Download Solvable Affine Term Structure Models

Solvable Affine Term Structure Models
Martino Grasselli
Università di Padova∗, ESILV†and SAFE‡
Claudio Tebaldi
Università di Verona§and SAFE
June 2005
Abstract
An Affine Term Structure Model (ATSM) is said to be solvable if the pricing problem
has an explicit solution, i.e. the corresponding Riccati ordinary differential equations
have a regular globally integrable flow. We identify the parametric restrictions which
are necessary and sufficient for an ATSM with continuous paths, to be solvable in a
state space D+ ×Rn−m where D+ , the domain of positive factors, has the geometry of a
symmetric cone. This class of state spaces includes as special cases those introduced by
Duffie and Kan (1996), and Wishart term structure processes discussed by Gourieroux
and Sufana (2003). For all solvable models we provide the procedure to find the explicit
solution of the Riccati ODE.
Keywords: Affine Terms Structure Models, Riccati ODE, Lie algebra, Symmetric Cone.
1
Introduction
The class of multifactor Affine Term Structure Models (ATSM hereafter), has the following
financial appealing properties:
1. The sensitivities of the zero coupon yield curve to the stochastic factors are deterministic, as discussed in Brown and Schaefer (1994);
2. For a given state space D it is possible to identify explicit parametric restrictions, called
admissibility conditions, granting the existence of a regular affine process with state
space D. The problem of admissibility has been introduced by Duffie and Kan (1996),
Dai and Singleton (2000), Duffie, Filipović and Schachermayer (2003) and Filipović
(2005);
3. The pricing problem can be reduced to the solution of a system of Ordinary Differential
Equations (ODE) as discussed in Duffie, Pan and Singleton (2000). In fact, the explicit
expression of the conditional discounted characteristic function of the factors can be
specified in terms of the solutions of Riccati ODE for any admissible ATSM (see e.g.
Duffie et al. 2003).
∗ Dipartimento
di Matematica Pura ed Applicata, via Belzoni 7, Padova. Email: [email protected]
Supérieure d’Ingénieur Léonard de Vinci, Département Mathématiques et Ingénierie Financière,
92916 Paris La Défense
‡ Center for Studies in Actuarial and Financial Engineering - Economics, Via Giardino Giusti 2, 37129
Verona, Italy.
§ Department of Economics, Via Giardino Giusti 2, 37129 Verona, Italy. Fax +39 0458054935. E-mail:
[email protected]
† Ecole
1
In this paper we characterize the class of admissible ATSM, with factors following continuous paths, for which the pricing problem has an explicit solution, i.e. the non linear
Riccati ODE can be linearized and integrated. We term this class of models Solvable Affine
Term Structure Models (SATSM). The main result of the paper is the characterization of
necessary and sufficient conditions for an ATSM to be solvable when the state space of
positive factors has the geometry of a symmetric cone. An excellent introduction to the
properties and applications of symmetric cones can be found in the book of Faraut and
Korányi (1994). This class of state spaces virtually includes all those previously considered
in financial applications and provides possible extensions yet unexplored.
The explicit solution for all SATSM can be obtained by solving linear ODE. Thus the
notion of solvability characterizes the largest class of multivariate dynamic diffusive factor
models which, while relaxing the constant volatility hypothesis, preserve the computational
tractability of Gaussian models.
The procedure to find the explicit solution is carried out for all solvable models within the
n−m
Duffie and Kan (1996) state space D= Rm
, and for Wishart models (see Gourieroux
+ ×R
and Sufana 2003) where D=P os+ (r, R).
The paper is organized as follows: in Section 2 we introduce the pricing problem in
ATSM and we discuss the admissibility and solvability conditions. In Section 3 we provide
the procedure to integrate the ODE when the model is solvable. In Section 4 we apply the
above results to the class of models introduced in Duffie and Kan (1996) as well as to the
class of Wishart term structure models. We leave in the Appendix the technical results on
Jordan algebras which are used in the proofs of our main results.
2
Pricing in an ATSM
We follow the definition of regular affine processes given in Duffie et al. (2003) in the case
where the Markovian process (Xt )t∈R+ ∈ D ⊆ V , where D is the state space, and V is a
finite dimensional real vector space, dim V = n, with the standard scalar product denoted
by h, i. With a slight abuse of notation we denote with V also the complex extension of
the vector space on C; correspondingly h, i will denote the Hermitian extension of the scalar
product.
Being interested in pricing contingent claims, we follow Duffie, Pan and Singleton (2000),
Bakshi and Madan (1999) and compute the discounted characteristic function of the factors
Xt , conditional on the information at time t ≤ T :
Definition 1 The regular (continuous) Markov process X is said to be affine if for every t ∈
R+ the ”discounted” conditional characteristic function has exponential-affine dependence
on the initial condition x. That is,
!
#
"
à Z
T
(η0 + hη, Xs i) ds exp (i hu, XT i)
ΨX (u, x, t, T ) = Et exp −
t
¡
¢
= exp V 0 (t, iu) − hV (t, iu) , xi ,
(1)
where (t, u) ∈ R+ × V , V 0 : R+ × V → C, V : R+ × V → V , and η0 ∈ R+ , η ∈ D.
Typically the term R (Xt ) = (η0 + hη, Xt i) represents the short term rate and consistently
the zero-coupon bond price is given by:
¡
¢
P (t, T ) = exp V 0 (t) − hV (t) , Xt i = ΨX (0, x, t, T )
(2)
V (T ) = 0.
2
A standard argument, see e.g. Duffie and Kan (1996), shows that the infinitesimal
generator for a regular affine diffusion process, has necessarily the following functional form:
A=
¤ ­
®
1 £
T r (Σ (x) + Σ0 ) DT D + Ω0 + Ω(x), DT − (η0 + hη, xi) ,
2
(3)
where D denotes the (row vector) gradient operator, T r the trace over Mn (V ), DT D the
Hessian matrix, Σ (x) , Σ0 ∈ Symn (V ) are positive semidefinite, Ω (x) , Ω0 ∈ V . The relevant
property is that Σ (x), Ω (x) are linear functions of x ∈ V , hence:
Σ (x) =
n
X
k
Ci,j
xk ,
Ω (x) =
k=1
n
X
Ωkl xk .
k=1
Following Duffie et al. (2003), we look for the conditions on the parameters which allow
to define uniquely a regular affine (continuous path) process ∀x ∈ D, ∀t ∈ R+ . By applying
the Feynman-Kač argument, we have
−
∂ΨX
+ AΨX = 0
∂t
ΨX (u, x, T, T ) = exp (i hu, xi) ,
that is
−
¤
­
®
∂ΨX
1 £
+ T r (Σ (x) + Σ0 ) DT D ΨX + Ω0 + Ω (x) , DT ΨX − (η0 + hη, xi) ΨX = 0
∂t
2
and by replacing the exponential form we obtain (notice that DT ΨX = V)
∂V 0
=
∂t
¿
À
¤ ­
®
∂V
1 £
, x + T r (Σ (x) + Σ0 ) VV T − Ω0 + Ω (x) , V − (η0 + hη, xi)
∂t
2
V 0 (T, iu) = 0
(4)
V (T, iu) = iu.
This equality implies that the vector of factor sensitivities V solves a (quadratic) ordinary differential equation. Notice that given the solution for V, the determination of V 0
is obtained by direct integration, thus we are going to concentrate our discussion on the
ODE for V. In the following section we are going to investigate the parametric restrictions
granting the existence of the Markov semigroup, or equivalently the global existence of a
non singular flow for the ODE (admissibility conditions).
2.1
Admissible ATSM in a symmetric cone state space
Definition 2 An ATSM model is admissible in a state space D ⊆ V if the generator (3)
and the corresponding regular affine Markov semigroup exist and are unique for any initial
condition in D, or equivalently if ΨX is uniquely defined by (1) ∀x ∈ D and ∀t ∈ R+ .
According to the definition, an ”Admissible ATSM” is no more than a ”regular affine
process with admissible parameters” in the terminology of Duffie et al.(2003). Notice that
our definition, however, is still implicit since it does not identify the parametric restrictions
granting admissibility. Theorem 2 and Proposition 3 will partially answer to this problem
by giving sufficient conditions on the parameters. In fact, our goal will be the complete
characterization of the subclass of admissible models for which the Riccati ODE can be
explicitly solved.
Now we recall the notion of symmetric cone domain, which generalizes the notion of
”positive factors’ state space”
3
Definition 3 A symmetric cone C ⊂ V is an open convex cone which has the two following
properties:
©
ª
i) C is self dual, i.e. C= y ∈ V ; hx, yi > 0,∀x ∈ C
ii) C is homogeneous, i.e. the automorphism group G (C) = {g ∈ GL (V ) ; g (C) = C}
operates transitively on C.
A symmetric cone is said to be irreducible if it cannot be decomposed in a Cartesian
product of two symmetric cones. Among (irreducible) symmetric cones C = P osr (R) ⊂
Symr (R) (m = r(r + 1)/2) is prototypical and will be the state space for the class of
Wishart models. The Duffie et al. (2003) state space C = Rm
+ is a reducible symmetric cone
and is obtained considering m copies of R+ , i.e. P os1 (R).
The geometry of a symmetric cone domain is directly related to the algebraic notion of
Euclidean Jordan Algebra (EJA):
Definition 4 (Faraut and Korányi 1994) (V, h, i , ∗) forms an Euclidean Jordan Algebra
(EJA) if V is a finite dimensional real vector space with a scalar product h, i and ∗ : V ×V →
V is a bilinear commutative (not necessarily associative) product with unity e such that the
following properties hold:
¡
¢
i) Jordan algebra property: u2 ∗ (v ∗ u) = u2 ∗ v ∗ u,
ii) Euclidean property hu ∗ v, wi = hv, u ∗ wi.
The following objects can be naturally defined within each EJA:
- the linear map L (u) ∈ Mn (V ) defined by L (u) v :=
¡ u¢∗ v,
- the quadratic representation: P (u) = 2L2 (u) − L u2 , which satisfies:
P (gu) = gP (u) g T ,
∀g ∈ G (C)
(5)
- the related bilinear form: P (u, v) = [P (u + v) − P (u) − P (v)] /2.
The following results reduce the classification of symmetric cones to the classification of
simple (without nontrivial ideals) EJA:
Proposition 1 (Faraut and Korányi 1994)
i) if C ⊆ V is a symmetric cone then V carries the structure of a Euclidean Jordan
Algebra and the closure C is the set of squares u ∗ u, u ∈ V ;
ii) the symmetric cone is irreducible if and only if the associated EJA is simple, i.e. it
has no non trivial ideals;
iii) there’s a one to one correspondence between the decomposition of a symmetric cone
in a product of irreducible symmetric cones and the decomposition of an EJA as a direct
sum of simple EJA.
All possible finite dimensional irreducible symmetric cones are:
1. The families of positive definite matrix cones P os (r, R), Herm+ (r, C), Herm+ (r, H)
(H indicates the quaternions field),
©
ª
Pn
2. The Lorentz cones Λn = x ∈ Rn : x21 − i=2 x2i > 0, x1 > 0 ,
3. The exceptional cone (27 dimensional cone of 3 × 3 ”positive definite” matrices over
the Cayley algebra).
For example, when C = P os (r, R), the corresponding EJA is given by Symr (R) with
(Jordan) product: A ∗ B = (AB + BA) /2.
Now we state our first result which proves a remarkable connection between the conditional covariance of an admissible ATSM (on an irreducible symmetric cone state space D)
and the relative quadratic representation P (V):
4
Theorem 2 Consider an irreducible symmetric cone state space D ⊂ V and let (V, h, i , ∗)
be the corresponding EJA. Then, for any positive definite covariance matrix Σα (.) of an
admissible ATSM in D, there exists αΣ ∈ D such that:
£
¤
T r Σα (x) VV T = hx, P (V) αΣ i ∀x ∈ D, ∀V ∈V
(6)
holds. Conversely, (6) defines the covariance matrix Σα (x) of an admissible ATSM in D.
Notice that relation (6) provides a one-to-one correspondence between the quadratic
term of the Riccati ODE and the related quadratic representation of the EJA: as we shall
see in the next proposition, this relation can be extended to the complete vector field, thus
defining a sufficient condition for an ATSM to be admissible.
Proposition 3 Consider the following Riccati ODE defined in an irreducible symmetric
cone D:
1
∂V
= − P (V) αΣ + ΩT (V) + η,
(7)
∂t
2
where αΣ ∈ D is given in (6), η ∈ D and
©
ª
ΩT ∈ g (D) = H ∈ MN (V ) | 2P (HV, V) = HP (V) + P (V) H T .
Moreover, let η0 be a positive constant, Σ0 ∈ Symn (V ) be positive semidefinite and Ω0 =
kαΣ , k ≥ nr − 1, where r is the rank of the EJA. Then the corresponding ATSM given by
(3) is admissible in the state space D.
The proof of Proposition 3 given in Appendix is grounded on a construction due to Kantor, Koecher and Tits (KKT) (see the Appendix for a short introduction and e.g. Walcher
1986), which shows that each Jordan Algebra can be represented as a particular (quadratic)
polynomial Lie algebra. While Theorem 2 states that there’s a strong relationship between
the Jordan Algebra structure and the covariance term, Proposition 3 shows that the vector
field of the Riccati ODE is an element of the polynomial Lie algebra obtained using the
KKT construction.
The extension of the above results to general (non irreducible) symmetric cone state
spaces is immediate. In fact, if the symmetric cone is not irreducible, it decomposes as
D = D1 × ... × Dk , where D1 , ..., Dk are the corresponding irreducible symmetric cones.
Then the system of Riccati ODE corresponding to an ATSM admissible on D separates into
a direct sum of independent systems of Riccati ODE, each corresponding to an admissible
ATSM in Di .
2.2
Solvable ATSM in a symmetric cone state space
The above analysis has shown that it is possible to provide a constructive test to select a class
of admissible ATSM in any symmetric cone state space D. The conditions in Proposition
3 are sufficient but not necessary, as it is easy to realize by comparing with the ones of
Duffie et al. (2003), in the special case of D = Rm
+ (see also Section 4 below). However,
in this subsection we shall see that the class of admissible models which are selected by
Proposition 3 is completely characterized by the important property that the solution flow
of the corresponding Riccati ODE can be linearized, and then explicitly determined through
exponentiation.
First of all let us denote by (A (V, V ) , [, ]) the Lie algebra of all analytic maps from V
into V , where the commutator is defined by:
[f (V) , g (V)] = ∇g (V) f (V) − ∇f (V) g (V) ,
V ∈V , f, g ∈ A (V, V ) .
Within Lie approach to the integration of an ODE, the ability to get an explicit solution
is equivalent to its linearizability, which is formally defined by the following:
5
Definition 5 (Walcher 1991) An ODE dV/dt = R (V, t) admits a linearizing map if:
i) There’s a Lie algebra M of linear vector fields operating on a finite dimensional vector
space W ,
PK
ii) R (V, t) =
i=1 αi (t) fi (V) ∀V ∈ V , where α1 , ..., αK : R+ → R are continuous
functions of time and L = {fi (V)}i=1..r is a family of autonomous vector fields L ⊂ A (V, V ),
iii) There’s an analytic map ΦR : G → V with G 6= ∅ an open subset of W , with dense
R
image and such that
is¢ a solution-preserving map i.e. ∀f ∈ L, ∃F ∈ M , such that
¡ Φ
R
R
DΦ (v) f (V) = F Φ (v) ∀v ∈ G.
Roughly speaking, M provides a matrix (linear) representation of the non linear vector
¡ ¢−1
fields f (V) and when ΦR
exists, it maps a non linear ODE into a linear one.
It is well known (see e.g. Walcher 1991, Proposition 8.7) that the existence of such a
change of coordinates that linearizes the ODE implies the existence of a finite dimensional
Lie subalgebra containing L, and this notion is strictly related to both the integrability of
the ODE and the existence of a fundamental system of solutions for the ODE.
The decomposition of the vector field R and the choice of L are not unique. In particular,
some finite dimensional subalgebras can arise if we impose additional parametric restrictions
beyond admissibility on the expression of R1 . We shall consider as acceptable only those
finite dimensional Lie subalgebras which are independent of the choice of (possibly time
dependent) parameters. In fact, for the purpose of model estimation, any restriction would
appear as an artificial constraint and nullify the advantages of an explicit solution. We
formalize the notion of admissible parameters using a definition in line with Filipović (2005):
Definition 6 Let Rα be the vector field of the Riccati ODE arising from an ATSM with state
space D, and {αi (t)}i=1,..,I be the vector of all real parameters appearing in Rα . The vector
field Rα is said to be (strongly) admissible if ∀i the functions αi : R+ → R are continuous
and for any fixed t the corresponding autonomous Riccati ODE has a differentiable flow on
the positive line.
We can now introduce the following
Definition 7 An ATSM is said to be solvable in a state space D if for any (strongly) admissible choice of the parameters, the conditional characteristic function can be determined
by solving a sequence of linear ODEs.
Hence, an ATSM in a symmetric cone state space D is solvable iff the corresponding
Riccati ODE is linearizable.
We now state our main result, stating that the parametric conditions introduced in
Proposition 3 granting admissibility are also necessary for solvability.
Theorem 4 An ATSM in an irreducible symmetric cone state space D is solvable iff the
corresponding Riccati ODE satisfies the conditions in Proposition 3.
The proof of the theorem, given in Appendix, uses again the KKT construction: in fact,
following Walcher (1986), we will show that the KKT approach classifies all linearizable
quadratic ODE.
The extension to a generic (possibly reducible) symmetric cone state space D is straightforward: in this case the subalgebra of vector fields separates into a direct sum of subalgebras, hence the solution reduces to the solution of an independent ODE in each irreducible
symmetric cone.
1 Consider for example an autonomous equation, which always admits the trivial one dimensional algebra
{αF (V)}, α ∈ R. We thank the referee for pointing out this fact.
6
2.3
Extension to the state space D = D+ × Rn−m
Let D = D+ × Rn−m , where D+ is a symmetric cone and Rn−m represents the state space
of evolution of conditionally Gaussian factors. As discussed in Duffie et al. (2003) in an
n−m
ATSM with state space D = Rm
, the factor sensitivities corresponding to Gaussian
+ ×R
factors can be determined solving an independent system of linear equations. In the next
proposition we extend this result to an ATSM with state space D+ × Rn−m , requiring that
D+ is an ideal for the Riccati ODE corresponding to the complete problem, i.e. R (V, t) ∈
D+ , ∀V ∈ D+ .
Proposition 5 Let D+ be an ideal of a quadratic algebra induced by the (quadratic) polynomial ODE:
d
V = R (V, t)
dt
n−m
and let π : D+ ×R
n−m
→R
V (t) ∈ D+ × Rn−m ,
µ
be the natural projection. Then the solution V =
V D+
VG
¶
can be obtained by solving (n − m) linear ODEs:
¡
¢
d G
V = RG V G , t
V G (0) = πV (0)
dt
¡
¢
RG V G , t = πR (V, t) , V G = πV,
(8)
and the quadratic differential equation in D+ :
µ D ¶
µµ D ¶ ¶
µ
¶
d
d
0
V +
V +
=R
,t −
0
VG
VG
dt
dt
µ D
¶
µ
¶
0
V + (0)
= V (0) −
,
0
V G (0)
where V G denotes the solution to (8).
Proof. The result follows directly from Theorem 2.5 of Walcher (1991) and by noting
from (4) that the presence of conditionally Gaussian factors leads to linear ODEs.
The parametric restrictions on the Riccati ODE implied by the condition that D+ is
an ideal for the whole Riccati ODE depend on the specific choice for D+ , and will not be
discussed here.
3
The explicit solution of a SATSM
The above definition of solvability ATSM is intended to provide an explicit path to compute the conditional characteristic function by mean of linear ODEs. The computational
algorithm to find the general integral is the following:
1. Solve the ODE (8) corresponding to conditionally Gaussian factors;
2. Decompose the remaining ODE as a direct sum of Riccati ODE corresponding to
factors living on irreducible symmetric cone state spaces;
3. Linearize each of these Riccati ODEs according to the classification of simple EJAs
(a detailed classification of these algebras can be found in Faraut and Korányi 1994).
In fact, we recall that each EJA is related with the Lie algebra g (TC ), whose linear
matrix realization M is uniquely identified. The map ΦR of Definition 8 is then the
natural isomorphism between the vector field realization and the matrix realization of
g (TC ) (see also the examples in Section 4)
7
4. Solve by exponentiation the linearized system in M and use the solution preserving
map ΦR to recover the solution for the original ODE.
All these steps can be easily implemented with any numerical software and the only
relevant task is the computation of the matrix exponential function which solves the linear
system. While this computation is straightforward when the ODE is autonomous, in the
presence of time dependent coefficients the solution to the linear system involves a so called
time ordered exponential, a symbolic expression whose analytic computation is problematic.
On the other hand, this situation is exactly that occurring for linear (non-autonomous) ODEs
and nothing better could be expected. For a recent discussion of the computational solutions
in the non autonomous case see e.g. Moan and Oteo (2001). In the non-autonomous case, an
alternative procedure to determine the general integral of the Riccati ODE is proposed by
Walcher (1991) (pg.134). Notice however that this expression is explicit up to the knowledge
of a particular solution, which has to be found case by case.
Now we specialize our results to the class of models most relevant for financial applications.
4
4.1
Applications
n−m
The Duffie and Kan (1996) state space Rm
+ ×R
Consider the familiar setup of Duffie and Kan (1996) ATSM in the specification proposed
by Dai and Singleton (2000), which is essentially the same of Duffie et al. (2000) and Duffie
et al. (2003), when factors are restricted to have continuous paths.
The dynamics of the factors are determined (up to a rescaling) by the following SDE:
h¡
¡
¢
¢1/2 i
dXt = ΩXt + Ω0 dt + diag Ci Xt + Ci0
dWt , t ≥ 0,
(9)
n−m
,
X0 = x ∈ Rm
+ ×R
(Ci denotes the i − th row of the matrix C, m = rank
¡ (C) ≤ n) where
¢ Wt is an n−
dimensional Brownian motion and the parameters φ = η, η 0 , Ω, Ω0 , C, C 0 are defined as
follows:
i) the drift matrix Ω is given by
µ DD
¶
Ωm×m
0m×(n−m)
Ω=
,
(10)
ΩBD
ΩBB
(n−m)×m
(n−m)×(n−m)
where the out of diagonal elements of ΩBB are restricted to be nonnegative;
n−m
ii) Ω0 ∈ Rm
;
+ ×R
0
iii) C and C are given by
¶
µ
Im×m
0m×(n−m)
,
C=
BD
C(n−m)×m
0(n−m)×(n−m)
µ
¶
0m×1
C0 =
.
1(n−m)×1
(11)
B
iv) η0 ≥ 0, η D ∈ Rm
+ , η = 0.
Observe that the state space of positive factors Rm
+ is a reducible symmetric cone: the
corresponding EJA is given by Rm endowed with (the natural scalar product and) the Jordan
product (a ∗ b)i = ai bi , whose quadratic representation corresponds to P (c) = diag (c).
We apply our main theorem and we get the following
8
n−m
Theorem 6 The ATSM (9) is solvable in the state space Rm
if and only if the
+ × R
DD
matrix Ω
is diagonal.
Proof: The statement is a consequence of the reducibility of the EJA, which in turns
implies the diagonality of both C DD and ΩDD .
In conclusion, a necessary condition for an ATSM to be solvable in the state space
n−m
Rm
is that the positive risk factors are uncorrelated.
+ ×R
Remark 7 Even if the previous theorem represents a No Go result, there are some nontrivial
SATSM (for example the Balduzzi et al. 1996 model, see Grasselli and Tebaldi 2004) which
can be solved explicitly using our methodology.
For a SATSM, the Riccati ODEs become
−
n
X
¡ BD ¢T B
d D
D
Vi (τ ) = ηiD + ΩDD
V
+
Ω
V
ii
i
ij j
dτ
j=m+1
m
n
1 X ¡ D ¢2 1 X ¡ BD ¢T ¡ B ¢2
V (τ ) −
C
Vj (τ ) ,
−
ij
2 i=1 i
2 j=m+1
−
n
X
¡ BB ¢T B
d B
V (τ ),
Ω
Vi (τ ) =
ij j
dτ
j=m+1
i = m + 1, .., n,
(12)
i = 1, .., m
(13)
with boundary condition V(T ) = iu ∈ Cn .
The procedure of Section 3 in order to find the explicit solution in the case of a SATSM
consists in the following two steps:
Step 1) Put τ = T − t and solve the n − m linear ODEs (13):
n ¡
¢T o B
V B (τ ) = exp −τ ΩBB
V (0) .
(14)
Step 2) Plug (14) into (12) and for all i = 1, ..., m solve the 1-dimensional timedependent Riccati equations
1 D
d D
2
D
ei (τ ) + ΩDD
i = 1, ..., m,
(15)
V (τ ) = γ
ii Vi (τ ) − Vi (τ ) ,
dτ i
2
³¡
´
¢T
¡
¢T
Pn
2
where γ
ei (τ ) = ηi + j=m+1 ΩBD ij VjB (τ ) − 12 C BD ij VjB (τ ) .
The only nontrivial step is the second one. However, now we show that although the coefficients γ
ei (τ ) are time dependent, the one dimensional Riccati ODE (15) can be linearized.
In fact, the vector field
µ
¶
1 2
R(τ ) = γ (τ ) + ωV − V ,
(16)
2
−
can be included in the Lie algebra sl (2, R) with generators:
L0 = γ ∈ R, L1 = V, L2 = V 2 .
obeying the commutation relations:
[L0 , L1 ] = −L0 , [L0 , L2 ] = −2L1 , [L2 , L1 ] = L2 .
The consequence is that we can now rewrite our differential equation (15) in a coordinate
free way, and we can move from the above non linear representation to a more convenient
space W = R2 , where the group is linearly represented.
9
In fact, the generators L0 , L1 , L2 have the following linear realization in M2 (R):
µ
¶
µ
¶
µ
¶
1
0 1
1 0
0
0
M2 (L0 ) =
, M2 (L1 ) =
, M2 (L2 ) =
,
0 0
0 −1
−1 0
2
and in the new space W the equation (16) can be written
¶ µ
µ
d
π (τ )
= γ (τ ) M2 (L0 ) + ωM2 (L1 ) −
−
λ (τ )
dτ
which becomes the linear ODE:
µ
¶ µ 1
d
π (τ )
2ω
−
=
λ (τ )
− 12
dτ
γ (τ )
− 12 ω
as
1
M2 (L2 )
2
¶µ
π (τ )
λ (τ )
¶µ
π (τ )
λ (τ )
¶
,
¶
.
(17)
The solutions to this linear system are isomorphic to the solution space of the original
non linear Riccati ODE through the map:
µ
¶
π (τ )
−1
R
Φ
= π (τ ) λ (τ ) .
λ (τ )
In conclusion, for all i = 1, ..., m we obtain the explicit solution for (15):
Vi (τ ) =
M1i (τ ) Vi (0) + M2i (τ )
,
M3i (τ ) Vi (0) + M4i (τ )
with
µ
M1i (τ ) M2i (τ )
M3i (τ ) M4i (τ )
¶
½µ
= T exp
−τ ΩDD
ii
−τ /2
−
0
Rτ
0
γ
ei (τ 0 ) dτ 0
¶¾
,
where T exp denotes the time ordered exponential.
4.2
Wishart Affine models: D=P os+ (r, R)
Wishart stochastic processes have been firstly investigated by Bru (1991), while Gourieroux
and Sufana (2003) showed their potentialities for financial and econometric applications and
their ability to model multivariate intertemporal correlations.
Within the Wishart framework, factors are assumed to evolve according to the following
(symmetric matrix valued) process:
¡
¢
1/2
1/2
T
dXt = kQQT + ΩXt + Xt ΩT dt + Xt dWt Q + QT (dWt ) Xt ,
(18)
with Ω, Q ∈ Mr (R) (Q invertible), Wt ∈ Mr (R) is composed by r2 independent Brownian
r(r+1)
motions and k ≥ nr − 1 = r−1
).
2 (here dim(V ) = n =
2
The infinitesimal generator of the matrix process Xt is given by (see e.g. Bru 1991):
£¡
¢
¤
A = T r kQQT + 2Ωx D + 2xDQT QD ,
(19)
´
³
with D = ∂x∂ij . The characteristic function is
"
ΨX (u, x, t, T ) = Et
à Z
exp −
!
T
#
(η0 + T r [ηXs ]) ds exp (T r [iuXT ])
(20)
t
¡
¢
= exp V 0 (t, iu) − T r [V (t, iu) x] ,
10
(21)
and the corresponding (Matrix) Riccati ODE are given by
−
∂
V(τ ) = η + ΩT V(τ ) + V(τ )Ω − 2V(τ )QT QV(τ ),
∂τ
V(0) = u ∈ Sym(r, R).
(22)
This class of ATSM corresponds to the choice of state space D=P os+ (r, R), which is
an irreducible symmetric cone. The corresponding (simple) EJA is obtained by considering
the vector space V = Sym(r, R), with scalar product hA, Bi = T r [AB] and Jordan product
A ∗ B = (AB + BA)/2 (AB indicates the matrix product). Observe that the quadratic
representation of this algebra is given by: P (A) S = ASA, with S ∈ Sym(r, R). According
to our main theorem, the model is then solvable for all QT Q, C ∈ P os+ (r, R) and Ω ∈
Mr (R). As a consequence, the (Matrix) Riccati ODE can be written as a linear combination
of the generators for the Lie algebra sl (2r, R) (see also Lafortune and Winternitz 1996).
The equation (22) for V (τ ) is a symmetric Riccati Matrix ODE, which corresponds to
a linear flow in homogeneous coordinates (the r dimensional generalization of projective
−1
spaces), so in analogy with the one dimensional case, let us put V (τ ) = G (τ ) F (τ ) with
F (τ ) ∈ GL(r, R), G(τ ) ∈ Mr (R), then we get the linear ODE:
¶
µ
¶ µ
T ¶µ
d
G (τ )
G (τ )
−Ω 2QQ
,
−
=
F (τ )
F (τ )
η
ΩT
dτ
which is solved by
µ
¶
· µ
¶¸ µ
¶
µ
¶µ
¶
G (τ )
−Ω 2QQT
G (0)
M1 (τ ) M2 (τ )
G (0)
= exp −τ
=:
,
F (τ )
F (0)
M3 (τ ) M4 (τ )
F (0)
η
ΩT
where Mi (τ ) , i = 1, ..4 are r × r matrices. In conclusion we get:
−1
V (τ ) = G (τ ) F (τ )
(23)
= (M1 (τ ) V (0) + M2 (τ )) (M3 (τ ) V (0) + M4 (τ ))
−1
.
Notice that the last expression is completely explicit and does not involve any integration:
it is therefore more efficient than the alternative expression one obtains using the method
proposed by Walcher (1991) (see e.g. Gourieroux and Sufana 2003).
A promising line for future research is to consider symmetric cone domains different
from the above examples, which could be used to produce new solvable ATSM models. In
particular, the Lorentz cone domain provides a class of solvable models yet unexplored which
could possibly be useful in financial applications.
References
[1] Balduzzi P., Das S. R., Foresi S. and R. K. Sundaram (1996) ”A Simple Approach to
Three-Factor Models of Interest Rates”. Journal of Fixed Income, 6(3).
[2] Brown, R. and S. Schaefer (1994) ”Interest rate volatility and the shape of the term
structure”. Philosophical Transactions of the Royal Society: Physical Sciences and Engineering, 347, 449-598.
[3] Bru, M. F. (1991) ”Wishart Processes”. Journal of Theoretical Probability, 4, 725-743.
[4] Dai, Q. and K. Singleton (2000) ”Specification Analysis of Affine Term Structure Models”. Journal of Finance, 55, 1943-1978.
[5] Duffie, D. and R. Kan (1996) ”A Yield-Factor Model of Interest Rates”. Mathematical
Finance, 6 (4), 379-406.
11
[6] Duffie, D., J. Pan and K. Singleton (2000) Transform analysis and asset pricing for
affine jump-diffusions”. Econometrica, 68, 1343-1376.
[7] Duffie, D., D. Filipović and W. Schachermayer (2003) ”Affine Processes and Applications in Finance”. Annals of Applied Probability, 13, no. 3, 984–1053.
[8] Faraut J. and A. Korányi (1994) ”Analysis on symmetric cones” Oxford Mathematical
Monographs, Oxford University Press, New York.
[9] Filipović, D. (2005) ”Time-inhomogeneous Affine Processes”. Stochastic Processes and
their Applications, 115, no.4, 639–659.
[10] Gourieroux, C. and R. Sufana (2003) ”Wishart Quadratic Term Structure Models”.
CREF 03-10, HEC Montreal.
[11] Grasselli, M. and C. Tebaldi (2004) ”Bond Price and Impulse Response Function for
the Balduzzi, Das, Foresi and Sundaram (1996) model”. Economic Notes, 33, no. 3,
359-374.
[12] Lafortune, S. and P. Winternitz (1996) ”Superposition formulas for pseudounitary Riccati equations”, Journal of Mathematical Physics, 37, 1539-1550.
[13] Lawson, J. and Y. Lim (2000) ”Lie semigroups with triple decompositions”. Pacific
Journal of Mathematics, 194, no. 2, 393-412.
[14] Lawson, J. and Y. Lim (2003) ”The sympletic semigroup and Riccati differential equations: a case study”. Preprint.
[15] Moan, P. C. and Oteo, J. A. (2001) ”Convergence of the exponential Lie series”. Journal
of Mathematical Physsics, 42, no. 1, 501–508.
[16] Walcher, S. (1991) ”Algebras and Differential Equations”. Hadronic Press.
[17] Walcher, S. (1986) ” Uber polynomiale, insbesondere Riccatische, Differentialgleichungen mit Fundamentallosungen”. Math. Ann.. 275, 269-280.
Appendix
Proof of Theorem 2
Under the assumption that Σα (x) is non singular ∀x ∈ D, it is immediate to observe
that the lineariy of Σα (x) with respect to the argument x implies the relation:
£
¤
hx, α (V)i = T r Σα (x) VV T
∀x ∈ D
£
¤
which defines uniquely a vector α (V). This vector belongs to D, since T r Σα (x) VV T > 0
for x ∈ D, and the domain is self dual. Now we claim that admissibility implies:
α (V) = P (V) αΣ ,
(24)
where P (V) is the (unique up to a proportionality factor) quadratic representation of the
EJA corresponding to the irreducible symmetric cone D and αΣ ∈ D.
12
Consider a coordinate transformation x → y = gx where g ∈ G (D), the group of linear
automorphism of D defined by
G (D) = {g ∈ GL (V ) | gD = D} .
Affinity and admissibility imply that the conditional covariance will transform like:
¡ ¢−1
−1
Σα (x) = (g) Σα (gx) g T
∀x ∈ D, ∀g ∈ G (D) ,
so that
·
³¡ ¢−1 ´ ³¡ ¢−1 ´T ¸
£
¤
T
T r Σα (x) VV = T r Σα (gx) g T
V
gT
V
,
which in turn implies
D
³¡ ¢−1 ´E D
³¡ ¢−1 ´E
hx, α (V)i = gx, α g T
V
= x, g T α g T
V .
Now let us define αΣ = α (e) , where e is the identity element of the EJA: applying the
previous relation with V = e yields
D
³¡ ¢−1 ´
E
x, g T α g T
e − αΣ = 0 ∀x ∈ D, ∀g ∈ G (D) .
Since P (e) = Id, using (5) we get g T P
D
x, g T α
³¡
gT
³¡ ¢
´
−1
gT
e g = Id ∀g ∈ G (D) and
³¡ ¢−1 ´
E
¢−1 ´
e − gT P gT
e gαΣ = 0 ∀x ∈ D, ∀g ∈ G (D) ,
and applying the above equality for a system x = ei ∈ D, i = 1, ..r, of orthogonal idempotents
for the EJA (which generate the whole EJA V ), we can conclude that ∀g ∈ G (D)
³¡ ¢−1 ´
³¡ ¢−1 ´
α gT
e = P gT
e gαΣ .
−1
Observe that P (u) ∈ G (D) for u ∈ D, therefore we can put g = P (u) in the above
¡ ¢−1
equality. By symmetry we obtain g T
= P (u) ∈ G (D) and using the fundamental
property (5) we get
α (P (u) e) = P (u) αΣ
∀u ∈ D.
Applying again the above identity to an orthogonal system of idempotents ei ∈ D,
i = 1, ..r for the EJA, and using the fact that both α (V) and P (V) are quadratic forms in
V we can conclude that (24) holds true. The converse result follows easily from the above
discussion.
Proof of Proposition 3
In order to prove Proposition 3, first we provide an essential review of the KKT (Kantor,
Koecher and Tits) construction and its relationship with the ATSM problem. An extensive
discussion of the geometric and algebraic aspects of these topics can be found in Faraut and
Korányi (1994), Walcher (1991), Walcher (1986).
Let C ⊂ V be a symmetric cone; observe that G (C) is a Lie subgroup of G (TC ), the
group of holomorphic automorphisms of the ”tube domain” TC = V + iC.
The whole group of affine linear transformations on the tube domain, z → gz + a,
g ∈ G (C), a ∈ V , can be obtained as the semidirect product of translations N + =
13
{a ∈ V | z → z + a ∀z ∈ TC } and automorphisms in G (C). Adding the inversion J : z →
−z −1 , the whole group G (TC ) is generated.
© ª
Let g (TC ) be the Lie algebra of G (TC ): each curve gtR t∈R+ ⊂ G (TC ) defines a vector
field R (u) on V through the relation:
Df (z) R (z) :=
¢
d ¡ R
f gt (z) |t=0
dt
∀f ∈ C 1 (TC ) .
¡
¢−1
−
For example, consider the subgroup N − = J◦N + ◦J with elements gtN (z) = z −1 − tv
when V = R, t ∈ R+ . Then direct computation shows:
¢−1 ´
d ³¡ −1
f z − tv
= f 0 (z) z 2 v,
dt
|t=0
thus giving R−1 (z) = z 2 v = P (z) v.
The vector fields generated by N + , G (C) , N − close in the so called KKT (finite dimensional Lie) algebra g (TC ):
Proposition 8 (see Faraut and Korányi 1994, pg. 208) g (TC ) can be represented as follows:
g (TC ) = g−1 (TC ) ⊕ g0 (TC ) ⊕ g+1 (TC ) ,
where
i) g−1 (TC ) = {R− (z) = −P (z) ν; z ∈ V, ν ∈ V } generates the subgroup N − with ele¡
¢−1
−
ments gtN (z) = z −1 − tv
;
ii) g0 (TC ) = {R0 (z) = T z : z ∈ V, T ∈ g (C)} generates G (C) given by:
©
ª
g (C) = H ∈ Mn : 2P (V, HV) = HP (V) + P (V) H T ;
+
iii) g+1 (TC ) = {R+ (z) = u, u ∈ V } generates N + , i.e. gtN (z) = z + tu.
Among the elements of g (TC ) , we will be interested in those which generate subgroups
preserving the domain C, i.e. gt (C) ⊆ C ∀t ∈ R+ ; remarkably the following proposition
provides the general expression of the Lie generators for a compression semigroup, i.e. a one
parameter semigroup which leaves the symmetric cone C invariant.
Proposition 9 (see e.g. Section 7 of Lawson and Lim 2000) Let G (TC ) be the group of
holomorphic automorphisms and g (TC ) its Lie Algebra. Consider the semigroup of compressions ΓC ⊆ G (ΓC ):
ΓC := {g ∈ G (TC ) : gC ⊆ C} .
Each g ∈ ΓC is generated by R ∈ g (TC ) of the form:
R (u) = −P (u) a + T u + b
a, b ∈ C,
T ∈ g0 (TC ) .
∀u ∈ TC
(25)
Given the above results we can now prove Proposition 3.
Proof of Prop. 3: Following the argument of Proposition 7.4 in Duffie et al. (2003),
an admissible regular ATSM exists provided that:
i) the characteristic function of the Markov semigroup is well defined and
ii) the flow of the Riccati ODE arising in (4) exists and is uniquely globally defined
∀t ≥ t0 ,∀x ∈ D.
14
Now if Ω0 = kaΣ , with k > nr − 1, where r is the rank of the EJA, then condition i)
is satisfied from Theorem VII.1.3 in Faraut and Korányi (1994), while the global existence
condition concerning the flow of the Riccati comes from Sections 6 and 7 of Lawson and
Lim (2000), where they provide the expression of the compression semigroup generators (25)
for a generic symmetric cone. Repeating the arguments of Lawson and Lim (2003), we can
conclude the global existence of the flow of the Riccati.
Proof of Theorem 4
An important by-product of the KKT construction is the classification of all possible
quadratic finite dimensional algebras, and in particular (the vector fields of) the linearizable
Riccati ODE. Let us now briefly recall this result in order to prove our Theorem 4.
Let us consider a particular subalgebra of A (V, V ), namely the (infinite dimensional)
algebra of polynomial functions P ol (V ). We can define Pk ⊂ P ol (V ) as the subspace of
homogeneous polynomials of degree k, for k ≥ 0. Since [Pj , Pk ] ⊂ Pj+k for all j, k ∈ N, it
follows that
P ol (V ) = ⊕k∈N Pk ,
and it is a graded subalgebra of A (V, V ) (see Walcher 1991, pg. 118). For example, the
subspace P0 contains all constant maps and can be identified with V , while P1 can be
identified with Hom (V, V ).
According to our definition of solvability and to Chapter 8 of Walcher (1991), the family
L of vector fields of the Riccati ODE in an EJA has to close in a finite dimensional subalgebra
of A (V, V ), and in particular in a graded subalgebra of the type
L = L0 ⊕ L1 ⊕ L2 ,
where L0 = V (i.e. the algebra is transitive), Li ⊂ Pi , i = 1, 2.
The following theorem shows that the classification of these subalgebras is easily related
with the Kantor-Koecher-Tits class of algebras g (TC ):
Theorem 10 (Walcher 1991, Theorem 8.6 b) Let L = L0 ⊕L1 ⊕L2 be a transitive subalgebra
of P ol (V ). Then L is a subalgebra of a Kantor-Koecher-Tits algebra g (TC ) with Li ⊂
gi−1 (TC ), i = 0, 1, 2.
In other terms, the KKT construction classifies all possible quadratic finite dimensional
subalgebras, and in particular (the vector fields of) the linearizable Riccati ODE.
Proof of Th. 4: Each admissible ATSM in an irreducible symmetric cone generates a
quadratic polynomial ODE. Solvability requires that the set fi (V) ∈ Pi , i = 0, 1, 2 generate
a graded transitive subalgebra, say L. From the previous theorem, g (TC ) is maximal among
graded transitive subalgebras, hence L ⊆ g (TC ) . On the other hand, since L must contain
the constant vector fields, we have that V = g−1 (TC ) ⊂ L, and by Theorem 2 above L
must also contain g+1 (TC ) = P (V) a, a ∈ V . Now we can conclude that L ⊇ g (TC ) since
g0 (TC ) ⊆ [g−1 (TC ) , g+1 (TC )]. Hence L = g (TC ) and the theorem is proved.
15