Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields

Review of Derivatives Research, 6, 129–155, 2003 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Finite Dimensional Affine Realisations of HJMModels in Terms of Forward Rates and Yields

CARL CHIARELLA [email protected] of Finance and Economics, University of Technology Sydney, PO Box 123, Broadway NSW 2007,Australia

OH KANG KWON ∗ [email protected] of Finance and Economics, University of Technology Sydney, PO Box 123, Broadway NSW 2007,Australia

Received April 30, 2002; Revised July 24, 2003

Abstract. Finite dimensional Markovian HJM term structure models provide ideal settings for the study of termstructure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractabilityof Markovian models coexist. Consequently, these models became the focus of a series of papers includingCarverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998),de Jong and Santa-Clara (1999), Björk and Svensson (2001) and Chiarella and Kwon (2001a). However, thesemodels usually required the introduction of a large number of state variables which, at first sight, did not appearto have clear links to the market observed quantities, and the explicit realisations of the forward rate curve interms of the state variables were unclear. In this paper, it is shown that the forward rate curves for these modelsare affine functions of the state variables, and conversely that the state variables in these models can be expressedas affine functions of a finite number of forward rates or yields. This property is useful, for example, in theestimation of model parameters. The paper also provides explicit formulae for the bond prices in terms of thestate variables that generalise the formulae given in Inui and Kijima (1998), and applies the framework to obtainaffine representations for a number of popular interest rate models.

Keywords: Markovian models, interest rate models, Heath–Jarrow–Morton, forward rates, yields.

JEL classification: E43, G12, G13

The Heath, Jarrow, and Morton (1992) term structure framework is widely regarded asthe most general and flexible setting for the study of interest rate dynamics, having thecapacity to generate a wide range of forward rate dynamics and the ability to incorpo-rate prevailing market conditions, such as the currently observed yield curve, with internalconsistency. Since the bond market in this framework is arbitrage free and complete sub-ject only to mild restrictions, the Heath–Jarrow–Morton (HJM) framework also provides aconvenient setting for the study of interest rate derivatives.

However, the generality and the flexibility of the HJM framework is often at the expenseof theoretical and numerical tractability, since this framework provides a family of modelsthat are non-Markovian in general. In particular, this means that the standard Feynman–

∗ Corresponding author.

130 CHIARELLA AND KWON

Kac theorem no longer applies, rendering inaccessible the well-developed tools from thetheory of partial differential equations; Monte Carlo methods for these models are oftencomputationally intensive and require large storage; and these models generally give riseto non-recombining trees. To overcome these problems, researchers have sought ways tofind the subclasses of HJM models that admit Markovian realisations. In these MarkovianHJM models the flexibility of the HJM framework and the tractability of Markovian mod-els coexist to provide an ideal setting under which to study term structure dynamics andinterest rate derivatives.

It turns out that an HJM model is completely determined by the initial forward ratecurve and the forward rate volatility process. However, since the initial forward rate curveis completely determined by the market, the only remaining flexibility for obtaining finitedimensional Markovian models within the HJM framework rests in the pertinent choice,or specification, of the volatility process. Various restrictions on the forward rate volatilityprocess that lead to finite dimensional Markovian HJM models were obtained in Carverhill(1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Ki-jima (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 simpleobservation that the components of the forward rate volatility must satisfy ordinary dif-ferential equations in the maturity variable, a common generalisation to all these modelswas obtained in Chiarella and Kwon (2001a). The results were subsequently generalisedfurther in Björk and Svensson (2001) and Björk and Landén (2000).

Although theoretically appealing, most of the Markovian HJM models considered in theliterature introduce a large number of state variables which, at first sight, do not appearto have direct links to market observed quantities. The main aim of this paper is to show,under suitable restrictions on the forward rate volatility process, that the state variables inthese models are affine functions of a finite number of forward rates or yields, and further-more that the forward rate curve is an affine function of a finite number of forward ratesor yields. This observation is useful, for example, in the calibration of model parameterssince the state variables for these models are directly observed in the market. It also leadsto an explicit formula for the bond price that is exponential affine in the state variables forthese models.

It should be noted that, in the case of the models considered in this paper, the volatilitystructure of the state variables is not necessarily square root affine in the state variables.Since the corresponding bond prices are exponential affine in the state variables, this mayappear to contradict the results obtained in Duffie and Kan (1996). However, the squareroot affine nature of the volatility structure is only a sufficient condition to ensure expo-nential affine bond prices, and is necessary only under certain assumptions that are notsatisfied by the models considered in this paper. A detailed discussion of the Duffie andKan (1996) characterisation of affine term structure models can be found in Kwon (2001).

The structure of the remainder of the paper is as follows. A brief review of the HJMframework is given in Section 1, the main results providing explicit affine realisations ofMarkovian HJM models are presented in Section 2, and the formulae for the bond pricesand yields, generalising those obtained in Inui and Kijima (1998), are derived in Section 3.

FINITE DIMENSIONAL AFFINE REALISATIONS OF HJM MODELS 131

Examples illustrating the techniques developed in the paper are given in Section 4, andthe issues of model calibration and derivative pricing are considered briefly in Section 6.Finally, the paper concludes with Section 7.

1. The Heath, Jarrow and Morton Framework

This section presents the key definitions and results from Heath, Jarrow, and Morton(1992), but under the parameterisation of the maturity variable introduced in Musiela(1993). Let (�,F , (Ft )t∈R+,P) be a complete filtered probability space satisfying theusual conditions, where the filtration (Ft )t∈R+ generated by a standard n-dimensionalP-Wiener process w(t). For any t, x ∈ R+ denote by b(t, x) the price of a (t + x)-maturity zero coupon bond at time t , and define the (t +x)-maturity instantaneous forwardrate r(t, x) by the equation

r(t, x) = −∂ ln b(t, x)

∂x. (1)

Then in the risk-neutral formulation of the HJM term structure models on (�,F ,

(Ft )t∈R+,P), the forward rates r(t, x) are assumed to satisfy the stochastic integralequation1

r(t, x) = r(0, t + x) +∫ t

0σ(s, t + x)τ σ (s, t + x) ds

+∫ t

0σ(s, t + x)τ dw(s), (2)

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

rate volatility process,

σ (t, t + x) =∫ t+x

t

σ (t, u) du (3)

and the superscript τ denotes matrix transposition. It can be shown, as, for example, inBrace and Musiela (1994), that the stochastic differential equation corresponding to (2) is

dr(t, x) = ∂

∂x

[r(t, x) + 1

2

∣∣σ (t, t + x)∣∣2] dt + σ(t, t + x)τ dw(t), (4)

and the corresponding stochastic differential equation for the bond price, b(t, x), is

db(t, x) = b(t, x)[r(t) − r(t, x)

]dt − b(t, x)σ(t, t + x)τ dwt , (5)

where r(t) = r(t, 0) is the short rate. Note that an HJM term structure model is completelydetermined by its dimension n, the initial forward rate curve r(0, x) and the forward ratevolatility process σ(t, x).


Definition 1. An HJM model M is said to admit a d-dimensional realisation, for somed ∈ N+, if there exists a d-dimensional process z(t), adapted to the filtration (Ft ), and adeterministic function � : R+ × R+ × R

d → R+ such that

r(t, x) = �(t, x, z(t)

), (6)

for all t, x ∈ R+.

At this point, the d-dimensional process z(t) is not restricted in any way except forbeing adapted to (Ft ). In order for this definition to be useful in practice, some additionalstructure must be placed on the process z(t).

Definition 2. An HJM model M is said to admit a d-dimensional Markovian realisationif it admits a d-dimensional realisation and the process z(t) is an (Ft )-Markov processsatisfying a stochastic differential equation of the form{

dz(t) = µz(t, z) dt + σz(t, z) dw(t),

z(0) = z0,(7)

where µz : R+×Rd → R

d and σz : R+×Rd → R

d×n are sufficiently regular and z0 ∈ Rd .

So if an HJM model admits a Markovian realisation, then the explicit dependence ofr(t, x) on t and x are deterministic and the dependence on the Wiener path enters onlythrough the (Ft )-Markov process z(t). The short rate models such as those considered inVasicek (1977), Cox, Ingersoll, and Ross (1985) and Ho and Lee (1986) are examples ofHJM models with 1-dimensional Markovian realisations where z(t) is the short rate r(t, 0).

In practice, it is very convenient to work with models in which the forward rate curve isan affine function of the process z(t).

Definition 3. An HJM model A with a d-dimensional realisation given by (6) is said to beaffine if � is an affine function in z. That is, the forward rate process can be written in theform

r(t, x) = �(t, x, z(t)

) = h0(t, x) + h(t, x)τ z(t), (8)

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

Note that in view of (1) the HJM model A is affine if and only if the bond price in A isexponential affine2 in the sense of Duffie and Kan (1996).

2. State Variables and the Affine Nature of Forward Rate Curves

In Chiarella and Kwon (2001a), a large class of HJM models with finite dimensionalMarkovian realisations were introduced. This section considers these models from a view-point which not only leads to a concrete characterisation of the process z(t) in (7) and


the function �(t, x, z) in (6), but also simplifies proofs to many of the results. As re-marked above, this class includes the models considered in Carverhill (1994), Ritchkenand Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998) andde Jong and Santa-Clara (1999) as special cases, and provides a unifying framework underwhich to study these models.

Let n ∈ N+ and let σ(t, x) = (σ1(t, x), . . . , σn(t, x)) be the forward rate volatilityprocess introduced in (1).

Assumption 1. For each i ∈ {1, . . . , n} there exists di ∈ N+ such that σi(t, x) is di timesdifferentiable with respect to x and satisfies a di-th order homogeneous linear differentialequation of the form3

Liσi (t, x) = 0, where Li = ∂di

∂xdi−

di−1∑j=0

κij (x)∂j

∂xj(9)

and κij (x) are continuous deterministic functions.

The volatility processes considered in Carverhill (1994), Ritchken and Sankarasubra-manian (1995), Inui and Kijima (1998) and de Jong and Santa-Clara (1999) satisfy theabove assumption with di = 1 for all i, and the volatility process considered in Bhar andChiarella (1997) satisfy the above assumptions with di = k ≥ 0 and Li = (∂/∂x − κi)

k .The following lemma is a well-known result from the theory of ordinary differential

equations.

Lemma 1. Let σi(t, x), for 1 ≤ i ≤ n, satisfy Assumption 1. Then there exist stochasticprocesses cij and deterministic functions σij such that

σi(t, x) =di∑

j=1

cij (t)σij (x). (10)

Proof: Since the coefficient functions κij in (9) are continuous by assumption, the resultfollows from Coddington and Levinson (1955), Theorem 5.1. �

Intuitively, the deterministic functions σij are the set of fundamental solutions to theordinary differential equation (9), and (10) expresses the particular solution σi as a linearcombination of the fundamental solutions σij with appropriate coefficients cij .

The notable feature of (10) is that it permits the time dependence and the maturity de-pendence in the components of the forward rate volatility process to be separated. Thisseparation is essential in the determination of the Markovian realisations obtained below.

Corollary 1. Let cij be the coefficient processes defined in (10). Then the cij are adaptedto the filtration (Ft ).


Proof: Let i ∈ {1, . . . , n}. Firstly, since {σi1, . . . , σidi } is a linearly independent set offunctions, there exist x1, . . . , xd1 such that the matrix

Mi(x1, . . . , xdi ) =

σi1(x1) σi2(x1) . . . σidi (x1)

σi1(x2) σi2(x2) . . . σidi (x2)...

.... . .

...

σi1(xdi ) σi2(xdi ) . . . σidi (xdi )

(11)

is invertible. Substituting x1, . . . , xdi successively into (10) and inverting the resultingequation gives[

c1i(t), . . . , cidi (t)]τ = Mi(x1, . . . , xdi )

−1[(σi(t, x1), . . . , σi(t, xdi )]τ.

Since σi(t, x) is adapted to (Ft ) for all x ∈ R+, the right-hand side is adapted to (Ft ) and,consequently, the processes cij are adapted to (Ft ) for all l ≤ j ≤ di . �

The above lemma allows a simplified proof to a generalisation of the result obtained inChiarella and Kwon (2001a). This generalisation expresses the forward rate curve as anaffine function of a certain set of state variables that summarises the history of the forwardrate curve evolution.

Proposition 1. Let σi(t, x), for 1 ≤ i ≤ n, satisfy Assumption 1. Then the correspondingHJM model admits a finite dimensional affine realisation

r(t, x) = r(0, t + x) +n∑

i=1

di∑j=1

σij (t + x)ψij (t)

+n∑

i=1

di∑j,k=1j≤k

[σij (t + x)σik(t + x) + εjkσik(t + x)σij (t + x)

]ϕijk(t), (12)

where

σij (x) =∫ x

0σij (s) ds, (13)

ψij (t) =

∫ t

0cij (s) dwi(s) −

di∑k=1

∫ t

0cij (s)cik(s)σik(s) ds, (14)

ϕijk(t) =

∫ t

0cij (s)cik(s) ds, (15)

εij ={

1, if j = k,

0, if j = k.(16)


Proof: Substituting the expressions for the σi(s, t + x) from (10) into the expression forr(t, x) in (2) gives

r(t, x) = r(0, t + x) +n∑

i=1

di∑j=1

σij (t + x)

∫ t

0cij (s) dwi(s)

+n∑

i=1

di∑j,k=1

σij (t + x)

∫ t

0cij (s)cik(s)

∫ t+x

s

σik(u) du ds

= r(0, t + x) +n∑

i=1

di∑j=1

σij (t + x)

∫ t

0cij (s) dwi(s)

+n∑

i=1

di∑j,k=1

σij (t + x)

∫ t

0cij (s)cik(s)

[σik(t + x) − σik(s)

]ds.

Collecting terms and using (14) and (15) gives

r(t, x) = r(0, t + x) +n∑

i=1

di∑j=1

σij (t + x)ψij (t)

+n∑

i=1

di∑j,k=1

σij (t + x)σik(t + x)

∫ t

0cij (s)cik(s) ds,

which is equivalent to (12). It is easily shown that ψij (t) and ϕi

jk(t) are well defined for allt ∈ R+, and it follows from Corollary 1 that they are adapted. �

Intuitively, Proposition 1 implies that the path history of the forward rate curve is cap-tured by the finite number of quantities ψi

j (t) and ϕijk(t), and that the forward rate curve

is an affine function of these quantities.In practice, however, some of these quantities turn out to be deterministic, and it is

useful to select the minimal non-deterministic subset of {ψij , ϕ

ijk} with which to describe

the forward rate curve.Let S = {ψi

j , ϕijk | i ≤ j ≤ n, 1 ≤ j, k ≤ di}, let S∗ ⊂ S be the subset con-

sisting of non-deterministic elements and denote by n∗ the size of the set S∗. Then theexpression (12) for the forward rate r(t, x) can be written

r(t, x) = a0(t, x) +∑χ∈S∗

aχ(t, x)χ(t), (17)


where a0 and aχ are deterministic functions, and χ is used to denote the elements of theset S∗. For example, if S∗ = S, then it follows from (12) that

aχ (t, x) =

r(0, t + x), if χ = 0,

σij (t + x) if χ = ψij ,

σij (t + x)σik(t + x) + εjkσij (t + x)σik(t + x), if χ = ϕijk.

Assumption 2. There exists m ∈ N+ such that σ(t, x) is a function of t , x and m forwardrates r(t, x∗

1 ), . . . , r(t, x∗m), so that

σ(t, x) = σ(t, x, r(t, x∗

1 ), . . . , r(t, x∗m)

). (18)

The intuition motivating the next proposition is that, in view of (17), each maturity x

gives rise to a linear equation in the corresponding forward rate r(t, x) and the elementsof S∗. Provided that some invertibility condition is satisfied, a suitable system of theseequations can then be inverted to express the elements of S∗ in terms of a certain setforward rates.

Proposition 2. Suppose Assumptions 1 and 2 are satisfied, and let n∗ = |S∗| as de-fined above. Write S∗ = {χ1, . . . , χn∗ } and let r(t, x) be as given in (17). If there existx1, x2, . . . , xn∗ ∈ R+ such that the matrix

M∗(t, x1, . . . , xn∗) =

aχ1(t, x1) aχ2(t, x1) . . . aχn∗ (t, x1)

aχ1(t, x2) aχ2(t, x2) . . . aχn∗ (t, x2)...

.... . .

...

aχ1(t, xn∗) aχ2(t, xn∗) . . . aχn∗ (t, xn∗)

(19)

is invertible for all t ∈ R+, then the corresponding HJM model admits an n∗-dimensionalaffine realisation. Furthermore, the realisation is Markovian and the components of theMarkov process z(t) can be assumed to be the forward rates r(t, xi) so that

z(t) = [r(t, x1), . . . , r(t, xn∗)

]τ. (20)

Proof: An argument similar to that used in Corollary 1 applied to (17) shows that[χ1(t), . . . , χn∗(t)

]τ = M∗(t, x1, . . . , xn∗)−1[r(t, x1), . . . , r(t, xn∗)]τ

(21)

for all t ∈ R+, where r(t, x) = r(t, x) − a0(t, x). In particular, each χ ∈ S∗ is an affinefunction in the forward rates r(t, xi), and so (17) can be written

r(t, x) = b0(t, x) +n∗∑i=1

bi(t, x)r(t, xi), (22)

where bi are the deterministic functions obtained by substituting the χi(t) from (21)into (17) and collecting the terms in r(t, xi). This shows that the underlying HJM modeladmits an n∗-dimensional affine realisation in terms of the forward rates r(t, xi). It now


remains to show that the set of forward rates {r(t, x1), . . . , r(t, xn∗)} form an (Ft )-Markovprocess.

Firstly, by use of (22) the forward rates r(t, x∗i ) in (18) can be replaced by the forward

rates r(t, xi) so that

σ(t, x) = σ(t, x, r(t, x1), . . . , r(t, xn∗)

). (23)

Next, from (4) each r(t, xk) satisfies the stochastic differential equation

dr(t, xk) =[b′

0(t, xk) +n∗∑i=1

b′i (t, xk)r(t, xi) + σ(t, t + xk)σ (t, t + xk)

]dt

+ σ(t, xk)∗ dw(t),

where b′i (t, x) = ∂bi(t, x)/∂x. In view of (23) these equations are of the form

dr(t, xk) = µ(t, r(t, x1), . . . , r(t, xn∗)

)dt

+ σ(t, xk, r(t, x1), . . . , r(t, xn∗)

)τdw(t), (24)

and it follows that z(t) = [r(t, x1), . . . , r(t, xn∗)]τ is an (Ft )-Markov process. �

Although it was shown in Chiarella and Kwon (2001a) that the HJM models satisfyingthe Assumptions 1 and 2 admit finite dimensional realisations, neither the nature of thestate variables nor the functional form of the realisations were determined. Proposition 2addresses this gap and provides a much simpler proof of the previous results.

3. Expressions for Bond Prices and Yields

The purpose of this short section is to derive the closed form formulae for the zero couponbond prices and yields in terms of the forward rates r(t, x1), . . . , r(t, xn∗) introducedin (19).

Proposition 3. The price of the (t + x)-maturity zero coupon bond b(t, x) is given by theformula

b(t, x) = exp

[−b0(t, x) −

n∗∑i=1

bi(t, x)r(t, xi)

], (25)

where

bi(t, x) =∫ x

0bi(t, u) du (26)

and bi(t, x) are as defined in (22).

Proof: Follows immediately from (1) and (22). �


Recall that for any 0 < x ∈ R+, the x-maturity yield y(t, x) at time t ∈ R+ is given bythe equation

y(t, x) = 1

x

∫ x

0r(t, u) du. (27)

Proposition 4. The x-maturity yield y(t, x) is given by the formula

y(t, x) = 1

x

[b0(t, x) +

n∗∑i=1

bi(t, x)r(t, xi)

], (28)

where bi(t, x) are as given in (26).

The invertibility condition in Proposition 2 allowed the forward rate curve to be ex-pressed in terms of a certain set of forward rates. It is possible to find the correspondinginvertibility condition that allows the forward rate curve to be expressed in terms of a setof zero coupon bonds or yields.

Corollary 2. Suppose there exist x1, . . . , xn∗ such that the matrix

M(t, x1, . . . , xn∗) =

b1(t, x1) b2(t, x1) . . . bn∗(t, x1)

b1(t, x2) b2(t, x2) . . . bn∗(t, x2)...

.... . .

...

b1(t, xn∗) b2(t, xn∗) . . . bn∗(t, xn∗)

(29)

is invertible for all t ∈ R+. Then the forward rate curve r(t, ·) can be expressed as anaffine function of the set of zero coupon bonds {b(t, x1, . . . , b(t, xn∗)}. If xi = 0 for all1 ≤ i ≤ n∗, then the forward rate curve can also be expressed as an affine function of theset of yields {y(t, x1), . . . , y(t, xn∗)}.

Since yields of various maturities are directly observed in the market, the last corollaryis useful in the estimation of model parameters.

4. Examples

This section illustrates the general framework developed in the previous sections with somesimple examples that yield many of the popular interest rate models. In particular, theresults from Sections 2 and 3 are applied to obtain explicit functional form for the forwardrate curve, bond prices and yields as affine functions of a set of forward rates.


4.1. Extended Vasicek Model

The Hull and White (1990) extension of the Vasicek (1977) model, Mvas, is a1-dimensional HJM model corresponding to the forward rate volatility

σ(t, x) = σ0e−κ(x−t ), (30)

where σ0 and κ are constants.4 Since ∂σ(t, x)/∂x+κσ(t, x) = 0 Assumption 1 is satisfied,and since σ(t, x) is deterministic Assumption 2 is trivially satisfied. In fact, Mvas is aspecial case of the Ritchken and Sankarasubramanian (1995) model. If the forward ratevolatility is rewritten in the form

σ(t, x) = σ0eκt · e−κx, (31)

then the coefficient and the deterministic function of x in (10) can be identified as

c(t) = σ0eκt and σ(x) = e−κx . (32)

The relevant state variables for Mvas are hence ϕ11(t) and ψ1(t), where the superscriptshave been omitted for this example. But

ϕ11(t) =∫ t

0c(s)2 ds = σ 2

0

2κ

(e2κt − 1

), (33)

and so ϕ11(t) is deterministic in this case while

ψ1(t) =∫ t

0c(s) dw(s) −

∫ t

0c(s)2σ (s) ds = σ0

∫ t

0eκs dw(s) − σ 2

0

2κ2

(eκt − 1

)2 (34)

is non-deterministic. Hence, S∗ = {ψ1(t)} in this case, and it follows that Mvas admits a1-dimensional affine realisation. The affine expression (22) for the forward rate curve inthis example is

r(t, x) = r(0, t + x) + e−κ(t+x)

[σ 2

0

2κ2

(e2κt − 1

)(1 − e−κ(t+x)

) + ψ1(t)

]. (35)

Rearranging this equation allows the state variable ψ1(t) to be expressed in terms of anyforward rate, r(t, x∗), of maturity x∗ ∈ R+, viz.

ψ1(t) = eκ(t+x∗)[r(t, x∗) − r(0, t + x∗)] − σ 2

0

2κ2

(e2κt − 1

)(1 − e−κ(t+x∗)). (36)

Substituting (36) into (35) gives the expression

r(t, x) = r(0, t + x) − e−κ(x−x∗)r(0, t + x∗)

+ σ 20

2κ2e−κ(t+x)

(eκt − e−κt

)(e−κx − e−κx∗)

+ e−κ(x−x∗)r(t, x∗) (37)


for the forward rate curve in terms of the forward rate r(t, x∗). The stochastic differentialequation governing the dynamics of r(t, x∗) can then be obtained from (24).

A popular version of this model is obtained by setting x∗ = 0, so that (36) expressesψ1(t) and (37) expresses the forward rate curve in terms of the short rate r(t).

The expressions for the yield and bond prices of any maturity in this case are easilycomputed and are given by

y(t, x)= γ (t, x) + 1

κx

(1 − e−κ(t+x)

)ψ1(t)

+ σ 20

4κ3x

(eκt − e−κt

)(1 − e−κx

)(2 − e−κt − e−κ(t+x)

), (38)

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

b(0, t)exp

[− 1

κ

(1 − e−κ(t+x)

)ψ1(t)

]

× exp

[− σ 2

0

4κ3

(eκt − e−κt

)(1 − e−κx

)(2 − e−κt − e−κ(t+x)

)], (39)

where

γ (t, x) = 1

x

∫ x

0r(0, t + u) du.

Once again, the expression (36) for ψ1(t) can be substituted in the previous two equationsto obtain the yields and bond prices in terms of the forward rate r(t, x∗).

Note that similar arguments using (38) instead of (35) allow the forward rate curve, theyield curve and bond prices to be expressed in terms of a yield rather than a forward rate.For example, rearranging (38) with x = x∗ gives

ψ1(t) = κx∗

1 − e−κ(t+x∗)

×[y(t, x∗)− γ

(t, x∗)− σ 2

0

4κ3x∗(eκt − e−κt

)(1 − e−κx∗)(

2 − e−κt − e−κ(t+x∗))],and substituting this expression for ψ1(t) into (35) gives

r(t, x) = r(0, t + x) + κx∗e−κ(t+x)

1 − e−κ(t+x∗)[y(t, x∗) − γ

(t, x∗)]

+ σ 20

2κ2(1 − e−κ(t+x∗))e−κ(t+x)

(eκt − e−κt

)×

[(eκt − 1

) + (e−κx∗ − e−κx

) + 1

2e−κt

(1 − e−2κx∗)]

. (40)

This final representation is important since the instantaneous forward rates are not directlyobserved in the market and are consequently inconvenient for practical implementation ofthese models. In contrast, the yields are directly observed in the market and hence moresuited for that purpose.


Figure 1. Forward rate curves for the Vasicek model with σ0 = 0.03, κ = 0.2 and a flat 5% initial curve. Thecurves correspond to r(t) being 4.9%, 5% and 5.1%.

Note also that (40) enables the unobserved instantaneous forward rates to be determinedexplicitly in terms of the observed yields.

Examples of the types of forward rate curves that can be generated by the Vasicek modelare shown in Figure 1. The short rate, r(t), was used as the state variable, and the parametervalues, σ0 = 0.03 and κ = 0.2, were chosen so that the short rate volatility is 3% and the10-year forward rate volatility is approximately 13.5% of the short rate volatility. Forsimplicity, a flat initial curve at 5% was assumed. The curves shown in the figure are at3 months time and correspond to r(t) taking the value 4.9%, 5% or 5.1% at that time.

4.2. Ritchken–Sankarasubramanian Model

The next example is the Ritchken and Sankarasubramanian (1995) model, Mrs, which is a1-dimensional HJM model corresponding to the forward rate volatility process

σ(t, x) = g(t, r(t)

)e− ∫ x

t κ(u)du, (41)

where κ and g is are deterministic functions. Note that since σ(t, x) satisfies the equation∂σ(t, x)/∂x + κ(x)σ (t, x) = 0, Assumption 1 is satisfied, and since the dependence ofσ(t, x) on the path enters only through the short rate r(t), Assumption 2 is also satisfied.Once again, rewriting the forward rate volatility in the form

σ(t, x) = g(t, r(t)

)e∫ t

0 κ(v)dv · e− ∫ x0 κ(v)dv (42)

allows the coefficient and the function of x in (10) to be identified as

c(t) = g(t, r(t)

)e∫ t

0 κ(v)dv and σ(x) = e− ∫ x0 κ(v)dv. (43)


The state variables ϕ11(t) and ψ1(t) for Mrs are given by

ϕ11(t) =∫ t

0g(s, r(s)

)2e2

∫ s0 κ(v)dv ds, (44)

ψ1(t) =∫ t

0g(s, r(s)

)e∫ s

0 κ(v)dvdw(s)

−∫ t

0g(s, r(s)

)2e2

∫ s0 κ(v)dv

∫ s

0e− ∫ u

0 κ(v)dv du ds, (45)

where the superscripts have once again been omitted from ϕ11(t) and ψ1(t). In contrast tothe extended Vasicek model, ϕ11(t) and ψ1(t) are both non-deterministic in this case andso S∗ = {ϕ11(t), ψ1(t)}. It follows that Mrs admits a 2-dimensional affine realisation. Infact, (22) can be written

r(t, x) = r(0, t + x) + α(t, x)ψ1(t) + β(t, x)ϕ11(t), (46)

where

α(t, x) = e− ∫ t+x0 κ(v)dv,

β(t, x) = e− ∫ t+x0 κ(v)dv

∫ t+x

0e− ∫ u

0 κ(v)dv du.

Now, since α(t, x) > 0 for all t and x, if x1 = x2 we have

-(t, x1, x2) = det

[α(t, x1) β(t, x1)

α(t, x2) β(t, x2)

]= α(t, x1)

2α(t, x2)2∫ t+x2

t+x1

α(t, u) du = 0

for all t ∈ R+, and equation (21) in this case becomes

ψ1(t) = β(t, x2)rδ(t, x1) − β(t, x1)rδ(t, x2)

-(t, x1, x2), (47)

ϕ11(t) = α(t, x1)rδ(t, x2) − α(t, x2)rδ(t, x1)

-(t, x1, x2), (48)

where rδ(t, x) = r(t, x)− r(0, t + x). The equations (47) and (48) can now be substitutedinto (46) to give

r(t, x) = r(0, t + x) + -(t, x, x2)

-(t, x1, x2)rδ(t, x1) + -(t, x1, x)

-(t, x1, x2)rδ(t, x2), (49)

which expresses the forward rate curve as an affine function of the two forward ratesr(t, x1) and r(t, x2).

To express the state variables ψ1(t) and ϕ11(t) in terms of yields, note from (27) and (46)that

y(t, x) = γ (t, x) + 1

xα(t, x)ψ1(t) + 1

xβ(t, x)ϕ11(t), (50)


where

α(t, x) =∫ x

0α(t, u) du, β(t, x) =

∫ x

0β(t, u) du.

If x1 = x2, then we have

-(t, x1, x2) = det

[α(t, x1) β(t, x1)

α(t, x2) β(t, x2)

]= 1

2α(t, x1)α(t, x2)

[α(t, x2) − α(t, x1)

] = 0,

and the expressions for ψ1(t) and ϕ11(t) in terms of the yields y(t, x1) and y(t, x2) are

ψ1(t) = x1β(t, x2)yδ(t, x1) − x2β(t, x1)yδ(t, x2)

-(t, x1, x2), (51)

ϕ11(t) = x2α(t, x1)yδ(t, x2) − x1α(t, x2)yδ(t, x1)

-(t, x1, x2), (52)

where yδ(t, x) = y(t, x)− γ (t, x). Substituting (51) and (52) into (46), for example, gives

r(t, x) = r(0, t + x) +[α(t, x)β(t, x2) − β(t, x)α(t, x2)

-(t, x1, x2)

]x1yδ(t, x1)

+[β(t, x)α(t, x1) − α(t, x)β(t, x1)

-(t, x1, x2)

]x2yδ(t, x2). (53)

Again, (53) enables the unobserved instantaneous forward rates to be determined explicitlyin terms of the observed yields. An expression of this form was first obtained in Bliss andRitchken (1996). Finally, the bond price is given by the equation

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

b(0, t)exp

[−α(t, x)ψ1(t) − β(t, x)ϕ11(t)]. (54)

Equations (47) and (48) can be used to express the bond price in terms of a set of forwardrates, and similarly (51) and (52) can be used to express the bond price in terms of a set ofyields.

Examples of the types of forward rate curves that can be generated by the Ritchken–Sankarasubramanian model are shown in Figure 2. The short rate, r(t), and the 10-yearforward rate, r(t, 10), were used as the state variables, and to enable comparison with theVasicek model, the parameter values, σ0 = 0.135 and κ = 0.2, were chosen to maintainthe short rate volatility at 3% and the volatility of the 10-year forward rate at approximately13.5% of the short rate volatility. Once again, the initial curve was assumed to be flat at5%. The curves shown in the figure are at 3 months time, and correspond to r(t) takingthe value 4.9%, 5% or 5.1%, and r(t, 10) taking the value 4.95%, 5% or 5.05% at thattime. The extra flexibility afforded by the existence of an additional state variable resultsin a richer class of forward rate curves for the Ritchken–Sankarasubramanian model whencompared to those possible under the Vasicek model.


Figure 2. Forward rate curves for the Ritchken–Sankarasubramanian model with σ0 = 0.135, κ = 0.2 and flat5% initial curve. The curves correspond to r(t) taking values 4.9%, 5% and 5.1%, and r(t, 10) taking values4.95%, 5% and 5.05%.

4.3. HJM Model with Humped Volatility

The next example is a 1-dimensional HJM model Mhv corresponding to the forward ratevolatility

σ(t, x) = rλ(t)[β0 + β1(x − t)

]e−κ(x−t ), (55)

where λ, κ, β0 and β1 are constants. This represents a term structure model in whichthe volatility is level dependent and contains a hump.5 Since (∂/∂x + κ)2σ(t, x) = 0,Assumption 1 is satisfied and since the dependence of σ(t, x) on the path enters onlythrough the short rate r(t) Assumption 2 is also satisfied. Rewriting σ(t, x) in the form

σ(t, x) = (β0 − tβ1)rλ(t)eκt · e−κx + β1r

λ(t)eκt · xe−κx, (56)

the quantities in the decomposition (10) can be identified as

c1(t) = (β0 − tβ1)rλ(t)eκt , σ1(x) = e−κx,

c2(t) = β1rλ(t)eκt , σ2(x) = xe−κx.

The state variables ψ1(t), ψ2(t), ϕ11(t), ϕ12(t) and ϕ22(t) for Mhv are given by

ϕ11(t) =∫ t

0(β0 − sβ1)

2r2λ(s)e2κs ds, (57)

ϕ12(t) =∫ t

0β1(β0 − sβ1)r

2λ(s)e2κs ds, (58)

ϕ22(t) =∫ t

0β2

1 r2λ(s)e2κs ds, (59)


ψ1(t) =∫ t

0(β0 − sβ1)r

λ(s)eκsdw(s)

− 1

κ

∫ t

0(β0 − sβ1)

2r2λ(s)e2κs(1 − e−κs)ds

− 1

κ2

∫ t

0β1(β0 − sβ1)r

2λ(s)e2κs[1 − (1 + κs)e−κs]ds, (60)

ψ2(t) =∫ t

0β1r

λ(s)eκsdw(s)

− 1

κ

∫ t

0β1(β0 − sβ1)r

2λ(s)e2κs(1 − e−κs)ds

− 1

κ2

∫ t

0β2

1r2λ(s)e2κs[1 − (1 + κs)e−κs

]ds, (61)

where the superscripts have once again been omitted from the state variables. In this caseall the state variables are non-deterministic so that

S∗ = {ψ1(t), ψ2(t), ϕ11(t), ϕ12(t), ϕ22(t)

}.

Now, equation (13) for σj (x) gives

σ1(x) = 1

κ

(1 − e−κx

), (62)

σ2(x) = 1

κ2

[1 − (1 + κx)e−κx

], (63)

and the expression (22) for the forward rate curve in Mhv is

r(t, x) = r(0, t + x) +2∑

i=1

σi(t + x)ψi(t) +2∑

i=1

σi(t + x)σi(t + x)ϕii(t)

+ [σ1(t + x)σ2(t + x) + σ2(t + x)σ1(t + x)

]ϕ12(t). (64)

Let c(t, x) be the R5-valued function defined by

c(t, x)= [σ1(t + x), σ2(t + x), σ1(t + x)σ1(t + x),

σ1(t + x)σ2(t + x) + σ2(t + x)σ1(t + x), σ2(t + x)σ2(t + x)]τ

so that (64) can be written more succinctly in the form

rδ(t, x) = c(t, x)τ[ψ1(t), ψ2(t), ϕ11(t), ϕ12(t), ϕ22(t)

]τ, (65)

and define the matrix M∗(t, x1, . . . , x5) by

M∗(t, x1, . . . , x5) = [c(t, x1), c(t, x2), c(t, x3), c(t, x4), c(t, x5)

]τ.


Then it follows from the linear independence of the functions e−κx , xe−κx , x2e−κx , e−2κx

and xe−2κx that

-∗(t, x1, . . . , x5) = detM∗(t, x1, . . . , x5) = 0

for any x1 < · · · < x5, and hence (64) can be inverted to express the state variables in termsof the forward rates r(t, x1), . . . , r(t, x5). That is, substituting x = x1, . . . , x5 into (65)and inverting the resulting system of equations gives[ψ1(t), ψ2(t), ϕ11(t), ϕ12(t), ϕ22(t)

]τ = M∗(t, x1, . . . , x5)−1[rδ(t, x1), . . . , rδ(t, x5)

]τ,

and substituting this expression into (65) gives

r(t, x) = r(0, t + x) + c(t, x)τM∗(t, x1, . . . , x5)−1[rδ(t, x1), . . . , rδ(t, x5)

]τ. (66)

Formulae for the corresponding yields and bond prices can be obtained by integrating (64)as in the previous examples.

Examples of the types of forward rate curves that can be generated by the humpedvolatility model are shown in Figure 3. The state variables in this case were the short rate,r(t), and the forward rates r(t, 2.25), r(t, 5), r(t, 7.5) and r(t, 10). To enable comparisonwith the previous examples, the parameter values, β0 = 0.135, β1 = 0.035, κ = 0.2 andλ = 0.5, were chosen which maintains the short rate volatility at 3% and the 10-year for-ward rate volatility at 13.5% of the short rate volatility. These parameters place the humpin the volatility curve at x = 1.25. As with the previous examples, the initial curve wasassumed to be flat at 5%. The curves in the figure are once again at 3 months time, and cor-respond to r(t) taking values 4.9%, 5% or 5.1%, and r(t, 10) taking values 4.95%, 5% or5.05%. The values of the remaining state variables were chosen so that the resulting curveswere smooth. An important feature of the forward rate curves allowed under the humpedvolatility specification is the possibility of sharp curvature changes which is absent in thecurves allowed under the previous two models.

Figure 3. Forward rate curves for the humped volatility model with β0 = 0.135, β1 = 0.036, κ = 0.2, λ = 0.5and flat 5% initial curve. The curves correspond to r(t) taking values 4.9%, 5% and 5.1%, and r(t, 10) takingvalues 4.95%, 5% and 5.05%.


4.4. A Three Dimensional HJM Model

Principal components analysis reveals that the shifting, tilting and bending account formost of the movements in the forward rate curve. With this in mind, the final exampleconsiders a 3-dimensional HJM model, M3d, in which each of these factors are representedas a component of the forward rate volatility process. That is, we consider a HJM modelwith

σ1(t, x) = β11(t, r(t)

), (67)

σ2(t, x) = β21(t, r(t)

)e−κ2(x−t ), (68)

σ3(t, x) = [β31

(t, r(t)

) + β32(t, r(t)

)(x − t)

]e−κ3(x−t ), (69)

where κ2 and κ3 are constants, r(t) = (r(t, x1), r(t, x2), . . . , r(t, xm))τ is a vector offorward rates with maturities x1 < x2 < · · · < xm, and βij (t, r(t)) are deterministic func-tions. Then since ∂/∂x(σ1(t, x)) = 0, (∂/∂x + κ2)σ2(t, x) = 0 and (∂/∂x + κ3)

2σ2(t, x)

= 0 Assumption 1 is satisfied, and since the dependence of σi(t, x) on the path enters onlythrough the forward rate vector r(t), Assumption 2 is also satisfied.

Decompositions of the σi(t, x) in the form (10) are easily seen to be

c11(t) = β11(t, r(t)

), σ11(x) = 1,

c21(t) = β21(t, r(t)

)eκ2t , σ21(x) = e−κ2x,

c31(t) = [β31

(t, r(t)

) − tβ32(t, r(t)

)]eκ3t , σ31(x) = e−κ3x,

c32(t) = β32(t, r(t)

)eκ3t , σ32(x) = xe−κ3x,

and the σij (x) in this case are

σ11(x) = x,

σi1(x) = 1

κi

(1 − e−κix

), i = 2, 3,

σ32(x) = 1

κ23

[1 − (1 + κ3x)e

−κ3x].

Denoting βij (s) = βij (s, r(s)) for brevity, the state variables for M3d are

ϕ111(t) =

∫ t

0β11(s)

2 ds,

ϕ211(t) =

∫ t

0β21(s)

2e2κ2s ds,

ϕ311(t) =

∫ t

0

[β31(s) − sβ32(s)

]2e2κ3s ds,

ϕ312(t) =

∫ t

0

[β31(s) − sβ32(s)

]β32(s)e

2κ3s ds,

ϕ322(t) =

∫ t

0β32(s)

2e2κ2s ds,


ψ11 (t) =

∫ t

0β11(s) dw1(s) −

∫ t

0sβ11(s)

2 ds,

ψ21 (t) =

∫ t

0β11(s)e

κ2s dw2(s) − 1

κ2

∫ t

0β21(s)

2e2κ2s(1 − e−κ2s

)ds,

ψ31 (t) =

∫ t

0

[β31(s) − sβ32(s)

]eκ3s dw3(s)

− 1

κ3

∫ t

0

[β31(s) − sβ32(s)

]2e2κ3s

(1 − e−κ3s

)ds

− 1

κ23

∫ t

0

[β31(s) − sβ32(s)

]β32(s)e

2κ3s[1 − (1 + κ3s)e

−κ3s]ds,

ψ32 (t) =

∫ t

0β32(s)e

κ3s dw3(s)

− 1

κ3

∫ t

0

[β31(s) − sβ32(s)

]β32(s)e

2κ3s(1 − e−κ3s

)ds

− 1

κ23

∫ t

0β32(s)

2e2κ3s[1 − (1 + κ3s)e

−κ3s]ds,

and the expression (12) for the forward rate curve in M3d is

r(t, x) = r(0, t + x) + σ11(t + x)ψ11 (t) + σ21(t + x)ψ2

1 (t)

+ σ31(t + x)ψ31 (t) + σ32(t + x)ψ3

2 (t)

+ σ11(t + x)σ11(t + x)ϕ111(t)

+ σ21(t + x)σ21(t + x)ϕ211(t)

+ σ31(t + x)σ31(t + x)ϕ311(t)

+ [σ31(t + x)σ32(t + x) + σ32(t + x)σ31(t + x)

]ϕ3

12(t)

+ σ32(t + x)σ32(t + x)ϕ322(t). (70)

Let c(t, x) = (c1(t, x), . . . , c9(t, x))τ be the R

9-valued function where

c1(t, x) = σ11(t + x),

c2(t, x) = σ21(t + x),

c3(t, x) = σ31(t + x),

c4(t, x) = σ32(t + x),

c5(t, x) = σ11(t + x)σ11(t + x),

c6(t, x) = σ21(t + x)σ21(t + x),

c7(t, x) = σ31(t + x)σ31(t + x),

c8(t, x) = σ31(t + x)σ32(t + x) + σ32(t + x)σ31(t + x),

c9(t, x) = σ32(t + x)σ32(t + x),


and let ψ = (ψ11 , ψ

21 , ψ

31 , ψ

32 , ϕ

111, ϕ

211, ϕ

311, ϕ

312, ϕ

322) be the R

9-valued process whosecomponents are the state variables. Then the forward rate curve can be written more suc-cinctly as

r(t, x) = r(0, t + x) + c(t, x)τχ(t).

Now, if κ2 = κ3, κ2 = 2κ3 and κ3 = 2κ2 then the functions 1, e−κ2x , e−κ3x , e−2κ2x , e−2κ3x ,x, xe−κ3x , xe−2κ3x and x2e−2κ3χ are independent, and there exist x1 < x2 < · · · < x9such that the matrix

M∗(t, x1, . . . , x9) = [c(t, x1), . . . , c(t, x9)

]τis invertible. Hence, the state variables can be written

χ(t) = M∗(t, x1, . . . , x9)−1[rδ(t, x1), . . . , rδ(t, x9)

]τ,

where rδ(t, xj ) = r(t, xj ) − r(0, t + xj ) as usual, and the forward rate equation (70) canbe rewritten in terms of the forward rates r(t, xj ) as

r(t, x) = r(0, t + x) + c(t, x)τM∗(t, x1, . . . , x9)−1[rδ(t, x1), . . . , rδ(t, x9)

]τ.

This equation can be integrated to give the yields and the bond prices as in the previousexamples.

Examples of the types of forward rate curves that can be generated by the 3-dimensionalHJM models is shown in Figure 4. The state variables in this case were the short rater(t) and the forward rates r(t, 0.125k), where k ∈ {1, 2, . . . , 8}. It was assumed that thethree volatility components made equal contributions, and the parameters, β11 = 0.0775,β21 = 0.0775, β31 = 0.0775, β32 = 0.02675, κ2 = 0.075 and κ3 = 0.2, λ = 0.5, werechosen so that the short rate volatility is 3%, the 10-year forward rate volatility is 13.5%of the short rate volatility and the hump in the volatility curve occurs at x = 1.25. Once

Figure 4. Forward rate curves for the 3-dimensional HJM model with β0 = 0.135, β1 = 0.036, κ = 0.2,λ = 0.5 and flat 5% initial curve. The curves correspond to r(t) taking values 4.9%, 5% and 5.1%, and r(t, 10)taking values 4.95%, 5% and 5.05%.


again, the initial curve was assumed to be flat at 5%, and the curves shown in the figureare after 3 months. The shapes of forward rate curves possible under the 3-dimensionalmodel include those from the previous models, and may also exhibit oscillatory patternswith sharp curvature changes.

5. Relationship to the Duffie–Kan Characterisation of Affine Models

An important feature of the interest rate models discussed in this paper is that they areaffine and yet the volatility structures of the state variables do not have the form specifiedin Duffie and Kan (1996). That is, the volatilities of the state variables are not square rootaffine in the state variables. The reason for this apparent inconsistency is that the squareroot affine characterisation of the volatilities requires additional conditions that are not metby the models considered in this paper.

More specifically, suppose that the bond market is arbitrage free, so that a martingalemeasure exists, and let X(t) = (X1(t), . . . , Xd(t)) be an Itô process satisfying a stochasticdifferential equation of the form

dX(t) = µ(t, X(t)

)dt + σ

(t, X(t)

)dwt , (71)

under the martingale measure. Now, if the bond price, b(t, x), has the exponential affineform

b(t, x) = exp

[−b0(t, x) −

d∑i=1

bi(t, x)Xi(t)

], (72)

then Duffie and Kan showed that σ(t,X(t)) is square root affine in X(t) provided thatµ(t,X(t)) and σ(t,X(t))∗σ(t,X(t)) are affine in X(t).

In broad terms, the bond price is exponential affine if and only if the expression

µ(t, X(t)

)∗B(t, x) − 1

2

∣∣σ (t, X(t)

)∗B(t, x)

∣∣2 (73)

is affine, where B(t, x) = (b1(t, x), . . . , bd (t, x))∗. However, in order for the vola-

tility, σ(t, x), to be square root affine, the additional conditions that µ(t,Xt) andσ(t,X(t))∗σ(t,X(t)) be affine must be imposed. For the models considered in thispaper, the expression given in (73) is indeed affine, but the terms µ(t,X(t)) andσ(t,X(t))∗σ(t,X(t)) fail to be so. Consequently, the bond prices are exponential affinebut the volatilities are not square root affine.

As an example, consider the Ritchken–Sankarasubramanian model from Section 4.2corresponding to g(t, r(t)) = σ0r(t) and κ(x) = κ , where σ0, κ ∈ R+, so that

σ(t, x) = σ0r(t)e−κ(x−t ).


For simplicity, take as the state variables ϕ11(t) and ψ1(t) defined in (44) and (45), respec-tively. Then the stochastic differential equations for ϕ11(t) and ψ1(t) under the martingalemeasure are

dϕ11(t) = σ 20 r(t)

2e2κt dt,

dψ1(t) = − 1

κσ 2

0 r(t)2e2κt(1 − e−κt

)dt + σ0r(t)e

κt dwt .

In particular, letting X(t) = (ϕ11(t), ψ1(t))∗, the drift and the volatility terms are

µ(t, X(t)

) =(σ 2

0 r(t)2e2κt ,− 1

κσ 2

0 r(t)2e2κt(1 − e−κt

))∗,

σ(t, X(t)

) = (0, σ0r(t)e

κt)∗.

From (54), the coefficients b1(t, x) and b2(t, x) in the bond price equation (72) are

b1(t, x) = 1

2κ2e−κt

(2 − e−kt

(1 + e−κx

))(1 − e−κx

),

b2(t, x) = 1

κe−κt

(1 − e−κx

).

Computing the expression (73) for this case gives

µ(t, X(t)

)∗B(t, x) − 1

2

∣∣σ (t, X(t)

)∗B(t, x)

∣∣2 ≡ 0,

which is trivially affine in X(t) and accounts for the exponential affine bond price. How-ever, µ(t,X(t)) and σ(t,X(t))∗σ(t,X(t)) are quadratic in r(t), and since r(t) is affinein X(t), it follows that µ(t,X(t)) and σ(t,X(t))∗σ(t,X(t)) are, in fact, quadratic in X(t).In particular, they are not affine in X(t) and so the Duffie–Kan square root affine character-isation of σ(t,X(t)) does not apply in this case. A more detailed discussion clarifying theDuffie–Kan characterisation of affine term structure models can be found in Kwon (2001).

6. Model Calibration and Derivative Pricing

This section outlines some practical applications of the framework developed in this paper,in particular to model calibration and derivative pricing.

To clarify the exposition, the Ritchken–Sankarasubramanian model with g(t, r(t)) =σ0r(t)

γ and κ(x) = κ0, where σ0, γ and κ0 are constants, will be used to illustrate theapplications. Note that the forward rate volatility process for the corresponding model,Mrs, is given by the equation

σ(t, x) = σ0r(t)γ e−κ(x−t ).

Now, the calibration of model parameters to observed market data is a difficult problemin the implementation of interest rate models. In cases where explicit formulae for deriva-tive prices are known, this difficulty can be overcome by computing the parameters as thosethat minimise the error in the model implied prices. So in view of the explicit expression


for the bond price (25), it may appear that the parameters for the models considered in thispaper can be calibrated by minimising the pricing error as explained above. Unfortunately,it turns out that, in general, only some of the model parameters appear explicitly in thepricing formulae. For example, in Mrs, the bond price is given by the expression

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

b(0, t)

[−a0(κ, t, x) − a1(κ, t, x)r(t) − a3(κ, t, x)r(t, x∗)],

where x∗ > 0, and the ai(κ, t, x) are deterministic functions of t, x and the parameter κ0only. Consequently, only a subset of the parameters can be calibrated using this approachat best. Furthermore, the possibility of using other derivatives to calibrate the remainingparameters is excluded by the fact that the explicit pricing formulae for these derivativesare often not available.

An alternative calibration procedure is to use the dynamics of the observed quantities.Since the stochastic differential equations for the derivatives prices are available, it mayappear possible to determine the parameters by first discretising these equations and thencalibrating the implied price changes against the market observed changes. For example,it follows from (5) that the stochastic differential equation for the zero coupon bond price,b(t, x), in Mrs is

db(t, x) = b(t, x)[r(t) − r(t, x)

]dt − σ0

κ0r(t)γ b(t, x)

[1 − e−κ0x

]dw(t).

So assuming that the bond prices and forward rates of all maturities are observable, it mayappear plausible to try and calibrate the changes in the bond prices implied by the aboveequation against the observed changes by setting

bobs(t + -t, x) − bobs(t, x) ≈ bobs(t, x)[robs(t) − robs(t, x)

]-t

− σ0

κ0robs(t)γ bobs(t, x)

[1 − e−κ0x

]-w(t),

where the superscript “obs” denotes observed quantities. The problem with this approachis that the left-hand side of the above equation is observed under the market measure whilethe right-hand side is given under the martingale measure as indicated by the term -w(t),where w(t) is a standard Wiener process under the martingale measure. In particular, thedistributional properties of the left-hand side is not the same as that for the right-handside, and the reconciliation of the two sides requires the knowledge of the so-called marketprice of risk which is not observable. Consequently, this approach is also limited, unlessan assumption is made for the functional form of the market price of risk.

An important observation from (15) is that the stochastic differential equation,

dϕijk(t) = cij (t)cik(t) dt,

for the state variable ϕijk(t) does not explicitly involve the increments of the Wiener process

w(t), and hence allows the measure inconsistency problem to be avoided. More specifi-cally, the above equation can be discretised to give

ϕijk(t + -t) − ϕi

jk(t) ≈ cij (t)cik(t)-t.


Next, as shown in Sections 2 and 3, the quantities ϕijk(t), cij (t) and cik(t) can be expressed

explicitly in terms of the model parameters and observable quantities, and so the previousequation can be written

ϕi,obsjk (t + -t; θ) − ϕ

i,obsjk (t; θ) ≈ cobs

ij (t; θ)cobsik (t; θ)-t,

where θ denotes the model parameters which are the only unknowns in the equation. Sofinally, the model parameters can be determined by minimising, for example, the sum ofthe squared errors

εijk(t) = (ϕi,obsjk (t + -t; θ) − ϕ

i,obsjk (t; θ) − cobs

ij (t; θ)cobsik (t; θ)-t

)2.

The calibration of interest rate models using precisely this procedure is undertaken in Kwon(2002).

Once the model parameters have been determined, numerical techniques such as MonteCarlo simulation method can be applied to compute the derivative prices, bearing in mindthat the dynamics of the underlying quantities must be generated under the martingalemeasure. But this does not pose any problems, since the stochastic differential equations,(14) and (15), for the state variables were specified under the martingale measure.

7. Conclusion

Markovian HJM models are ideal for the study of term structure dynamics and interestrate derivatives, but usually require the introduction of a large number of state variables.In many of the Markovian HJM models considered in the literature, the issues of relatingthe state variables to market observed quantities and the determination of the explicit func-tional form of the forward rate curve in terms of the state variables were left unaddressedto a large extent. The results of this paper resolve this gap by linking the state variables di-rectly, and explicitly, to the forward rates and market observed yields, and establishing thatthe models are, in fact, affine with respect to a finite number of forward rates or yields. Theapproach taken in this paper also simplifies and generalises the results previously obtainedin the literature.

The results of this paper provide a consistent framework for the systematic constructionof a wide range of affine term structure models within the HJM framework, and Chiarellaand Kwon (2001b) provides an example of such a construction. This framework can beextended in various ways, for example, to the stochastic volatility setting as in Chiarella,Colwell, and Kwon (2001) and to the jump diffusion setting as in Chiarella and Kwon(2002).

This paper also established an explicit formula for the bond price in terms of the statevariables, generalising the results of Ritchken and Sankarasubramanian (1995) and Inuiand Kijima (1998). In particular, it was shown that the bond price takes an exponentialaffine form for a much broader class of models than the square root affine volatility modelsconsidered in Duffie and Kan (1996).

The representations of the forward rate curve in terms of market observed yields ob-tained 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 modelsremains an on-going project.

Notes

1. It is a consequence of the HJM forward rate drift restriction that the forward rate drift under the risk neutralmeasure is σ(t, t + x)τ

∫ t+xt σ (t, u) du.

2. Recall that Duffie and Kan (1996) refer to b(t, x) as being exponential affine if there exists a d-dimensionalMarkov process z(t) such that

b(t, x) = exp[−k0(t, x) − k(t, x)τ z(t)],where k0(t, x) and k(t, x) are deterministic R- and R

d -valued functions, respectively.3. A similar condition was obtained in Björk and Svensson (2001) for the separable volatility case.4. The extension where σ0 and κ are deterministic functions of t can be handled in a similar manner.5. For a detailed discussion of a simpler case of this model with λ = 0, see Ritchken and Chuang (1999).

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Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields