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STATE VARIABLES AND THE AFFINE NATURE OF MARKOVIAN HJM

TERM STRUCTURE MODELS

CARL CHIARELLA1 AND OH KANG KWON2

1 School of Finance and Economics

University of Technology, Sydney

PO Box 123

Broadway NSW 2007

Australia

[email protected]

2 School of Banking and Finance

University of New South Wales

Sydney NSW 2052

Australia

[email protected]

ABSTRACT. Finite dimensional Markovian HJM term structure models provide an ideal

setting for the study of term structure dynamics and interest rate derivatives where the

flexibility of the HJM framework and the tractability of Markovian models coexist. Conse-

quently, these models became the focus of a series of papers including Carverhill (1994),

Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima

(1998) and de Jong and Santa-Clara (1999). In Chiarella and Kwon (2001b), a common

generalisation of these models was obtained in which the components of the forward rate

volatility process satisfied ordinary differential equations in the maturity variable. How-

ever, the generalised models require the introduction of a large number of state variables

which, at first sight, do not appear to have clear links to market observed quantities. In

this paper, it is shown that the forward rate curves for these models can often be expressed

as affine functions of the state variables, and conversely that the state variables in these

models can often be expressed as affine functions of a finite number of benchmark forward

rates. Consequently, for these models, the entire forward rate curve is not only Markov butaffine with respect to a finite number of benchmark forward rates. It is also shown that the

forward rate curve can be expressed as an affine function of a finite number of yields which

are directly observed in the market. This property is useful, for example, in the estimation

of model parameters. Finally, an explicit formula for the bond price in terms of the state

variables, generalising the formula given in Inui and Kijima (1998), is provided for the

models considered in this paper.

1. INTRODUCTION

The Heath, Jarrow and Morton (1992) term structure framework is widely regarded as

the most general and flexible setting for the study of interest rate dynamics, having the

capacity to generate a wide range of forward rate dynamics and the ability to incorpo-rate any prevailing market conditions with internal consistency. Since the bond market in

this framework is arbitrage free and complete subject only to mild restrictions, the Heath-

Jarrow-Morton (HJM) framework also provides a convenient setting for the study of inter-

est rate derivatives.

However, the generality and the flexibility of the HJM framework is often at the ex-

pense of theoretical and numerical tractability, since the HJM models are non-Markovian

in general. In particular, this means that the standard Feynman-Kac theorem no longer

applies, rendering inaccessible the well-developed tools from the theory of partial differ-

ential equations; Monte Carlo methods for these models are often inefficient and require

large storage; and these models generally give rise to non-recombining trees. To overcome

these problems, researchers turned to the subclass of HJM models that admit Markovian

Date: First draft June 8, 1999. Current revision April 11, 2001. Printed May 8, 2001.

1

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2 CARL CHIARELLA AND OH KANG KWON

realisations. In these Markovian HJM models the flexibility of the HJM framework and the

tractability of Markovian models coexist to provide an ideal setting under which to study

term structure dynamics and interest rate derivatives.It turns out that an HJM model is completely determined by the initial forward rate

curve and the forward rate volatility process. However, since the initial forward rate curve

is completely determined by the market, the only way to obtain finite dimensional Mar-

kovian models within the HJM framework rests in the pertinent choice, or specification,

of the volatility process. Various restrictions on the forward rate volatility process that

lead to finite dimensional Markovian HJM models were obtained in Carverhill (1994),

Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima

(1998) and de Jong and Santa-Clara (1999). Although the models in Inui and Kijima

(1998) were higher dimensional analogues of the models from Ritchken and Sankarasub-

ramanian (1995), the links between the other models remained unclear. Based on a simple

observation that the components of the forward rate volatility must satisfy ordinary differ-

ential equations in the maturity variable, a common generalisation to all these models was

obtained in Chiarella and Kwon (2001b).

Although theoretically appealing, the Markovian HJM models obtained in Chiarella and

Kwon (2001b) involve a large number of state variables which, at first sight, do not appear

to have direct links to market observed quantities. The main aim of this paper is to show

under suitable restrictions that the state variables are, in fact, affine functions of benchmark

forward rates or yields. This observation is useful, for example, in the calibration of model

parameters since the state variables for these models are directly observed in the market.

This observation also leads to an explicit formula for the bond price in terms of the state

variables for these models.

The structure of the remainder of the paper is as follows. The HJM framework is briefly

reviewed in Section 2, and the class of Markovian HJM models introduced in Chiarella

and Kwon (2001b) is reviewed in Section 3. Additional state variables for the Markovian

HJM models are then introduced in Section 4 together with results that relate them to thevariables introduced in Chiarella and Kwon (2001b). Sections 5 and 6 contain the main

results which express the state variables in terms of benchmark forward rates and yields.

Section 7 applies these results to simple examples, and the paper finally concludes with

Section 8.

2. THE HEATH, JARROW AND MORTON FRAMEWORK

Fix > 0 and let (, F, (Ft)t[0,],P) be a complete filtered probability space sat-isfying the usual conditions, where the filtration (Ft)t[0,] is generated by a standardn-dimensional P-Wiener process w(t). For t [0, ] denote by p(t, ) the price ofa -maturity zero coupon bond at time t, and define the -maturity instantaneous forward

rate f(t, ) by the equation

f(t, ) = lnp(t, )

. (2.1)

Then in the risk-neutral formulation of Heath, Jarrow and Morton (1992) term structure

models on (, F, (Ft)t[0,],P), the forward rates are assumed to satisfy the stochastic

integral equation1

f(t, ) = f(0, ) +

t0

(s, )(s, ) ds +

t0

(s, ) dw(s), (2.2)

1It is a consequence of the HJM forward rate drift restriction that the forward rate drift under the risk neutral

measure is (t, )

t (t, u) du.

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STATE VARIABLES AND AFFINE NATURE OF MARKOVIAN HJM MODELS 3

where f(0, ) is the initial forward rate curve, (t, ) is an adapted Rn+-valued forward

rate volatility process, (t, ) = t (t, u) du and the superscript denotes matrix trans-position. Note that a Heath, Jarrow and Morton, henceforth HJM, term structure model iscompletely determined by its dimension n, the initial forward rate curve f(0, ) and theforward rate volatility process (t, ).

Let d N+. Then an HJM model M is said to be Markovian with d factors if there ex-ists a d-dimensional (Ft)-Markov process z(t) such that the forward rates can be expressedin the form f(t , , z(t)), where the explicit dependence on t and are deterministic andthe dependence on the Wiener path enters only through z(t). Many of the short rate mod-els such as Vasicek (1977), Cox, Ingersoll and Ross (1985) and Ho and Lee (1986) are

examples of Markovian HJM models.

A Markovian HJM model A is said to be affine if the forward rate process can be writtenin the form

f(t , , z(t)) = h0(t, ) + h(t, )z(t), (2.3)

where h0(t, ) and h(t, ) are deterministic R and Rd-valued functions respectively. Note

that in view of (2.1) A is affine if and only if the bond price in A is exponential affine2 inthe sense of Duffie and Kan (1996).

Throughout this paper, the Musiela (1993) parameterisation, = t + x, of the maturityvariable will often be used. Under this parameterisation the fundamental quantities are the

benchmark forward rates r(t, x) defined by the equation

r(t, x) = f(t, t + x). (2.4)

The benchmark forward rates satisfy the stochastic integral equation

r(t, x) = f(0, t + x) +

t0

(s, t + x)(s, t + x) ds +

t0

(s, t + x)dw(s), (2.5)

and the stochastic differential equation

dr(t, x) =

x

r(t, x) +

1

2|(t, t + x)|2

dt + (t, t + x)dw(t), (2.6)

where | | denotes the standard Euclidean norm. If the corresponding benchmark bondprices b(t, x) are defined by b(t, x) = p(t, t + x), then it follows from (2.1) that theysatisfy the equation

b(t, x) = exp

x0

r(t, u) du

. (2.7)

3. REVIEW OF MARKOVIAN HJM MODELS

In this section, the class of Markovian HJM models introduced in Chiarella and Kwon(2001b) is briefly reviewed. As indicated above, this class includes the models considered

in Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997),

Inui and Kijima (1998) and de Jong and Santa-Clara (1999) as special cases.

Let m N, and assume given d1, . . . , dn N and x1 < x2 < < xm R+. Thenin addition to the standard HJM assumptions, suppose further that:

[A1] (t, ) is a function oft, and m benchmark forward rates r(t, x1), . . . , r(t, xm),so that

(t, ) = (t,,r(t, x1), . . . , r(t, xm)), (3.1)

2Recall that Duffie and Kan (1996) refer to p(t, ) as being exponential affine if there exists a d-dimensionalMarkov process z(t) such that p(t, ) = exp[k0(t, ) k(t, )z(t)], where k0(t, ) and k(t, ) are deter-

ministic R and Rd-valued functions respectively.

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[A2] for each 1 i n, i(t, ) is di times differentiable with respect to and satisfiesa di-th order homogeneous linear differential equation of the form

3

Lii(t, ) = 0, where Li =di

di

di1j=0

i,j()j

j(3.2)

and i,j() are deterministic functions.

Note that the volatility processes considered in Carverhill (1994), Ritchken and Sankara-

subramanian (1995), Inui and Kijima (1998) and de Jong and Santa-Clara (1999) satisfy the

above assumptions with di = 1 for all i, and the volatility process considered in Bhar and

Chiarella (1997) satisfy the above assumptions with di = k 0 and Li = (/t i)k

.

In order to establish that the HJM models satisfying assumptions [A1] and [A2] admit

finite dimensional Markovian realisations, a finite set of state variables that capture the

history of the Wiener path must be identified. Thus, for each 1 i n, 0 p q < di,0 l < di and x [0, ), define the stochastic processes

p,qi,x (t) and

li,x(t) by

p,qi,x (t) =

t0

pi(s, t + x)

xpqi(s, t + x)

xqds, (3.3)

li,x(t) =

t0

li(s, t + x)

xl

t+xs

i(s, u) duds +

t0

li(s, t + x)

xldwi(s). (3.4)

Theorem 3.1 (Chiarella and Kwon (2001b, Theorem 2.3)). Suppose that the forward rate

volatility process (t, ) satisfies the assumptions [A1] and [A2]. Then the correspondingHJM model is Markovian with respect to the set

{r(t, xk), pi,qii,xk

(t), lii,xk(t)} (3.5)

where 1 i n, 1 k m, 0 pi qi < di and1 li < di.

The proof of the above theorem relies on noting the interdependence of the dynamics

of r(t, xk), pi,qii,xk

(t), and lii,xk(t) on one another, and recognising that when assumption

[A2] is satisfied then pi,qii,xk (t) and lii,xk

(t) are expressible in terms of the processes in (3.5)ifpi di, qi di, or li di.

At this stage, the economic significance of the state variables p,qi,xk(t) and li,xk

(t) intro-duced in (3.3) and (3.4) is unclear, and it is not immediately obvious if they are expressible

in terms of the quantities directly observed in the market. This problem is resolved in

Section 5, in which a clear link is established between the state variables and the market

forward rate curve.

4. ALTERNATIVE SET OF STATE VARIABLES

In this section, the HJM models that satisfy the assumptions [A1] and [A2] of Section 3

are considered with respect to an alternative set of state variables. These variables providean easier passage to linking the variables p,qi,xk(t) and li,xk

(t) in (3.5) to quantities that areeconomically meaningful.

If it is assumed that i,j() in (3.2) are continuous, then it is known from the theory

of ordinary differential equations4 that there exist di linearly independent deterministicfunctions i,j() such that Lii,j() = 0 for all 1 j di, and stochastic coefficientprocesses ci,j(t) = ci,j(t, r(t, x1), . . . , r(t, xm)) such that

i(t, ) =

dij=1

ci,j(t) i,j(). (4.1)

3A similar condition was obtained in the deterministic volatility case by Bj ork and Gombani (1999), and

generalised to the separable volatility case by Bjork and Svensson (1999).

4See for example Coddington and Levinson (1955, Theorem 5.1).

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Equation (4.1) simply expresses the general solution i(t, ) of the ordinary differentialequation (3.2) as a linear combination of di linearly independent solutions i,j() with

coefficients ci,j(t). The significance of this expression is that the dependence of i(t, )on the current time t and the maturity time are essentially separated.

As a motivation for the variables to be introduced, note that the benchmark forward

rates corresponding to the volatility process (4.1) satisfy the stochastic integral equation

r(t, x) = f(0, t + x) +ni=1

t0

i(s, t + x) dwi(s)

+

ni=1

t0

i(s, t + x)

t+xs

i(s, u) duds

= f(0, t + x) +n

i=1di

j=1i,j(t + x)

t

0

ci,j(s) dwi(s)

+

ni=1

dij,j=1

i,j(t + x)

t0

ci,j(s)ci,j(s)

t+xs

i,j (u) duds.

= f(0, t + x) +ni=1

dij=1

i,j(t + x)

t0

ci,j(s) dwi(s)

+n

i=1

dij,j=1

i,j(t + x)

t0

ci,j(s)ci,j(s)

t+x0

i,j(u) du

s0

i,j(u) du

ds.

= f(0, t + x) +ni=1

di

j=1

i,j(t + x)

t0

ci,j(s) dwi(s) di

j=1

t0

ci,j(s)ci,j (s)

s0

i,j (u) duds

+

ni=1

dij,j=1

i,j(t + x)

t+x0

i,j(u) du

t0

ci,j(s)ci,j (s) ds.

(4.2)

From this it can be seen that the quantities which play an important role in describing the

dynamic evolution of the benchmark forward rates are the deterministic functions i,j(t)defined by

i,j(t) =

t

0

i,j(u) du, (4.3)

and the stochastic processes j,j

i (t), j,j

i (t) and ji (t) defined by

j,j

i (t) =

t0

ci,j(s) ci,j(s) ds, (4.4)

ji (t) =

t0

ci,j(s) dwi(s) di

j=1

t0

ci,j(s) ci,j (s) i,j(s) ds, (4.5)

for 1 i n and 1 j, j di. One of the consequences of (4.2) is that the pathdependence of the forward rate curve is completely captured by the stochastic processes

j,j

i (t) and ji (t).

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The next lemma establishes the connection between the state variables p,qi,x (t) and

li,x(t) as defined in (3.3) and (3.4) and the stochastic processes j,j

i (t) and ji (t).

Lemma 4.1. Letp,qi,x (t) andli,x(t) be defined as in (3.3) and (3.4). Then

p,qi,x (t) =

1j,jdi

pi,j(t + x)

xp

qi,j(t + x)

xqj,j

i (t), (4.6)

li,x(t) =

1jdi

li,j(t + x)

xlji (t)

+

1j,jdi

i,j(t + x)li,j(t + x)

xlj,j

i (t).

(4.7)

In particular, p,qi,x (t) andli,x(t) are linear in

j,j

i (t) andji (t).

Proof. Consider firstly (4.6). Using the definition of p,qi,x (t) from (3.3)

p,qi,x (t) =

t0

pi(s, t + x)

xp

qi(s, t + x)

xqds

=

1j,jdi

pi,j(t + x)

xp

qi,j(t + x)

xq

t0

ci,j(s)ci,j(s) ds

=

1j,jdi

pi,j(t + x)

xp

qi,j(t + x)

xqj,j

i (t).

where (4.4) was used in the last equality. Similarly, using the definition of li,x(t) from(3.4)

li,x(t) =t0

l

i(s, t + x)xl

t+xs

i(s, u) duds +t0

l

i(s, t + x)xl

dwi(s)

=

1j,jdi

li,j(t + x)

xl

t0

ci,j(s)ci,j (s)

t+xs

i,j(u) duds

+

1jdi

li,j(t + x)

xl

t0

ci,j(s) dwi(s)

=

1j,jdi

li,j(t + x)

xl

t0

ci,j(s)ci,j (s) [i,j(t + x) i,j (s)] ds

+ 1jdi

li,j(t + x)

xl t

0

ci,j(s) dwi(s)

=

1jdi

li,j(t + x)

xl

t

0

ci,j(s) dwi(s)

1jdi

t0

ci,j(s) ci,j (s) i,j(s) ds

+

1j,jdi

li,j(t + x)

xli,j(t + x)

t0

ci,j(s) ci,j (s) ds

=

1jdi

li,j(t + x)

xlji (t) +

1jjdi

i,j (t + x)li,j(t + x)

xll,j

i (t)

where the last equality follows from (4.4) and (4.5).

Inversion of (4.6) and (4.7) to express j,j

i (t) and j

i (t) in terms ofp,qi,x (t) and

li,x(t)

relies on the invertibility of a certain coefficient matrix whose precise definition requires

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the introduction of some notation. Firstly, let i =12

di(di + 1),

Pi = {(j, j) N N | 1 j j di},

and define i : Pi {1, 2, . . . , i} to be the lexicographical ordering given by

i(j, j) =

1

2(j 1)(2di j) + j

, (4.8)

so that for example, i(1, j) = j and i(2, j) = di 1 + j and i(di, di) = i. Notethat 1i is well defined since i is a bijection. Next define the coordinate projectionsi : N N N, for i = 1, 2, so that 1(j, j

) = j and 2(j, j) = j, and for notational

convenience denote j,i = j 1i . For example, since i(1, j

) = j for 1 j di,it follows that 1,i(j

) = 1 and 2,i(j) = j for 1 j di, and similarly since

i(2, j) = di 1 + j

for 2 j di, it follows that 1,i(di 1 + j) = 2 and

2,i(di 1 + j) = j for 2 j di. Finally, define j,j by

j,j = 12 , j = j

1, j = j . (4.9)

Now, for any x [0, ), define a di di matrix A(i,x), a di i matrix B(

i,x) and an

i i matrix C(i,x) by

A(i,x)j,j =

j1i,j(t, x)

xj1, (4.10)

B(i,x)j,j = 1,i(j),2,i(j)i,2,i(j)

j1i,1,i(j)xj1

+ 1,i(j),2,i(j)i,1,i(j)j1i,2,i(j)

xj1

(4.11)

C(i,x)j,j = 1,i(j),2,i(j)

1,i(j)1i,1,i(j)

x1,i(j)1

2,i(j)1i,2,i(j)

x2,i(j)1

+ 1,i(j),2,i(j)1,i(j)1i,2,i(j)

x1,i(j)1

2,i(j)1i,1,i(j)

x2,i(j)1.

(4.12)

and let M(i,x) be the (di + i) (di + i) matrix given by

M(i,x) =

A(i,x) B(i,x)

0 C(i,x)

(4.13)

Then (4.6) and (4.7) can be written in the matrix form0i,x(t),

1i,x(t), . . . ,

di1i,x (t),

0,0i,x (t),

0,1i,x (t), . . . ,

di1,di1i,x (t)

= M(i,x)

1i (t),

2i (t), . . . ,

dii (t),

1,1i (t),

1,2i (t), . . . ,

di,dii (t)

(4.14)where the superscript denotes matrix transposition and p,qi,x (t) and j,j

i (t) have beenwritten in lexicographical order. The next lemma gives the conditions under which (4.6)

and (4.7) can be inverted.

Lemma 4.2. Suppose for each 1 i n there exists 1 ki, ki m such that A

(i,xki)

and C(i,xk

i)

are invertible. Then j,j

i (t) and ji (t) can be expressed as linear functions

ofp,qi,xki

(t) andli,xki(t) with deterministic coefficients. In this case, the processes j,j

i (t)

and j

i (t), where 1 j j di, 1 j

di and 1 i n, are contained in thelinear span of the Markovian system given in Theorem 3.1.

Proof. This is clear from (4.13) and (4.14).

Proposition 4.3. Let M be an HJM model satisfying the assumptions [A1] and [A2]. If

the conditions of Lemma 4.2 are satisfied then M is an affine term structure model.

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Proof. Firstly, the set {r(t, xk), pi,qii,xk

(t), lii,xk} forms a Markovian system by Theorem 3.1.Next, from (2.5) and (3.4), the benchmark forward rate r(t, x) can be written in the form

r(t, x) = f(0, t + x) +ni=1

0i,x(t).

Using (4.7) to substitute for 0i,x(t) gives

r(t, x) = f(0, t + x) +n

i=1

diji=1

i,ji(t + x) jii (t)

+n

i=1

diji,j

i=1jij

i

ji,j

i

i (t + x) ji,j

i

i (t)

(4.15)

where

ji,j

i

i (t + x) =

i,ji(t + x) i,ji(t + x) ifji = j

i,

i,ji(t + x) i,ji(t + x) + i,ji(t + x) i,ji(t + x) ifji < ji.

(4.16)

Since i,ji(t + x) and i,ji(t + x) are deterministic functions, r(t, x) is an affine function

of ji,j

i

i (t) and jii (t) for all x R+, where 1 i n and 1 ji j

i di. But

since the conditions of Lemma 4.2 are satisfied by assumption, j,j

i (t) and ji (t) can be

expressed as linear functions of p,qi,xki

(t) and i,xki (t) with deterministic coefficients. This

establishes that r(t, x) is an affine function of p,qi,xki

(t) and i,xki (t), and hence that M is

affine.

The next corollary extends the Inui-Kijima bond price formula to the more generalMarkovian models introduced in Chiarella and Kwon (2001b).

Corollary 4.4. Let M be an HJM model satisfying the assumptions of Proposition 4.3.Then the benchmark bond prices in M are exponential affine and are given by the formula

b(t, x) =b(0, t + x)

b(0, t)exp

n

i=1

diji=1

i,ji(t, x) jii (t) +

ni=1

diji,j

i=1jij

i

ji,j

i

i (t, x) ji,j

i

i (t)

,

(4.17)

where

i,ji(t, x) = x

0

i,ji(t + u) du,

ji,j

ii (t, x) =

x0

ji,j

ii (t + u) du.

Proof. From (2.7), the determination of b(t, x) involves the computation of the integralx0

r(t, u) du. But from (4.15) this, in turn, requires the integrals of deterministic functions

f(0, u), i,ji(u) and ji,j

i

i (u), which results in (4.17).

Note that the bond price given in (4.17) is exponential affine in the sense of Duffie and

Kan (1996). However, note also that the functional form of the forward rate volatility has

not been restricted beyond (3.1) and (3.2). In particular, the components of (t, ) arenot restricted to be square root affine in the state variables, which is a restriction obtained

in Duffie and Kan (1996) for the class of affine models they consider. Thus the class of

affine models considered in this paper include those which do not necessarily fall under

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the Duffie-Kan characterisation. For a more complete discussion on the assumptions un-

derlying the Duffie-Kan characterisation of affine models see Kwon (2001).

5. STATE VARIABLES AND THE FORWARD RATE CURVE

The purpose of this section is to determine the explicit links between the state variables

and the forward rate curve for an HJM model M which satisfies the assumptions [A1] and[A2] of Section 3 and the conditions of Lemma 4.2. Firstly, recall from Theorem 3.1 that

the models which satisfy these assumptions are Markov with respect to p,qi,x (t) and li,x(t).

5

But since these variables are expressible in terms of ji,j

i

i (t) and jii (t) by Lemma 4.1, it

suffices to determine the links between the processes ji,j

ii (t) and

jii (t) and the forward

rate curve. Now, (4.15) gives the forward rates as affine functions of ji,j

i

i (t) and jii (t),

and so the processes jj ,l

ii (t) and

jii (t) can be expressed in terms of the forward rates by

inverting (4.15). The details of this inversion procedure and the sufficient conditions under

which this inversion is possible are given in the remainder of this section.For x R+ define r(t, x) = r(t, x) f(0, t + x). Then (4.15) implies that r(t, x)

satisfies the equation

r(t, x) =ni=1

diji=1

i,ji(t + x) jii (t)

+

ni=1

diji,j

i=1jij

i

ji,j

i

i (t + x) ji,j

i

i (t)

(5.1)

and consequently each x R+ gives rise to a linear equation for r(t, x) in terms of

ji,j

i

i (t) and jii (t) with deterministic coefficients. It is possible that some of the processes

ji,j

ii (t) and

jii (t) are linearly dependent on others or even deterministic. In order to

obtain a minimal set of stochastic processes with which to capture the state of the forward

rate curve, define the set

S, = {ji,j

i

i (t), jii (t) | 1 i n, 1 ji j

i di, 1 j

i di}, (5.2)

let S, be the subset of deterministic elements in S, and let S, be a maximal linearly

independent subset ofS, S,. Choose an ordering for S

, and write j(t) for the

elements of this ordered set, where 1 j |S,|. Then by moving the deterministicterms on the righthand side of (5.1) to the lefthand side, (5.1) can be rewritten in the form

r(t, x) =

|S,

|

j=1

j

(t + x) j

(t), (5.3)

where j(t + x) are the appropriate deterministic coefficient functions and

r(t, x) = r(t, x) n

i=1

jii (t)S

,

i,ji(t + x) jii (t)

ni=1

ji,j

ii (t)S

,

ji,j

i

i (t + x) ji,j

i

i (t).

(5.4)

For notational convenience, set n = |S,|.

5Along with a finite number benchmark forward rates.

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Proposition 5.1. For any 1, . . . , n R+, define the matrix (t, 1, . . . , n) by

(t, 1, . . . , n) =

1(t + 1) 2(t + 1) n(t + 1)1(t + 2) 2(t + 2) n(t + 2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1(t + n) 2(t + n) n(t + n)

, (5.5)and suppose there exist 1, . . . , n R+ such that det (t, 1, . . . , n) = 0 for all t.Then the state variables j(t) can be written as affine functions of the benchmark forwardrates r(t, 1), . . . , r(t, n).

Proof. Let 1, . . . , n R+ be as given. Then (5.3) holds for each x = i, and writingthe resulting equations in matrix form gives

[r(t, 1), . . . , r(t, n)]

= (t, 1, . . . , S) [1(t), . . . , n(t)] . (5.6)

Now, since det(t, 1, . . . , n) = 0 the last equation can be inverted and results in ex-

pressions of the form

j(t) =nj=1

j,j(t + 1, . . . , t + n) r(t, j ) (5.7)

for 1 j n, where j,j are the entries of (t, 1, . . . , n)1. Since j,j are deter-

ministic, and r(t, j ) and r(t, j) are related by (5.4), this completes the proof.

The above result establishes that the variables jii (t) and ji,j

ii (t), and hence also

p,qi,x (t) and li,x(t), are affine functions of a finite number of benchmark forward rates

r(t, 1), . . . , r(t, n). Furthermore, since the forward rate curve is affine in jii (t) and

ji,j

i

i (t) by (4.15), the entire forward rate curve is affine in r(t, 1), . . . , r(t, n). This isthe content of the next corollary.

Corollary 5.2. If the conditions of Proposition 5.1 are satisfied, then the forward rate

curve is affine with respect to a finite number of benchmark forward rates.

Note that substituting (5.7) into (4.17) allows the bond price to be expressed as an

exponential affine function of a finite number of benchmark forward rates. Although the

notation is somewhat cumbersome for the general case, the expression for the bond price

is easily obtained in specific cases as will be shown in Section 7.

6. STATE VARIABLES AS YIELDS

It was established in the previous section that the state variables in a large class of Mar-

kovian HJM models can be expressed as affine functions of a finite number of forward

rates. However, the instantaneous forward rates are not directly observable in the mar-

ket, and for the numerical implementation of these models, it is convenient to express the

state variables in terms of directly observed market quantities. Fortunately, for the models

considered in Section 4 and Section 5, it is possible to express the forward rates as affine

functions of the market observed yields and this, in turn, allows these models to be affine

with respect to a finite number of yields.

Let x R++. Then the x-tenor yield, y(t, x), is defined by the equation

y(t, x) =1

x

x0

r(t, u) du. (6.1)

From (5.3), the x-tenor yield satisfies the equation

y(t, x) =n

j=1

j(t, x) j(t), (6.2)

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where

y(t, x) =

1

xx0 r

(t, u) du, (6.3)

j(t, x) =1

x

x0

j(t + u) du. (6.4)

The next result establishes a sufficient condition under which the variables j(t) are ex-pressible as affine functions of yields and is the direct analogue of Proposition 5.1.

Proposition 6.1. For1, . . . , n R+, define the matrix (t, 1, . . . , n) by

(t, 1, . . . , n) =

1(t, 1) 2(t, 1) n(t, 1)1(t, 2) 2(t, 2) n(t, 2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1(t, n) 2(t, n) n(t, n)

, (6.5)

and suppose there exist 1, . . . , n R+ such that det (t, 1, . . . , n) = 0 for all t.Then the state variables j(t) are affine in the yields y(t, 1), . . . , y(t, n).

Proof. Apply the arguments in the proof of Proposition 5.1 to (6.2).

Corollary 6.2. If the conditions of Proposition 6.1 are satisfied, then the forward rate

curve is affine with respect to a finite number of benchmark yields.

As was the case with benchmark forward rates, the above results allow the bond price

to be expressed as an exponential affine function of a finite number of benchmark yields.

7. EXAMPLES

This section illustrates the general framework developed in the previous sections withsome simple examples. In particular, the results from Sections 5 and 6 are applied to

express the state variables for these examples in terms of a finite number of benchmark

forward rates or yields.

7.1. Extended Vasicek Model. The Hull and White (1990) extended Vasicek model MVasis a 1-dimensional HJM model corresponding to the forward rate volatility

(t, ) = 0e(t), (7.1)

where 0 and are constants. Since (t, )/ + (t, ) = 0, MVas is a special caseof the Ritchken and Sankarasubramanian (1995) model and is consequently Markov. If the

forward rate volatility is rewritten in the form

(t, ) =

0et

e

, (7.2)then the coefficient and the deterministic function of in (4.1) can be identified as

c(t) = 0et and () = e. (7.3)

The relevant state variables for MVas are hence 11(t) and 1(t), where the subscriptshave been omitted for this example. But

11(t) =

t0

c(s)2 ds =1

220

e2t 1

(7.4)

and so 11(t) is deterministic in this case while

1(t) = t

0

c(s) dw(s) t

0

c(s)2(s) ds = 0 t

0

es dw(s) 20

2(et 1)2 (7.5)

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is stochastic. So for MVas, S, = {11(t)} and S, = {

1(t)}. It follows that MVas is

a 1-factor Markovian model with 1(t) being the sole state variable, and the forward rate

in (4.15) can be written

r(t, x) = f(0, t + x) + e(t+x)

1

220

e2t 1

1 e(t+x)

+ 1(t)

. (7.6)

The conditions of Proposition 5.1 are trivially satisfied, and

1(t) = e(t+x) [r(t, x) f(0, t + x)] 1

220

e2t 1

1 e(t+x)

(7.7)

is the expression of 1(t) in terms of any benchmark forward rate r(t, x). In particular,setting x = 0 allows 1(t) to be expressed in terms of the spot rate r(t)

1(t) = et

r(t) f(0, t) 1

220

e2t 1

1 et

. (7.8)

The expressions for the yield and bond prices of any maturity in this case are easily com-

puted and are given by

y(t, x) = 0(t, x) +1

22x20

et et

1 ex

+

1

xet

1 ex

1(t)

(7.9)

b(t, x) =b(0, t + x)

b(0, t)exp

1

2220

et et

1 ex

exp

1

et

1 ex

1(t)

(7.10)

Once again, the expression (7.8) for 1(t) can be substituted to obtain the yields andbond prices in terms of the spot rate. Note that similar arguments using (7.9) instead

of (7.6) allows the bond price to be expressed in terms of a benchmark yield rather than a

benchmark forward rate.

7.2. Ritchken-Sankarasubramanian Model. The next example is a generic Ritchken

and Sankarasubramanian (1995) model MRS in which the forward rate volatility processis given by

(t, ) = g(t, r(t))e(t), (7.11)

where is a constant and g is a deterministic function. Note that MRS is a 1-dimensionalHJM model and that (t, ) satisfies the equation (t, )/+(t, ) = 0. Once again,rewriting the forward rate volatility in the form

(t, ) = g(t, r(t))et e (7.12)

allows the coefficient and the function of in (4.1) to be identified as

c(t) = g(t, r(t))et and () = e. (7.13)

The state variables for MRS are 11(t) and 1(t) given by

11(t) =

t0

g(s, r(s))2e2s ds, (7.14)

1(t) =

t0

g(s, r(s))es dw(s) 1

t0

g(s, r(s))2e2s(1 es) ds, (7.15)

where the subscripts have been omitted from 11(t) and 1(t) . In contrast to the situation

in the extended Vasicek model, 11(t) and 1(t) are both stochastic in this case and so

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S, = and S, = {

11(t), 1(t)}. It follows that MRS is a 2-factor Markovianmodel and the forward rate in (4.15) can be written

r(t, x) = f(0, t + x) + (t, x) 1(t) + (t, x) 11(t), (7.16)

where (t, x) = e(t+x) and (t, x) =1

e(t+x)

1 e(t+x)

.

Now, if1 = 2 then

det

(t, 1) (t, 1)

(t, 2) (t, 2)

=

1

e(3t+1+2)

e2 e1

= 0, (7.17)

and Proposition 5.1 gives

1(t) =(t, 2)r(t, 1) (t, 1)r(t, 2)

(t, 1)(t, 2) (t, 1)(t, 2), (7.18)

11(t) =(t, 1)r(t, 2) (t, 2)r(t, 1)

(t, 1)(t, 2) (t, 1)(t, 2)

(7.19)

as the expressions for the state variables in terms of the forward rates in this case. The

standard Markovian state variables for this model, 0,0(t) and 0(t), can now be obtainedfrom (4.6) and (4.7).

To express the state variables in terms of yields, note from (6.1) and (7.6) that

y(t, x) = 0(t, x) + (t, x) 1(t) + (t, x) 11(t), (7.20)

where (t, x) and (t, x) are given by

(t, x) =1

xet(1 ex),

(t, x) =1

22xet(1 ex)

2 et e(t+x)

.

Once again, if1 = 2 then it is easily shown that

det

(t, 1) (t, 1)

(t, 2) (t, 2)

=

e3t

2312(1 e1)(1 e2)(e2 e1) = 0,

and so Proposition 6.1 gives

1(t) =(2)y(t, 1) (1)y(t, 2)

(1)(2) (1)(2), (7.21)

11(t) =(1)y(t, 2) (2)y(t, 1)

(1)(2) (1)(2). (7.22)

where y(t, x) = y(t, x) 0(t, x). It should be noted that an expression of this form wasfirst obtained in Bliss and Ritchken (1996). Finally, the bond price is given by

b(t, x) = b(0, t + x)b(0, t)

exp

x(t, x) 1(t) x(t, x) 11(t)

. (7.23)

Equations (7.18) and (7.19) can be used to express the bond price in terms of benchmark

forward rates, and similarly (7.21) and (7.22) can be used to express the bond price in terms

of benchmark yields.

7.3. HJM Model with Humped Volatility. The next example considered is a 1-dimensionalHJM model MHV corresponding to the forward rate volatility

(t, ) = r(t) [0 + 1( t)] e(t), (7.24)

where , , 0 and 1 are constants. This represents a term structure model in which thevolatility is level dependent and contains a hump.6 Since (/ + )2(t, ) = 0, MHV

6For a detailed discussion of this model with = 0, see Ritchken and Chuang (1999).

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is an example of the Markovian models considered in Chiarella and Kwon (2001 b) and the

framework developed in the previous sections apply. Rewriting (t, ) in the form

(t, ) = (0 t1)r(t)et e + 1r(t)et e, (7.25)

the quantities in the decomposition (4.1) can be identified as

c1(t) = (0 t1)r(t)et,

c2(t) = 1r(t)et,

1() = e,

2() = te.

The state variables for MHV are 1(t), 2(t), 11(t), 12(t) and 22(t) given by

11(t) =

t0

(0 s1)2r2(s)e2s ds, (7.26)

12(t) =

t0

1(0 s1)r2(s)e2s ds, (7.27)

22(t) =t0

21r2(s)e2s ds, (7.28)

1(t) =

t0

(0 s1)r(s)es dw(s)

1

t0

(0 s1)2r2(s)e2s(1 es) ds

1

2

t0

1(0 s1)r2(s)e2s[1 (1 + s)es] ds,

(7.29)

2(t) =

t0

1r(s)es dw(s)

1

t0 1(0 s1)r

2

(s)e2s

(1 es

) ds

1

2

t0

21r2(s)e2s[1 (1 + s)es] ds,

(7.30)

where the subscripts have once again been omitted from the state variables. In this case

S, = and S, = {

1(t), 2(t), 11(t), 12(t), 22(t)}. Now, equation (4.3) for

j(t) gives

1(t) =1

(1 et), (7.31)

2(t) =1

2[1 (1 + t)et], (7.32)

and the expression (4.15) for the forward rate in MHV is

r(t, x) = f(0, t + x) +2

i=1

i(t + x) i(t) +

2i=1

i(t + x)i(t + x) i,i(t)

+ [1(t + x)2(t + x) + 2(t + x)1(t + x)] 1,2(t).

(7.33)

It follows from the linear independence of the functions ex, xex, x2ex, e2x andxe2x that if1 < < 5, then the matrix

=

1(1) 2(1) 1(1)1(1) 12(1) 2(1)2(1)1(2) 2(2) 1(2)1(2) 12(2) 2(2)2(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1(5) 2(5) 1(5)1(5) 12(5) 2(5)2(5)

(7.34)

is non-singular, where 12() = 1()2() + 2()1() and i = t + i. Hence,

(7.33) can be inverted to express the state variables in terms of the forward rates r(t, i),

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1 i 5. Formulae for the corresponding yields and bond prices can be obtained byintegrating (7.33) as in the previous examples.

7.4. Two-Dimensional Extended Vasicek Model. The final example considered is a 2-dimensional HJM model M2D in which the volatility is given by

i(t, ) = iei(t) (7.35)

for i = 1, 2, where i and i are constants and 1 = 2. This is the 2-dimensionalextension of MVas considered earlier. Since i(t, )/ + i(t, ) = 0, M2D fallsunder the Inui-Kijima framework and is consequently Markov. Rewriting i(t, ) for eachi gives

i(t, ) = ieit ei, (7.36)

and so the quantities in (3.2) are easily identified as

ci,1(t) = ieit and i,1() = e

i. (7.37)

The state variables for M2D are 111 (t), 11(t),

112 (t) and

12(t). However, as was the case

for MVas the variables 111 (t) and 112 (t) are deterministic since

11i (t) =1

22i

e2it 1

, (7.38)

while the variables 1(t) and 2(t) are stochastic since from (4.5)

11i (t) = i

t0

eis dw(s) 2i

2i(eit 1)2. (7.39)

So S, = {111 (t), 112 (t)} and S, = {11(t), 12(t)} for M2D. Hence M2D is a 2-factor Markovian HJM model with state variables 11(t) and

12(t), and (4.15) implies that

the forward rate in M2D is given by

r(t, x) = f(0, t + x) +2

i=1

ei(t+x)

1

2i2i

e2it 1

1 ei(t+x)

+ 1i (t)

.

Since 1 = 2, there exist 1 and 2 such that (11 + 22) (12 + 21) = 0. Forsuch a choice of 1 and 2

= det

e1(t+1) e2(t+2)

e1(t+2) e2(t+1)

= e(1+2)t

e(11+22) e(12+21)

= 0,

and so Proposition 5.1 applies to give11(t)12(t)

=

1

e2(t+2) e2(t+1)

e1(t+2) e1(t+1)

r(t, 1)r(t, 2)

, (7.40)

where

r(t, j) = r(t, j) f(0, t + j)

2

i=1

1

2i

2i ei(t+j) e2it 1 1 ei(t+j) .

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The yields and bond prices are given by

y(t, x) = 0(t, x) +

2i=1

122i x

2i

eit eit

1 eix

+2

i=1

1

ixeit

1 eix

1i (t)

(7.41)

b(t, x) =b(0, t + x)

b(0, t)exp

2i=1

1

22i2i

eit eit

1 eix

exp

2i=1

1

ieit

1 eix

1i (t)

.

(7.42)

As noted previously, expressions for 11(t) and 12(t) from (7.40) can be substituted into

(7.42) to obtain a formula for the bond price in terms of two benchmark forward rates,and similar arguments using yields in place of forward rates allows the bond price to be

expressed in terms of two benchmark yields.

8. CONCLUSION

The Markovian HJM models introduced in Chiarella and Kwon (2001b) generalised

many of the Markovian HJM models previously considered in the literature, but did not

address the problem of relating the state variables to market observed quantities, nor in-

vestigate in depth the nature of these models. The results of this paper resolve this gap by

linking the state variables directly, and explicitly, to the forward rates and market observed

yields, and establishing that the models are in fact affine with respect to a finite number of

benchmark forward rates or yields.

Consequently, the setup in Chiarella and Kwon (2001b) provides a consistent frame-

work for the systematic construction of a wide range of affine term structure models within

the HJM framework, and Chiarella and Kwon (2001a) provides an example of such a con-

struction.

This paper also established an explicit formula for the bond price in terms of the state

variables which generalises the results of Ritchken and Sankarasubramanian (1995) and

Inui and Kijima (1998). In particular, it was shown that the bond price takes an exponential

affine form in a much broader class of models than the square root affine volatility models

considered in Duffie and Kan (1996).

The results obtained in this paper are of significant value in implementing the models

in practice, and the research into the practical implementation, calibration and evaluation

of these models remains an on-going project.

REFERENCES

Bhar, R. and Chiarella, C. (1997), Transformation of Heath-Jarrow-Morton Models to Markovian Systems,

European Journal of Finance 3, 126.

Bjork, T. and Gombani, A. (1999), Minimal Realizations of Interest Rate Models, Finance and Stochastics

3, 413432.

Bjork, T. and Svensson, L. (1999), On the Existence of Finite Dimensional Realizations for Nonlinear Forward

Rate Models, Working paper, Stockholm School of Economics.

Bliss, R. and Ritchken, P. (1996), Empirical Tests of Two State-Variable Heath-Jarrow-Morton Models, Journal

of Money, Credit, and Banking 28(3), 452481.

Carverhill, A. (1994), When is the Short Rate Markovian?, Mathematical Finance 4(4), 305312.

Chiarella, C. and Kwon, O. (2001a), Formulation of Popular Interest Models under the HJM Framework, Asia

Pacific Financial Markets (Forthcoming).

Chiarella, C. and Kwon, O. (2001b), Forward Rate Dependent Markovian Transformations of the Heath-Jarrow-

Morton Term Structure Model, Finance and Stochastics 5(2), 237257.

7/28/2019 10.1.1.28.5998

17/17


Coddington, E. and Levinson, N. (1955), Intoduction to Ordinary Differential Equations, McGraw-Hill Publish-

ing Inc., New York.

Cox, J., Ingersoll, J. and Ross, S. (1985), A Theory of the Term Structure of Interest Rates, Econometrica53(2), 385407.

de Jong, F. and Santa-Clara, P. (1999), The Dynamics of the Forward Interest Rate Curve: A Formulation with

State Variables, Journal of Financial and Quantative Analysis 34(1), 131157.

Duffie, D. and Kan, R. (1996), A Yield Factor Model of Interest Rates, Mathematical Finance 6(4), 379406.

Heath, D., Jarrow, R. and Morton, A. (1992), Bond Princing and the Term Structure of Interest Rates: A New

Methodology for Contingent Claim Valuation, Econometrica 60(1), 77105.

Ho, T. and Lee, S. (1986), Term Structure Movements and Pricing Interest Rate Contingent Claims, Journal of

Finance 41, 10111028.

Hull, J. and White, A. (1990), Pricing Interest Rate Derivative Securities, Review of Financial Studies 3, 573

592.

Inui, K. and Kijima, M. (1998), A Markovian Framework in Multi-Factor Heath-Jarrow-Morton Models, Jour-

nal of Financial and Quantitative Analysis 33(3), 423440.

Kwon, O. (2001), Characterisation of Affine Term Structure Models, Working Paper, School of Banking and

Finance, University of New South Wales.

Musiela, M. (1993), Stochastic PDEs and Term Structure Models, Journees Internationales des Finance, IGR-AFFI, La Baule .

Ritchken, P. and Chuang, Y. (1999), Interest Rate Option Pricing with Volatility Humps, Review of Derivatives

Research 3, 237262.

Ritchken, P. and Sankarasubramanian, L. (1995), Volatility Structures of Forward Rates and the Dynamics of the

Term Structure, Mathematical Finance 5(1), 5572.

Vasicek, O. (1977), An Equilibrium Characterisation of the Term Structure, Journal of Financial Economics

5, 177188.

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