Consistency Problems for Jump-Diffusion Models

0.25 MB·English

by  Erhan Bayraktar

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
WeareindebtedtoErhanC¸inlarandDamirFilipovi´cforhelpfuldiscussions.
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, Bj¨ork 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 for-
ward rate dynamics under this measure specified by (1.1) satisfy the following rela-
tion for ∀ 0≤t<T.
T
(1.3) α(t,T)=σ(t,T) σ(t,s)ds− ρ(t,T,y)e− tTρ(t,u,y)duν(t,dy).
Zt ZΘ R
Proof. See [5], Theorem 3.13 and Proposition 3.14. (cid:3)
Lemma 1.1 gives the drift condition for a generalized HJM model, which gener-
alizesthetraditionalarbitrage-freeconditionfordiffusionHJMmodels. Thisresult
providesawayforustoderiveconsistencyconditionsforamulti-factormodelwith
jumps.
In this paper, we consider the consistency issue for jump-diffusion models. In
particular,in Section 2,we bring a jump-diffusion model into the generalizedHJM
CONSISTENCY OF JUMP-DIFFUSION MODELS 3
framework,and derive the consistency condition for the coefficient functions of the
model. In Section 3, we discuss a class of separable term structure models, which
includes affine andquadratic models as specialcases. Inparticular,the affine term
structureisinvestigatedandsufficientandnecessaryconditionsforajump-diffusion
model to be affine are derived. A typical non-separable term structure model,
namelytheNelson-Siegeltermstructure,isexaminedinSection4. Underregularity
conditions, we conclude there exists no jump-diffusion model consistent with the
Nelson-Siegel forward curves. Brief concluding remarks are made in Section 5.
1.1. Basic Notation. Table1definesnotationthatwillbe usedfrequentlyinthis
paper.
2. Consistency Conditions for Multi-factor Jump-Diffusion Models
Consideramulti-factortermstructuremodelMwithforwardratesofthe form:
r(t,T)=G(T −t,X ),
t
where the state process X follows a jump-diffusion in which the jump part fol-
lowsa multivariate point processwith state space (D,D) under a complete filtered
probability space (Ω,F,(F ),P):
t
(2.1) dXt =b(Xt)dt+c(Xt)dWt+ N(dt,dz)k(Xt−,z), X0 =x,
ZE
whereG:R ×D 7→Risadeterministicfunction,W isann-dimensionalstandard
+ t
Brownian motion and N(·,·) is a Poisson random measure on R ×E with mean
+
measure Leb×ψ. P denotes the equivalent martingale measure.
We will take X in (2.1) to be piece-wise continuous. This implies that ψ is a
t
finite measure on E, i.e., ψ(E) < ∞ and that 0 ds Eψ(dz)k(Xs−,z) < ∞ a.s.,
for each t ∈ R . Moreover, later in this section and in the following sections we
+ R R
willrestrictX to be a jump diffusion processwhose jumps havea fixed probability
distributionQonJ andarrivewithintensity{λ(X ):t≥0}(seeAssumption2.2).
t
Definition2.1. A multi-factorjump-diffusion model M is said beconsistent if the
induced dynamics of the forward rates G satisfies the relation (1.3), for all x∈D.
Assumption 2.1. For the sake of analytical tractability, we make the following
assumptions.
• The function G∈C1,2(R ×D);
+
• the functions b : D 7→ Rn, c : D 7→ Rn×n and k : D×E 7→ J ⊂ Rn are
deterministic and continuous; and
• (2.1) has a unique strong solution X in D, for each x∈D.
t
4 ERHANBAYRAKTAR,LICHEN,ANDH.VINCENTPOOR
By Itoˆ’s formula, the dynamics of the forward rates G(·,X ) can be derived as
t
follows.
t n t
(2.2)G(·,Xt) = G(·,x0)+ ∂τG(·,Xs)ds+ ∂xiG(·,Xs−)bi(Xs−)ds
Z0 i=1Z0
X
n t n
+ ∂xiG(·,Xs−) ci,j(Xs−)dWtj
i=1Z0 j=1
X X
n t
+ ai,j(Xs−)∂xi∂xjG(·,Xs−)ds
i,j=1Z0
X
t
+ N(ds,dz)[G(·,Xs− +k(Xs−,z))−G(·,Xs−)],
Z0 ZE
where
1
(2.3) a(X )= c(X )c(X )T
t t t
2
is a semi-positive definite matrix.
Lemma 2.1. If we define N′(·,·) as a random measure on (R ,J) by
+
t
(2.4) N′([0,t],B,ω)= N(ds,dz,ω)1 , ∀B ∈J, t>0,
{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 condi-
tion 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 L´evy 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.
1ThisclassofmodelshasbeeninvestigatedbyFilipovi´c[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, Filipovi´c [15] demonstrated that affine and qua-
dratic models are the only two possible consistent models that can produce sepa-
rable 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 L´evy 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