CONSISTENCY PROBLEMS FOR JUMP-DIFFUSION MODELS ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR 5 Abstract. Inthispaperconsistencyproblemsformulti-factorjump-diffusion 0 models, where the jump parts follow multivariate point processes are exam- 0 2 ined. First the gap between jump-diffusion models and generalized Heath- Jarrow-Morton(HJM)modelsisbridged. Byapplyingthedriftconditionfor n a generalized arbitrage-free HJM model, the consistency condition for jump- a J diffusion models is derived. Then we consider a case in which the forward 3 ratecurvehas aseparablestructure, andobtainaspecific versionofthegen- 2 eral consistency condition. In particular, a necessary and sufficient condition for ajump-diffusionmodel to be affine isprovided. Finallythe Nelson-Siegel ] T type of forward curve structures is discussed. It is demonstrated that under I regularitycondition, thereexistsnojump-diffusionmodelconsistentwiththe s. Nelson-Siegelcurves. c [ 1 v 5 5 0 1 0 5 0 / s c : v i X r a WeareindebtedtoErhanÇinlarandDamirFilipovićforhelpfuldiscussions. 1 2 ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR 1. The Arbitrage-free Condition for Generalized HJM Models Thepurposeofthispaperistostudyconsistencyproblemsformulti-factorjump- diffusiontermstructuremodelsofinterestrates. Theconceptofconsistencyinthis context was first introduced and studied in [4]. Previous works ([13], [14], [15]) in this area have focused on diffusion models without considering jumps. Because jump-diffusion models usually provide a better characterizationof the randomness in financial markets than do diffusion models (see [1], [19]), there has been an upsurge in the modeling of interest rate dynamics with jumps (e.g. [3], [12], [17], [20]). Therefore it is of interest to clarify the consistency conditions for jump- diffusion models. Consider a Heath-Jarrow-Morton (HJM) model ([16]) incorporating a marked point process. The dynamics of the forward curve for such a model can be given by (1.1) dr(t,T)=α(t,T)dt+σ(t,T)dB + ρ(t,T,y)µ(dt,dy), t ZΘ where B is a standard Brownian motion and µ(dt,dy) is a random measure on R ×Θwiththecompensatorν(t,dy)dt. Thusthepriceofazero-couponbondcan + be written as (1.2) P(t,T)=e− tTr(t,u)du. R A measure Q is said to be a local martingale measure if the discounted bond price P(t,T) D(t,T)= e− 0tr(s,s)ds is a Q-local martingale, for each T ∈R . ItRis well known that the existence of an + equivalent local martingale measure implies the absence of arbitrage (e.g. see [9]). Under regularity conditions, Björk et al. [5] give the following lemma for the arbitrage-free condition of a generalized HJM model defined by (1.1). Lemma 1.1. An equivalent local martingale measure exists if and only if the forward rate dynamics under this measure specified by (1.1) satisfy the following rela- tion for ∀ 0≤t0, {z:k(Xs−(ω),z)∈B} Z0 ZE then N′(dt,dξ) has a compensator L(Xt−,dξ)dt, where L(·,·) is a transition kernel from (D,D) to (J,J), which is defined by (2.5) L(x,B)=ψ{z ∈E :k(x,z)∈B}, ∀ B ∈J. Furthermore, L(·,·) can be rewritten as (2.6) L(x,dξ)=λ(x)Q(x,dξ), where λ(x) represents the jump intensity of the process X, and Q(x,·) is the jump measure that is a probability kernel for the distribution of jump magnitudes. Proof. For each B ∈J and s,t∈R , we have + s+t Ex (N′(du,B)−duL(Xu−,B)) Fs (cid:20)Z0 (cid:12) (cid:21) s (cid:12) = (N′(du,B)−duL(Xu−(cid:12)(cid:12),B)) Z0 s+t +EXs N(du,dz)1{z:k(Xu−,z)∈B}−duL(Xu−,B) (cid:20)Zs (cid:18)ZE (cid:19)(cid:21) s = [N′(du,B)−duL(Xu−,B)]. Z0 By the monotone class theorem, we can deduce that for each f ∈J , + t Mf := [N′(du,dξ)−duL(X ,dξ)]f(ξ), t s− Z0 ZJ CONSISTENCY OF JUMP-DIFFUSION MODELS 5 is a martingale, and since L(Xt−,·) is predictable, the first part of Lemma 2.1 has been proved. Sinceψ(E)<∞,thenL(x,J)<∞. Thereforebysimplydefiningλ(x)=L(x,J) we obtain L(x,·) λ(x)>0 (2.7) Q(x,·)= λ(x) (δx(·) λ(x)=0, for each x ∈ D, where δ (·) denotes the Dirac measure at x. This completes the x proof. (cid:3) Now using (2.2) and Lemma 1.1, we can derive the following consistency condition for a jump diffusion model. Theorem 2.1. Under Assumption 2.1, a multi-factor jump-diffusion model M is consistent if, and only if, for each (τ,x) ∈ R × D, given the forward rates + curve G(τ,X ), the coefficients a(x), b(x), λ(x) and Q(x,·) satisfy the following t constraint. n n (2.8) −∂ G(τ,x)+ ∂ G(τ,x)b (x)+ a (x)∂ ∂ G(τ,x) τ xi i i,j xi xj i=1 i,j=1 X X n τ =2 a (x)∂ G(τ,x) ∂ G(u,x)du i,j xi xj i,j=1 Z0 X −λ(x) δ (x,τ,ξ)Q(x,dξ), 0 ZJ where δ0(x,τ,ξ)=[G(τ,x+ξ)−G(τ,x)]e− 0τ(G(u,x+ξ)−G(u,x))du. R Proof. Using Lemma 1.1 and Lemma 2.1, it follows from (2.2) that n t n t (2.9) ∂xiG(τ,Xs−)bi(Xs−)ds+ ai,j(Xs−)∂xi∂xjG(τ,Xs−)ds i=1Z0 i,j=1Z0 X X n t τ =2 ai,j(Xs−)∂xiG(τ,Xs−) ∂xjG(u,Xs−)duds i,j=1Z0 Z0 X t t − ds δ0(Xs−,τ,ξ)L(Xs−,dξ)+ ∂τG(τ,Xs)ds. Z0 ZJ Z0 Notice that there are at most countably many jumps for the process X during the time 0 to t, for each t > 0. Therefore, without loss of generality, we do not distinguish between Xs− and Xs in (2.9). By Assumption 2.1, we can obtain (2.8) by differentiating both sides of (2.9) with respect to t. (cid:3) Assumption 2.2. The jump intensity λ(·) is a continuous function on D, and the probability kernel Q(x,·) defined by (2.7) does not depend on x, i.e., (2.10) L(x,dξ)=λ(x)Q(dξ). 6 ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR Remark2.1. Themodels withthejumpmeasuredefinedby(2.10)includetwospe- cific models: pure diffusion models (λ(·)≡0), and models driven by Lévy processes or more precisely, compound Poisson processes when the intensity λ is independent of the state X. Now we can derive the following characterization theorem for a multi-factor jump-diffusion model M. Theorem 2.2. Under Assumptions 2.1 and 2.2, the jump measure Q(·) can be freely chosen subject only to the regularity condition: (2.11) δ (x,τ,ξ)Q(dξ)<∞, ∀ (τ,x)∈R ×D. 0 + ZJ Furthermore, if the forward rate curve G(·,x) satisfies the condition that the func- · tions∂ G(·,x), ∂ ∂ G(·,x), δ (x,·,ξ)Q(dξ)and∂ G(·,x) ∂ G(u,x)duare xi xi xj J 0 xi 0 xj linearly independent for all 1≤i,j ≤n and for all x in some dense subsetD ⊂D, R R 0 then the drift a(·), diffusion b(·) and jump intensity λ(·) of the state process X are uniquely determined by G. Proof. Set M = (n+1)2 and choose a sequence 0 ≤ τ < τ < ··· < τ , such 1 2 M that by the linear independence condition, the M ×M matrix with the ith row constructed as ∂ G(τ ,x),...,∂ G(τ ,x),∂ ∂ G(τ ,x),...,∂ ∂ G(τ ,x), x1 i xn i x1 x1 k xn xn i (cid:18) τi τi ∂ G(τ ,x) ∂ G(u,x)du,...,∂ G(τ ,x) ∂ G(u,x)du, δ (x,τ ,ξ)Q(dξ) , x1 i x1 xn i xn 0 i Z0 Z0 ZJ (cid:19) for each i = 1,2,...,M, is invertible. Therefore a(x), b(x) and λ(x) are uniquely determined by the arbitrage-free condition (2.8), for each x ∈ D . Because of the 0 continuity of a, b and λ, the extensions to the state space D are unique. This completes the proof of Theorem 2.2. (cid:3) Now by applying Theorem 2.2, we can discuss several specific cases. For sim- plicity, throughout the following sections Assumptions 2.1 and 2.2 are satisfied. 3. Separable Term Structure Models In this section, we consider the case in which the forwardrate curve G(τ,x) has a separable structure1 m (3.1) G(τ,x)= h (τ)φ (x), k k k=1 X where h :R 7→R is C1, and φ 7→R is C2 for k =1,...,m. Therefore according k + k to Theorem 2.1, we have the following consistency conditions. 1ThisclassofmodelshasbeeninvestigatedbyFilipović[15]inthediffusioncase. CONSISTENCY OF JUMP-DIFFUSION MODELS 7 Proposition3.1. A separable term structuremodel (3.1) is consistent if, and only if, the following equation holds: m n (3.2) (h (τ)−h (0))φ (x) = Γ (τ,x)b (x) k k k i i k=1 i=1 X X n + a (x)(Λ (τ,x)−2Γ (τ,x)Γ (τ,x)) i,j i,j i j i,j=1 X +λ(x)Ψ(H(τ),x), ∀ (τ,x)∈R ×D, + where for ∀ 1≤i,j ≤n and v ∈Rm, m ∂φ (x) (3.3) Γ (τ,x) := H (τ) k , i k ∂x i k=1 X m ∂2φ (x) (3.4) Λ (τ,x) := H (τ) k , i,j k ∂x ∂x i j k=1 X (3.5) Ψ(v,x) := 1−e−hv,φ(x+ξ)−φ(x)i Q(dξ) ZJ(cid:16) τ (cid:17) (3.6) with H (τ):= h (u)du, k k Z0 (3.7) H(τ):=(H (τ),H (τ),...,H (τ))T, 1 2 m (3.8) φ(x):=(φ (x),φ (x),...,φ (x))T. 1 2 m Moreover, if we assume that the functions Ψ(H(·),x), Γ (·,x), Λ (·,x)−2Γ (·,x)Γ (·,x), ∀ 1≤i,j ≤n i i,j i j are linearly independent for all x in some dense subset D ⊂ D, then a, b, and λ 0 are uniquely determined by h, φ and the measure Q. Proof. The result follows from Theorems 2.1 and 2.2. (cid:3) Now on setting n ∂φ (x) n ∂2φ (x) B (x) := b (x) k + a (x) k , k i i,j ∂x ∂x ∂x i i j i=1 i,j=1 X X n ∂φ (x)∂φ (x) k l and A (x)=A (x) := 2 a (x) , ∀1≤k,l≤m, k,l l,k i,j ∂x ∂x i j i,j=1 X it follows from (3.2) that m m m (3.9) (h (τ)−h (0))φ (x) = H (τ)B (x)− H (τ)H (τ)A (x) k k k k k k l k,l k=1 k=1 k,l=1 X X X +λ(x)Ψ(H(τ),x), ∀(τ,x)∈R ×D. + Therefore, once we know (a(·),b(·),λ(·),Q(·),(h (0) ) and φ(·), we can derive i 0≤i≤n the representationfor H(·). This is clarified by the following proposition. 8 ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR Proposition 3.2. Suppose that the functions φ ,...,φ are linearly independent. 1 m Then the coefficient functions H ,...,H solve a system of ordinary differential 1 m equations (ODEs): dH (τ) (3.10) k =R (H(τ)), 1≤k ≤m, k dτ where R has the form k (3.11) R (v)=θ +hβ ,vi−hα v,vi+γ (v), ∀v ∈Rm k k k k k with θ =h (0), β ∈Rm, α is an m×m symmetric matrix, and γ (v) is a linear k k k k k combination of terms of the form Ψ(v,x), where Ψ(v,x) is introduced in (3.5). Proof. Choose m mutually distinct points (xl) in D such that the m×m 1≤l≤m matrix φ (xl) isinvertible. Thenbymultiplyingtheinversematrixonbothsides k of (3.9) and setting θ = h (0), we obtain (3.10). It is easy to see R (·) has the (cid:0) (cid:1) k k i form of (3.11) by appropriately setting α , β and γ . (cid:3) k k k Remark3.1. Noticethatspecifying(a(·),b(·),λ(·),Q(·),(h (0) ),(φ (·) )) k 0≤k≤m k 0≤i≤m is equivalent to specifying a multi-factor short rate model. Therefore Proposition 3.2provides awaytosolvetheforwardratestructurebyapplyingtheconsistencyre- quirements. Moreover it implies a necessary condition for a model to be consistent: the existence of the solution of the ODE system (3.10). 3.1. Affine Term Structure Models. Now we will take a look at the simplest class of separable term structure models, namely the affine term structure models. The next proposition gives necessary and sufficient condition for a jump-diffusion model to be affine. Proposition 3.3. If the functions a(x), b(x), λ(x), G(0,x) are affine, and Q(·) satisfies (2.11), then the term structure of forward rates G(·,x) is affine, that is, n (3.12) G(τ,x)=h (τ)+ h (τ)x , ∀ (τ,x)∈R ×D, 0 i i + i=1 X for a continuously differentiable function h. Conversely if G(τ,x) is affine and H ,...,H ,H H , H H ,...,H H , 1−Ψ(H) 1 n 1 1 1 2 n n are linearly independent functions, then the functions a(·), b(·) and λ(·) are affine. Proof. The first part of the proposition is well established (e.g. [5], [12]). In particular,onecanshowthatgivenajump-diffusionmodelwiththedrift,diffusion, jump intensity and the short rate being affine functions of the state, the price of a zero-couponbondhasanexponentialaffineformandthe coefficientfunctions solve a series of generalized Riccati equations. Thus the term structure is affine. For the second part of the proof we will use Proposition 3.1. If we set φ (0)=1, φ (x)=x , 1≤k ≤n, 0 k k CONSISTENCY OF JUMP-DIFFUSION MODELS 9 in(3.2)thenwecanderivethefollowingconsistencyconditionforaffinetermstruc- ture models: (3.13) n n n h (τ)−h (0)+ (h (τ)−h (0))x = H (τ)b (x)− a (x)H (τ)H (τ) 0 0 i i i i i i,j i j i=1 i=1 i,j=1 X X X +λ(x)(1−Ψ(H(τ)), where H(·)=(H (·),...,H (·))T and Ψ(v)= e−hv,ξiQ(dξ), which is the Laplace 1 n J transform of the probability measure Q. Since the left hand side of (3.13) is affine R and the coefficient matrix is invertible and independent of x, a(x), b(x) and λ(x) must be affine functions of x. This completes the proof. (cid:3) Now it is assumed that a(·), b(·), λ(·) and Q(·) are given as affine functions, and (h (0)) are known. The following corollary can be derived directly from i 0≤i≤n Proposition 3.2. Corollary 3.1. Under the consistency condition (3.13), if a(·), b(·), λ(·) and Q(·) satisfy Assumptions 2.1 and 2.2, the coefficient functions (H (·)) can be de- i 0≤i≤n termined from a system of ODEs shown as follows. For k=0,1,...n, dH (τ) (3.14) k =R (H(τ)), k dτ where R has the form k (3.15) Rk(v)=θk+hβk,vi+hαkv,vi+γk 1−e− nj=1vjξj Q(dξ), ZJ(cid:16) P (cid:17) with θ =h (0). k k Remark 3.2. The above system of ODEs (3.14) and (3.15) are known as general- izedRiccatiequations (GREs),which have alsobeenderived in[10]usingadifferent approach. TheexistenceanduniquenessofthesolutionstotheGREshavealsobeen studied in [10]. Under regularity conditions, Filipović [15] demonstrated that affine and quadratic models are the only two possible consistent models that can produce separable polynomial term structure in the diffusion case. We have already seen that affine term structure models allow for jumps in the state process X. However, importing jumps into the state process is generally difficult with polynomial term structure models2 if not impossible. For example, it was shown in [7] that with a quadratictermstructure,thestateprocessX canonlyfollowaso-calledquadratic t process that does not allow jumps. For this reason, several alternative approaches have been adopted by researchers. One approach is to accommodate the exist- ing affine jump-diffusion and quadratic models into one framework, namely the 2Thismeansthatφk(x)definedin(3.1)isapolynomialinx,foreach1≤k≤m. 10 ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR linear-quadratic jump-diffusion (LQJD) class originally proposed by Piazzesi [20] and further developed by Cheng and Scaillet [8]. Another approach is to apply a special Lévy process to drive the dynamics of the state variables (see [2]). Then pricingbondsandotherderivativescanbeachievedbyapproximation(see[6],[17]). 4. The Nelson-Siegel Curves Inthissection,wediscussapopularnon-separabletermstructuremodel,namely theNelson-Siegelcurvefamily(see[18]). Thiscurvefamilyhasbeenstudiedin[13], andas we will show,there exists no non-trivialdiffusion model consistent with the Nelson-Siegel forward curve. The Nelson-Siegel forward curves are given by the form: (4.1) G(τ,x)=x +(x +x τ)e−x4τ, (x ,x ,x )∈R3, x >0. 1 2 3 1 2 3 4 Since X4 > 0, we restrict the support of the jump measure Q(dξ) to {ξ ≥ 0}. 4 Thus we redefine D :=R3×R and J :=R3×R in this section. We also make ++ + the following regularity assumption. Assumption 4.1. The measure Q of the jump magnitude satisfies the following regularity condition: (4.2) (1+|ξ |3)e−r2ξ2−r3ξ3Q(dξ)<∞, ∀ r ,r ∈R . 4 2 3 + ZJ If we move the term 2 n a (x)∂ G(τ,x) τ ∂ G(u,x)du to the LHS of i,j=1 i,j xi 0 xj (2.8), then using (A.19)-(A.23) of the Appendix, we canwrite the consistency con- P R dition for the Nelson-Siegel curve as (4.3) q (τ,x)+q (τ,x)e−x4τ +q (τ,x)e−2x4τ =λ(x) δ (x,τ,ξ)Q(dξ), 0 1 2 0 ZJ where q (τ,·), q (τ,·) and q (τ,·) are polynomial functions of τ with degrees less 0 1 2 than or equal to 1, 3, and 4, respectively (shown in the Appendix). The following theorem asserts that under regularity conditions, there exists no non-trivial multi-factor model consistent with the Nelson-Siegel forward curve in the jump-diffusion case. The proof is included in the Appendix. Theorem 4.1. Under Assumptions 2.1, 2.2 and 4.1, a jump-diffusion model with the Nelson-Siegel-type forward curves is consistent if, and only if, the state process X is deterministic upon x; i.e., for ∀ x∈D, a(x) =0, and either λ(x)=0 or the jump measure Q is the Dirac measure at 0. 5. Conclusion Motivated by previous work on consistency problems for diffusion models, this article investigated this issue in the jump-diffusion case. Unlike the diffusion case, here we have four elements to consider: the drift, diffusion, jump intensity and

Consistency Problems for Jump-Diffusion Models

Similar Consistency Problems for Jump-Diffusion Models