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Interest rate models on Lie groupsF. C. Parka; C. M. Chunb; C. W. Hanc; N. Webberd

a Institute for Research in Finance & Economics, Seoul National University, Seoul 151-742, Korea b

Center for Cognitive Robotics Research, Korea Institute of Science and Technology, Seoul 136-791,Korea c Capital Markets & Portfolio Research, Inc., Seoul 151-742, Korea d Warwick Business School,Warwick, UK

First published on: 11 May 2010

To cite this Article Park, F. C. , Chun, C. M. , Han, C. W. and Webber, N.(2011) 'Interest rate models on Lie groups',Quantitative Finance, 11: 4, 559 — 572, First published on: 11 May 2010 (iFirst)To link to this Article: DOI: 10.1080/14697680903468963URL: http://dx.doi.org/10.1080/14697680903468963

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Quantitative Finance, Vol. 11, No. 4, April 2011, 559–572

Interest rate models on Lie groups

F. C. PARK*y, C. M. CHUNz, C. W. HANx and N. WEBBER{

yInstitute for Research in Finance & Economics, Seoul National University, Seoul 151-742, KoreazCenter for Cognitive Robotics Research, Korea Institute of Science and Technology,

Seoul 136-791, KoreaxCapital Markets & Portfolio Research, Inc., Seoul 151-742, Korea

{Warwick Business School, Warwick, UK

(Received 26 June 2008; in final form 29 October 2009)

This paper examines an alternative approach to interest rate modeling, in which the nonlinearand random behavior of interest rates is captured by a stochastic differential equationevolving on a curved state space. We consider as candidate state spaces the matrix Lie groups;these offer not only a rich geometric structure, but—unlike general Riemannian manifolds—also allow for diffusion processes to be constructed easily without invoking the machinery ofstochastic calculus on manifolds. After formulating bilinear stochastic differential equationson general matrix Lie groups, we then consider interest rate models in which the short rate isdefined as linear or quadratic functions of the state. Stochastic volatility is also augmented tothese models in a way that respects the Riemannian manifold structure of symmetricpositive-definite matrices. Methods for numerical integration, parameter identification,pricing, and other practical issues are addressed through examples.

Keywords: Financial mathematics; Financial engineering; Interest rate modelling; Affine termstructure models

1. Introduction

The various interest rate term structure models that havebeen proposed in the literature are motivated in large partby a desire to capture the inherent nonlinear behavior ofinterest rates. Most involve a nonlinear stochastic statedynamics evolving on a vector space, together with aspecification of the mathematical relation between thestate vector and the economic variables of interest (e.g.,the short rate, forward rates, various spot rates). Themyriad issues involved in choosing a vector space interestrate model are well-documented by, for example, Jamesand Webber (2000) and Brigo and Mercurio (2001).

In this paper we examine an alternative approach tocapturing the inherent nonlinear stochastic behavior ofthe interest rate term structure. Unlike past approaches,in which the drift and noise volatility terms of thestochastic state equations are carefully crafted to reflectvarious observed phenomena (e.g., the mean reversioneffect, the positivity of rates, the qualitatively differentbehavior at low versus high interest rate regimes,matching with the observed yield curve), we try to keep

the drift and volatility terms simple, and instead choose

an underlying state space that is curved.The idea of constructing interest rate models on general

differentiable manifolds, and more generally the problem

of valuing financial models evolving on nonlinear state

spaces, was first explored by Nunes and Webber (1997),

James and Webber (2000) and Webber (2000). In these

works they construct a general coordinate-invariant

modeling framework, and develop a concrete example

of a short rate model on the circle S1. A principal

component analysis of historical data is also undertaken

to try to uncover any manifold structure in the observa-

tions. Earlier work by Hughston (1994) also presents a

framework for arbitrage-free pricing on general

Riemannian manifolds.Other notable work on interest rate models that makes

explicit contact with geometry includes Bjork (2000), who

examines the Heath–Jarrow–Morton framework in a

geometric setting and—by extending results from non-

linear controllability—determines the family of yield

curves that can be propagated in a consistent fashion;

the main result is a set of conditions under which

finite-dimensional realizations of the forward rate process

are possible. Flipovic and Teichmann (2004) examine the*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2011 Taylor & Francis

http://www.informaworld.comDOI: 10.1080/14697680903468963

Downloaded By: [Park, Frank C.] At: 13:42 26 March 2011

closely related question of when a general stochasticdifferential equation evolving on a separable Hilbertspace admits invariant manifolds, and characterize allfinite-dimensional Heath–Jarrow–Morton models thatadmit arbitrary initial yield curves.

One appeal of a geometric model evolving on a curvedstate space is the possibility of a global description ofinterest rates valid over a wide range of regimes. There isno a priori reason, apart from our intuitive familiaritywith reasoning and calculating in vector spaces, to insistthat the actual state space for interest rates necessarily bea vector space; the earlier empirical studies reported byWebber (2000) support this view. Furthermore, thevarious observed behavioral properties of interest rates,for example the mean reversion effect, positivity andboundedness of rates, and regime shifts, may potentiallybe captured more naturally by a curved state space,leading to simpler characterizations of the drift andvolatility.

Paradoxically, what has slowed the development of thegeometric interest rate framework laid out by Nunes andWebber (1997) and James and Webber (2000) has in factbeen its purported advantage of generality: the set ofdifferentiable manifolds encompasses an untenably largeclass of state spaces with very little structure to exploit.Moreover, when the dynamics are eventually expressed inlocal coordinates, the ensuing formulas become sounwieldy as to offer little if any insight. Unless the classof admissible curved state spaces is further narrowed, onecan become easily bogged down just in the construction ofgeometrically well-defined noise processes (i.e. that arecoordinate-invariant), which requires the machinery ofstochastic calculus on manifolds.

In this paper we carry out the geometric programoriginally initiated by Nunes and Webber (1997) andWebber (2000), but instead restrict our attention to aspecial class of differentiable manifold, the matrix Liegroups, as the underlying state space. We focus on Liegroups for the following reasons.

(1) Diffusions on matrix Lie groups are straightfor-ward to construct and intuitive to understand. Bythe same logic, bilinear stochastic differentialequations on Lie groups—in a sense to be madeprecise below—are also intuitive and easily manip-ulable. This can be traced to the fact that the rulesfor stochastic calculus on matrix Lie groups arestraightforward, unlike stochastic calculus ongeneral differentiable manifolds, where the con-struction of diffusions requires complicated math-ematical machinery (Emery 1989).

(2) There exists a wide variety of Lie groups that canserve as candidate state spaces, and a wealth ofresults and methods for analysis and computationon general Lie groups. For example, the SpecialOrthogonal group SO(n), which is compact, ispotentially well-suited as a state space for boundedpositive rate models. The case study analysed byJames and Webber (2000) is in fact developed forthe circle S1, which is equivalent to SO(2).

We also show how the above models can be augmentedto have stochastic volatility in a way that respects thegeometry of the space of covariance matrices. Thecovariance matrices, which by definition are required tobe symmetric positive-definite, also happen to possess thestructure of a Riemannian manifold, P(n), with a natu-rally defined metric. We show how to construct stochasticdynamics on the manifold P(n), and use these as astochastic covariance associated with a Lie group diffu-sion process.

The paper is organized as follows. We begin in section 2with some geometric preliminaries on matrix Lie groups,and review how the standard construction of diffusions inRn can be extended to matrix Lie groups. We thenformulate bilinear stochastic differential equations onmatrix Lie groups, and consider short rates that are linearor quadratic functions on the group. The correspondingshort rate dynamics are then explicitly worked out via anextension of Ito’s formula to matrix Lie groups, andgeometric discretization schemes for Lie group stochasticequations are briefly discussed. Section 3 discusses thegeometry of covariance matrices in the context ofstochastic volatility models. Section 4 provides examplesof short rate models constructed on the circle S1, therotation group SO(3), and other representative two- andthree-dimensional Lie groups. We conclude in section 5by mentioning some possible directions for furtherinvestigation.

2. Stochastic differential equations on Lie groups

2.1. Motivational example

Before proceeding to the formal construction of stochas-tic differential equations on Lie groups, we begin with asimple and intuitive example that illustrates some of thepractical issues that arise when a state space becomescurved. For this purpose consider a unit sphere centeredat the origin. It can be characterized implicitly by theequation x2þ y2þ z2¼ 1, or explicitly in terms of spher-ical coordinates (�,�) as

x ¼ cos � sin�,

y ¼ sin � sin�,

z ¼ cos�:

ð1Þ

As is well-known, the shortest path lying on the sphereconnecting two arbitrary points is given by the arc of agreat circle. The most direct way to show this is todetermine the curve of minimum length (or minimalgeodesic) between two given points on the sphere, whichwe now do. Parametrize the spherical curve C(t)¼ (x(t),y(t), z(t)), t2 [0, 1], in spherical coordinates by (�(t), �(t)).Setting the incremental arclength ds2 ¼ dx2 þ dy2þdz2 ¼ ð _x2 þ _y2 þ _z2Þdt2, the length of C(t) can now beevaluated via the integral

L ¼

Z 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_�2 þ _�2 sin2 �

qdt, ð2Þ

560 F. C. Park et al.


subject to endpoint boundary conditions. This is astandard problem in the calculus of variations, involvingthe solution of a pair of nonlinear second-order differen-tial equations in �(t) and �(t) with split boundaryconditions. We make three remarks about this problem.

. In general, the minimal geodesics on the spheredo not correspond to straight lines in the �–�plane.

. Other parametrizations of the sphere are pos-sible, e.g. stereographic projections of the form�¼ x/(1� z), �¼ y/(1� z). Different parametri-zations clearly lead to different formulations ofthe variational problem.

. Defining length in R3 in different ways, e.g.setting the incremental arclength to ds2¼ezdx2þ e�z dy2þ ex

2þy2 dz2, also leads to differ-ent formulations of the variational problem.

Once a notion of length in R3 has been set, then thelength of any curve on the sphere should be invariant withrespect to the parametrization of the sphere. Moreover, adistance metric on the sphere can now be naturallydefined: given two points p and q on the sphere, thedistance between p and q is defined to be the length of theminimal geodesic connecting p and q. The sample mean ofa collection of N points {p1, . . . , pN} on the sphere canthen be determined as the point �p on the sphere thatminimizes

Lð �pÞ ¼1

N

Xki¼1

d ð pi, �pÞ: ð3Þ

Such a definition of sample mean clearly respects thegeometry of the sphere, and does not depend on, forexample, the choice of parametrization of the sphere.

Consider now a state dynamics on the sphere, expressedas the following pair of stochastic differential equations:

d� ¼ f ð�,�Þdtþ d!, ð4Þ

d� ¼ gð�,�Þdtþ d�, ð5Þ

where d! and d� are independent Wiener processes. Ifanother parametrization were chosen for the sphere, e.g.the stereographic projection coordinates (�,�) mentionedearlier, then the stochastic state equations would betransformed in the usual Ito fashion to the form

d� ¼ uð�,�Þdtþ rð�,�Þd!, ð6Þ

d� ¼ vð�,�Þdtþ sð�,�Þd�, ð7Þ

where the functions u, v, r and s are obtained from thefirst and second derivatives of f and g. Two remarks are inorder.

. Clearly, there must be a plausible physicalrationale for the additive Wiener noise intro-duced to d� and d�. It may, for example, bemore physically plausible to introduce additivenoise to d� and d� instead, or another coordi-nate parametrization for the sphere.

. Applying the standard Euler discretization to

the equations for d� and d� will clearly result in

different paths (on the sphere) than applying

the same discretization to the equations for d�and d�. Any geometrically well-defined discre-

tization of the state equations should not

depend on the choice of coordinate parame-

trization for the sphere. A natural remedy for

this coordinate dependence is to discretize the

equations in such a way that they are propa-

gated along minimal geodesics.

The example on the sphere illustrates the main

challenges in formulating stochastic state dynamics on

arbitrary curved spaces. For example, taking an

m-dimensional surface S embedded in Rn (m5n) to be

the underlying state space, the appropriate notion of

incremental arclength (or more formally, the choice of a

Riemannian metric) on S must be first be chosen in a

meaningful way. Noting that S will have many possible

coordinate parametrizations (or local coordinates), any

state dynamics on S should be formulated such that the

noise model is physically meaningful in the chosen

coordinates. Discretization of the state dynamics should

also not depend on the choice of local coordinates, but

rather evolve along minimal geodesics—in this case the

minimal geodesics should preferably be easily computa-

ble, without having to solve a nonlinear two-point

boundary value problem.We now consider one final related example involving

the space P(n) of n� n symmetric positive-definite matri-

ces; the obvious connection with interest rate models is

that covariance matrices belong to P(n). To begin, let

P2P(2), i.e.

P ¼x y

y z

� �, ð8Þ

where xz� y240, x40. If we consider P as a point

p¼ (x, y, z)2R3, then the above conditions describe the

interior of a cone in R3. Given two points p, q2R3, the

straight line connecting p and q will typically fall outside

the cone as prescribed. More generally, if we wish to

determine the average of two matrices P1,P22P(n), then

the arithmetic mean �P ¼ ðP1 þ P2Þ=2 clearly lies in P(n).

However, �P typically does not preserve first-order

properties, e.g. even if det P1¼ detP2, det �P may be

larger. Second-order statistics are also not generally

preserved; for example, standard principal component

analysis will fail to preserve positive-definiteness (the

‘straight lines’ defined by the modes of variation do not

remain within P(n), because P(n) is a curved space)

(Fletcher and Joshi 2004, Lenglet et al. 2006).The above discussions highlight two things: (i) the

importance of taking into account the geometry of the

curved space when doing analysis; and (ii) the need to

restrict the admissible curved state spaces to a more

manageable class. The latter point is the primary moti-

vation behind considering the matrix Lie groups as

candidate state spaces for interest rate models. In the

Interest rate models on Lie groups 561


next subsection we review the main features and

constructs of matrix Lie groups.

2.2. Matrix Lie group basics

A Lie group is essentially a differentiable manifold thatalso possesses the structure of a continuous algebraicgroup. Our focus in this paper will be exclusively onmatrix Lie groups.

If G is a matrix Lie group, then

(1) the identity matrix I is an element of G;(2) for every X2G, its inverse X�1 is well-defined and

also an element of G; and(3) (X1X2)X3¼X1(X2X3) for every X1,X2,X32G.

Associated with every matrix Lie group G is its Liealgebra g. The Lie algebra can be identified with thetangent space to G at the identity element. More precisely,let X(t) be a differentiable curve on G such that X(0)¼ I.Then the derivative of X(t) at t¼ 0, denoted _Xð0Þ, isclearly a tangent vector to G at I. The collection of allsuch vectors _Xð0Þ—obtained as the time derivative of allpossible differentiable curves X(t) on G such thatX(0)¼ I—constitutes the lie algebra g. As an example,the general linear group GL(n) consisting of real n� nnon-singular matrices is a Lie group of dimension n2. TheLie algebra of GL(n), denoted gl(n), is given by the vectorspace of n� n real matrices. Our primary interest will bein matrix Lie groups that are subgroups of GL(n).

Because the tangent space always has the structure of avector space, in the case of Lie algebras a furtheroperation, the matrix commutator, is identified as oneof its defining properties: given A,B2 g, then

½A,B� ¼ AB� BA ð9Þ

is denoted the Lie bracket of A and B. It can be verifiedthat the Lie bracket satisfies the following properties: forany A,B,C2 g and scalars �,�2R,

(1) [�Aþ �B,C ]¼�[A,C ]þ �[B,C ] and [C,�Aþ�B]¼�[C,A]þ�[C,B],

(2) [A, B]¼�[B, A],(3) [A, [B,C ]]þ [C, [A,B]]þ [B, [C,A]]¼ 0.

One of the primary connections between a matrix Liegroup G and its Lie algebra g is given by the exponentialmap exp : g!G; this is defined by the familiar series

eA ¼ Iþ Aþ1

2!A2 þ

1

3!A3 þ � � � : ð10Þ

The series is always guaranteed to converge for finite A,just like the familiar exponential of a scalar, e� (�2R).The exponential map has a smooth inverse in a neigh-borhood of the identity I2G, and thus defines a set oflocal coordinates for G over this neighborhood: if{�1, . . . ,�N} is a basis for g, and U an open set of Gover which the smooth inverse is well-defined, then

exp : <N! U

ðq1, . . . , qNÞ� e�1q1þ��þ�NqNð11Þ

defines a set of local coordinates for U, with inverse map

denoted log :U! g. Local coordinates constructed in this

fashion are referred to as exponential coordinates.Consider once again a differentiable curve X(t) on G,

with _XðtÞ its time derivative. One of the remarkable

features of Lie groups is that both _XX�1 and X�1 _X are

elements of g. These left and right translations of the

tangent vector _X to g offer a a natural way to identify the

tangent space at any point X2G with the Lie algebra.

Left and right translations also lead to a natural way of

constructing smoothly varying inner products (or

Riemannian metrics) over the entire collection of tangent

spaces on G, simply by constructing an inner product on

the Lie algebra g. Specifically, let h�, �ig denote an inner

product on g. The inner product of _X with itself can then

be defined via left translation as

h _X, _X iL ¼ hX�1 _X,X�1 _X ig, ð12Þ

or via right translation as

h _X, _X iR ¼ h _XX�1, _XX�1ig: ð13Þ

h _X, _X iL is left-invariant in the sense that if the original

curve X(t) is now multiplied from the left by some

constant T2G, to Y(t)¼TX(t), then

h _Y, _Y iL ¼ h _X, _X iL: ð14Þ

That is, the resulting inner product is invariant to left

translations. Similarly, right-invariant metrics are invar-

iant with respect to right translations of the form X(t)T.The physical significance of translation-invariance is

that the resulting inner product does not depend on the

choice of identity element. That is, given a set of elements

{X1, . . . ,XN}2G, choosing a different identity element

then implies that each element Xi is translated by some

constant T2G, to either TXi or XiT. Translation-

invariant Riemannian metrics are thus invariant in the

sense of not depending on the choice of identity element.

Any metric that is both left- and right-invariant is

clearly preferable, but such metrics only exist for a

restricted class of Lie groups. In all other cases there

are compelling reasons to choose a translation-

invariant Riemannian metric (typically of the left-

invariant variety).Given a choice of left-invariant Riemannian metric,

minimal geodesics on G can now be formulated as the

curves that minimize the squared length criterion

JðU Þ ¼

Z 1

0

hUðtÞ,UðtÞigdt, ð15Þ

subject to the matrix differential equation

_XðtÞ ¼ XðtÞUðtÞ, ð16Þ

with boundary conditions X(0) and X(1) specified; in the

above X(t)2G, U2 g, h�, �ig denotes an inner product on

g, and the choice of left-invariant Riemannian metric is

implicit in the way the differential equation is expressed.

It can be shown through variational arguments that



geodesics on G must satisfy the following set of differen-tial equations:

_X ¼ XU, ð17Þ

_U ¼ ½UT,U �, ð18Þ

with boundary conditions X(0) and X(1) given, leading toa classical two-point boundary value problem.

We now consider some specific and well-knownexamples of matrix Lie groups (other examples can befound in, e.g., Choquet-Bruhat et al. 1982).

2.2.1. The special orthogonal group. The SpecialOrthogonal Group SO(n), commonly referred to as thegroup of proper rotations, consists of the n� n realmatrices R that satisfy

RR> ¼ R>R ¼ I, detR ¼ 1: ð19Þ

SO(2) can be readily identified with the circle S1, sinceelements of SO(2) are of the form

SOð2Þ ¼cos � � sin �

sin � cos �

� �, ð20Þ

with � 2 [0, 2�]. The Lie algebra of SO(n), denoted so(n),consists of the n� n real skew-symmetric matrices�2Rn�n, i.e. that satisfy

�þ�> ¼ 0: ð21Þ

In the case of so(3), its standard basis is defined asfollows:

E1 ¼

0 �1 0

1 0 0

0 0 0

0B@

1CA, E2 ¼

0 0 1

0 0 0

�1 0 0

0B@

1CA,

E3 ¼

0 0 0

0 0 �1

0 1 0

0B@

1CA: ð22Þ

The inner product on so(n) given by

h�1,�2i ¼ Trð�>1 �2Þ, ð23Þ

for any �1,�22 so(n), is the unique choice for which theleft- and right-invariant Riemannian metrics coincide.This metric is referred to as the bi-invariant, or natural,Riemannian metric on SO(n). Minimal geodesics withrespect to this bi-invariant metric are given by left andright translations of exponentials of the form exp(�t),where �2 so(n) is constant.

In the case of SO(3), both the exponential andlogarithm formulas can be derived in closed form: iflogR¼�, then exp�¼R, with �2 so(3) and R2SO(3)given by

exp� ¼ Iþsin k�k

k�k��þ

1� cos k�k

k�k2��2, ð24Þ

logR ¼�

2 sin �ðR� R>Þ, ð25Þ

where k�k2¼ tr(��>) and � 2 [0,�] satisfies 1þ2 cos �¼Tr(R).

2.2.2. The special linear group. The Special LinearGroup SL(n), also known as the group of volume-preserving transformations, consists of the real n� nmatrices X2Rn�n that satisfy det X¼ 1. The associatedLie group sl(n) consists of the real n� n matrices of tracezero. In the case of sl(2), its standard basis is defined asfollows:

E1 ¼1 0

0 �1

� �, E2 ¼

0 1

0 0

� �, E3 ¼

0 0

1 0

� �:

ð26Þ

It is known that any element X of the general linear groupGL(n) can be decomposed via the Iwasawa decomposition(Bridson and Haefliger 1999) as X¼KAN, whereK2O(n), A is diagonal with positive entries, and N isupper-triangular with unit diagonal entries. Restricting Xto have positive determinant in turn forces K to be anelement of SO(n). Restricting the above Iwasawa decom-position to SL(n), an element X2SL(n) can be decom-posed as X¼RBN, where R2SO(n), B is diagonal withunit determinant, and N is upper-triangular with unitdiagonal entries. For example, any volume-preservingtransformation in two dimensions can be decomposed as

X ¼cos

�

2sin

�

2

� sin�

2cos

�

2

0B@

1CA et=2 0

0 e�t=2

� �1 �

0 1

� �: ð27Þ

Three parameters corresponding to rotation (�), stretch-ing (t), and shearness (�) can therefore be used as localcoordinates to parametrize any element of SL(2).Analytic formulas for the exponential on sl(2) can alsobe derived (see, e.g., Bernstein and So 1993).

2.2.3. Other matrix Lie groups of dimension 3. Wemention three additional non-compact matrix Liegroups of dimension 3. The Heisenberg group H3 isgiven by

H3 ¼

1 x y

0 1 z

0 0 1

0B@

1CA : x, y, z 2 <

8><>:

9>=>;, ð28Þ

with Lie algebra h3 given by

h3 ¼

0 � �

0 0

0 0 0

0B@

1CA : �,�, 2 <

8><>:

9>=>;: ð29Þ

The group E(1, 1), which is a subgroup of SL(3), isgiven by

Eð1, 1Þ ¼

e� 0 x

0 e�� y

0 0 1

0B@

1CA : �, x, y 2 <

8><>:

9>=>;, ð30Þ



with Lie algebra e(1, 1) given by

eð1, 1Þ ¼

� 0 �

0 ��

0 0 0

0B@

1CA : �,�,� 2 <

8><>:

9>=>;: ð31Þ

The group SE(2) of planar Euclidean motions is given by

SEð2Þ ¼R p

0 1

� �: R 2 SOð2Þ, p 2 <2

� �, ð32Þ

with Lie algebra se(2) given by

seð2Þ ¼

0 �! �

! 0 �

0 0 0

0B@

1CA : !,�,� 2 <

8><>:

9>=>;: ð33Þ

2.3. Bilinear stochastic differential equations onLie groups

We first consider the following general setting. Let G bean m-dimensional matrix Lie group and g its correspond-ing matrix Lie algebra, with basis elements E, . . . ,Em2 g.The state equations are assumed to be in left-invariantform (the development for right-invariant systems isanalogous, and will not be repeated):

dX ¼ X � AðX Þdtþ XXmi¼1

biðX ÞEi dwi, ð34Þ

where X2G is the state, the maps A :G! g, bi : G!Rare assumed C2, and dwi2R denote Wiener processes.The above can alternatively be written

X�1dX ¼ AðX ÞdtþXmi¼1

biðX ÞEi dwi: ð35Þ

Jump processes can also be included into either of theabove equations in a straightforward fashion.

In the event that both A(X ) and the bi (X ) in (34) areconstant, the resulting state equation then becomes linearin the state, and also linear in the driving term, but notjointly linear; such systems are referred to as bilinearsystems in the literature, and we shall adopt this sameterminology for the case of constant A(X ) and bi (X ). Inthis case we can rewrite, without loss of generality, thebilinear state equations (34) as

dX ¼ X � Adtþ X dW, ð36Þ

where dW2 g is a Wiener process on the Lie algebra withcovariance S¼ bb>2Rm�m rather than I (the bi terms arenow reflected in the covariance of the diffusion dW ratherthan appearing explicitly in the state equations).

Our interest in the bilinear case can be traced to severalreasons.

(1) It is the simplest non-trivial example of a stochasticsystem on G.

(2) In the case of systems on vector spaces, arbitrarilygood bilinear approximations to deterministicnonlinear systems can be obtained (see, e.g.,

Sastry 1999 and the references cited therein);presumably such approximations can in principlealso be constructed for nonlinear systems evolvingon matrix Lie groups as well.

(3) Both the Fokker–Planck equations for the proba-bility density (X, t), as well as the dynamics ofquadratic and linear functions defined on G(derived further below), can be analytically char-acterized. The Fokker–Planck operator, denotedL, is given by (Chirikjian 2008)

L ¼ �Xmi¼1

aiERi þ

1

2

Xmj¼1

Xmk¼1

bjbkERj E

Rk , ð37Þ

where ERi is defined as the Lie derivative

ERi ðX, tÞ ¼

d

dtðX � expðtEiÞ, tÞjt¼0, ð38Þ

with respect to the basis element Ei of g.

2.3.1. Ito’s rule for quadratic and linear functions. Forthe bilinear state equation (36), we now consider a qua-dratic function on G of the form f ðX Þ ¼ 1

2 TrðX>QXN )

for some given symmetric Q,N2Rn�n, and derive thestochastic dynamics for f. Given X02G, we parametrize aneighborhood of X0 by X0 e

�, with �2 g:

X ¼ X0 Iþ�þ�2

2!þ � � �

� �: ð39Þ

f(X ) is then expanded to second order in �, leading to

f ðX Þ ¼ f ðX0Þ þ1

2Tr X>0 QX0ð2�Nþ�N�> þ�2NÞ� �

þ oð�2Þ, ð40Þ

where we make use of the matrix trace identitiesTr(A)¼Tr(A>) and Tr(ABC )¼Tr(CAB)¼Tr(BCA).

To derive the stochastic dynamics for f, f(X )� f(X0) isset to df, � is replaced by X�1 � dX¼Adtþ dW and X0 byX, the higher-order o(�2) terms are ignored, and thestandard Ito rules are applied, i.e. dt � dt¼ 0, dwi � dt¼dt � dwi¼ 0, and dwi � dwj¼ ijdt, where dwi, i¼ 1, . . . ,m,denote the independent entries of dW2 g, and ij denotesthe correlation between !i and !j. Manipulating theresulting equation leads to the following proposition.

Proposition 2.1: Given the bilinear stochastic equationdX¼XA dtþX � dW, where X2G, A, dW2 g with Aconstant, and dW is a diffusion process on g whoseindependent entries are denoted by dwi. Given the quadraticfunction

f ðX Þ ¼1

2Tr X>QXN� �

, ð41Þ

where Q, N2Rn�n are assumed symmetric, the dynamicsfor f are given by

df ¼ Tr

X>QX

AN dtþ dW �Nþ

1

2dW �N � dW>

þ1

2dW � dW �N

!!, ð42Þ



where the standard Ito rules dt � dt¼ dwi � dt¼ dt � dwi¼ 0,

dwi � dwj¼ ij dt, with ij the correlation between wi and wj,

are applied in the evaluation of the final two terms.

The corresponding results for the case when f is linear

in X can also be derived in a similar fashion.

Proposition 2.2: Under the same conditions as those given

in proposition 2.1, but with f(X )¼Tr(MX ), M symmetric,

the corresponding dynamics for f are given by

df ¼ Tr MX Adtþ dWþ1

2dW � dW

� �� : ð43Þ

We now explicitly work out the quadratic SO(3) case of

the former proposition, under the assumption that the

Wiener noise terms are uncorrelated, i.e. dwi � dwj¼ 0 for

i 6¼ j, and dwi � dwi¼ 1 for all i. First,

A ¼

0 �a3 a2

a3 0 �a1

�a2 a1 0

0B@

1CA, ð44Þ

dW ¼

0 �dw3 dw2

dw3 0 �dw1

�dw2 dw1 0

0B@

1CA, ð45Þ

from which it easily follows that dW � dW¼�2I dt. The

dynamics for f is thus given by

d f ¼Tr X>QX ðA� I ÞNdtþ dW �N�1

2dW �N �dW

� �� :

ð46Þ

2.4. Numerical integration on Lie groups

The numerical integration of state equations on Lie

groups requires a discretization of the equations. The

discretization must be performed in such a way that, at

each iteration, Xt always remain on the group. Beginning

with the work of Crouch and Grossman (1993), consid-

erable literature exists on the subject (see also the recent

work of Munthe-Kaas 1999 and Iserles and Munthe-

Kaas 2000). The primary motivation in these works is to

generalize Runge–Kutta and other numerical integration

methods developed for ODEs on Rn to general Lie

groups. The simplest first-order discretization, and one

that is for the most part sufficient for our purposes, is the

exponential Euler discretization given by

Xiþ1 ¼ Xi exp AðX, tÞDtþXmj¼1

bj ðX ÞEj

ffiffiffiffiffiDtp

�iþ1, j

!, ð47Þ

where each �i¼ (�i,1, . . . , �i,m) is an m-dimensional

zero-mean Gaussian with specified covariance matrix S.

Higher-order discretization methods, and other aspects of

the numerical integration of stochastic differential equa-

tions, are discussed by Kloeden and Platen (1999).

3. Stochastic volatility and the geometry of covariance

matrices

Recall that covariance matrices are characterized by beingreal symmetric and positive-definite. As before we denotethe space of such matrices by P(n):

PðnÞ ¼ X 2 <n�n j X ¼ XT,X4 0

: ð48Þ

Note that P(n) is not a group because it is not closedunder matrix multiplication. It is, however, a differenti-able manifold that can be endowed with a naturalRiemannian structure in the following sense. Any elementof P(n) can be identified with a symmetric positive-definite quadratic form h�, �,i on Rn. A choice of basis forRn then leads to a P(n) matrix representation for h�, �i.Any minimal geodesic between two points on P(n) shouldthen be invariant with respect to the choice of basis forRn. In coordinates, if Q(t)2P(n) is the minimal geodesicconnecting Q(0)¼Q0 and Q(1)2Q(1), then the minimalgeodesic connecting Q0 ¼ JQ0J

T and Q1 ¼ JQ1JT, where

J2GLþ(n) (the group of non-singular matrices withpositive determinant—to keep the coordinate transfor-mation orientation-preserving) should be QðtÞ ¼ JQðtÞJT.

The unique Riemannian metric on P(n) that meets theabove criteria is given by the GLþ(n) invariant metric, andcan be found by identifying P(n) with GLþ(n)/O(n), i.e.the orthogonal part of the polar decomposition. Thetangent space at a point X2P(n) can be identified withthe n� n symmetric matrices S(n). A basis for S(n) can beconstructed in the usual way, i.e. the basis elementEij2S(n), where i� j, is a symmetric matrix whose ij and jielements are one, and the remaining elements zero.Geometric and algorithmic details of the computationalprocedure for determining minimal geodesics on P(n) inthe above sense can be found in Fletcher and Joshi (2004).

3.0.1. Stochastic differential equations on P(n). In termsof the standard basis for S(n), general stochasticdifferential equations on P(n) can be constructed asfollows:

dP ¼ AðPÞdtþXni¼1

Xij¼1

ijðPÞEij dwij, ð49Þ

where A :P(n)!S(n), ij :P(n)!R are assumed twicedifferentiable, and the dwij denote independent Wienerprocesses.

The simplest non-trivial case of the above equationoccurs when A is a linear function of P, and the ij areconstant. To ensure that A always remains symmetric, theequation assumes the form

dP ¼ ðAPþ PAT þQÞdtþXni¼1

Xij¼1

ijEijdwij, ð50Þ

where A2Rn�n,Q2S(n) are constant. In the event thatthe ij are zero, closed-form solutions to the aboveinvolving the matrix exponential eAt and its transpose



are readily available. If A further admits the diagonaliza-tion A¼PDP�1, the matrix exponential in this casesimplifies to

eAt ¼ PeDtP�1P0: ð51Þ

A slightly more general case occurs when A(P) isquadratic in P:

dP ¼ ðAPþ PAT þ PBPT þQÞdtþXni¼1

Xij¼1

ijEij dwij,

ð52Þ

where A, Q, and P are as before, and B2S(n) is constant.

3.0.2. Numerical integration on P(n). As with thepropagation of the state equation on Lie groups, thepropagation of equation (49) should also always remainon the manifold P(n). One means of doing so withoutresorting to, for example, ad hoc projection methods, is toexploit the fact that P(n) is a Riemannian symmetric spacethat is geodesically complete (see Fletcher and Joshi(2004) for details); as such the minimal geodesics providea natural way of discretizing general stochastic differ-ential equations on P(n).

Given a tangent vector X2S(n), the minimal geodesicemanating from some P2P(n) in the direction of X can becomputed as follows.

(1) Evaluate the orthogonal decompositionP¼UDUT, where U2SO(n) and D2Rn�n isdiagonal.

(2) Set G ¼ UffiffiffiffiDp

, so that P¼GGT.(3) Set Y¼G�1XG�T, and evaluate its orthogonal

decomposition Y¼V�VT, where V2SO(n) and�2Rn�n is diagonal.

(4) The desired geodesic is parametrized as

Pt ¼ ðGV Þe�tðGV ÞT: ð53Þ

The integration of either the linear or quadraticequations on P(n) described previously can now beachieved by setting the tangent vector X to the right-hand side of the stochastic differential equation (e.g.,X¼APþPAT

þQ in the linear case, and X¼APþPATþPBPT

þQ in the quadratic case), and followingthe above procedure to evaluate Pt.

4. Short rate models on Lie groups

In this section we construct short rate models on somespecific Lie groups, in some cases generating sampleforward paths and examining the relationship betweentheir qualitative behavior and the model parameters. Thebasic idea is to first construct a stochastic bilinear statedynamics on the Lie group, and then to define the short asthe trace of a linear or quadratic function defined on theLie group.

To develop intuition we begin with the SO(2) casestudy, which corresponds to a one-factor short rate model

with linear dynamics evolving on the circle. A reasonably

detailed analysis of a quadratic three-factor model on

SO(3) is then provided, in which the model parameters

are fitted to historical U.S. interest data, and the state

dynamics are augmented with geometrically defined

stochastic volatilities on P(n). We conclude this section

by briefly examining the structure of short rate models on

other groups, i.e. the Heisenberg groupH3 and the Special

Linear Group SL(2).

4.1. A short rate model on the circle

We first consider a short rate model on the circle S1,

which as shown earlier is equivalent to the matrix group

SO(2). The short rate r is defined as a linear function of

the state; that is, r(R)¼Tr(MR), where R2SO(2) and

M is symmetric positive-definite:

M ¼m11 m12

m21 m22

� �4 0, ð54Þ

with m12¼m21. Recall that R2SO(2) is of the form

R ¼cos � � sin �

sin � cos �

� �: ð55Þ

� 2 [0, 2�] thus acts as local coordinates for SO(2). The

state equation is of the form

dR ¼ RAdtþ RdW, ð56Þ

where A, dW2 so(2) are given by

A ¼0 ��

� 0

� �, ð57Þ

dW ¼0 �dw

dw 0

� �, ð58Þ

with �2R constant. The dynamics for (r, �) can then be

written jointly as

dr

�

� �¼�� sin � � �

2 cos �

�

� �dtþ

�� sin �

1

� �dw,

ð59Þ

where �¼m11þm22 (note that the positive-definiteness

requirement on M makes � positive). It can readily be

verified that r¼Tr(MR)¼� cos � is restricted to lie in the

range [��,�]; r can therefore be ensured to be positive by

adding a positive offset �.Since r¼� cos � or cos �¼ r/�, from basic trigono-

metric relationships we can identify

sin � ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � r2

p�

, ð60Þ

provided �24r2. Using this relation the short rate

dynamics for r can be expressed as

dr ¼ ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � r2

p�

r

2

� �dt�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � r2

pdw: ð61Þ



Figure 1 shows 50 sample paths for the case when

�¼ 0.001, m11¼m22¼ 3.0, dw has variance 1.0362, and

r0¼ 0.05125. Figure 2 shows 50 sample paths when the

variance for dw is quadrupled to 4.1448. In both cases it

can be observed that, after the constant offset � is added

to ensure non-negative rates, the short rate remains

bounded between 0 and 0.12.

As far as we are aware short rate models of the form

suggested by equation (61) have yet to be examined in the

literature. While we only considered constant values for

the parameter � (or, alternatively, the 2� 2 matrix M ), in

principle it is possible to consider time-varying �t (orMt) to

allow for yield curve fitting, as is commonly done in many

one-factor short rate models like the Hull–White model.

Figure 2. Fifty short rate sample paths for the linear SO(2) model with increased volatility: ¼ 4.1448.

Figure 1. Fifty short rate sample paths for the linear SO(2) model: ¼ 1.0362.



4.2. A short rate model on SO(3)

We now consider a three-factor short rate model on SO(3)

of the form

dR ¼ RAdtþ RdW, ð62Þ

in which A, dW2 so(3). Define the short rate r to be a

quadratic function of the state R, i.e.

rðRÞ ¼1

2TrðQRNR>Þ, ð63Þ

where both Q,N2Rn�n are symmetric positive-definite,

thereby ensuring that r always remains positive. The

corresponding stochastic differential equation for r can be

derived from our earlier equation (46).The model parameters in this example are A2 so(3) in

the underlying state equations, the symmetric positive-

definite matrices Q,N2R3�3 used to define the short rate,

and the 3� 3 covariance matrix S associated with the

Wiener process dW2 so(3). These parameters are esti-

mated using historical U.S. Treasury interest rate data

from January 1990 to December 1999 (see figure 3). The

published interest rates are the spot rates R(t, tþ �),denoting the (annualized) interest rate at time t for length

of maturity �. Recall that the spot rates can be obtained

from the short rate via the expectation

Rðt, tþ �Þ ¼ E1

�

Z tþ�

t

rðsÞds

� � �, ð64Þ

or from the corresponding pricing partial differential

equation on SO(3) discussed earlier; for our study we use

Monte Carlo simulation to directly evaluate the above

expectation.

The following simple least-squares estimation proce-

dure is used for parameter estimation.

(1) Choose an initial set of parameters A2 so(3),

Q2R3�3,N2R3�3, with both Q and N subject to

the constraint that they are symmetric positive

definite. The covariance matrix for the Wiener

process is assumed to be S2R3�3, which is also

assumed symmetric positive definite. Choose the

initial value R(0)2SO(3) to be the identity for

convenience.(2) Generate a sufficient number of sample paths for

the short rate, and determine the time series for all

the spot rates; denote the estimated spot rate data

by Rðt, tþ �Þ.(3) Determine if the choice of model parameters is an

optimizer for the objective function

LðA,Q,N,S Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðRðt, tþ �Þ � Rðt, tþ �ÞÞ2

q, ð65Þ

where R(t, tþ �)’s are historical time series data of

spot rates for the following maturities: 3 months,

6 months, 1 year, 2 years, 3 years, 5 years,

7 years, 10 years, 20 years, and 30 years. If the

optimization convergence criterion is not satisfied,

update the parameter values, and go to Step 2.

Note that a wide range of highly sophisticated methods

are available for estimating the model parameters (see, for

example, the various methods described by James and

Webber 2000).Using our simple least-squares fitting procedure, and

performing the optimization over a wide range of

regularly spaced starting points, the following set of

Figure 3. Spot rates from Jan. 1990 to Dec. 1999. Data from the U.S. Treasury Department.



parameter values are obtained:

A ¼

0 �0:3864 0:3789

0:3864 0 �0:6088

�0:3789 0:6088 0

0BB@

1CCA,

S ¼

1:3642 0 0

0 0:6054 0

0 0 1:0831

0BB@

1CCA,

Q ¼

1:0088 1:2867 0:9744

1:2867 2:0259 1:5095

0:9744 1:5095 1:1694

0BB@

1CCA,

N ¼

0:8185 1:0145 0:7084

1:0145 1:2824 0:8626

0:7084 0:8626 0:6469

0BB@

1CCA: ð66Þ

To examine the goodness of fit of the model to the

actual data, we calculate the skewness and kurtosis of the

error between the estimated and actual short rates, using

the 3-month maturity spot rate as a proxy. Figure 4 shows

the distribution of the estimation errors for the models

given in equation (66). The skewness and kurtosis of the

errors are determined to be �0.4267 and 3.1947,

respectively. Figure 5 shows 50 sample paths of the

short rate for the given set of parameter values.Finally, figure 6 depicts 50 sample paths for the

SO(3) model with stochastic volatility; here the covar-

iance S of the noise term dW2 so(3) appearing in

equation (62) is assumed to be driven by linear

dynamics of the form

dS ¼ FSþ SF> þ dZ, dZ ¼

dz1 0 0

0 dz2 0

0 0 dz3

0B@

1CA,ð67Þ

where dZ S(n) is assumed to be diagonal, and

F¼diag[�0.001,�0.001,�0.001] (choosing F such thatthe real parts of all its eigenvalues are negative ensures

that S will not eventually diverge to infinity as

t increases).

4.3. Other Lie groups as state spaces

In this section we briefly consider some other matrix Liegroups as candidate state spaces. Our purpose will be to

qualitatively examine the structure of the short rate

equations, pointing out any similarities with existing short

rate models.We first take the three-dimensional Heisenberg group

H3 to be the state space, and consider a state space

dynamics of the form dX ¼ XA dtþ XP3

i¼1 biEi dwi,

bi2R, where

X ¼

1 x y

0 1 z

0 0 1

0B@

1CA, A ¼

0 � �

0 0

0 0 0

0B@

1CA, ð68Þ

the �, �, 2R are constant, the Ei are the standard basisfor the Lie algebra h3, and the dwi(t) are independent

Wiener processes. Writing the dynamics for the x, y, z

Figure 4. Distribution of estimation error of short rates.



explicitly, we obtain

dx ¼ � dtþ b1 dw1, ð69Þ

dy ¼ ð�þ xÞdtþ b2 dw2 þ b3x1dw1, ð70Þ

dz ¼ dtþ b3 dw3: ð71Þ

A short rate linear in (x, y, z) bears some structuralsimilarities to the one-factor Brennan and Schwartz(1980) model

dr ¼ ð�þ �rÞdtþ r dw, ð72Þ

where �, �, are scalar constants. If b2 is assumed zeroand the short rate is taken to be r¼ y, then the resultingshort rate model becomes a two-factor model with

Figure 5. Fifty short rate sample paths for the SO(3) model with deterministic volatility.

Figure 6. Fifty short rate sample paths for the SO(3) model with stochastic volatility.



stochastic drift and volatility introduced via the dynamics

of x (which is simply Brownian motion with a constant

drift term). For the same reason that the Brennan and

Schwartz model does not lead to analytic bond pricing

formulas, the Heisenberg model also likely will not in the

general case.We next take the three-dimensional group E(1, 1) to be

the state space, in which case the X2E(1, 1) and

A2 e(1, 1) in the state equations are of the form

X ¼

e� 0 x

0 e�� y

0 0 1

0B@

1CA, A ¼

0 �

0 � �

0 0 0

0B@

1CA: ð73Þ

Setting z¼ e�, the state equations dX¼XA dtþX dW,

dW2 e(1, 1), can be written explicitly in terms of

(x, y, z) as

dx ¼ �z dtþ z dw1, ð74Þ

dy ¼ �z�1 dtþ z�1 dw2, ð75Þ

dz ¼ z dtþ z dw3: ð76Þ

z is a geometric Brownian motion (and therefore always

positive), while x and y are Brownian motions with

stochastic drift and volatility terms driven by z.

Examining the structure of the equations, analytic bond

pricing formulas do not seem likely, even for short rates

that are linear in (x, y, z).Finally, we consider a short rate model on the Special

Linear Group SL(2). Here X2SL(2) and A2 sl(2) are of

the form

X ¼x1 x2

x3 x4

� �, A ¼

a1 a2

a3 �a1

� �, ð77Þ

and the state equations dX¼XA dtþX dW can be

written in vector form as

d

x1

x2

x3

x4

0BBB@

1CCCA ¼

a1 a3 0 0

a2 �a1 0 0

0 0 a1 a3

0 0 a2 �a1

0BBB@

1CCCA

x1

x2

x3

x4

0BBB@

1CCCA

þ

x1 0 x2

�x2 x1 0

x3 0 x4

�x4 x3 0

0BBB@

1CCCA

dw1

dw2

dw3

0B@

1CA: ð78Þ

Note that the eigenvalues of A are �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ a2a3

q; one

implication of this is that if a21 þ a2a3 � 0, then the

expected value of X(t) will become unbounded as t goes to

infinity. If, on the other hand, a21 þ a2a3 5 0, then the

expected value of X(t) will be periodic of the form

E ½XðtÞ� ¼cos �tþ

a1�sin �t

a2�sin �t

a3�sin �t cos �t�

a1�sin �t

0B@

1CAE ½Xð0Þ�,

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ða21 þ a2a3Þ

q: ð79Þ

5. Conclusions

This paper has introduced a new class of geometricinterest rate models evolving on matrix Lie groups. Toour knowledge, the Lie group interest rate frameworkproposed in this paper has not been addressed in theliterature beyond the works cited earlier. While we do notexpect that such models will displace anytime soon any ofthe better known and widely used vector space interestrate models—the models investigated in this papergenerally do not yield closed-form bond pricing formulas,which seems to be an important priority for practi-tioners—we believe that the evolution of the study ofinterest rate models to curved state spaces is a natural andprofitable one, offering hope of a much richer, globalcharacterization of interest rate phenomena.

Our Lie group framework offers a practical balancebetween the geometric richness provided by differentiablemanifolds, without having to invoke the complex machin-ery and involved formulas of stochastic calculus onmanifolds. The case studies on SO(2) and SO(3) providea compact description of a short rate model that is positiveand bounded, with stochastic volatility introduced in ageometrically well-defined way in the latter case.

These case studies offer only a hint of the possible Liegroups upon which interest rate models can be con-structed; identifying other Lie groups as candidate statespaces, as well as more thorough statistical studies toidentify any manifold structure in financial data, are topicsclearly worth investigating. A further natural extension isto develop an analogous Lie group framework for theHeath–Jarrow–Morton forward term structure model.

As mentioned earlier, the Lie group models consideredsuggest that closed-form bond pricing formulas willgenerally be the exception rather than the rule, so thatone will likely have to turn to numerical methods forpricing. For this and other reasons this paper has notexamined in detail further practical issues such as yieldcurve fitting, model parameter identification, or a cross-comparison of the performance of various models withextant vector space models. These are clearly topics worthinvestigating further, e.g. numerical methods for solvingthe corresponding pricing partial differential equations,and extending Monte Carlo filtering algorithms to Liegroups for the purposes of model parameter estimation.

Acknowledgements

Research funding from the following institutions andprograms is gratefully acknowledged: SIRFE, IAMD,CBMS, KIST-CIR, ROSAEC-ERC and MKE-AIM.

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