)=(re-1)+2r,and I=VK2 +2o2.(A(T)is also known as a relatively complicated function of the underlying parameters.) Although the QG model is driven by one risk factor,it can be viewed equivalently as a degenerate two-factor model(with two state variables driven by the same Brownian motion).To see this,note that,from Ito's lemma, dr:=[(2ka+n08+Yo2)+2k0yY:-2kYr:]dt +(B+2yYi)odw. Using the fact that r is affine in Y and Y2,we see that the instantaneous conditional means and covariance of r:and Yi are affine in (rt,Yi). One can build up multi-factor DTSMs from these one factor examples by simply assuming that the short rate is the sum of N independent risk factors,with eachY following one of the preceding one-factor models for which a solution for zero prices is known (see Cox,Ingersoll, and Ross [1985],Chen and Scott [1993],Pearson and Sun [1994],Duffie and Singleton [1997],and Jagannathan,Kaplan,and Sun 2001]for multi-factor versions of the square-root model).In this case,the discount function is given by D(t,T)=IIND(t,T),where D(t,T)i is the discount function in a single-factor model with the short rate given by r=Y.This approach leads,however,to rather restrictive formulations of multi-factor models,par- ticularly with regard to the assumption of zero correlations among the risk factors.We turn next to multi-factor models with correlated risk factors. 3.2 Multi-factor DTSMs A quite general formulation of multi-factor models has rt=g(Yi,t),where Y=(Y:1<i<N)'and these risk factors may be mutually correlated. Specifications of the function g(,t)and the dynamics of Yi are constrained only by the so-called admissibility conditions which stipulate that (i)Yi must be a well-defined stochastic process;and (ii)the conditional expectation in (3)exists and is finite (equivalently,the PDE(4)has a well-defined and finite solution).In practice,however,model specifications are often influenced by their computational tractability in pricing FIS. Two classes of diffusion-based multi-factor models have been the focal points of much of the literature on pricing default-free bonds:affine models (see,e.g.,Duffie and Kan 1996 and Dai and Singleton 2000)and quadratic 12
where B(τ ) = C(τ ) 2κ(θ+ β 2γ ) Γ eΓτ−1 eΓτ+1 + β γ , and C(τ ) = γ(e2Γτ −1) (Γ+κ)(e2Γτ −1)+2Γ, and Γ = pκ2 + 2γσ2. (A(τ ) is also known as a relatively complicated function of the underlying parameters.) Although the QG model is driven by one risk factor, it can be viewed equivalently as a degenerate two-factor model (with two state variables driven by the same Brownian motion). To see this, note that, from Ito’s lemma, drt = (2κα + κθβ + γσ2 )+2κθγYt − 2κγrt dt + (β + 2γYt)σdWQ t . Using the fact that r is affine in Y and Y 2, we see that the instantaneous conditional means and covariance of rt and Yt are affine in (rt, Yt). One can build up multi-factor DTSMs from these one factor examples by simply assuming that the short rate is the sum of N independent risk factors, rt = PN i=1 Y i t , with each Y following one of the preceding one-factor models for which a solution for zero prices is known (see Cox, Ingersoll, and Ross [1985], Chen and Scott [1993], Pearson and Sun [1994], Duffie and Singleton [1997], and Jagannathan, Kaplan, and Sun [2001] for multi-factor versions of the square-root model). In this case, the discount function is given by D(t, T) = QN i=1 D(t, T)i , where D(t, T)i is the discount function in a single-factor model with the short rate given by rt = Y i t . This approach leads, however, to rather restrictive formulations of multi-factor models, particularly with regard to the assumption of zero correlations among the risk factors. We turn next to multi-factor models with correlated risk factors. 3.2 Multi-factor DTSMs A quite general formulation of multi-factor models has rt = g(Yt, t), where Yt = (Y i t : 1 ≤ i ≤ N)0 and these risk factors may be mutually correlated. Specifications of the function g(·, t) and the dynamics of Yt are constrained only by the so-called admissibility conditions which stipulate that (i) Yt must be a well-defined stochastic process; and (ii) the conditional expectation in (3) exists and is finite (equivalently, the PDE (4) has a well-defined and finite solution). In practice, however, model specifications are often influenced by their computational tractability in pricing FIS. Two classes of diffusion-based multi-factor models have been the focal points of much of the literature on pricing default-free bonds: affine models (see, e.g., Duffie and Kan [1996] and Dai and Singleton [2000]) and quadratic 12
Gaussian models (see,e.g.,Beaglehole and Tenney [1991],Ahn,Dittmar, and Gallant [2002]and Leippold and Wu [2001]).These models have the common feature that the discount function has the exponential form: D(t,T)=eGm4TD,T≥t, (12) where G(Yr,T;T)=0 for all Yr,and limrt Gr(Yi,t;T)exists and is finite for all Y:and t≤T. Heuristically,the affine and quadratic Gaussian models are "derived" from the requirement that G(Y,;)be,respectively,an affine and quadratic function of the state vector Y.Naturally,such a requirement restricts the functional form of g(Yi,t)and the Q-dynamics of Yi.By definition,rt= limfrom which it follows that r=g(Yi,t)=lim G(Yi,t;T) T→tT-t (13) Thus,rt must be affine in Y in affine models and quadratic in Y in quadratic Gaussian models.Furthermore,substituting (12)into (4)yields G+(Y,t)'Gr+Trace [o(Y,t)a(Y,ty(Gyy-GyGy+(Y,t)=0, (14) which may be viewed as a restriction on the risk-neutral drift u(Y,t)and diffusion o(Y,t)for the state vector Y. Affine Models Affine term structure models are characterized by the requirement that G(Yi,t;T)be affine in Y;i.e., G(Yi,t;T)=A(T-t)+B(T-t)'Yi. In this case,r:must also be affine in Yi:r:=a+B'Yi,where a=A'(0)and B=B(0).Furthermore,equation (14)reduces to A()+B(r)Y:=u(Y,t)'B(r)-5[B(r)'o(Vi,t)a(Yi,t)'B(r)+(a+B'Y:), (15) 13
Gaussian models (see, e.g., Beaglehole and Tenney [1991], Ahn, Dittmar, and Gallant [2002] and Leippold and Wu [2001]). These models have the common feature that the discount function has the exponential form: D(t, T) = e−G(Yt,t; T) , T ≥ t, (12) where G(YT , T; T) = 0 for all YT , and limT→t GT (Yt, t; T) exists and is finite for all Yt and t ≤ T. Heuristically, the affine and quadratic Gaussian models are “derived” from the requirement that G(Y, ·; ·) be, respectively, an affine and quadratic function of the state vector Y . Naturally, such a requirement restricts the functional form of g(Yt, t) and the Q-dynamics of Yt. By definition, rt = − limT→t log Dt,T T −t from which it follows that rt = g(Yt, t) = limT→t G(Yt, t; T) T − t . (13) Thus, rt must be affine in Y in affine models and quadratic in Y in quadratic Gaussian models. Furthermore, substituting (12) into (4) yields Gt + µ(Y, t) 0 GY + 1 2 Trace [σ(Y, t)σ(Y, t) 0 (GY Y 0 − GY GY 0)] + g(Y, t)=0, (14) which may be viewed as a restriction on the risk-neutral drift µ(Y, t) and diffusion σ(Y, t) for the state vector Y . Affine Models Affine term structure models are characterized by the requirement that G(Yt, t; T) be affine in Yt; i.e., G(Yt, t; T) = A(T − t) + B(T − t) 0 Yt. In this case, rt must also be affine in Yt: rt = α + β0 Yt, where α = A0 (0) and β = B0 (0). Furthermore, equation (14) reduces to A˙(τ ) + B˙(τ )Yt = µ(Yt, t) 0 B(τ ) − 1 2 [B(τ ) 0 σ(Yt, t)σ(Yt, t) 0 B(τ )] + (α + β0 Yt), (15) 13
where r=T-t,A(T)=4()=-4(T-,and B(T)is similarly defined. Or Duffie and Kan [1996 show that,in order for (15)to hold for any Yi,it is sufficient that6 1.u(Yi,t)be affine in Yi:u(Yi,t)=a+by:,where a be a N x 1 vector and b be a N x N matrix. 2.a(Y,t)a(Y,t)be affine in :(t)a(,t)=ho+,where ho and h,j=1,2,...,N,are N x N matrices. 3.A(T)and B(T)satisfy the following ODEs: Aa+d'B(T)-jB(T)'hoB(T), (16) B=B+6B(r)-2, (17) where vi(T)=B(T)'hB(T). For suitable choices of (a,B;a,b;ho,h:1 <j<N),the Ricatti equations (16)-(17)admit a unique solution (A(T),B(T))under the initial conditions A(0)=0 and B(0)=ONx1.It is easy to verify that the solution has the property A'(0)=a and B'(0)=B. Dai and Singleton 2000 examine multi-factor affine models with the following structure: Tt 80+6Yi, dYi K (0-Y:)dt+vS:dw, where St is a diagonal matrix with [Stli=+Bi.Letting B be the Nx N matrix with ith column given by B,they construct admissible affine models- models that give unique,well-defined solutions for D(t,T)-by restricting the parameter vector(o,d,K,Θ，∑，a,B).Specifically,for given m=rank(B), they introduce the canonical model with the structure BB mxm 0mx(N-m） DB (N-m)×m K@-m)x(N-m)] 6A more general mathematical characterization of affine models is presented in Duffie, Filipovic,and Schachermayer [2001].See Gourieroux,Monfort,and Polimenis [2002]for a formal development of multi-factor affine models in discrete time. 14
where τ ≡ T − t, A˙(τ ) = ∂A(τ) ∂τ = −∂A(T −t) ∂t , and B˙(τ ) is similarly defined. Duffie and Kan [1996] show that, in order for (15) to hold for any Yt, it is sufficient that6 1. µ(Yt, t) be affine in Yt: µ(Yt, t) = a + bYt, where a be a N × 1 vector and b be a N × N matrix. 2. σ(Yt, t)σ(Yt, t)0 be affine in Yt: σ(Yt, t)σ(Yt, t)0 = h0+PN j=1 hj 1Y j t , where h0 and hj 1, j = 1, 2,... ,N, are N × N matrices. 3. A(τ ) and B(τ ) satisfy the following ODEs: A˙ = α + a0 B(τ ) − 1 2 B(τ ) 0 h0B(τ ), (16) B˙ = β + b0 B(τ ) − 1 2 v(τ ), (17) where vj (τ ) ≡ B(τ )0 hj 1B(τ ). For suitable choices of (α, β; a, b; h0, hj 1 : 1 ≤ j ≤ N), the Ricatti equations (16)–(17) admit a unique solution (A(τ ), B(τ )) under the initial conditions A(0) = 0 and B(0) = 0N×1. It is easy to verify that the solution has the property A0 (0) = α and B0 (0) = β. Dai and Singleton [2000] examine multi-factor affine models with the following structure: rt = δ0 + δ0 Yt, dYt = K (Θ − Yt) dt + ΣpSt dWQ t , where St is a diagonal matrix with [St]ii = αi+Y 0 t βi. Letting B be the N ×N matrix with i th column given by βi, they construct admissible affine models– models that give unique, well-defined solutions for D(t, T)– by restricting the parameter vector (δ0, δ, K, Θ, Σ, α, B). Specifically, for given m = rank(B), they introduce the canonical model with the structure K = KBB m×m 0m×(N−m) KDB (N−m)×m KDD (N−m)×(N−m) , 6A more general mathematical characterization of affine models is presented in Duffie, Filipovic, and Schachermayer [2001]. See Gourieroux, Monfort, and Polimenis [2002] for a formal development of multi-factor affine models in discrete time. 14