Description
In economics, market structure is the number of firms producing identical products which are homogeneous.
Bond Market Structure in the
Presence of Marked Point Processes
?
Tomas Bj¨ ork
Department of Finance
Stockholm School of Economics
Box 6501, S-113 83 Stockholm SWEDEN
Yuri Kabanov
Central Economics and Mathematics Institute
Russian Academy of Sciences
and
Laboratoire de Math´ematiques
Universit´e de Franche-Comt´e
16 Route de Gray, F-25030 Besan¸con Cedex FRANCE
Wolfgang Runggaldier
Dipartimento di Matematica Pura et Applicata
Universit´a di Padova
Via Belzoni 7, 35131 Padova ITALY
February 28, 1996
Submitted to
Mathematical Finance
?
The ?nancial support and hospitality of the University of Padua, the Isaac New-
ton Institute, Cambridge University, and the Stockholm School of Economics are
gratefully acknowledged.
1
Abstract
We investigate the term structure of zero coupon bonds when
interest rates are driven by a general marked point process as
well as by a Wiener process. Developing a theory which allows for
measure-valued trading portfolios we study existence and unique-
ness of a martingale measure. We also study completeness and its
relation to the uniqueness of a martingale measure. For the case
of a ?nite jump spectrum we give a fairly general completeness
result and for a Wiener–Poisson model we prove the existence of
a time- independent set of basic bonds. We also give su?cient
conditions for the existence of an a?ne term structure.
Key words: bond market, term structure of interest rates, jump-
di?usion model, measure-valued portfolio, arbitrage, market complete-
ness, martingale operator, hedging operator, a?ne term structure.
1 Introduction
One of the most challenging mathematical problems arising in the theory
of ?nancial markets concerns market completeness, i.e. the possibility of
duplicating a contingent claim by a self-?nancing portfolio. Informally,
such a possibility arises whenever there are as many risky assets available
for hedging as there are independent sources of randomness in the market.
In bond markets as well as in stock markets it seems reasonable to
take into account the possible occurrence of jumps, considering not only
the simple Poisson jump models, but also marked point process models
allowing a continuous jump spectrum. However, introducing a continuous
jump spectrum also introduces a possibly in?nite number of independent
sources of randomness and, as a consequence, completeness may be lost.
In traditional stock market models there are usually only a ?nite
number of basic assets available for hedging, and in order to have com-
pleteness one usually assumes that their prices are driven by a ?nite
number (equaling the number of basic assets) of Wiener processes. More
realistic jump-di?usion models seem to encounter some skepticism pre-
cisely due to the completeness problems mentioned above.
There is, however, a fundamental di?erence between stock and bond
markets: while in stock markets portfolios are naturally limited to a ?nite
number of basic assets, in bond markets there is at least the theoretical
possibility of having portfolios with an in?nite number of assets, namely
bonds with a continuum of possible maturities. Since all modern contin-
uous time models of bond markets assume the existence of bonds with a
2
continuum of maturities, it seems reasonable to require that a coherent
theory of bond markets should allow for portfolios consisting of uncount-
ably many bonds. We also see from the discussion above that, in models
with a continuous jump spectrum, such portfolios are indeed necessary
if completeness is not to be lost.
It is worth noticing that also in stock market models one may con-
sider a continuum of derivative securities, such as e.g. options parame-
terized by maturities and/or strikes.
The purpose of our paper is to present an approach which, on one
hand, allows bond prices to be driven also by marked point processes
while, on the other hand, admitting portfolios with an in?nite number
of securities. As such, this approach appears to be new and leads to the
two mathematical problems of:
• an appropriate modeling of the evolution of bond prices and their
forward rates;
• a correct de?nition of in?nite-dimensional portfolios of bonds and
the corresponding value processes by viewing trading strategies as
measure-valued processes.
A further point of interest in this context is that, in stock markets
and under general assumptions, completeness of the market is equiva-
lent to uniqueness of the martingale measure. The question now arises
whether this fact remains true also in bond markets when marked point
processes with continuous mark spaces, i.e. an in?nite number of sources
of randomness, are allowed? One of the main results of this paper is that,
at this level of generality, uniqueness of the martingale measure implies
only that the set of hedgeable claims is dense in the set of all contin-
gent claims. This phenomenon is not entirely unexpected and has been
observed by di?erent authors (see, e.g., de?nition of quasicompleteness
in [24]); its nature is transparent on the basis of elementary functional
analysis which we rely upon in Section 4.
The main results of the paper are as follows.
• We give conditions for the existence of a martingale measure in
terms of conditions on the coe?cients for the bond- and forward
rate dynamics. In particular we extend the Heath–Jarrow–Morton
“drift condition” to point process models.
• We show that the martingale measure is unique if and only if certain
integral operators of the ?rst kind (the “martingale operators”) are
injective.
3
• We show that a contingent claim can be replicated by a self-?nancing
portfolio if and only if certain integral equations of the ?rst kind
(the “hedging equations”) have solutions. Furthermore, the integral
operators appearing in these equations (the “hedging operators”)
turn out to be adjoint of the martingale operators.
• We show that uniqueness of the martingale measure is equivalent to
the denseness of the image space of the hedging operators. In partic-
ular, it turns out that in the case with a continuous jump spectrum,
uniqueness of the martingale measure does not imply completeness
of the bond market. Instead, uniqueness of the martingale mea-
sure is shown to be equivalent to approximate completeness of the
market.
• Under additional conditions on the forward rate dynamics we can
give a rather explicit characterization of the set of hedgeable claims
in terms of certain Laplace transforms.
• In particular, we study the model with a ?nite mark space (for the
jumps) showing that in this case one may hedge an arbitrary claim
by a portfolio consisting of a ?nite number of bonds, having essen-
tially arbitrary but di?erent maturities. This considerably extends
and clari?es a previous result by Shirakawa [28].
• We give su?cient conditions for the existence of a so-called a?ne
term structure (ATS) for the bond prices.
The paper has the following structure. In Section 2 we lay the foun-
dations and we present a “toolbox” of propositions which explain the
interrelations between the dynamics of the forward rates, the bond prices
and the short rate of interest.
In Section 3 we de?ne our measure-valued portfolios with their value
processes and investigate the existence and uniqueness of a martingale
measure. We also give the martingale dynamics of the various objects,
leading among other things to a HJM-type “drift condition”.
In a stock market, the current state of a portfolio is a vector of
quantities of securities held at time t which can be identi?ed with a linear
functional; it gives the portfolio value being applied to the current asset
price vector. In a bond market, the latter is substituted by a price curve
which one can consider as a vector in a space of continuous functions. By
analogy, it is natural to identify a current state of a portfolio with a linear
functional, i.e. with an element of the dual space, a signed ?nite measure.
So, our approach is based on a kind of stochastic integral with respect
4
to the price curve process though we avoid a more technical discussion
of this aspect here (see [4]).
In Section 4 we study uniqueness of the martingale measure and its
relation to the completeness of the bond market. Section 5 is devoted to
a more detailed study of two cases when we can characterize the set of
hedgeable claims. In 5.1 we consider a class of models with in?nite mark
space which leads us to Laplace transform theory and in 5.2 we explore
the case of a ?nite mark space. We end by discussing the existence of
a?ne term structures in Section 6.
For the case of Wiener-driven interest rates there is an enormous
number of papers. For general information about arbitrage free markets
we refer to the book [13] by Du?e. Basic papers in the area are Harrison–
Kreps [17], Harrison–Pliska [18]. For interest rate theory we recommend
Artzner–Delbaen [1] and some other important references can be found
in the bibliography; the recent book by Dana and Jeanblanc-Picqu´e [10]
contains a comprehensive account of main models.
Very little seems to have been written about interest rate models
driven by point processes. Shirakawa [28], Bj¨ ork [3], and Jarrow–Madan
[23] all consider an interest rate model of the type to be discussed below
for the case when the mark space is ?nite, i.e. when the model is driven
by a ?nite number of counting processes. (Jarrow–Madan also consider
the interplay between the stock- and the bond market). In the present
paper we focus primarily on the case of an in?nite mark space, but the
interest rate models above are included as special cases of our model,
and our results for the ?nite case amount to a considerable extension of
those in[28].
In an interesting preprint, Jarrow–Madan [24] consider a fairly gen-
eral model of asset prices driven by semimartingales. Their mathemati-
cal framework is that of topological vector spaces and, using a concept
of quasicompleteness, they obtain denseness results which are related to
ours.
Babbs and Webber [2] study a model where the short rate is driven by
a ?nite number of counting processes. The counting process intensities are
driven by the short rate itself and by an underlying di?usion-type process.
Lindberg–Orszag–Perraudin [25] consider a model where the short rate
is a Cox process with a squared Ornstein–Uhlenbeck process as intensity
process. Using Karhunen–Lo`eve expansions they obtain quasi-analytic
formulas for bond prices.
Structurally the present paper is based on Bj¨ ork [3] where only the
?nite case is treated. The working paper Bj¨ ork–Kabanov–Runggaldier
5
[5] contains some additional topics not treated here. In particular some
pricing formulas are given, and the change of num´eraire technique de-
veloped by Geman et. al. in [16] is applied to the bond market. In a
forthcoming paper [4] we develop the theory further by studying models
driven by rather general L´evy processes, and this also entails a study of
stochastic integration with respect to C-valued processes. In the present
exposition we want to focus on ?nancial aspects, so we try to avoid, as
far as possible, details and generalizations (even straightforward ones)
if they lead to mathematical sophistications. For the present paper the
main reference concerning point processes and Girsanov transformations
are Br´emaud [7] and Elliott [15]. For the more complicated paper [4], the
excellent (but much more advanced) exposition by Jacod and Shiryaev
[22] is the imperative reference.
Throughout the paper we use the Heath–Jarrow–Morton parameter-
ization, i.e. forward rates and bond prices are parameterized by time of
maturity T. In certain applications it is more convenient to parameterize
forward rates by instead using the time to maturity, as is done in Brace-
Musiela [6]. This can easily be accomplished, since there exists a simple
set of translation formulae between the two ways of parametrization.
2 Relations between df(t, T), dp(t, T), and
dr
t
We consider a ?nancial market model “living” on a stochastic basis (?l-
tered probability space) (?, T, F, P) where F = ¦T
t
¦
t?0
. The basis is
assumed to carry a Wiener process W as well as a marked point process
µ(dt, dx) on a measurable Lusin mark space (E, c) with compensator
?(dt, dx). We assume that ?([0, t] E) < ? P-a.s. for all ?nite t, i.e. µ
is a multivariate point process in the terminology of [22].
The main assets to be considered on the market are zero coupon
bonds with di?erent maturities. We denote the price at time t of a bond
maturing at time T (a “T-bond”) by p(t, T).
Assumption 2.1 We assume that
1. There exists a (frictionless) market for T-bonds for every T > 0.
2. For every ?xed T, the process ¦p(t, T); 0 ? t ? T¦ is an optional
stochastic process with p(t, t) = 1 for all t.
6
3. For every ?xed t, p(t, T) is P-a.s. continuously di?erentiable in the
T-variable. This partial derivative is often denoted by
p
T
(t, T) =
?p(t, T)
?T
.
We now de?ne the various interest rates.
De?nition 2.2 The instantaneous forward rate at T, contracted at t,
is given by
f(t, T) = ?
? log p(t, T)
?T
.
The short rate is de?ned by
r
t
= f(t, t).
The money account process is de?ned by
B
t
= exp
__
t
0
r
s
ds
_
,
i.e.
dB
t
= r
t
B
t
dt, B
0
= 1.
For the rest of the paper we shall, either by implication or by as-
sumption, consider dynamics of the following type.
Short rate dynamics
dr(t) = a
t
dt + b
t
dW
t
+
_
E
q(t, x)µ(dt, dx), (1)
Bond price dynamics
dp(t, T) = p(t, T)m(t, T)dt + p(t, T)v(t, T)dW
t
+ p(t?, T)
_
E
n(t, x, T)µ(dt, dx), (2)
Forward rate dynamics
df(t, T) = ?(t, T)dt + ?(t, T)dW
t
+
_
E
?(t, x, T)µ(dt, dx). (3)
7
In the above formulas the coe?cients are assumed to meet stan-
dard conditions required to guarantee that the various processes are well
de?ned.
We shall now study the formal relations which must hold between
bond prices and interest rates. These relations hold regardless of the
measure under consideration, and in particular we do not assume that
markets are free of arbitrage. We shall, however, need a number of tech-
nical assumptions which we collect below in an “operational” manner.
Assumption 2.3
1. For each ?xed ?, t and, (in appropriate cases) x, all the objects
m(t, T), v(t, T), n(t, x, T), ?(t, T), ?(t, T), and ?(t, x, T) are as-
sumed to be continuously di?erentiable in the T -variable. This
partial T-derivative sometimes is denoted by m
T
(t, T) etc.
2. All processes are assumed to be regular enough to allow us to di?er-
entiate under the integral sign as well as to interchange the order
of integration.
3. For any t the price curves p(?, t, .) are bounded functions for almost
all ?.
This assumption is rather ad hoc and one would, of course, like to
give conditions which imply the desired properties above. This can be
done but at a fairly high price as to technical complexity. As for the
point process integrals, these are made trajectorywise, so the standard
Fubini theorem can be applied. For the stochastic Fubini theorem for the
interchange of integration with respect to dW and dt see Protter [26] and
also Heath–Jarrow–Morton [19] for a ?nancial application.
Proposition 2.4
1. If p(t, T) satis?es (2), then for the forward rate dynamics we have
df(t, T) = ?(t, T)dt + ?(t, T)dW
t
+
_
E
?(t, x, T)µ(dt, dx),
where ?, ? and ? are given by
_
¸
_
¸
_
?(t, T) = v
T
(t, T) v(t, T) ?m
T
(t, T),
?(t, T) = ?v
T
(t, T),
?(t, x, T) = ?n
T
(t, x, T) [1 +n(t, x, T)]
?1
.
(4)
8
2. If f(t, T) satis?es (3) then the short rate satis?es
dr
t
= a
t
dt + b
t
dW
t
+
_
E
q(t, x)µ(dt, dx),
where
_
¸
_
¸
_
a
t
= f
T
(t, t) +?(t, t),
b
t
= ?(t, t),
q(t, x) = ?(t, x, t).
(5)
3. If f(t, T) satis?es (3) then p(t, T) satis?es
dp(t, T) = p(t, T)
_
r
t
+ A(t, T) +
1
2
S
2
(t, T)dt
_
+ p(t, T)S(t, T)dW
t
+ p(t?, T)
_
E
_
e
D(t,x,T)
?1
_
µ(dt, dx),
where
_
¸
_
¸
_
A(t, T) = ?
_
T
t
?(t, s)ds,
S(t, T) = ?
_
T
t
?(t, s)ds,
D(t, x, T) = ?
_
T
t
?(t, x, s)ds.
(6)
Proof. The ?rst part of the Proposition follows immediately if we apply
the Itˆ o formula to the process log p(t, T), write this in integrated form
and di?erentiate with respect to T.
For the second part we integrate the forward rate dynamics to get
r
t
= f(0, t) +
_
t
0
?(s, t)ds +
_
t
0
?(s, t)dW
s
(7)
+
_
t
0
_
E
?(s, x, t)µ(ds, dx).
Now we can write
?(s, t) = ?(s, s) +
_
t
s
?
T
(s, u)du,
?(s, t) = ?(s, s) +
_
t
s
?
T
(s, u)du,
?(s, x, t) = ?(s, x, s) +
_
t
s
?
T
(s, x, u)du,
and, inserting this into (7) we have
r
t
= f(0, t) +
_
t
0
?(s, s)ds +
_
t
0
_
t
s
?
T
(s, u)duds
+
_
t
0
?(s, s)dW
s
+
_
t
0
_
t
s
?
T
(s, u)dudW
s
+
_
t
0
_
E
?(s, x, s)µ(ds, dx) +
_
t
0
_
E
_
t
s
?
T
(s, x, u)duµ(ds, dx).
9
Changing the order of integration and identifying terms gives us the
result.
For the third part we adapt a technique from Heath–Jarrow–Morton
[19]. Using the de?nition of the forward rates we may write
p(t, T) = exp ¦Z(t, T)¦ (8)
where Z is given by
Z(t, T) = ?
_
T
t
f(t, s)ds. (9)
Writing (3) in integrated form, we obtain
f(t, s) = f(0, s)+
_
t
0
?(u, s)du+
_
t
0
?(u, s)dW
u
+
_
t
0
_
E
?(u, x, s)µ(du, dx).
Inserting this expression into (9), splitting the integrals and changing the
order of integration gives us
Z(t, T) = ?
_
T
t
f(0, s)ds ?
_
t
0
_
T
t
?(u, s)dsdu ?
_
t
0
_
T
t
?(u, s)dsdW
u
?
_
t
0
_
T
t
_
E
?(u, x, s)dsµ(du, dx)
= ?
_
T
0
f(0, s)ds ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx)
+
_
t
0
f(0, s)ds +
_
t
0
_
t
u
?(u, s)dsdu +
_
t
0
_
t
u
?(u, s)dsdW
u
+
_
t
0
_
t
u
_
E
?(u, x, s)dsµ(du, dx)
= Z(0, T) ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx)
+
_
t
0
f(0, s)ds +
_
t
0
_
s
0
?(u, s)duds +
_
t
0
_
s
0
?(u, s)dW
u
ds
+
_
t
0
_
s
0
_
E
?(u, x, s)µ(du, dx)ds.
Now we can use the fact that r
s
= f(s, s) and, integrating the forward
rate dynamics (3) over the interval [0, s], we see that the last two lines
10
above equal
_
t
0
r
s
ds so we ?nally obtain
Z(t, T) = Z(0, T) +
_
t
0
r
s
ds ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx).
Thus, with A, S and D as in the statement of the proposition, the sto-
chastic di?erential of Z is given by
dZ(t, T) = ¦r
t
+ A(t, T)¦ dt + S(t, T)dW
t
+
_
E
D(t, x, T)µ(dt, dx),
and an application of the Itˆ o formula to the process p(t, T) = exp ¦Z(t, T)¦
completes the proof.
Remark 2.5 To ?t reality, a “good” model of bond price dynamics or in-
terest rates must satisfy other important conditions. A bond price process
“should” e.g. take values in the interval [0, 1] and forward rates “ought”
to be positive (see [27]). We do not restrict ourselves to the class of “re-
alistic models” (obviously the most important ones) since we also want
to treat generalizations of “bad” models (like the various Gaussian mod-
els for the short rate) which are useful because their simplicity leads to
instructive explicit formulas.
3 Absence of arbitrage
3.1 Generalities
The purpose of this section is to give the appropriate de?nitions of self-
?nancing measure-valued portfolios, contingent claims, arbitrage possi-
bilities and martingale measures. We then proceed to show that the ex-
istence of a martingale measure implies absence of arbitrage, and we
end the section by investigating existence and uniqueness of martingale
measures.
We make the following standing assumption for the rest of the sec-
tion.
Assumption 3.1 We assume that
(i) There exists an asset (usually referred to as locally risk-free) with the
price process
B
t
= exp
__
t
0
r
s
ds
_
.
11
(ii) The ?ltration F = (T
t
) is the natural ?ltration generated by W and
µ, i.e.
T
t
= ?¦W
s
, µ([0, s] A), B; 0 ? s ? t, A ? c, B ? ^¦
where ^ is the collections of P-null sets from T.
(iii) The point process µ has an intensity ?, i.e. the P-compensator ?
has the form
?(dt, dx) = ?(t, dx)dt
where ?(t, A) is a predictable process for all A ? c.
(iv) The stochastic basis has the predictable representation property: any
local martingale M is of the form
M
t
= M
0
+
_
t
0
f
s
dW
s
+
_
t
0
_
E
?(s, x)(µ(ds, dx) ??(ds, dx))
where f is a process measurable with respect to the predictable ?-
algebra T and ? is a
˜
T-measurable function (
˜
T = T ?c) such that
for all ?nite t
_
t
0
[f
s
[
2
ds < ?,
_
t
0
_
E
[?(s, x)[?(ds, dx) < ?.
We need (ii) and (iv) above in order to have control over the class of
absolute continuous measure transformations of the basic (“objective”)
probability measure P. These assumptions are made largely for conve-
nience, but if we omit them, some of the equivalences proved below will
be weakened to one-side implications. See [4] for further information. The
assumption (iii) is not really needed at all from a logical point of view,
but it makes some of the formulas below much easier to read.
3.2 Self-?nancing portfolios
De?nition 3.2 A portfolio in the bond market is a pair ¦g
t
, h
t
(dT)¦,
where
1. The component g is a predictable process.
2. For each ?, t, the set function h
t
(?, ) is a signed ?nite Borel mea-
sure on [t, ?).
3. For each Borel set A the process h
t
(A) is predictable.
12
The intuitive interpretation of the above de?nition is that g
t
is the
number of units of the risk-free asset held in the portfolio at time t. The
object h
t
(dT) is interpreted as the “number” of bonds, with maturities
in the interval [T, T + dT], held at time t.
We will now give the de?nition of an admissible portfolio.
De?nition 3.3
1. The discounted bond prices Z(t, T) are de?ned by
Z(t, T) =
p(t, T)
B(t)
.
2. A portfolio ¦g, h¦ is said to be feasible if the following conditions
hold for every t:
_
t
0
[g
s
[ds < ?, (10)
_
t
0
_
?
s
[m(s, T)[[h
s
(dT)[ds < ?, (11)
_
t
0
_
E
_
?
s
[n(s, x, T)[[h
s
(dT)[?(ds, dx) < ?, (12)
_
t
0
__
?
s
[v(s, T)[[h
s
(dT)[
_
2
ds < ?. (13)
3. The value process corresponding to a feasible portfolio ¦g, h¦ is
de?ned by
V
t
= g
t
B
t
+
_
?
t
p(t, T)h
t
(dT). (14)
4. The discounted value process is
V
Z
t
= B
?1
t
V
t
. (15)
5. A feasible portfolio is said to be admissible if there is a number
a ? 0 such that V
Z
t
? ?a P ?a.s. for all t.
6. A feasible portfolio is said to be self-?nancing if the corresponding
value process satis?es
V
t
= V
0
+
_
t
0
g
s
dB
s
+
_
t
0
_
?
s
m(s, T)p(s, T)h
s
(dT)ds
+
_
t
0
_
?
s
v(s, T)p(s, T)h
s
(dT)dW
s
(16)
+
_
t
0
_
?
s
_
E
n(s, x, T)p(s?, T)h
s
(dT)µ(ds, dx).
13
There are obvious modi?cations of these de?nitions like “admissible
on the interval [0, T
0
]”.
The relation (16) is a way of making mathematical sense out of the
expression
dV
t
= g
t
dB
t
+
_
?
t
h
t
(dT)dp(t, T) (17)
which is the formal generalization of the standard self-?nancing condi-
tion. We shall sometimes use equation (17) as a shorthand notation for
the equation (16). It seems natural that the adequate stochastic calculus
for the theory of bond market has to include an integration of measure-
valued processes with respect to jump-di?usion processes with values in
some Banach space of continuous functions. Some versions of such a cal-
culus are given in our paper [4].
We shall as usual be working much with discounted prices, and the
following lemma shows that the self-?nancing condition is the same for
the discounted bond prices Z(t, T) as for the undiscounted ones.
Lemma 3.4 For an admissible portfolio the following conditions are equiv-
alent.
(i) dV
t
= g
t
dB
t
+
_
?
t
h
t
(dT)dp(t, T),
(ii) dV
Z
t
=
_
?
t
h
t
(dT)dZ(t, T).
Proof. The Itˆo formula.
Notice that for a self-?nancing portfolio the g-component is auto-
matically de?ned by the initial endowment V
0
and the h-component; the
pair (V
0
, h) is sometimes called the investment strategy of a self-?nancing
portfolio.
For technical purposes it is sometimes convenient to extend the de-
?nition of the bond price process p(t, T) (as well as other processes)
from the interval [0, T] to the whole half-line. It is then natural to put
Z(t, T) = 1, A(t, T) = 0 etc. for t ? T, i.e. one can think that after the
time of maturity the money is transferred to the bank account.
Remark 3.5 From the point of view of economics, discounting means
that the locally risk-free asset is chosen as the “num´eraire”, i.e. the
prices of all other assets are evaluated in the units of this selected one.
Some mathematical properties may however change under a change of
the num´eraire, see [11].
14
We now go on to de?ne contingent claims and arbitrage portfolios,
modifying somewhat the standard concepts.
De?nition 3.6
1. A contingent T-claim is a random variable X ? L
0
+
(T
T
, P)
(i.e. an arbitrary non-negative T
T
-measurable random variable).
We shall use the notation L
0
++
(T
T
, P) for the set of elements X of
L
0
+
(T
T
, P) with P(X > 0) > 0.
2. An arbitrage portfolio is an admissible self-?nancing portfolio
¦g, h¦ such that the corresponding value process has the properties
(a) V
0
= 0,
(b) V
T
? L
0
++
(T
T
, P).
If no arbitrage portfolios exist for any T ? R
+
we say that the
model is “free of arbitrage” or “arbitrage-free” (AF).
We now want to tie absence of arbitrage to the existence of a martin-
gale measures. Since we do not ?x a (?nite deterministic) time horizon,
it turns out to be convenient to consider a martingale density process as
a basic object (rather than a martingale measure).
De?nition 3.7 Take the measure P as given. We say that a positive
martingale L = (L
t
)
t?0
with E
P
[L
t
] = 1 is a martingale density if for
every T > 0 the process ¦Z(t, T)L
t
; 0 ? t ? T¦ is a P-local martingale.
If, moreover, L
t
> 0 for all t ? R
+
we say that L is a strict martingale
density.
De?nition 3.8 We say that a probability Q on (?, T) is a martingale
measure if Q
t
? P
t
(where Q
t
= Q[T
t
, P
t
= P[T
t
) and the process
¦Z(t, T); 0 ? t ? T¦ is a Q-local martingale for every T > 0 .
In other words, Q is a martingale measure if it is locally equivalent
to P and the density process dQ
t
/dP
t
is a strict martingale density.
Proposition 3.9 Suppose that there exists a strict martingale density
L. Then the model is arbitrage-free.
15
Proof. Fix any admissible self-?nancing portfolio ¦g, h¦ and assume
that for some ?nite T the corresponding value process is such that V
T
?
L
0
++
(T
T
, P). By admissibility, V
Z
? ?a for some a > 0. The process
(V
Z
+a)L is a positive local martingale hence a supermartingale. As L is a
martingale, V
Z
L is a supermartingale. Thus, E
P
_
V
Z
0
L
0
_
? E
P
_
V
Z
T
L
T
_
>
0, which is impossible because we assume that V
Z
0
= 0.
Remark 3.10 Notice that for the model restricted to some ?nite time
horizon T, a strict martingale density de?nes an equivalent martin-
gale measure Q
T
= L
T
P, i.e. a probability which is equivalent to P on
T
T
(in symbols: Q
T
T
? P
T
) such that all discounted bond prices are mar-
tingales on [0, T]. If E
P
[L
?
] = 1, there exists an equivalent martingale
measure also for the in?nite horizon and the above proposition can be
easily extended to this case in an obvious way. In general, when L is not
uniformly integrable, a measure Q on T such that L
t
= dQ
t
/dP
t
, may
not exist. The following simple example when a martingale density does
not de?ne Q explains the situation.
Let the stochastic basis be the coordinate space of counting functions
N = (N
t
) equipped with the measure of the unit rate Poisson process. Let
us modify this space by excluding only one point: the function which is
identically zero. It is clear that the process L
t
= I
{Nt=0}
e
t
is a martingale
density de?ning Q
T
for every ?nite T (under Q
T
the coordinate process
has the intensity zero on I
[T,?]
) but the measure Q such that Q[T
T
=
Q
T
[T
T
for all T does not exist.
This example reveals that the origin of such an undesirable property
lies in a certain pathology of the stochastic basis while Proposition 3.9
shows that one can work with a strict martingale density without any
reference to the martingale measure. Facing the choice between an in-
signi?cant supplementary requirement and a perspective to be far away
from the traditional language we prefer the ?rst option. So we impose
Assumption 3.11 For any positive martingale L = (L
t
) with E
P
[L
t
] =
1 there exists a probability measure Q on T such that L
t
= dQ
t
/dP
t
.
Remark 3.12 In numerous papers devoted to the term structure of in-
terest rates one can observe a rather confusing terminology : the model
is said to be arbitrage-free if there exists a martingale measure. The ori-
gin of this striking di?erence with the theory of stock markets (where
arbitrage means the possibility to get a pro?t which in some sense is
riskless) is clear, because in continuous-time bond market models there
16
are uncountably many basic securities and the key question is : what are
portfolios of bonds ? The discussion of the latter problem is avoided since
the straightforward use of ?nite-dimensional stochastic integrals does not
allow to de?ne a general portfolio in a correct way (see the apparent di?-
culties with the basic bonds in [28]). Interesting mathematical problems
concerning relations between di?erent de?nitions of arbitrage are almost
untouched in the theory of bond markets; this subject is beyond the scope
of the present paper as well.
3.3 Existence of martingale measures
Suppose that the bond prices and forward rates have P-dynamics given
by the equations (2) and (3). We now ask how various coe?cients in these
equations must be related in order to ensure the existence of a martin-
gale measure (or, in view of the Assumption 3.11, of a strict martingale
density). The main technical tool is, as usual, a suitable version of the
Girsanov theorem, which we now recall. The ?rst (direct) part (I) below
holds true regardless of how large the ?ltration is chosen to be, but the
converse part (II) depends heavily on the fact that we have assumed the
predictable representation property.
Theorem 3.13 (Girsanov)
I. Let ? be a predictable process and ? = ?(?, t, x) be a strictly positive
˜
T-measurable function such that for ?nite t
_
t
0
[?
s
[
2
ds < ?,
_
t
0
_
E
[?(s, x)[?(s, dx)ds < ?.
De?ne the process L by
log L
t
=
_
t
0
?
s
dW
s
?
1
2
_
t
0
[?
s
[
2
dW
s
+
_
t
0
_
E
log ?(s, x)µ(ds, dx) +
_
t
0
_
E
(1 ??(s, x))?(ds, dx), (18)
or, equivalently, by
dL
t
= L
t
?
t
dW
t
+ L
t?
_
E
(?(t, x) ?1) ¦µ(dt, dx) ??(dt, dx)¦ , L
0
= 1,
(19)
and suppose that for all ?nite t
E
P
[L
t
] = 1. (20)
17
Then there exists a probability measure Q on T locally equivalent to P
with
dQ
t
= L
t
dP
t
(21)
such that:
(i) We have
dW
t
= ?
t
dt + d
˜
W
t
, (22)
where
˜
W is a Q-Wiener process.
(ii) The point process µ has a Q-intensity, given by
?
Q
(t, dx) = ?(t, x)?(t, dx). (23)
II. Every probability measure Q locally equivalent to P has the structure
above.
We now come to the main results concerning the existence of a mar-
tingale measure. They generalize the corresponding results of Heath–
Jarrow–Morton and can be easily extended to the case of a multidimen-
sional Wiener process. The identities between processes are understood
dPdt-a.e.
Theorem 3.14
I. Let the bond price dynamics be given by (2). Assume that n(t, x, T) for
any ?xed T is bounded by a constant (depending on T). Then there exists
a martingale measure Q if and only if the following conditions hold:
(i) There exists a predictable process ? and a
˜
T-measurable function
?(t, x) with ? > 0 satisfying the integrability conditions of Theorem
3.13 and such that E
P
[L
t
] = 1 for all ?nite t, where L is de?ned
by (19).
(ii) For all T > 0 on [0, T] we have
m(t, T) + ?
t
v(t, T) +
_
E
?(t, x)n(t, x, T)?(t, dx) = r
t
. (24)
II. Let the forward rate dynamics be given by (3). Assume that e
D(t,x,T)
for any ?xed T is bounded by a constant (depending on T). Then there
exists a martingale measure if and only if the following conditions hold:
18
(iii) There exist a predictable process ? and a
˜
T-measurable function
?(t, x) with ? > 0 satisfying the integrability conditions of Theorem
3.13 and such that E
P
[L
t
] = 1 for all ?nite t where L is de?ned by
(19).
(iv) For all T > 0, on [0, T] we have
A(t, T) +
1
2
S
2
(t, T) + ?
t
S(t, T) +
_
E
?(t, x)?(t, dx) = 0, (25)
where
?(t, dx) =
_
e
D(t,x,T)
?1
_
?(t, dx)
and A, S and D are de?ned as in (6).
Proof.
I. First of all it is easy to see (using the Itˆ o formula) that a measure Q
is a martingale measure if and only if the bond dynamics under Q are of
the form
dp(t, T) = r
t
p(t, T)dt + dM
Q
t
, (26)
where M
Q
is a Q-local martingale. Using the Girsanov theorem we see
that under any equivalent measure Q, the bond dynamics have the fol-
lowing form, where we have compensated µ under Q.
dp(t, T) = p(t, T)m(t, T)dt + p(t, T)v(t, T)(?
t
dt + d
˜
W
t
)
+ p(t?, T)
_
E
n(t, x, T)?(t, x)?(t, dx)dt
+ p(t?, T)
_
E
n(t, x, T) ¦µ(dt, dx) ??(t, x)?(t, dx)dt¦ .
Thus we have
dp(t, T) = p(t, T)
_
m(t, T) +v(t, T)?
t
_
E
n(t, x, T)?(t, x)?(t, dx)
_
dt +
+ dM
Q
t
.
Comparing this with the equation (26) gives the result.
II. If the forward rate dynamics are given by (3) then the corresponding
bond price dynamics are given by Proposition 2.4. We can then apply
part 1 of the present theorem.
We now turn to the issue of so called “martingale modelling”, and re-
mark that one of the main morals of the martingale approach to arbitrage-
free pricing of derivative securities can be formulated as follows.
19
• The dynamics of prices and interest rates under the objective prob-
ability measure P are, to a high degree, irrelevant. The important
objects to study are the dynamics of prices and interest rates under
the martingale measure Q.
When building a model it is thus natural, and in most cases ex-
tremely time saving, to specify all objects directly under a martingale
measure Q. This will of course impose restrictions on the various para-
meters in, e.g., the forward rate equations, and the main results are as
follows.
Proposition 3.15 Assume that we specify the forward rate dynamics
under a martingale measure Q by
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
_
E
?(t, x, T)µ(dt, dx). (27)
Then the following relation holds
?(t, T) = ?(t, T)
_
T
t
?(t, s)ds ?
_
E
?(t, x, T)e
D(t,x,T)
?
Q
(t, dx). (28)
Furthermore, the bond price dynamics under Q are given by
dp(t, T) = p(t, T)r
t
dt + p(t, T)S(t, T)d
˜
W
t
+ p(t?, T)
_
E
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx), (29)
where ˜ µ is the Q-compensated point process
˜ µ(dt, dx) = µ(dt, dx) ??
Q
(t, dx)dt.
Here ?
Q
is the Q-intensity of µ whereas D and S are de?ned by (6).
Proof. Since we are working under Q we may use Theorem 3.14 with
? = 0 and ? = 1 to obtain
A(t, T) +
1
2
S
2
(t, T) +
_
E
_
e
D(t,x,T)
?1
_
?
Q
(t, dx) = 0,
and di?erentiating this equation with respect to T gives us the equation
(28). The result on bond prices now follows immediately from the result
above and from Proposition 2.4.
The single most important formula in this section is the relation (28)
which is the point process extension of the Heath–Jarrow–Morton “drift
20
condition”. We see that if we want to model the forward rates directly
under the martingale measure Q, then the drift ? is uniquely determined
by the di?usion volatility ?, the jump volatility ? and by the Q-intensity
?
Q
. This has important implications when it comes to parameter estima-
tion, since we are modelling under Q while our concrete observations, of
course, are made under an objective measure P. As far as volatilities are
concerned they do not change under an equivalent measure transforma-
tion, so “in principle” we can determine ? and ? from actual observations
of the forward rate trajectories. The intensity measure however presents
a totally di?erent problem. Suppose for simplicity that µ is a standard
Poisson process (under Q) with Q-intensity ?
Q
. If we could observe the
forward rates under Q then we would, of course, have access to a vast ma-
chinery of statistical estimation theory for the determination of a point
estimate of ?
Q
, but the problem here is that we are not making observa-
tions under Q, but under P. Thus the estimation of the Q-intensity ?
Q
is not a statistical estimation problem to be solved with standard sta-
tistical techniques. This fact may be regarded as a piece of bad news or
as an interesting problem. We opt for the latter interpretation, and one
obvious way out is to estimate ?
Q
by using market data for bond prices
(which contain implicit information concerning ?
Q
).
4 Uniqueness of Q and market complete-
ness
4.1 Uniqueness of the martingale measure
Throughout this section we shall work with a model speci?ed by the
forward rate dynamics under
Assumption 4.1 The coe?cient D(t, x, T) is uniformly bounded.
The main issue to be dealt with below is the relation between unique-
ness of the martingale measure and completeness of the bond market.
Using Theorem 3.14 we immediately have the following result.
Proposition 4.2 Let the forward rate dynamics be given by (3) and as-
sume that the assumptions (iii) and (iv) of Theorem 3.14 (equivalent to
existence of a martingale measure Q) are satis?ed. Then the martingale
measure Q is unique if and only if dPdt-a.e.
Ker /
t
(?) = 0 (30)
21
where the linear operator
/
t
(?) : R L
2
(E, c, ?(?, t, dx)) ?C[0, ?[ (31)
is de?ned by
/
t
(?) : (?, ?) ?S(?, t, .)? +
_
E
?(x)?(?, t, dx, T) (32)
with
?(?, t, dx, T) =
_
e
D(?,t,x,T)
?1
_
?(?, t, dx).
The important thing to note here is that the operators /
t
(?) are
integral operators of the ?rst kind. We shall refer to / as “the martingale
operators”.
Corollary 4.3 Suppose that the forward rate dynamics is given by (3),
that the model coe?cients ?(t, T), ?(t, T), ?(t, x, T), and ?(t, dx) are
deterministic and that the martingale measure Q is unique. Then the
Girsanov transformation parameters ? and ? are deterministic functions,
i.e. under Q the process
˜
W is a Wiener process with constant drift, and
µ is a Poisson measure.
Proof. It is su?cient to notice that the operators /
t
do not depend of ?
and hence (outside the exclusive dPdt-null sets) values of the Girsanov
transformation parameters corresponding to a ?xed t but di?erent ? must
satisfy the same equation (25), which has a unique solutions because of
(30).
Corollary 4.4 If we add to the hypotheses of Corollary 4.3 the assump-
tion that ?(t, T) = ?(T ?t), ?(t, T) = ?(T ?t), ?(t, x, T) = ?(T ?t, x),
and ?(t, dx) = ?(dx) then ? and ? do not depend on t, i.e. under Q the
process
˜
W is a Wiener process with a constant drift and µ is a Poisson
measure invariant under time translations.
Of course, the above assertions are almost trivial but they can be
considered as an overture to a more systematic use of classical functional
analysis, which appears to be an adequate tool in the considered set-
ting. Clearly, instead of considering families of operators as we do, one
can chose slightly di?erent de?nitions and e.g. consider a single operator
acting from one space of random processes to another.
22
In spite of the simplicity of the de?nition (31)-(32), it has a drawback
because it uses C[0, ?[ with its associated complicated dual. In order
to be able to work with a more manageable dual space it is therefore
natural to modify the de?nition of the martingale operators and impose
the following constraint on the model:
Assumption 4.5 For any t, x
lim
T??
Z(t?, T)S(t, T) = 0, lim
T??
Z(t?, T)
_
e
D(t,x,T)
?1
_
= 0
where D and S are given by (6), and Z is the discounted price process.
Let C
0
[0, ?[ be the space of continuous functions on [0, ?[ converg-
ing to zero at in?nity. Notice that here we have the well known duality
C
0
[0, ?[= /[0, ?[, where /[0, ?[ is the space of measures on [0, ?[.
The formula
/
Z
t
(?) : (?, ?) ?Z(?, t?, .)S(?, t, .)? +Z(?, t?, .)
_
E
?(x)?(?, t, dx, .)
de?nes a linear operator
/
Z
t
(?) : R L
2
(E, c, ?(?, t, dx)) ?C
0
[0, ?[.
In other words, /
Z
t
(?) is the product of the operator /
t
(?) and the
operator of multiplication by the function Z(?, t?, .) and one may write
/
Z
t
= Z
t
/
t
. Clearly, the above results hold also with / substituted by
/
Z
but the modi?ed de?nition leads to some nice duality arising in the
context of market completeness.
As an alternative, to avoid problems with the dual space, one can
suppose that there is a ?nite time horizon T
f
and all traded bonds have
maturities T ? T
f
. In this case, in section 5.1 below, we have to restrict
ourselves to measures G
t
(dT) with support in [T, T
f
].
4.2 Completeness
Let the forward rate dynamics under a martingale measure Q be given
by
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
_
E
?(t, x, T)µ(dt, dx), (33)
where
˜
W is a Q-Wiener process and µ has the Q-intensity ?
Q
. Our aim
is now to investigate the possibility of hedging contingent claims.
23
De?nition 4.6 Consider a contingent claim X ? L
?
(T
T
0
) expressed in
terms of the num´eraire. We say that it can be replicated or that we
can hedge against X if there exists a self-?nancing portfolio with the
bounded, discounted value process V
Z
such that
V
Z
T
0
= X. (34)
If every such X ? L
?
(T
T
0
) (for every T
0
) can be replicated, the
model is said to be complete.
If for every such X ? L
?
(T
T
0
) there exists a sequence of uniformly
bounded hedgeable claims converging to X in probability, we say that the
model is approximately complete.
It is important to notice that the spaces L
?
and L
0
are invariant
under an equivalent change of probability measures (recall also that con-
vergence in probability can be expressed in terms of convergence a.s.
along a subsequence).
Suppose now that we want to ?nd a self-?nancing portfolio ¦g, h¦
which replicates X ? L
?
+
(T
T
0
). Using Lemma 3.4 we see that the problem
is reduced to ?nding a portfolio strategy with an initial endowment V
Z
0
and a bond investment process h such that
dV
Z
t
=
_
?
t
h
t
(dT)dZ(t, T), (35)
V
Z
T
0
= X, (36)
Proposition 3.15 gives us the Q-dynamics of the bond prices and a simple
calculation shows that for Z we have the dynamics
dZ(t, T) = Z(t, T)S(t, T)d
˜
W
t
+ Z(t?, T)
_
E
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx),
(37)
where S and D are de?ned as usual. We are thus looking for a pair
_
V
Z
0
, h
_
such that
V
Z
T
0
= X, (38)
dV
Z
t
=
_
?
t
h(t, dT)Z(t, T)S(t, T)d
˜
W
t
+
_
E
_
?
t
h(t, dT)Z(t?, T)
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx), (39)
with the integrability conditions
_
T
0
0
__
?
s
[h(s, dT)[ [Z(s, T)S(s, T)[
_
2
ds < ?, (40)
24
_
T
0
0
_
E
_
?
s
[h(s, dT)[ [Z(s, T)
_
e
D(t,x,T)
?1
_
[?(ds, dx) < ?. (41)
Now, since X ? L
?
+
(T
T
0
) the process
M
t
= E
Q
[X[T
t
] (42)
is a Q-martingale. By the assumptions it has an integral representation,
that is there are ? and ? such that
dM
t
= ?
t
d
˜
W
t
+
_
E
?(t, x)˜ µ(dt, dx), (43)
with
E
Q
_
_
T
0
0
?
2
t
dt
_
< ?
and
E
Q
_
_
T
0
0
_
E
?
2
(t, x)d?(dt, dx)
_
< ?.
Now we may formulate our ?rst proposition concerning hedging.
Proposition 4.7 We can replicate a claim X ? L
?
+
(T
T
0
) if and only if
there exists a predictable measure-valued process h(t, dT) which satis?es
the integrability conditions (40) and (41) and solves on [0, T
0
] (dPdt-a.e.)
the equations
/
Z
t
h =
_
?
t
?(t, .)
_
(44)
where ? and ? are de?ned as above and where the “hedging operators”
/
Z
t
(acting on measures) are de?ned by
/
Z
t
(?) : m ?
_
¸
¸
_
_
?
t
Z(?, t?, T)S(?, t, T)m(dT)
_
?
t
Z(?, t?, T)
_
e
D(?,t,.,T)
?1
_
m(dT)
_
¸
¸
_
. (45)
Proof. Su?ciency. Assume that h(t, dT) is solution of (44). Then we
have
dM
t
=
_
?
t
h(t, dT)Z(t, T)S(t, T)d
˜
W
t
+
_
E
_
?
t
h(t, dT)Z(t?, T)
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx). (46)
Now we de?ne g by
g
t
= M
t
?
_
?
t
h(t, dT)Z(t, T). (47)
25
We see from (47) that the value process corresponding to the portfolio
¦g, h¦ is given by V
Z
t
= M
t
. Furthermore, it follows from (46) that the
portfolio is self-?nancing. Finally we see from the de?nition of M that
V
Z
T
0
= M
T
0
= X. Thus we have found a hedge against X and su?ciency
is proved.
Necessity. The discounted value process V
Z
of a hedging portfolio ¦g, h¦
is a bounded martingale with V
Z
T
0
= X. Thus V
Z
is indistinguishable
from M given by (42), and the uniqueness considerations yield (44).
Note that the hedging operators
/
Z
t
: /[0, ?[?R L
2
(E, c, ?
Q
(t, dx)) (48)
are indeed the adjoint of the martingale operators /
Z
t
.
To sum up we have the following conclusions.
Proposition 4.8
1. The martingale measure is unique if and only if the mappings /
Z
are injective (a.e.).
2. The market is complete if and only if the mappings /
Z?
are surjec-
tive (a.e.).
The proof of a natural extension of the second assertion which we
give below involves a measurable selection technique.
Proposition 4.9 The following conditions are equivalent.
(i) The market is approximately complete.
(ii) cl (Im/
Z?
t
(?)) = R L
2
(E, c, ?
Q
(?, t, dx)) (a.e.).
Proof. (i) ?(ii) Let X be a bounded discounted contingent T
0
-claim to
be approximated. Using truncation arguments we can suppose, without
loss of generality, that X is such that ? and ? in the representation (43)
are bounded. For ? > 0 put
F
?
(t, m) = [/
Z,1
t
(m) ??
t
[
2
+|/
Z,2
t
(m) ??(t, .)|
2
L
2
(?
Q
(t,dx))
where we use superscripts to denote the ?rst and the second “coordi-
nates” in (45). Let us consider on /[0, ?[ the ?-algebra ¼ generated
26
by the weak topology (more precisely, by ?(/[0, ?[, C
0
[0, ?[)). Recall
that balls in /[0, ?[ are metrizable compacts, hence (/[0, ?[, ¼) is
a Lusin space as a countable union of Polish spaces. The function F
?
,
being T-measurable in (?, t) and continuous in m, is jointly measurable.
Therefore, the set-valued mapping
(?, t) ?¦m ? /[0, ?[: F
?
(?, t, m) ? ?¦
has a T ?¼-measurable graph and, hence, admits a T-measurable a.e.-
selector m
?
(t, dT) (see, e.g. [12]), which “almost” solves the problem: the
terminal values of the processes, de?ned by I
[0,t]
m
?
(t, dT) and the initial
endowment E
Q
[X], converge to X in L
2
(T
T
0
, Q), hence in probability,
as ? ? 0. One can notice, however, that the construction is not accom-
plished since the strategy generates a value process which is not bounded
(and even admissible). A standard truncation and localization arguments
?nally lead to the desired goal.
(i) ?(ii) Assume that the market is approximately complete. Then
there exists a countable set H = ¦X
n
¦ of bounded hedgeable random
variables, dense in the Hilbert space L
2
(T
T
0
, Q) and closed under linear
combinations with rational coe?cients; let g
n
= (?
n
, ?
n
) be the pair of
functions in the integral representation of X
n
given by (43). Without
loss of generality, we may assume that for all (?, t) one has |g
n
|
?,t
< ?
where |.|
?,t
and (., .)
?,t
are, respectively, the norm and the scalar product
in R L
2
(E, c, ?
Q
(?, t, dx)). Let us denote by H
?,t
the closure in this
norm of the set ¦g
n
(?, t)¦, which is, evidently, a linear subspace, and by
H
?
?,t
its orthogonal complement.
It is easy to show that there exists a pair of functions g = (?, ?) such
that ? is T-measurable, ? is
˜
T-measurable, and |g|
?,t
= 1 if H
?
?,t
,= 0.
Indeed, let ¦I(j)¦ be a sequence of indicator functions generating c and
k(?, t) = inf
_
j : inf
n
|I(j, .) ??
n
(?, t, .)|
L
2
(?
Q
(?,t,dx))
> 0
_
.
Put ˜ ?
t
(?) = I
{sup
n
|?
n
t
(?)|>0}
, ˜ ?(?, t, x) = I(k(?, t), x) if k(?, t) < ? and
˜ ?(?, t) = 0 otherwise. The pair of functions ˜ g = (˜ ?, ˜ ?) meets the neces-
sary measurability requirements. Furthermore, there is ˜ g
?
= (˜ ?
?
, ˜ ?
?
)
which is measurable in the same way and such that all the sections
˜ g
?
(?, t) are representatives of the projections of ˜ g(?, t) onto H
?,t
(one
can orthogonalize ¦g
n
(?, t)¦ preserving measurability and notice that in
this case the Fourier coe?cients are obviously predictable). Normalizing
the di?erence ˜ g ? ˜ g
?
we get g with the required properties.
The pair g = (?, ?) de?nes by (43) with M
0
= 0, a random variable
M
T
= X ? L
2
(T
T
0
, Q), orthogonal, by construction, to all X
n
. If (ii) does
not hold then X is nontrivial; this leads to an apparent contradiction.
27
By experience from the theory of ?nancial markets with ?nitely many
assets one could expect that the market is complete if and only if the
martingale measure is unique, but in our in?nite dimensional setting this
is no longer true. Due to the duality relation (Ker /)
?
= cl (Im/
) we
obtain instead from the above assertions
Theorem 4.10 The following statements are equivalent:
(i) The martingale measure is unique.
(ii) The market is approximately complete.
For a model with a ?nite mark space E, where the hedging problem
is reduced to a ?nite dimensional system of equations (for each (?, t)),
the duality relation is simpler: (Ker /)
?
= Im/
, so in this case we have
Corollary 4.11 Suppose that the mark space E is ?nite. Then the bond
market is complete if and only if the martingale measure is unique.
The same conclusion holds if for almost all (?, t) the measures ?
t
(?, dx)
are concentrated in a ?nite number of points. In general, for an in?nite
E the “principle” that uniqueness of Q is equivalent to completeness of
the market fails: the set of hedgeable claims may be a strict subset in the
set of all claims L
?
(T
T
0
). Clearly, it is the case when D is continuous
in x and bounded (hence the image contains only continuous functions);
typically, /
Z?
t
is a compact operator and, hence, has no bounded inverse.
Thus the case with an in?nite mark space introduces some com-
pletely new features into the theory, and we also encounter some new
problems when it comes to the numerical computation of hedging port-
folios. The formal result is as follows.
Corollary 4.12 Suppose that the mark space E is in?nite. Then the
hedging equation (44) is ill-posed in the sense of Hadamard, i.e. the in-
verse of /
Z
t
restricted to Im/
Z
is not bounded.
Proof. This follows immediately from the fact that /
Z
is compact.
The main content of this result is that the hedging equation is nu-
merically ill-conditioned, in the sense that a small disturbance of the
right-hand side (e.g. due to a small round-o? error) gives rise to large
?uctuations in the solution. Thus, a naive approximation scheme for the
calculation of a concrete hedge may very well lead to great numerical
problems. Fortunately, there exists a large literature on stable solutions
of ill-posed problems but we will not pursue this topic here.
28
5 Characterization of hedgeable claims
5.1 Laplace transforms
In this section we suppose that Q is unique and that E is in?nite. Assume
for simplicity that we have no driving Wiener process. One can also think
that the model coe?cients in (50) below are deterministic.
From the general theory developed in the previous section it follows
that the hedging equation, symbolically written as
/
?
G = ? (49)
with the measure G
t
(dT) = Z(t?, T)h
t
(dT), can only be solved for a
right-hand side ? in a dense subset of the image space. The purpose
of this section is to present a class of models, for which we can give an
explicit characterization of the class of hedgeable claims.
Assumption 5.1 The forward rate dynamics under Q is given by
df(t, T) = ?(t, T)dt +
_
E
?(t, x, T) µ(dt, dx) (50)
with ? of the form
?(t, x, T) = ?c
t
(x) ?
0
(t, T) (51)
where the functions c and ?
0
are such that
(i) For each t the mapping c
t
: E ?R is injective.
(ii) For each t the set c
t
(E) is an interval [l
t
, ?[ or ]l
t
, ?[ (i.e. the left
endpoint l
t
may or may not belong to c
t
(E).
(iii) ?
0
> 0.
The important restriction introduced by this assumption is the volatil-
ity structure given by (51) (see, however, Remark 6.8 of Section 6). Con-
dition (i) simply means that di?erent points in E really give rise to dif-
ferent behavior of the forward rates. Assumption (ii) guarantees that we
have an in?nite mark space and that c
t
(E) has a limit point at in?nity;
assumption (iii) does not seem to be severe.
Suppose now that we want to hedge against a particular bounded
discounted T
0
-claim X. The martingale representation result, see (43),
will then provide us with a function ?(t, x), and the hedging problem
29
reduces, modulo a measurable selection, to the problem of ?nding a
measure-valued process G such that for almost all t ? [0, T
0
]
_
?
t
_
e
D(t,x,T)
?1
_
G(t, dT) = ?(t, x) ?
Q
(t, dx)-a.e. (52)
Using Assumption 5.1 we obtain
_
?
t
_
e
?ct(x)?(t,T)
?1
_
G(t, dT) = ?(t, x) ?
Q
(t, dx)-a.e. (53)
where ? is given by
?(t, T) =
_
T
t
?
0
(t, s) ds. (54)
Since c
t
is assumed to be injective we can write (53) as
_
?
t
_
e
?y?(t,T)
?1
_
G(t, dT) = ?
c
(t, y). (55)
where ?
c
(t, y) = ?(t, c
?1
t
), y ? c
t
(E), and the equality is understood
modulo ?
c
Q
(t, dy), the image of the measure ?
Q
(t, dx) under the mapping
c
t
.
Since the right-hand side of (55) is a class of equivalence, the rigorous
formulation of the following necessary condition concerns, in fact, the
properties of a representative of this class.
Lemma 5.2 Necessary conditions for the existence of a solution to the
hedging equation (52) are that
(i) For each t the function ?
c
(t, y) can be extended to an analytic func-
tion for all complex z such that Re z > l
t
.
(ii) For each t the limit lim
y??
?
c
(t, y) exists.
Proof Analyticity follows from the fact that, because ? > 0, the left-
hand side of (55) is analytic. Existence of the limit then follows directly
from (55).
We thus see that it is only in rather special cases that we can solve
the hedging equation, and this fact is in complete accordance with the
denseness result of the preceding section.
30
We continue our investigation, assuming that we can actually solve
the hedging equation. Then we may write (55) as
_
?
t
e
?y?(t,T)
G(t, dT) = ?
c
(t, y) ??
c
(t, ?) (56)
and (remember that t is ?xed in each equation) change the integration
variable from T to u by the substitution u = ?(t, T). Thus the measure
G(t, dT) will be pushed to a measure G
?
(t, du), de?ned by
G
?
(t, du) = G(t, ?
?1
(t, dT)). (57)
We now have
_
?
0
e
?yu
G
?
(t, du) = ?
c
(t, y) ??
c
(t, ?), ?y ? c
t
(E). (58)
This equation is a Laplace transformation for each t, and we have the
following characterization of the set of ?’s for which we can solve the
hedging equation.
Theorem 5.3 Consider a ?xed claim X and its corresponding ?. Then
we can hedge exactly against X if and only if the following conditions
hold for each (?, t):
(i) The function ?
c
(t, y) can be extended to an entire analytic function
for all complex z such that Re z > l
t
.
(ii) For each t the limit lim
y??
?
c
(t, y) exists.
(iii) The function ?
c
(t, y)??
c
(t, ?) is the Laplace transform of a signed
(?nite) Borel measure on [0, ?).
For a particular claim we may also want to know if the replicating
portfolio contains only bonds with maturities in a prespeci?ed set. If X
corresponds to an expiration time T
0
one can, e.g., ask whether it can
be hedged with a portfolio consisting entirely of bonds with maturities
greater than T
0
. Properties of this kind can, in fact, be read o? immedi-
ately from the structure of the predictable representation, i.e. of ?.
Proposition 5.4 Consider a ?xed hedgeable T
0
-claim X and its cor-
responding ?. Also ?x a real number (maturity) T
1
. Then the hedging
portfolio can be composed entirely of bonds with maturities greater than
T
1
if and only if the following condition holds for every t ? T
0
.
lim
y??
e
y?(t,T
1
)
(?
c
(t, y) ??
c
(t, ?)) = 0. (59)
31
Proof The h-component of a portfolio consists entirely of bonds with
maturities greater than T
1
if and only if the measure G(t, dT) has its
support in [T
1
, ?[, which is equivalent to the property that G
?
has its
support in [?(t, T
1
), ?[. Thus we can rewrite (58) as
_
?
?(t,T
1
)
e
?yu
G
?
(t, du) = ?
c
(t, y) ??
c
(t, ?),
which, after the change of variables v = u ??(t, T
1
), becomes
_
?
0
e
?y[v+?(t,T
1
)]
G
?,T
1
(t, dv) = ?
c
(t, y) ??
c
(t, ?),
where G
?,T
1
is the translation of G
?
. Thus we obtain
_
?
0
e
?yv
G
?,T
1
(t, dv) = e
y?(t,T
1
)
(?
c
(t, y) ??
c
(t, ?)),
and the result follows.
5.2 The case of a ?nite mark space
In the case when the mark space E is ?nite, we can write the forward
rate dynamics as
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
n
i=1
?
i
(t, T)dN
i
t
(60)
where N
1
, , N
n
are counting processes with predictable intensity processes
?
1
, , ?
n
. The process
˜
W is supposed to be m-dimensional standard
Wiener, so ?(t, T) is an m-dimensional (row) vector process. In this case it
is reasonable to look for a hedging portfolio with the h-component instan-
taneously consisting of n+m bonds with di?erent maturities T
1
, , T
n
,
T
n+1
, , T
n+m
(i.e. h(t, dT) is a discrete measure concentrated in these
points), and the hedging equation can be written in the following matrix
form (where m may be equal to zero).
A(t, T
1
, , T
n+m
)
_
¸
¸
_
G
1
t
.
.
.
G
n+m
t
_
¸
¸
_
=
_
?
t
?(t)
_
(61)
where
?
t
=
_
¸
¸
_
?
1
t
.
.
.
?
m
t
_
¸
¸
_
, ?(t) =
_
¸
¸
_
?(t, 1)
.
.
.
?(t, n)
_
¸
¸
_
, (62)
32
A(t, T
1
, , T
n+m
) =
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
S
m
(t, T
1
) S
m
(t, T
n+m
)
.
.
.
.
.
.
.
.
.
S
1
(t, T
1
) S
1
(t, T
n+m
)
e
D
1
(t,T
1
)
?1 e
D
1
(t,T
n+m
)
?1
.
.
.
.
.
.
.
.
.
e
Dn(t,T
1
)
?1 e
Dn(t,T
n+m
)
?1
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
, (63)
S
i
(t, T
j
) = ?
_
T
j
t
?
i
(t, s) ds , D
i
(t, T
j
) = ?
_
T
j
t
?
i
(t, s) ds. (64)
Here the (?, ?) -process, as usual, comes from the martingale rep-
resentation theorem, with ?
i
as the integrand corresponding to
˜
W
i
and
?(i) as the integrand corresponding to the N
i
-process. The process G
i
is
(see comment after (49)) the discounted amount invested in the portfolio
corresponding to the bonds with maturity T
i
.
The main problem in this section is to give conditions that guarantee
completeness of the bond market. In concrete terms this means that we
want to give conditions on the forward rate dynamics implying the exis-
tence of maturities T
1
, , T
n+m
such that the matrix A(t, T
1
, , T
n+m
)
is invertible. From a practical point of view it would be particularly pleas-
ing if these maturities can be chosen in such a way that they stay ?xed
when the time t is running. Intuitively, it is also natural to expect that
the maturities can be chosen arbitrarily, as long as they are distinct from
one another.
The main result in this section says that, given smoothness of S and
D in the maturity variable T, we can choose maturities almost arbitrarily.
If, furthermore the volatilities are deterministic and S and D are also
smooth in the t-variable, then the maturities can be chosen ?xed over
time, i.e. maturities do not change with the running time t.
We start with a general mathematical observation in the following
Proposition 5.5 Let f
1
, , f
M
be a set of real-valued functions such
that
33
(i) For each i the function f
i
is real-valued analytic, i.e. it can be ex-
tended to a holomorphic function in the complex plane.
(ii) The functions f
1
, , f
M
are linearly independent.
For each choice of reals T
1
, , T
M
consider the matrix B de?ned by
B(T
1
, , T
M
) = ¦f
i
(T
j
)¦
i,j
. (65)
Then, given any ?nite interval [I
L
, I
R
] of a positive length, we can choose
T
1
, , T
M
in [I
L
, I
R
] such that B is invertible. Furthermore, apart from
a ?nite set of points, we can choose T
1
, , T
M
arbitrarily in [I
L
, I
R
] as
long as they are distinct.
Proof. We ?x the interval [I
L
, I
R
] and prove the result by induction on
the number of functions. For M = 1 the statement is obviously true, since
by analyticity the function f
1
can have at most ?nitely many zeroes on a
compact set. Suppose therefore that the statement is true for M = n?1,
and consider the matrix function B(t) de?ned by
B(t) =
_
¸
¸
¸
¸
_
f
1
(t) f
1
(T
2
) f
1
(T
n
)
f
2
(t) f
2
(T
2
) f
2
(T
n
)
.
.
.
.
.
.
.
.
.
f
n
(t) f
n
(T
2
) f
n
(T
n
)
_
¸
¸
¸
¸
_
(66)
where, by the induction hypothesis, we have chosen T
2
, , T
n
in such a
way that all (n ?1)-dimensional quadratic submatrices of the last n ?1
columns are invertible. Our task is now to prove that we can choose
a point t such that B(t) is invertible and to do this we consider the
determinant det B(t). Expanding det B(t) along the ?rst column we see
that
det B(t) =
n
i=1
a
i
f
i
(t) (67)
where the a
i
’s are subdeterminants of the last n?1 columns and hence (by
the induction hypothesis) nonzero. Thus we see from (67) that det B(t) is
an analytic function and, because of the assumed linear independence, it
is not identically equal to zero. Thus it has at most ?nitely many zeroes
in the interval [I
L
, I
R
] . If we choose T
1
as any number in [I
L
, I
R
] , except
for the ?nite set of “forbidden” values, we get the result.
Applying this result to the bond market situation we have
Theorem 5.6 Assume that
34
(i) For each ?, t all functions ?
i
(t, T) and ?
j
(t, T) are analytic in the
T-variable.
(ii) For each ?, t the following functions of the argument T are linearly
independent:
e
D
i
(t,T)
?1, S
j
(t, T), i = 1, , n, j = 1, , m. (68)
Then the market is complete. Furthermore, for each t we can choose the
distinct bond maturities arbitrarily, apart from a ?nite number of values
on every compact interval. If all functions above are deterministic and
analytic also in the t-variable, then the maturities can be chosen to be
the same for every t.
Proof. The main part of the statement follows immediately from Propo-
sition 5.5. The last statement follows from the fact that, if we ?x the
maturities at t = 0 such that the corresponding det B(t) ,= 0, then,
again by the assumed analyticity in the t-variable, det B(t) is zero only
for ?nitely many t-values. Furthermore, in the replicating portfolio we
are integrating compensated Poisson processes having intensities, so the
strategies can be chosen arbitrarily on the zero set of B, since this (de-
terministic) set has Lebesgue measure zero, while outside this set they
have to satisfy the system (61).
As an easy corollary we immediately have the following extension
of a result of Shirakawa (see [28]). Note that we allow for more than
one Wiener process, whereas the proof in [28] depends critically on an
assumption of only one Wiener process. In addition, in [28] the maturities
of the bonds in the hedging portfolio cannot be chosen freely, and the
maturities also vary with running time t. In contrast, we can prespecify
arbitrary maturities (as long as they are distinct) and these maturities are
allowed to stay ?xed as t varies. For practical purposes this is extremely
important, since in real life we only have access to a ?nite set of maturities
for traded bonds.
Corollary 5.7 Assume that the forward rate volatilities have the form
_
?
j
(t, T) = q
j?1
(T ?t), j = 1, , m,
?
i
(t, T) = ?
i
, i = 1, , n,
(69)
where ?
1
, , ?
n
are constants and q
j?1
(s) is a polynomial of degree j ?1
with a non-vanishing leading term. Then the market is complete. Fur-
thermore, the maturities can be chosen arbitrarily.
35
Proof. Follows immediately from Theorem 5.6.
The next result and its proof explain Shirakawa’s idea of using the
Vandermonde matrix to construct the “basic bonds”.
Corollary 5.8 Let m = 0 and ?
i
(t, T) = ?
i
?(T ?t) where ? is a strictly
positive function and ?
i
are distinct non-zero constants. Then the market
is complete.
Proof. One can always choose a number a > 0 and a monotone sequence
of u
k
such that
_
u
k
0
?(s)ds = ka, k = 1, . . . n.
Take maturities T
k
= t + u
k
. Since D
i
(t, T
k
) = ka?
i
we have, putting
?
i
= e
a?
i
, that
det A(t, T
1
, . . . , T
n
) = det (?
k
i
?1) ,= 0.
Indeed, the linear dependence condition can be written as
f(?
i
) :=
n
k=1
?
k
?
k
i
?
n
k=1
?
k
= 0, i = 1, ..., n,
with a nontrivial vector ?; this is impossible: since also f(1) = 0 the
coe?cients of the polynomial f(?) of degree n must be equal to zero.
Remark 5.9 There is an important practical as well as mathematical
di?erence between the situation when the maturities of bonds in hedging
portfolios depend or do not depend on the current time t. In the for-
mer case the portfolio contains only instantaneously a ?nite set of bonds
(“basic bonds” at t) but when t varies, then the union of these sets of
securities may happen to be in?nite and even non-countable, and hence
one can not apply the classical theory of stochastic integration. As can be
seen from Corollary 5.8, the system of “basic bonds” constructed in [28]
depends unfortunately on t. We note again that in our results above we
may, in fact, chose maturities which stay ?xed during the entire trading
period.
36
6 A?ne term structures
As soon as one moves from abstract theory to practical applications, and
in particular to algorithms which have to be executed in real time on a
computer, the need emerges of easily manageable analytical formulas. In
the case of interest rate derivatives one is particularly fortunate if the
models possess a so-called a?ne term structure.
In this section the starting point is that we take as given the dynam-
ics of the short rate. The notations are a bit di?erent from those of the
others sections and we omit certain somewhat boring mathematical de-
tails, such as e.g. technical conditions ensuring that solutions to certain
equations below actually exist and have desirable properties (integrabil-
ity etc.).
De?nition 6.1 An interest rate model is said to have an a?ne term
structure if bond prices can be described as
p(t, T) = F(t, r
t
, T), (70)
where
log F(t, r, T) = A(t, T) ?B(t, T)r, (71)
and where A and B are deterministic functions. We sometimes use the
notation
F(t, r, T) = F
T
(t, r).
A model exhibiting an a?ne term structure occurs naturally only
in a Markovian environment and so the starting point in this section is
that we consider the dynamics of the short rate of interest given a priori
as a Markov process. Furthermore, we choose to specify the r-dynamics
directly under the martingale measure Q.
Assumption 6.2 We assume that under Q all bounded discounted price
processes are martingales, the short rate is assumed to be the solution of
a stochastic di?erential equation of the form
dr
t
= a(t, r
t
)dt + b(t, r
t
)d
˜
W
t
+
_
E
q(t, r
t
, x)µ(dt, dx), (72)
where a(t, r), b(t, r), and q(t, r, x) are given deterministic functions. The
process
˜
W is Q-Wiener and µ has a predictable Q-intensity
?(?, t, dx) = ?(t, r
t?
, dx), (73)
where ?(t, r, dx) is a deterministic measure for each t and r.
37
The main problem here is that of ?nding su?cient conditions on a, b, q,
and ? for the existence of an a?ne term structure. We start by presenting
the fundamental partial di?erential-di?erence equation in this context
concerning the pricing of simple claims in a general Markovian setting.
Proposition 6.3 Suppose that the short rate is given by (72) and con-
sider, for a ?xed T, any bounded (discounted) contingent claim X, to be
paid at T, of the form
X = ?(r
T
). (74)
Then the arbitrage-free price process ?(t; X) of this asset is given by
?(t; X) = F(t, r
t
), (75)
where F is a (su?ciently regular) function which is the a solution of the
Cauchy problem
_
¸
_
¸
_
?F
?t
(t, r) +/F(t, r) ?rF(t, r) = 0,
F(T, r) = ?(r),
(76)
with
/F(t, r) = a(t, r)
?F
?r
(t, r) +
1
2
b
2
(t, r)
?
2
F
?r
2
(t, r)
+
_
E
¦F(t, r + q(t, r, x)) ?F(t, r)¦ ?(t, r, dx).
(77)
Proof. By the Itˆ o formula we have the following representation:
F(t, r
t
) exp
_
?
_
t
0
r
s
ds
_
= F
0
+
_
t
0
?F(s, r
s
)
?r
exp
_
?
_
s
0
r
u
du
_
b(s, r
s
)d
˜
W
s
+
_
t
0
_
E
¦F(s, r
s?
+ q(s, r
s?
, x)) ?F(s, r
s?
)¦ exp
_
?
_
s
0
r
u
du
_
˜ µ(ds, dx)
where F
0
= F(0, r
0
) and ˜ µ(ds, dx) = µ(ds, dx) ? ?(s, r
s?
, dx)ds. The
right-hand side of this representation de?nes a local martingale. By As-
sumption 6.2 it is in fact a true martingale so, using the boundary con-
dition, we see that it is the discounted price process of the contingent
claim X and (75) follows.
Notice that, in general,
?(t; X) = E
Q
_
?(r
T
) exp
_
?
_
T
t
r
s
ds
_
[T
t
_
, (78)
38
and, because of the Markovian setting, we have in fact (75) with
F(t, r) = E
Q
_
?(r
T
) exp
_
?
_
T
t
r
s
ds
_
[r
t
= r
_
. (79)
Since / is easily seen to be the in?nitesimal operator of r, the relation
(76) is nothing but the Kolmogorov backward equation.
Corollary 6.4 Given the short rate dynamics (72) – (73), bond prices
are given by (70), where
_
¸
_
¸
_
?F
T
?t
(t, r) +/F
T
(t, r) ?rF
T
(t, r) = 0.
F
T
(T, r) = 1.
(80)
We now turn to the existence of the a?ne term structure. The as-
sertion below is an extension of a result by Du?e ([13], see also [8]).
Proposition 6.5 Suppose that the r-dynamics under Q is given by (72)
and the model parameters a, b, q, and ? have the following structure
a(t, r) = ?
1
(t) +?
2
(t)r,
b(t, r) =
_
?
1
(t) +?
2
(t)r,
q(t, r, x) = q(t, x),
?(t, r, dx) = l
1
(t, dx) +l
2
(t, dx)r,
(81)
Suppose that the functions A(., T) and B(., T) on [0, T] solve the following
system of ODE’s
?B
?t
(t, T) +?
2
(t)B(t, T) ?
1
2
?
2
(t)B
2
(t, T) + ?
2
(t, B(t, T)) = ?1,
B(T, T) = 0,
(82)
?A
?t
(t, T) +?
1
(t)B(t, T) +
1
2
?
1
(t)B
2
(t, T) + ?
1
(t, B(t, T)) = 0,
A(T, T) = 0,
(83)
where
?
i
(t, y) =
_
E
_
1 ?e
?yq(t,x)
_
l
i
(t, dx), i = 1, 2. (84)
Then the model has an a?ne term structure of the form (70) – (71).
39
We see that, for ?xed T, the system (82) – (83) has a nice recursive
structure (and, in particular, when l
2
= 0 (82) is a Riccati-type equation
for B). Given a solution to (82), we can then easily determine A by a
simple integration.
Notice also that one needs some caution to ensure ? to be positive.
This is the case if l
1
? 0 and l
2
= 0 or l
1
? 0, l
2
? 0, and the process r
is positive (recall that, in principle, we admit negative values of interest
rates).
Remark 6.6 Notice that the class of models satisfying (81) general-
izes some well-known term structure models such as Vasi?cek [29], Cox–
Ingersoll–Ross [9], Ho–Lee [20], Hull–White [21].
Remark 6.7 If a model possesses an a?ne term structure, then, always
under the assumption 6.2 and that of Proposition 6.5, the forward rate
dynamics are easily obtained as
df(t, T) = ¦B
t,T
(t, T)r
t
?A
t,T
(t, T) +B
T
(t, T)a(t, r)¦dt
+B
T
(t, T)b(t, r)d
˜
W
t
+
_
E
B
T
(t, T)q(t, x)µ(dt, dx) (85)
where B
T
(t, T) is the partial derivative with respect to T and B
t,T
the
partial with respect to t and T.
Remark 6.8 We close this section by pointing out that, if a model pos-
sesses an a?ne term structure, then (see Remark 6.7) the forward rate
dynamics satisfy (85) from which it is immediately seen that assump-
tion 5.1, in particular the decomposition property (51), is satis?ed with
c
t
(x) = ?q(t, x) and ?
0
(t, T) = B
T
(t, T). This implies that for the a?ne
term structure models the hedging problem can be approached by means
of a Laplace transform inversion (see section 5.1), which makes this prob-
lem considerably easier compared to the general setting of section 4.2.
References
[1] Artzner, P. & Delbaen, F. (1989) Term structure of interest rates.
Advances in Applied Mathematics 10, 95-129.
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an o?cial short rate. Working paper, University of Warwick.
[3] Bj¨ ork, T. (1995) On the term structure of discontinuous interest
rates. Surveys in Industrial and Applied Mathematics 2 No.4, 626-
657.
40
[4] Bj¨ ork, T. & Di Masi, G. & Kabanov, Y. & Runggaldier, W. (1995)
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41
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tegrals in the theory of continuous trading. Stochastic Processes &
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terest rate contingent claims. Journal of Finance 41, 1011-1029.
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42
doc_183392638.pdf
In economics, market structure is the number of firms producing identical products which are homogeneous.
Bond Market Structure in the
Presence of Marked Point Processes
?
Tomas Bj¨ ork
Department of Finance
Stockholm School of Economics
Box 6501, S-113 83 Stockholm SWEDEN
Yuri Kabanov
Central Economics and Mathematics Institute
Russian Academy of Sciences
and
Laboratoire de Math´ematiques
Universit´e de Franche-Comt´e
16 Route de Gray, F-25030 Besan¸con Cedex FRANCE
Wolfgang Runggaldier
Dipartimento di Matematica Pura et Applicata
Universit´a di Padova
Via Belzoni 7, 35131 Padova ITALY
February 28, 1996
Submitted to
Mathematical Finance
?
The ?nancial support and hospitality of the University of Padua, the Isaac New-
ton Institute, Cambridge University, and the Stockholm School of Economics are
gratefully acknowledged.
1
Abstract
We investigate the term structure of zero coupon bonds when
interest rates are driven by a general marked point process as
well as by a Wiener process. Developing a theory which allows for
measure-valued trading portfolios we study existence and unique-
ness of a martingale measure. We also study completeness and its
relation to the uniqueness of a martingale measure. For the case
of a ?nite jump spectrum we give a fairly general completeness
result and for a Wiener–Poisson model we prove the existence of
a time- independent set of basic bonds. We also give su?cient
conditions for the existence of an a?ne term structure.
Key words: bond market, term structure of interest rates, jump-
di?usion model, measure-valued portfolio, arbitrage, market complete-
ness, martingale operator, hedging operator, a?ne term structure.
1 Introduction
One of the most challenging mathematical problems arising in the theory
of ?nancial markets concerns market completeness, i.e. the possibility of
duplicating a contingent claim by a self-?nancing portfolio. Informally,
such a possibility arises whenever there are as many risky assets available
for hedging as there are independent sources of randomness in the market.
In bond markets as well as in stock markets it seems reasonable to
take into account the possible occurrence of jumps, considering not only
the simple Poisson jump models, but also marked point process models
allowing a continuous jump spectrum. However, introducing a continuous
jump spectrum also introduces a possibly in?nite number of independent
sources of randomness and, as a consequence, completeness may be lost.
In traditional stock market models there are usually only a ?nite
number of basic assets available for hedging, and in order to have com-
pleteness one usually assumes that their prices are driven by a ?nite
number (equaling the number of basic assets) of Wiener processes. More
realistic jump-di?usion models seem to encounter some skepticism pre-
cisely due to the completeness problems mentioned above.
There is, however, a fundamental di?erence between stock and bond
markets: while in stock markets portfolios are naturally limited to a ?nite
number of basic assets, in bond markets there is at least the theoretical
possibility of having portfolios with an in?nite number of assets, namely
bonds with a continuum of possible maturities. Since all modern contin-
uous time models of bond markets assume the existence of bonds with a
2
continuum of maturities, it seems reasonable to require that a coherent
theory of bond markets should allow for portfolios consisting of uncount-
ably many bonds. We also see from the discussion above that, in models
with a continuous jump spectrum, such portfolios are indeed necessary
if completeness is not to be lost.
It is worth noticing that also in stock market models one may con-
sider a continuum of derivative securities, such as e.g. options parame-
terized by maturities and/or strikes.
The purpose of our paper is to present an approach which, on one
hand, allows bond prices to be driven also by marked point processes
while, on the other hand, admitting portfolios with an in?nite number
of securities. As such, this approach appears to be new and leads to the
two mathematical problems of:
• an appropriate modeling of the evolution of bond prices and their
forward rates;
• a correct de?nition of in?nite-dimensional portfolios of bonds and
the corresponding value processes by viewing trading strategies as
measure-valued processes.
A further point of interest in this context is that, in stock markets
and under general assumptions, completeness of the market is equiva-
lent to uniqueness of the martingale measure. The question now arises
whether this fact remains true also in bond markets when marked point
processes with continuous mark spaces, i.e. an in?nite number of sources
of randomness, are allowed? One of the main results of this paper is that,
at this level of generality, uniqueness of the martingale measure implies
only that the set of hedgeable claims is dense in the set of all contin-
gent claims. This phenomenon is not entirely unexpected and has been
observed by di?erent authors (see, e.g., de?nition of quasicompleteness
in [24]); its nature is transparent on the basis of elementary functional
analysis which we rely upon in Section 4.
The main results of the paper are as follows.
• We give conditions for the existence of a martingale measure in
terms of conditions on the coe?cients for the bond- and forward
rate dynamics. In particular we extend the Heath–Jarrow–Morton
“drift condition” to point process models.
• We show that the martingale measure is unique if and only if certain
integral operators of the ?rst kind (the “martingale operators”) are
injective.
3
• We show that a contingent claim can be replicated by a self-?nancing
portfolio if and only if certain integral equations of the ?rst kind
(the “hedging equations”) have solutions. Furthermore, the integral
operators appearing in these equations (the “hedging operators”)
turn out to be adjoint of the martingale operators.
• We show that uniqueness of the martingale measure is equivalent to
the denseness of the image space of the hedging operators. In partic-
ular, it turns out that in the case with a continuous jump spectrum,
uniqueness of the martingale measure does not imply completeness
of the bond market. Instead, uniqueness of the martingale mea-
sure is shown to be equivalent to approximate completeness of the
market.
• Under additional conditions on the forward rate dynamics we can
give a rather explicit characterization of the set of hedgeable claims
in terms of certain Laplace transforms.
• In particular, we study the model with a ?nite mark space (for the
jumps) showing that in this case one may hedge an arbitrary claim
by a portfolio consisting of a ?nite number of bonds, having essen-
tially arbitrary but di?erent maturities. This considerably extends
and clari?es a previous result by Shirakawa [28].
• We give su?cient conditions for the existence of a so-called a?ne
term structure (ATS) for the bond prices.
The paper has the following structure. In Section 2 we lay the foun-
dations and we present a “toolbox” of propositions which explain the
interrelations between the dynamics of the forward rates, the bond prices
and the short rate of interest.
In Section 3 we de?ne our measure-valued portfolios with their value
processes and investigate the existence and uniqueness of a martingale
measure. We also give the martingale dynamics of the various objects,
leading among other things to a HJM-type “drift condition”.
In a stock market, the current state of a portfolio is a vector of
quantities of securities held at time t which can be identi?ed with a linear
functional; it gives the portfolio value being applied to the current asset
price vector. In a bond market, the latter is substituted by a price curve
which one can consider as a vector in a space of continuous functions. By
analogy, it is natural to identify a current state of a portfolio with a linear
functional, i.e. with an element of the dual space, a signed ?nite measure.
So, our approach is based on a kind of stochastic integral with respect
4
to the price curve process though we avoid a more technical discussion
of this aspect here (see [4]).
In Section 4 we study uniqueness of the martingale measure and its
relation to the completeness of the bond market. Section 5 is devoted to
a more detailed study of two cases when we can characterize the set of
hedgeable claims. In 5.1 we consider a class of models with in?nite mark
space which leads us to Laplace transform theory and in 5.2 we explore
the case of a ?nite mark space. We end by discussing the existence of
a?ne term structures in Section 6.
For the case of Wiener-driven interest rates there is an enormous
number of papers. For general information about arbitrage free markets
we refer to the book [13] by Du?e. Basic papers in the area are Harrison–
Kreps [17], Harrison–Pliska [18]. For interest rate theory we recommend
Artzner–Delbaen [1] and some other important references can be found
in the bibliography; the recent book by Dana and Jeanblanc-Picqu´e [10]
contains a comprehensive account of main models.
Very little seems to have been written about interest rate models
driven by point processes. Shirakawa [28], Bj¨ ork [3], and Jarrow–Madan
[23] all consider an interest rate model of the type to be discussed below
for the case when the mark space is ?nite, i.e. when the model is driven
by a ?nite number of counting processes. (Jarrow–Madan also consider
the interplay between the stock- and the bond market). In the present
paper we focus primarily on the case of an in?nite mark space, but the
interest rate models above are included as special cases of our model,
and our results for the ?nite case amount to a considerable extension of
those in[28].
In an interesting preprint, Jarrow–Madan [24] consider a fairly gen-
eral model of asset prices driven by semimartingales. Their mathemati-
cal framework is that of topological vector spaces and, using a concept
of quasicompleteness, they obtain denseness results which are related to
ours.
Babbs and Webber [2] study a model where the short rate is driven by
a ?nite number of counting processes. The counting process intensities are
driven by the short rate itself and by an underlying di?usion-type process.
Lindberg–Orszag–Perraudin [25] consider a model where the short rate
is a Cox process with a squared Ornstein–Uhlenbeck process as intensity
process. Using Karhunen–Lo`eve expansions they obtain quasi-analytic
formulas for bond prices.
Structurally the present paper is based on Bj¨ ork [3] where only the
?nite case is treated. The working paper Bj¨ ork–Kabanov–Runggaldier
5
[5] contains some additional topics not treated here. In particular some
pricing formulas are given, and the change of num´eraire technique de-
veloped by Geman et. al. in [16] is applied to the bond market. In a
forthcoming paper [4] we develop the theory further by studying models
driven by rather general L´evy processes, and this also entails a study of
stochastic integration with respect to C-valued processes. In the present
exposition we want to focus on ?nancial aspects, so we try to avoid, as
far as possible, details and generalizations (even straightforward ones)
if they lead to mathematical sophistications. For the present paper the
main reference concerning point processes and Girsanov transformations
are Br´emaud [7] and Elliott [15]. For the more complicated paper [4], the
excellent (but much more advanced) exposition by Jacod and Shiryaev
[22] is the imperative reference.
Throughout the paper we use the Heath–Jarrow–Morton parameter-
ization, i.e. forward rates and bond prices are parameterized by time of
maturity T. In certain applications it is more convenient to parameterize
forward rates by instead using the time to maturity, as is done in Brace-
Musiela [6]. This can easily be accomplished, since there exists a simple
set of translation formulae between the two ways of parametrization.
2 Relations between df(t, T), dp(t, T), and
dr
t
We consider a ?nancial market model “living” on a stochastic basis (?l-
tered probability space) (?, T, F, P) where F = ¦T
t
¦
t?0
. The basis is
assumed to carry a Wiener process W as well as a marked point process
µ(dt, dx) on a measurable Lusin mark space (E, c) with compensator
?(dt, dx). We assume that ?([0, t] E) < ? P-a.s. for all ?nite t, i.e. µ
is a multivariate point process in the terminology of [22].
The main assets to be considered on the market are zero coupon
bonds with di?erent maturities. We denote the price at time t of a bond
maturing at time T (a “T-bond”) by p(t, T).
Assumption 2.1 We assume that
1. There exists a (frictionless) market for T-bonds for every T > 0.
2. For every ?xed T, the process ¦p(t, T); 0 ? t ? T¦ is an optional
stochastic process with p(t, t) = 1 for all t.
6
3. For every ?xed t, p(t, T) is P-a.s. continuously di?erentiable in the
T-variable. This partial derivative is often denoted by
p
T
(t, T) =
?p(t, T)
?T
.
We now de?ne the various interest rates.
De?nition 2.2 The instantaneous forward rate at T, contracted at t,
is given by
f(t, T) = ?
? log p(t, T)
?T
.
The short rate is de?ned by
r
t
= f(t, t).
The money account process is de?ned by
B
t
= exp
__
t
0
r
s
ds
_
,
i.e.
dB
t
= r
t
B
t
dt, B
0
= 1.
For the rest of the paper we shall, either by implication or by as-
sumption, consider dynamics of the following type.
Short rate dynamics
dr(t) = a
t
dt + b
t
dW
t
+
_
E
q(t, x)µ(dt, dx), (1)
Bond price dynamics
dp(t, T) = p(t, T)m(t, T)dt + p(t, T)v(t, T)dW
t
+ p(t?, T)
_
E
n(t, x, T)µ(dt, dx), (2)
Forward rate dynamics
df(t, T) = ?(t, T)dt + ?(t, T)dW
t
+
_
E
?(t, x, T)µ(dt, dx). (3)
7
In the above formulas the coe?cients are assumed to meet stan-
dard conditions required to guarantee that the various processes are well
de?ned.
We shall now study the formal relations which must hold between
bond prices and interest rates. These relations hold regardless of the
measure under consideration, and in particular we do not assume that
markets are free of arbitrage. We shall, however, need a number of tech-
nical assumptions which we collect below in an “operational” manner.
Assumption 2.3
1. For each ?xed ?, t and, (in appropriate cases) x, all the objects
m(t, T), v(t, T), n(t, x, T), ?(t, T), ?(t, T), and ?(t, x, T) are as-
sumed to be continuously di?erentiable in the T -variable. This
partial T-derivative sometimes is denoted by m
T
(t, T) etc.
2. All processes are assumed to be regular enough to allow us to di?er-
entiate under the integral sign as well as to interchange the order
of integration.
3. For any t the price curves p(?, t, .) are bounded functions for almost
all ?.
This assumption is rather ad hoc and one would, of course, like to
give conditions which imply the desired properties above. This can be
done but at a fairly high price as to technical complexity. As for the
point process integrals, these are made trajectorywise, so the standard
Fubini theorem can be applied. For the stochastic Fubini theorem for the
interchange of integration with respect to dW and dt see Protter [26] and
also Heath–Jarrow–Morton [19] for a ?nancial application.
Proposition 2.4
1. If p(t, T) satis?es (2), then for the forward rate dynamics we have
df(t, T) = ?(t, T)dt + ?(t, T)dW
t
+
_
E
?(t, x, T)µ(dt, dx),
where ?, ? and ? are given by
_
¸
_
¸
_
?(t, T) = v
T
(t, T) v(t, T) ?m
T
(t, T),
?(t, T) = ?v
T
(t, T),
?(t, x, T) = ?n
T
(t, x, T) [1 +n(t, x, T)]
?1
.
(4)
8
2. If f(t, T) satis?es (3) then the short rate satis?es
dr
t
= a
t
dt + b
t
dW
t
+
_
E
q(t, x)µ(dt, dx),
where
_
¸
_
¸
_
a
t
= f
T
(t, t) +?(t, t),
b
t
= ?(t, t),
q(t, x) = ?(t, x, t).
(5)
3. If f(t, T) satis?es (3) then p(t, T) satis?es
dp(t, T) = p(t, T)
_
r
t
+ A(t, T) +
1
2
S
2
(t, T)dt
_
+ p(t, T)S(t, T)dW
t
+ p(t?, T)
_
E
_
e
D(t,x,T)
?1
_
µ(dt, dx),
where
_
¸
_
¸
_
A(t, T) = ?
_
T
t
?(t, s)ds,
S(t, T) = ?
_
T
t
?(t, s)ds,
D(t, x, T) = ?
_
T
t
?(t, x, s)ds.
(6)
Proof. The ?rst part of the Proposition follows immediately if we apply
the Itˆ o formula to the process log p(t, T), write this in integrated form
and di?erentiate with respect to T.
For the second part we integrate the forward rate dynamics to get
r
t
= f(0, t) +
_
t
0
?(s, t)ds +
_
t
0
?(s, t)dW
s
(7)
+
_
t
0
_
E
?(s, x, t)µ(ds, dx).
Now we can write
?(s, t) = ?(s, s) +
_
t
s
?
T
(s, u)du,
?(s, t) = ?(s, s) +
_
t
s
?
T
(s, u)du,
?(s, x, t) = ?(s, x, s) +
_
t
s
?
T
(s, x, u)du,
and, inserting this into (7) we have
r
t
= f(0, t) +
_
t
0
?(s, s)ds +
_
t
0
_
t
s
?
T
(s, u)duds
+
_
t
0
?(s, s)dW
s
+
_
t
0
_
t
s
?
T
(s, u)dudW
s
+
_
t
0
_
E
?(s, x, s)µ(ds, dx) +
_
t
0
_
E
_
t
s
?
T
(s, x, u)duµ(ds, dx).
9
Changing the order of integration and identifying terms gives us the
result.
For the third part we adapt a technique from Heath–Jarrow–Morton
[19]. Using the de?nition of the forward rates we may write
p(t, T) = exp ¦Z(t, T)¦ (8)
where Z is given by
Z(t, T) = ?
_
T
t
f(t, s)ds. (9)
Writing (3) in integrated form, we obtain
f(t, s) = f(0, s)+
_
t
0
?(u, s)du+
_
t
0
?(u, s)dW
u
+
_
t
0
_
E
?(u, x, s)µ(du, dx).
Inserting this expression into (9), splitting the integrals and changing the
order of integration gives us
Z(t, T) = ?
_
T
t
f(0, s)ds ?
_
t
0
_
T
t
?(u, s)dsdu ?
_
t
0
_
T
t
?(u, s)dsdW
u
?
_
t
0
_
T
t
_
E
?(u, x, s)dsµ(du, dx)
= ?
_
T
0
f(0, s)ds ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx)
+
_
t
0
f(0, s)ds +
_
t
0
_
t
u
?(u, s)dsdu +
_
t
0
_
t
u
?(u, s)dsdW
u
+
_
t
0
_
t
u
_
E
?(u, x, s)dsµ(du, dx)
= Z(0, T) ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx)
+
_
t
0
f(0, s)ds +
_
t
0
_
s
0
?(u, s)duds +
_
t
0
_
s
0
?(u, s)dW
u
ds
+
_
t
0
_
s
0
_
E
?(u, x, s)µ(du, dx)ds.
Now we can use the fact that r
s
= f(s, s) and, integrating the forward
rate dynamics (3) over the interval [0, s], we see that the last two lines
10
above equal
_
t
0
r
s
ds so we ?nally obtain
Z(t, T) = Z(0, T) +
_
t
0
r
s
ds ?
_
t
0
_
T
u
?(u, s)dsdu ?
_
t
0
_
T
u
?(u, s)dsdW
u
?
_
t
0
_
T
u
_
E
?(u, x, s)dsµ(du, dx).
Thus, with A, S and D as in the statement of the proposition, the sto-
chastic di?erential of Z is given by
dZ(t, T) = ¦r
t
+ A(t, T)¦ dt + S(t, T)dW
t
+
_
E
D(t, x, T)µ(dt, dx),
and an application of the Itˆ o formula to the process p(t, T) = exp ¦Z(t, T)¦
completes the proof.
Remark 2.5 To ?t reality, a “good” model of bond price dynamics or in-
terest rates must satisfy other important conditions. A bond price process
“should” e.g. take values in the interval [0, 1] and forward rates “ought”
to be positive (see [27]). We do not restrict ourselves to the class of “re-
alistic models” (obviously the most important ones) since we also want
to treat generalizations of “bad” models (like the various Gaussian mod-
els for the short rate) which are useful because their simplicity leads to
instructive explicit formulas.
3 Absence of arbitrage
3.1 Generalities
The purpose of this section is to give the appropriate de?nitions of self-
?nancing measure-valued portfolios, contingent claims, arbitrage possi-
bilities and martingale measures. We then proceed to show that the ex-
istence of a martingale measure implies absence of arbitrage, and we
end the section by investigating existence and uniqueness of martingale
measures.
We make the following standing assumption for the rest of the sec-
tion.
Assumption 3.1 We assume that
(i) There exists an asset (usually referred to as locally risk-free) with the
price process
B
t
= exp
__
t
0
r
s
ds
_
.
11
(ii) The ?ltration F = (T
t
) is the natural ?ltration generated by W and
µ, i.e.
T
t
= ?¦W
s
, µ([0, s] A), B; 0 ? s ? t, A ? c, B ? ^¦
where ^ is the collections of P-null sets from T.
(iii) The point process µ has an intensity ?, i.e. the P-compensator ?
has the form
?(dt, dx) = ?(t, dx)dt
where ?(t, A) is a predictable process for all A ? c.
(iv) The stochastic basis has the predictable representation property: any
local martingale M is of the form
M
t
= M
0
+
_
t
0
f
s
dW
s
+
_
t
0
_
E
?(s, x)(µ(ds, dx) ??(ds, dx))
where f is a process measurable with respect to the predictable ?-
algebra T and ? is a
˜
T-measurable function (
˜
T = T ?c) such that
for all ?nite t
_
t
0
[f
s
[
2
ds < ?,
_
t
0
_
E
[?(s, x)[?(ds, dx) < ?.
We need (ii) and (iv) above in order to have control over the class of
absolute continuous measure transformations of the basic (“objective”)
probability measure P. These assumptions are made largely for conve-
nience, but if we omit them, some of the equivalences proved below will
be weakened to one-side implications. See [4] for further information. The
assumption (iii) is not really needed at all from a logical point of view,
but it makes some of the formulas below much easier to read.
3.2 Self-?nancing portfolios
De?nition 3.2 A portfolio in the bond market is a pair ¦g
t
, h
t
(dT)¦,
where
1. The component g is a predictable process.
2. For each ?, t, the set function h
t
(?, ) is a signed ?nite Borel mea-
sure on [t, ?).
3. For each Borel set A the process h
t
(A) is predictable.
12
The intuitive interpretation of the above de?nition is that g
t
is the
number of units of the risk-free asset held in the portfolio at time t. The
object h
t
(dT) is interpreted as the “number” of bonds, with maturities
in the interval [T, T + dT], held at time t.
We will now give the de?nition of an admissible portfolio.
De?nition 3.3
1. The discounted bond prices Z(t, T) are de?ned by
Z(t, T) =
p(t, T)
B(t)
.
2. A portfolio ¦g, h¦ is said to be feasible if the following conditions
hold for every t:
_
t
0
[g
s
[ds < ?, (10)
_
t
0
_
?
s
[m(s, T)[[h
s
(dT)[ds < ?, (11)
_
t
0
_
E
_
?
s
[n(s, x, T)[[h
s
(dT)[?(ds, dx) < ?, (12)
_
t
0
__
?
s
[v(s, T)[[h
s
(dT)[
_
2
ds < ?. (13)
3. The value process corresponding to a feasible portfolio ¦g, h¦ is
de?ned by
V
t
= g
t
B
t
+
_
?
t
p(t, T)h
t
(dT). (14)
4. The discounted value process is
V
Z
t
= B
?1
t
V
t
. (15)
5. A feasible portfolio is said to be admissible if there is a number
a ? 0 such that V
Z
t
? ?a P ?a.s. for all t.
6. A feasible portfolio is said to be self-?nancing if the corresponding
value process satis?es
V
t
= V
0
+
_
t
0
g
s
dB
s
+
_
t
0
_
?
s
m(s, T)p(s, T)h
s
(dT)ds
+
_
t
0
_
?
s
v(s, T)p(s, T)h
s
(dT)dW
s
(16)
+
_
t
0
_
?
s
_
E
n(s, x, T)p(s?, T)h
s
(dT)µ(ds, dx).
13
There are obvious modi?cations of these de?nitions like “admissible
on the interval [0, T
0
]”.
The relation (16) is a way of making mathematical sense out of the
expression
dV
t
= g
t
dB
t
+
_
?
t
h
t
(dT)dp(t, T) (17)
which is the formal generalization of the standard self-?nancing condi-
tion. We shall sometimes use equation (17) as a shorthand notation for
the equation (16). It seems natural that the adequate stochastic calculus
for the theory of bond market has to include an integration of measure-
valued processes with respect to jump-di?usion processes with values in
some Banach space of continuous functions. Some versions of such a cal-
culus are given in our paper [4].
We shall as usual be working much with discounted prices, and the
following lemma shows that the self-?nancing condition is the same for
the discounted bond prices Z(t, T) as for the undiscounted ones.
Lemma 3.4 For an admissible portfolio the following conditions are equiv-
alent.
(i) dV
t
= g
t
dB
t
+
_
?
t
h
t
(dT)dp(t, T),
(ii) dV
Z
t
=
_
?
t
h
t
(dT)dZ(t, T).
Proof. The Itˆo formula.
Notice that for a self-?nancing portfolio the g-component is auto-
matically de?ned by the initial endowment V
0
and the h-component; the
pair (V
0
, h) is sometimes called the investment strategy of a self-?nancing
portfolio.
For technical purposes it is sometimes convenient to extend the de-
?nition of the bond price process p(t, T) (as well as other processes)
from the interval [0, T] to the whole half-line. It is then natural to put
Z(t, T) = 1, A(t, T) = 0 etc. for t ? T, i.e. one can think that after the
time of maturity the money is transferred to the bank account.
Remark 3.5 From the point of view of economics, discounting means
that the locally risk-free asset is chosen as the “num´eraire”, i.e. the
prices of all other assets are evaluated in the units of this selected one.
Some mathematical properties may however change under a change of
the num´eraire, see [11].
14
We now go on to de?ne contingent claims and arbitrage portfolios,
modifying somewhat the standard concepts.
De?nition 3.6
1. A contingent T-claim is a random variable X ? L
0
+
(T
T
, P)
(i.e. an arbitrary non-negative T
T
-measurable random variable).
We shall use the notation L
0
++
(T
T
, P) for the set of elements X of
L
0
+
(T
T
, P) with P(X > 0) > 0.
2. An arbitrage portfolio is an admissible self-?nancing portfolio
¦g, h¦ such that the corresponding value process has the properties
(a) V
0
= 0,
(b) V
T
? L
0
++
(T
T
, P).
If no arbitrage portfolios exist for any T ? R
+
we say that the
model is “free of arbitrage” or “arbitrage-free” (AF).
We now want to tie absence of arbitrage to the existence of a martin-
gale measures. Since we do not ?x a (?nite deterministic) time horizon,
it turns out to be convenient to consider a martingale density process as
a basic object (rather than a martingale measure).
De?nition 3.7 Take the measure P as given. We say that a positive
martingale L = (L
t
)
t?0
with E
P
[L
t
] = 1 is a martingale density if for
every T > 0 the process ¦Z(t, T)L
t
; 0 ? t ? T¦ is a P-local martingale.
If, moreover, L
t
> 0 for all t ? R
+
we say that L is a strict martingale
density.
De?nition 3.8 We say that a probability Q on (?, T) is a martingale
measure if Q
t
? P
t
(where Q
t
= Q[T
t
, P
t
= P[T
t
) and the process
¦Z(t, T); 0 ? t ? T¦ is a Q-local martingale for every T > 0 .
In other words, Q is a martingale measure if it is locally equivalent
to P and the density process dQ
t
/dP
t
is a strict martingale density.
Proposition 3.9 Suppose that there exists a strict martingale density
L. Then the model is arbitrage-free.
15
Proof. Fix any admissible self-?nancing portfolio ¦g, h¦ and assume
that for some ?nite T the corresponding value process is such that V
T
?
L
0
++
(T
T
, P). By admissibility, V
Z
? ?a for some a > 0. The process
(V
Z
+a)L is a positive local martingale hence a supermartingale. As L is a
martingale, V
Z
L is a supermartingale. Thus, E
P
_
V
Z
0
L
0
_
? E
P
_
V
Z
T
L
T
_
>
0, which is impossible because we assume that V
Z
0
= 0.
Remark 3.10 Notice that for the model restricted to some ?nite time
horizon T, a strict martingale density de?nes an equivalent martin-
gale measure Q
T
= L
T
P, i.e. a probability which is equivalent to P on
T
T
(in symbols: Q
T
T
? P
T
) such that all discounted bond prices are mar-
tingales on [0, T]. If E
P
[L
?
] = 1, there exists an equivalent martingale
measure also for the in?nite horizon and the above proposition can be
easily extended to this case in an obvious way. In general, when L is not
uniformly integrable, a measure Q on T such that L
t
= dQ
t
/dP
t
, may
not exist. The following simple example when a martingale density does
not de?ne Q explains the situation.
Let the stochastic basis be the coordinate space of counting functions
N = (N
t
) equipped with the measure of the unit rate Poisson process. Let
us modify this space by excluding only one point: the function which is
identically zero. It is clear that the process L
t
= I
{Nt=0}
e
t
is a martingale
density de?ning Q
T
for every ?nite T (under Q
T
the coordinate process
has the intensity zero on I
[T,?]
) but the measure Q such that Q[T
T
=
Q
T
[T
T
for all T does not exist.
This example reveals that the origin of such an undesirable property
lies in a certain pathology of the stochastic basis while Proposition 3.9
shows that one can work with a strict martingale density without any
reference to the martingale measure. Facing the choice between an in-
signi?cant supplementary requirement and a perspective to be far away
from the traditional language we prefer the ?rst option. So we impose
Assumption 3.11 For any positive martingale L = (L
t
) with E
P
[L
t
] =
1 there exists a probability measure Q on T such that L
t
= dQ
t
/dP
t
.
Remark 3.12 In numerous papers devoted to the term structure of in-
terest rates one can observe a rather confusing terminology : the model
is said to be arbitrage-free if there exists a martingale measure. The ori-
gin of this striking di?erence with the theory of stock markets (where
arbitrage means the possibility to get a pro?t which in some sense is
riskless) is clear, because in continuous-time bond market models there
16
are uncountably many basic securities and the key question is : what are
portfolios of bonds ? The discussion of the latter problem is avoided since
the straightforward use of ?nite-dimensional stochastic integrals does not
allow to de?ne a general portfolio in a correct way (see the apparent di?-
culties with the basic bonds in [28]). Interesting mathematical problems
concerning relations between di?erent de?nitions of arbitrage are almost
untouched in the theory of bond markets; this subject is beyond the scope
of the present paper as well.
3.3 Existence of martingale measures
Suppose that the bond prices and forward rates have P-dynamics given
by the equations (2) and (3). We now ask how various coe?cients in these
equations must be related in order to ensure the existence of a martin-
gale measure (or, in view of the Assumption 3.11, of a strict martingale
density). The main technical tool is, as usual, a suitable version of the
Girsanov theorem, which we now recall. The ?rst (direct) part (I) below
holds true regardless of how large the ?ltration is chosen to be, but the
converse part (II) depends heavily on the fact that we have assumed the
predictable representation property.
Theorem 3.13 (Girsanov)
I. Let ? be a predictable process and ? = ?(?, t, x) be a strictly positive
˜
T-measurable function such that for ?nite t
_
t
0
[?
s
[
2
ds < ?,
_
t
0
_
E
[?(s, x)[?(s, dx)ds < ?.
De?ne the process L by
log L
t
=
_
t
0
?
s
dW
s
?
1
2
_
t
0
[?
s
[
2
dW
s
+
_
t
0
_
E
log ?(s, x)µ(ds, dx) +
_
t
0
_
E
(1 ??(s, x))?(ds, dx), (18)
or, equivalently, by
dL
t
= L
t
?
t
dW
t
+ L
t?
_
E
(?(t, x) ?1) ¦µ(dt, dx) ??(dt, dx)¦ , L
0
= 1,
(19)
and suppose that for all ?nite t
E
P
[L
t
] = 1. (20)
17
Then there exists a probability measure Q on T locally equivalent to P
with
dQ
t
= L
t
dP
t
(21)
such that:
(i) We have
dW
t
= ?
t
dt + d
˜
W
t
, (22)
where
˜
W is a Q-Wiener process.
(ii) The point process µ has a Q-intensity, given by
?
Q
(t, dx) = ?(t, x)?(t, dx). (23)
II. Every probability measure Q locally equivalent to P has the structure
above.
We now come to the main results concerning the existence of a mar-
tingale measure. They generalize the corresponding results of Heath–
Jarrow–Morton and can be easily extended to the case of a multidimen-
sional Wiener process. The identities between processes are understood
dPdt-a.e.
Theorem 3.14
I. Let the bond price dynamics be given by (2). Assume that n(t, x, T) for
any ?xed T is bounded by a constant (depending on T). Then there exists
a martingale measure Q if and only if the following conditions hold:
(i) There exists a predictable process ? and a
˜
T-measurable function
?(t, x) with ? > 0 satisfying the integrability conditions of Theorem
3.13 and such that E
P
[L
t
] = 1 for all ?nite t, where L is de?ned
by (19).
(ii) For all T > 0 on [0, T] we have
m(t, T) + ?
t
v(t, T) +
_
E
?(t, x)n(t, x, T)?(t, dx) = r
t
. (24)
II. Let the forward rate dynamics be given by (3). Assume that e
D(t,x,T)
for any ?xed T is bounded by a constant (depending on T). Then there
exists a martingale measure if and only if the following conditions hold:
18
(iii) There exist a predictable process ? and a
˜
T-measurable function
?(t, x) with ? > 0 satisfying the integrability conditions of Theorem
3.13 and such that E
P
[L
t
] = 1 for all ?nite t where L is de?ned by
(19).
(iv) For all T > 0, on [0, T] we have
A(t, T) +
1
2
S
2
(t, T) + ?
t
S(t, T) +
_
E
?(t, x)?(t, dx) = 0, (25)
where
?(t, dx) =
_
e
D(t,x,T)
?1
_
?(t, dx)
and A, S and D are de?ned as in (6).
Proof.
I. First of all it is easy to see (using the Itˆ o formula) that a measure Q
is a martingale measure if and only if the bond dynamics under Q are of
the form
dp(t, T) = r
t
p(t, T)dt + dM
Q
t
, (26)
where M
Q
is a Q-local martingale. Using the Girsanov theorem we see
that under any equivalent measure Q, the bond dynamics have the fol-
lowing form, where we have compensated µ under Q.
dp(t, T) = p(t, T)m(t, T)dt + p(t, T)v(t, T)(?
t
dt + d
˜
W
t
)
+ p(t?, T)
_
E
n(t, x, T)?(t, x)?(t, dx)dt
+ p(t?, T)
_
E
n(t, x, T) ¦µ(dt, dx) ??(t, x)?(t, dx)dt¦ .
Thus we have
dp(t, T) = p(t, T)
_
m(t, T) +v(t, T)?
t
_
E
n(t, x, T)?(t, x)?(t, dx)
_
dt +
+ dM
Q
t
.
Comparing this with the equation (26) gives the result.
II. If the forward rate dynamics are given by (3) then the corresponding
bond price dynamics are given by Proposition 2.4. We can then apply
part 1 of the present theorem.
We now turn to the issue of so called “martingale modelling”, and re-
mark that one of the main morals of the martingale approach to arbitrage-
free pricing of derivative securities can be formulated as follows.
19
• The dynamics of prices and interest rates under the objective prob-
ability measure P are, to a high degree, irrelevant. The important
objects to study are the dynamics of prices and interest rates under
the martingale measure Q.
When building a model it is thus natural, and in most cases ex-
tremely time saving, to specify all objects directly under a martingale
measure Q. This will of course impose restrictions on the various para-
meters in, e.g., the forward rate equations, and the main results are as
follows.
Proposition 3.15 Assume that we specify the forward rate dynamics
under a martingale measure Q by
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
_
E
?(t, x, T)µ(dt, dx). (27)
Then the following relation holds
?(t, T) = ?(t, T)
_
T
t
?(t, s)ds ?
_
E
?(t, x, T)e
D(t,x,T)
?
Q
(t, dx). (28)
Furthermore, the bond price dynamics under Q are given by
dp(t, T) = p(t, T)r
t
dt + p(t, T)S(t, T)d
˜
W
t
+ p(t?, T)
_
E
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx), (29)
where ˜ µ is the Q-compensated point process
˜ µ(dt, dx) = µ(dt, dx) ??
Q
(t, dx)dt.
Here ?
Q
is the Q-intensity of µ whereas D and S are de?ned by (6).
Proof. Since we are working under Q we may use Theorem 3.14 with
? = 0 and ? = 1 to obtain
A(t, T) +
1
2
S
2
(t, T) +
_
E
_
e
D(t,x,T)
?1
_
?
Q
(t, dx) = 0,
and di?erentiating this equation with respect to T gives us the equation
(28). The result on bond prices now follows immediately from the result
above and from Proposition 2.4.
The single most important formula in this section is the relation (28)
which is the point process extension of the Heath–Jarrow–Morton “drift
20
condition”. We see that if we want to model the forward rates directly
under the martingale measure Q, then the drift ? is uniquely determined
by the di?usion volatility ?, the jump volatility ? and by the Q-intensity
?
Q
. This has important implications when it comes to parameter estima-
tion, since we are modelling under Q while our concrete observations, of
course, are made under an objective measure P. As far as volatilities are
concerned they do not change under an equivalent measure transforma-
tion, so “in principle” we can determine ? and ? from actual observations
of the forward rate trajectories. The intensity measure however presents
a totally di?erent problem. Suppose for simplicity that µ is a standard
Poisson process (under Q) with Q-intensity ?
Q
. If we could observe the
forward rates under Q then we would, of course, have access to a vast ma-
chinery of statistical estimation theory for the determination of a point
estimate of ?
Q
, but the problem here is that we are not making observa-
tions under Q, but under P. Thus the estimation of the Q-intensity ?
Q
is not a statistical estimation problem to be solved with standard sta-
tistical techniques. This fact may be regarded as a piece of bad news or
as an interesting problem. We opt for the latter interpretation, and one
obvious way out is to estimate ?
Q
by using market data for bond prices
(which contain implicit information concerning ?
Q
).
4 Uniqueness of Q and market complete-
ness
4.1 Uniqueness of the martingale measure
Throughout this section we shall work with a model speci?ed by the
forward rate dynamics under
Assumption 4.1 The coe?cient D(t, x, T) is uniformly bounded.
The main issue to be dealt with below is the relation between unique-
ness of the martingale measure and completeness of the bond market.
Using Theorem 3.14 we immediately have the following result.
Proposition 4.2 Let the forward rate dynamics be given by (3) and as-
sume that the assumptions (iii) and (iv) of Theorem 3.14 (equivalent to
existence of a martingale measure Q) are satis?ed. Then the martingale
measure Q is unique if and only if dPdt-a.e.
Ker /
t
(?) = 0 (30)
21
where the linear operator
/
t
(?) : R L
2
(E, c, ?(?, t, dx)) ?C[0, ?[ (31)
is de?ned by
/
t
(?) : (?, ?) ?S(?, t, .)? +
_
E
?(x)?(?, t, dx, T) (32)
with
?(?, t, dx, T) =
_
e
D(?,t,x,T)
?1
_
?(?, t, dx).
The important thing to note here is that the operators /
t
(?) are
integral operators of the ?rst kind. We shall refer to / as “the martingale
operators”.
Corollary 4.3 Suppose that the forward rate dynamics is given by (3),
that the model coe?cients ?(t, T), ?(t, T), ?(t, x, T), and ?(t, dx) are
deterministic and that the martingale measure Q is unique. Then the
Girsanov transformation parameters ? and ? are deterministic functions,
i.e. under Q the process
˜
W is a Wiener process with constant drift, and
µ is a Poisson measure.
Proof. It is su?cient to notice that the operators /
t
do not depend of ?
and hence (outside the exclusive dPdt-null sets) values of the Girsanov
transformation parameters corresponding to a ?xed t but di?erent ? must
satisfy the same equation (25), which has a unique solutions because of
(30).
Corollary 4.4 If we add to the hypotheses of Corollary 4.3 the assump-
tion that ?(t, T) = ?(T ?t), ?(t, T) = ?(T ?t), ?(t, x, T) = ?(T ?t, x),
and ?(t, dx) = ?(dx) then ? and ? do not depend on t, i.e. under Q the
process
˜
W is a Wiener process with a constant drift and µ is a Poisson
measure invariant under time translations.
Of course, the above assertions are almost trivial but they can be
considered as an overture to a more systematic use of classical functional
analysis, which appears to be an adequate tool in the considered set-
ting. Clearly, instead of considering families of operators as we do, one
can chose slightly di?erent de?nitions and e.g. consider a single operator
acting from one space of random processes to another.
22
In spite of the simplicity of the de?nition (31)-(32), it has a drawback
because it uses C[0, ?[ with its associated complicated dual. In order
to be able to work with a more manageable dual space it is therefore
natural to modify the de?nition of the martingale operators and impose
the following constraint on the model:
Assumption 4.5 For any t, x
lim
T??
Z(t?, T)S(t, T) = 0, lim
T??
Z(t?, T)
_
e
D(t,x,T)
?1
_
= 0
where D and S are given by (6), and Z is the discounted price process.
Let C
0
[0, ?[ be the space of continuous functions on [0, ?[ converg-
ing to zero at in?nity. Notice that here we have the well known duality
C
0
[0, ?[= /[0, ?[, where /[0, ?[ is the space of measures on [0, ?[.
The formula
/
Z
t
(?) : (?, ?) ?Z(?, t?, .)S(?, t, .)? +Z(?, t?, .)
_
E
?(x)?(?, t, dx, .)
de?nes a linear operator
/
Z
t
(?) : R L
2
(E, c, ?(?, t, dx)) ?C
0
[0, ?[.
In other words, /
Z
t
(?) is the product of the operator /
t
(?) and the
operator of multiplication by the function Z(?, t?, .) and one may write
/
Z
t
= Z
t
/
t
. Clearly, the above results hold also with / substituted by
/
Z
but the modi?ed de?nition leads to some nice duality arising in the
context of market completeness.
As an alternative, to avoid problems with the dual space, one can
suppose that there is a ?nite time horizon T
f
and all traded bonds have
maturities T ? T
f
. In this case, in section 5.1 below, we have to restrict
ourselves to measures G
t
(dT) with support in [T, T
f
].
4.2 Completeness
Let the forward rate dynamics under a martingale measure Q be given
by
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
_
E
?(t, x, T)µ(dt, dx), (33)
where
˜
W is a Q-Wiener process and µ has the Q-intensity ?
Q
. Our aim
is now to investigate the possibility of hedging contingent claims.
23
De?nition 4.6 Consider a contingent claim X ? L
?
(T
T
0
) expressed in
terms of the num´eraire. We say that it can be replicated or that we
can hedge against X if there exists a self-?nancing portfolio with the
bounded, discounted value process V
Z
such that
V
Z
T
0
= X. (34)
If every such X ? L
?
(T
T
0
) (for every T
0
) can be replicated, the
model is said to be complete.
If for every such X ? L
?
(T
T
0
) there exists a sequence of uniformly
bounded hedgeable claims converging to X in probability, we say that the
model is approximately complete.
It is important to notice that the spaces L
?
and L
0
are invariant
under an equivalent change of probability measures (recall also that con-
vergence in probability can be expressed in terms of convergence a.s.
along a subsequence).
Suppose now that we want to ?nd a self-?nancing portfolio ¦g, h¦
which replicates X ? L
?
+
(T
T
0
). Using Lemma 3.4 we see that the problem
is reduced to ?nding a portfolio strategy with an initial endowment V
Z
0
and a bond investment process h such that
dV
Z
t
=
_
?
t
h
t
(dT)dZ(t, T), (35)
V
Z
T
0
= X, (36)
Proposition 3.15 gives us the Q-dynamics of the bond prices and a simple
calculation shows that for Z we have the dynamics
dZ(t, T) = Z(t, T)S(t, T)d
˜
W
t
+ Z(t?, T)
_
E
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx),
(37)
where S and D are de?ned as usual. We are thus looking for a pair
_
V
Z
0
, h
_
such that
V
Z
T
0
= X, (38)
dV
Z
t
=
_
?
t
h(t, dT)Z(t, T)S(t, T)d
˜
W
t
+
_
E
_
?
t
h(t, dT)Z(t?, T)
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx), (39)
with the integrability conditions
_
T
0
0
__
?
s
[h(s, dT)[ [Z(s, T)S(s, T)[
_
2
ds < ?, (40)
24
_
T
0
0
_
E
_
?
s
[h(s, dT)[ [Z(s, T)
_
e
D(t,x,T)
?1
_
[?(ds, dx) < ?. (41)
Now, since X ? L
?
+
(T
T
0
) the process
M
t
= E
Q
[X[T
t
] (42)
is a Q-martingale. By the assumptions it has an integral representation,
that is there are ? and ? such that
dM
t
= ?
t
d
˜
W
t
+
_
E
?(t, x)˜ µ(dt, dx), (43)
with
E
Q
_
_
T
0
0
?
2
t
dt
_
< ?
and
E
Q
_
_
T
0
0
_
E
?
2
(t, x)d?(dt, dx)
_
< ?.
Now we may formulate our ?rst proposition concerning hedging.
Proposition 4.7 We can replicate a claim X ? L
?
+
(T
T
0
) if and only if
there exists a predictable measure-valued process h(t, dT) which satis?es
the integrability conditions (40) and (41) and solves on [0, T
0
] (dPdt-a.e.)
the equations
/
Z
t
h =
_
?
t
?(t, .)
_
(44)
where ? and ? are de?ned as above and where the “hedging operators”
/
Z
t
(acting on measures) are de?ned by
/
Z
t
(?) : m ?
_
¸
¸
_
_
?
t
Z(?, t?, T)S(?, t, T)m(dT)
_
?
t
Z(?, t?, T)
_
e
D(?,t,.,T)
?1
_
m(dT)
_
¸
¸
_
. (45)
Proof. Su?ciency. Assume that h(t, dT) is solution of (44). Then we
have
dM
t
=
_
?
t
h(t, dT)Z(t, T)S(t, T)d
˜
W
t
+
_
E
_
?
t
h(t, dT)Z(t?, T)
_
e
D(t,x,T)
?1
_
˜ µ(dt, dx). (46)
Now we de?ne g by
g
t
= M
t
?
_
?
t
h(t, dT)Z(t, T). (47)
25
We see from (47) that the value process corresponding to the portfolio
¦g, h¦ is given by V
Z
t
= M
t
. Furthermore, it follows from (46) that the
portfolio is self-?nancing. Finally we see from the de?nition of M that
V
Z
T
0
= M
T
0
= X. Thus we have found a hedge against X and su?ciency
is proved.
Necessity. The discounted value process V
Z
of a hedging portfolio ¦g, h¦
is a bounded martingale with V
Z
T
0
= X. Thus V
Z
is indistinguishable
from M given by (42), and the uniqueness considerations yield (44).
Note that the hedging operators
/
Z
t
: /[0, ?[?R L
2
(E, c, ?
Q
(t, dx)) (48)
are indeed the adjoint of the martingale operators /
Z
t
.
To sum up we have the following conclusions.
Proposition 4.8
1. The martingale measure is unique if and only if the mappings /
Z
are injective (a.e.).
2. The market is complete if and only if the mappings /
Z?
are surjec-
tive (a.e.).
The proof of a natural extension of the second assertion which we
give below involves a measurable selection technique.
Proposition 4.9 The following conditions are equivalent.
(i) The market is approximately complete.
(ii) cl (Im/
Z?
t
(?)) = R L
2
(E, c, ?
Q
(?, t, dx)) (a.e.).
Proof. (i) ?(ii) Let X be a bounded discounted contingent T
0
-claim to
be approximated. Using truncation arguments we can suppose, without
loss of generality, that X is such that ? and ? in the representation (43)
are bounded. For ? > 0 put
F
?
(t, m) = [/
Z,1
t
(m) ??
t
[
2
+|/
Z,2
t
(m) ??(t, .)|
2
L
2
(?
Q
(t,dx))
where we use superscripts to denote the ?rst and the second “coordi-
nates” in (45). Let us consider on /[0, ?[ the ?-algebra ¼ generated
26
by the weak topology (more precisely, by ?(/[0, ?[, C
0
[0, ?[)). Recall
that balls in /[0, ?[ are metrizable compacts, hence (/[0, ?[, ¼) is
a Lusin space as a countable union of Polish spaces. The function F
?
,
being T-measurable in (?, t) and continuous in m, is jointly measurable.
Therefore, the set-valued mapping
(?, t) ?¦m ? /[0, ?[: F
?
(?, t, m) ? ?¦
has a T ?¼-measurable graph and, hence, admits a T-measurable a.e.-
selector m
?
(t, dT) (see, e.g. [12]), which “almost” solves the problem: the
terminal values of the processes, de?ned by I
[0,t]
m
?
(t, dT) and the initial
endowment E
Q
[X], converge to X in L
2
(T
T
0
, Q), hence in probability,
as ? ? 0. One can notice, however, that the construction is not accom-
plished since the strategy generates a value process which is not bounded
(and even admissible). A standard truncation and localization arguments
?nally lead to the desired goal.
(i) ?(ii) Assume that the market is approximately complete. Then
there exists a countable set H = ¦X
n
¦ of bounded hedgeable random
variables, dense in the Hilbert space L
2
(T
T
0
, Q) and closed under linear
combinations with rational coe?cients; let g
n
= (?
n
, ?
n
) be the pair of
functions in the integral representation of X
n
given by (43). Without
loss of generality, we may assume that for all (?, t) one has |g
n
|
?,t
< ?
where |.|
?,t
and (., .)
?,t
are, respectively, the norm and the scalar product
in R L
2
(E, c, ?
Q
(?, t, dx)). Let us denote by H
?,t
the closure in this
norm of the set ¦g
n
(?, t)¦, which is, evidently, a linear subspace, and by
H
?
?,t
its orthogonal complement.
It is easy to show that there exists a pair of functions g = (?, ?) such
that ? is T-measurable, ? is
˜
T-measurable, and |g|
?,t
= 1 if H
?
?,t
,= 0.
Indeed, let ¦I(j)¦ be a sequence of indicator functions generating c and
k(?, t) = inf
_
j : inf
n
|I(j, .) ??
n
(?, t, .)|
L
2
(?
Q
(?,t,dx))
> 0
_
.
Put ˜ ?
t
(?) = I
{sup
n
|?
n
t
(?)|>0}
, ˜ ?(?, t, x) = I(k(?, t), x) if k(?, t) < ? and
˜ ?(?, t) = 0 otherwise. The pair of functions ˜ g = (˜ ?, ˜ ?) meets the neces-
sary measurability requirements. Furthermore, there is ˜ g
?
= (˜ ?
?
, ˜ ?
?
)
which is measurable in the same way and such that all the sections
˜ g
?
(?, t) are representatives of the projections of ˜ g(?, t) onto H
?,t
(one
can orthogonalize ¦g
n
(?, t)¦ preserving measurability and notice that in
this case the Fourier coe?cients are obviously predictable). Normalizing
the di?erence ˜ g ? ˜ g
?
we get g with the required properties.
The pair g = (?, ?) de?nes by (43) with M
0
= 0, a random variable
M
T
= X ? L
2
(T
T
0
, Q), orthogonal, by construction, to all X
n
. If (ii) does
not hold then X is nontrivial; this leads to an apparent contradiction.
27
By experience from the theory of ?nancial markets with ?nitely many
assets one could expect that the market is complete if and only if the
martingale measure is unique, but in our in?nite dimensional setting this
is no longer true. Due to the duality relation (Ker /)
?
= cl (Im/
) we
obtain instead from the above assertions
Theorem 4.10 The following statements are equivalent:
(i) The martingale measure is unique.
(ii) The market is approximately complete.
For a model with a ?nite mark space E, where the hedging problem
is reduced to a ?nite dimensional system of equations (for each (?, t)),
the duality relation is simpler: (Ker /)
?
= Im/
, so in this case we have
Corollary 4.11 Suppose that the mark space E is ?nite. Then the bond
market is complete if and only if the martingale measure is unique.
The same conclusion holds if for almost all (?, t) the measures ?
t
(?, dx)
are concentrated in a ?nite number of points. In general, for an in?nite
E the “principle” that uniqueness of Q is equivalent to completeness of
the market fails: the set of hedgeable claims may be a strict subset in the
set of all claims L
?
(T
T
0
). Clearly, it is the case when D is continuous
in x and bounded (hence the image contains only continuous functions);
typically, /
Z?
t
is a compact operator and, hence, has no bounded inverse.
Thus the case with an in?nite mark space introduces some com-
pletely new features into the theory, and we also encounter some new
problems when it comes to the numerical computation of hedging port-
folios. The formal result is as follows.
Corollary 4.12 Suppose that the mark space E is in?nite. Then the
hedging equation (44) is ill-posed in the sense of Hadamard, i.e. the in-
verse of /
Z
t
restricted to Im/
Z
is not bounded.
Proof. This follows immediately from the fact that /
Z
is compact.
The main content of this result is that the hedging equation is nu-
merically ill-conditioned, in the sense that a small disturbance of the
right-hand side (e.g. due to a small round-o? error) gives rise to large
?uctuations in the solution. Thus, a naive approximation scheme for the
calculation of a concrete hedge may very well lead to great numerical
problems. Fortunately, there exists a large literature on stable solutions
of ill-posed problems but we will not pursue this topic here.
28
5 Characterization of hedgeable claims
5.1 Laplace transforms
In this section we suppose that Q is unique and that E is in?nite. Assume
for simplicity that we have no driving Wiener process. One can also think
that the model coe?cients in (50) below are deterministic.
From the general theory developed in the previous section it follows
that the hedging equation, symbolically written as
/
?
G = ? (49)
with the measure G
t
(dT) = Z(t?, T)h
t
(dT), can only be solved for a
right-hand side ? in a dense subset of the image space. The purpose
of this section is to present a class of models, for which we can give an
explicit characterization of the class of hedgeable claims.
Assumption 5.1 The forward rate dynamics under Q is given by
df(t, T) = ?(t, T)dt +
_
E
?(t, x, T) µ(dt, dx) (50)
with ? of the form
?(t, x, T) = ?c
t
(x) ?
0
(t, T) (51)
where the functions c and ?
0
are such that
(i) For each t the mapping c
t
: E ?R is injective.
(ii) For each t the set c
t
(E) is an interval [l
t
, ?[ or ]l
t
, ?[ (i.e. the left
endpoint l
t
may or may not belong to c
t
(E).
(iii) ?
0
> 0.
The important restriction introduced by this assumption is the volatil-
ity structure given by (51) (see, however, Remark 6.8 of Section 6). Con-
dition (i) simply means that di?erent points in E really give rise to dif-
ferent behavior of the forward rates. Assumption (ii) guarantees that we
have an in?nite mark space and that c
t
(E) has a limit point at in?nity;
assumption (iii) does not seem to be severe.
Suppose now that we want to hedge against a particular bounded
discounted T
0
-claim X. The martingale representation result, see (43),
will then provide us with a function ?(t, x), and the hedging problem
29
reduces, modulo a measurable selection, to the problem of ?nding a
measure-valued process G such that for almost all t ? [0, T
0
]
_
?
t
_
e
D(t,x,T)
?1
_
G(t, dT) = ?(t, x) ?
Q
(t, dx)-a.e. (52)
Using Assumption 5.1 we obtain
_
?
t
_
e
?ct(x)?(t,T)
?1
_
G(t, dT) = ?(t, x) ?
Q
(t, dx)-a.e. (53)
where ? is given by
?(t, T) =
_
T
t
?
0
(t, s) ds. (54)
Since c
t
is assumed to be injective we can write (53) as
_
?
t
_
e
?y?(t,T)
?1
_
G(t, dT) = ?
c
(t, y). (55)
where ?
c
(t, y) = ?(t, c
?1
t
), y ? ct
(E), and the equality is understood
modulo ?
c
Q
(t, dy), the image of the measure ?
Q
(t, dx) under the mapping
c
t
.
Since the right-hand side of (55) is a class of equivalence, the rigorous
formulation of the following necessary condition concerns, in fact, the
properties of a representative of this class.
Lemma 5.2 Necessary conditions for the existence of a solution to the
hedging equation (52) are that
(i) For each t the function ?
c
(t, y) can be extended to an analytic func-
tion for all complex z such that Re z > l
t
.
(ii) For each t the limit lim
y??
?
c
(t, y) exists.
Proof Analyticity follows from the fact that, because ? > 0, the left-
hand side of (55) is analytic. Existence of the limit then follows directly
from (55).
We thus see that it is only in rather special cases that we can solve
the hedging equation, and this fact is in complete accordance with the
denseness result of the preceding section.
30
We continue our investigation, assuming that we can actually solve
the hedging equation. Then we may write (55) as
_
?
t
e
?y?(t,T)
G(t, dT) = ?
c
(t, y) ??
c
(t, ?) (56)
and (remember that t is ?xed in each equation) change the integration
variable from T to u by the substitution u = ?(t, T). Thus the measure
G(t, dT) will be pushed to a measure G
?
(t, du), de?ned by
G
?
(t, du) = G(t, ?
?1
(t, dT)). (57)
We now have
_
?
0
e
?yu
G
?
(t, du) = ?
c
(t, y) ??
c
(t, ?), ?y ? c
t
(E). (58)
This equation is a Laplace transformation for each t, and we have the
following characterization of the set of ?’s for which we can solve the
hedging equation.
Theorem 5.3 Consider a ?xed claim X and its corresponding ?. Then
we can hedge exactly against X if and only if the following conditions
hold for each (?, t):
(i) The function ?
c
(t, y) can be extended to an entire analytic function
for all complex z such that Re z > l
t
.
(ii) For each t the limit lim
y??
?
c
(t, y) exists.
(iii) The function ?
c
(t, y)??
c
(t, ?) is the Laplace transform of a signed
(?nite) Borel measure on [0, ?).
For a particular claim we may also want to know if the replicating
portfolio contains only bonds with maturities in a prespeci?ed set. If X
corresponds to an expiration time T
0
one can, e.g., ask whether it can
be hedged with a portfolio consisting entirely of bonds with maturities
greater than T
0
. Properties of this kind can, in fact, be read o? immedi-
ately from the structure of the predictable representation, i.e. of ?.
Proposition 5.4 Consider a ?xed hedgeable T
0
-claim X and its cor-
responding ?. Also ?x a real number (maturity) T
1
. Then the hedging
portfolio can be composed entirely of bonds with maturities greater than
T
1
if and only if the following condition holds for every t ? T
0
.
lim
y??
e
y?(t,T
1
)
(?
c
(t, y) ??
c
(t, ?)) = 0. (59)
31
Proof The h-component of a portfolio consists entirely of bonds with
maturities greater than T
1
if and only if the measure G(t, dT) has its
support in [T
1
, ?[, which is equivalent to the property that G
?
has its
support in [?(t, T
1
), ?[. Thus we can rewrite (58) as
_
?
?(t,T
1
)
e
?yu
G
?
(t, du) = ?
c
(t, y) ??
c
(t, ?),
which, after the change of variables v = u ??(t, T
1
), becomes
_
?
0
e
?y[v+?(t,T
1
)]
G
?,T
1
(t, dv) = ?
c
(t, y) ??
c
(t, ?),
where G
?,T
1
is the translation of G
?
. Thus we obtain
_
?
0
e
?yv
G
?,T
1
(t, dv) = e
y?(t,T
1
)
(?
c
(t, y) ??
c
(t, ?)),
and the result follows.
5.2 The case of a ?nite mark space
In the case when the mark space E is ?nite, we can write the forward
rate dynamics as
df(t, T) = ?(t, T)dt + ?(t, T)d
˜
W
t
+
n
i=1
?
i
(t, T)dN
i
t
(60)
where N
1
, , N
n
are counting processes with predictable intensity processes
?
1
, , ?
n
. The process
˜
W is supposed to be m-dimensional standard
Wiener, so ?(t, T) is an m-dimensional (row) vector process. In this case it
is reasonable to look for a hedging portfolio with the h-component instan-
taneously consisting of n+m bonds with di?erent maturities T
1
, , T
n
,
T
n+1
, , T
n+m
(i.e. h(t, dT) is a discrete measure concentrated in these
points), and the hedging equation can be written in the following matrix
form (where m may be equal to zero).
A(t, T
1
, , T
n+m
)
_
¸
¸
_
G
1
t
.
.
.
G
n+m
t
_
¸
¸
_
=
_
?
t
?(t)
_
(61)
where
?
t
=
_
¸
¸
_
?
1
t
.
.
.
?
m
t
_
¸
¸
_
, ?(t) =
_
¸
¸
_
?(t, 1)
.
.
.
?(t, n)
_
¸
¸
_
, (62)
32
A(t, T
1
, , T
n+m
) =
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
S
m
(t, T
1
) S
m
(t, T
n+m
)
.
.
.
.
.
.
.
.
.
S
1
(t, T
1
) S
1
(t, T
n+m
)
e
D
1
(t,T
1
)
?1 e
D
1
(t,T
n+m
)
?1
.
.
.
.
.
.
.
.
.
e
Dn(t,T
1
)
?1 e
Dn(t,T
n+m
)
?1
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
, (63)
S
i
(t, T
j
) = ?
_
T
j
t
?
i
(t, s) ds , D
i
(t, T
j
) = ?
_
T
j
t
?
i
(t, s) ds. (64)
Here the (?, ?) -process, as usual, comes from the martingale rep-
resentation theorem, with ?
i
as the integrand corresponding to
˜
W
i
and
?(i) as the integrand corresponding to the N
i
-process. The process G
i
is
(see comment after (49)) the discounted amount invested in the portfolio
corresponding to the bonds with maturity T
i
.
The main problem in this section is to give conditions that guarantee
completeness of the bond market. In concrete terms this means that we
want to give conditions on the forward rate dynamics implying the exis-
tence of maturities T
1
, , T
n+m
such that the matrix A(t, T
1
, , T
n+m
)
is invertible. From a practical point of view it would be particularly pleas-
ing if these maturities can be chosen in such a way that they stay ?xed
when the time t is running. Intuitively, it is also natural to expect that
the maturities can be chosen arbitrarily, as long as they are distinct from
one another.
The main result in this section says that, given smoothness of S and
D in the maturity variable T, we can choose maturities almost arbitrarily.
If, furthermore the volatilities are deterministic and S and D are also
smooth in the t-variable, then the maturities can be chosen ?xed over
time, i.e. maturities do not change with the running time t.
We start with a general mathematical observation in the following
Proposition 5.5 Let f
1
, , f
M
be a set of real-valued functions such
that
33
(i) For each i the function f
i
is real-valued analytic, i.e. it can be ex-
tended to a holomorphic function in the complex plane.
(ii) The functions f
1
, , f
M
are linearly independent.
For each choice of reals T
1
, , T
M
consider the matrix B de?ned by
B(T
1
, , T
M
) = ¦f
i
(T
j
)¦
i,j
. (65)
Then, given any ?nite interval [I
L
, I
R
] of a positive length, we can choose
T
1
, , T
M
in [I
L
, I
R
] such that B is invertible. Furthermore, apart from
a ?nite set of points, we can choose T
1
, , T
M
arbitrarily in [I
L
, I
R
] as
long as they are distinct.
Proof. We ?x the interval [I
L
, I
R
] and prove the result by induction on
the number of functions. For M = 1 the statement is obviously true, since
by analyticity the function f
1
can have at most ?nitely many zeroes on a
compact set. Suppose therefore that the statement is true for M = n?1,
and consider the matrix function B(t) de?ned by
B(t) =
_
¸
¸
¸
¸
_
f
1
(t) f
1
(T
2
) f
1
(T
n
)
f
2
(t) f
2
(T
2
) f
2
(T
n
)
.
.
.
.
.
.
.
.
.
f
n
(t) f
n
(T
2
) f
n
(T
n
)
_
¸
¸
¸
¸
_
(66)
where, by the induction hypothesis, we have chosen T
2
, , T
n
in such a
way that all (n ?1)-dimensional quadratic submatrices of the last n ?1
columns are invertible. Our task is now to prove that we can choose
a point t such that B(t) is invertible and to do this we consider the
determinant det B(t). Expanding det B(t) along the ?rst column we see
that
det B(t) =
n
i=1
a
i
f
i
(t) (67)
where the a
i
’s are subdeterminants of the last n?1 columns and hence (by
the induction hypothesis) nonzero. Thus we see from (67) that det B(t) is
an analytic function and, because of the assumed linear independence, it
is not identically equal to zero. Thus it has at most ?nitely many zeroes
in the interval [I
L
, I
R
] . If we choose T
1
as any number in [I
L
, I
R
] , except
for the ?nite set of “forbidden” values, we get the result.
Applying this result to the bond market situation we have
Theorem 5.6 Assume that
34
(i) For each ?, t all functions ?
i
(t, T) and ?
j
(t, T) are analytic in the
T-variable.
(ii) For each ?, t the following functions of the argument T are linearly
independent:
e
D
i
(t,T)
?1, S
j
(t, T), i = 1, , n, j = 1, , m. (68)
Then the market is complete. Furthermore, for each t we can choose the
distinct bond maturities arbitrarily, apart from a ?nite number of values
on every compact interval. If all functions above are deterministic and
analytic also in the t-variable, then the maturities can be chosen to be
the same for every t.
Proof. The main part of the statement follows immediately from Propo-
sition 5.5. The last statement follows from the fact that, if we ?x the
maturities at t = 0 such that the corresponding det B(t) ,= 0, then,
again by the assumed analyticity in the t-variable, det B(t) is zero only
for ?nitely many t-values. Furthermore, in the replicating portfolio we
are integrating compensated Poisson processes having intensities, so the
strategies can be chosen arbitrarily on the zero set of B, since this (de-
terministic) set has Lebesgue measure zero, while outside this set they
have to satisfy the system (61).
As an easy corollary we immediately have the following extension
of a result of Shirakawa (see [28]). Note that we allow for more than
one Wiener process, whereas the proof in [28] depends critically on an
assumption of only one Wiener process. In addition, in [28] the maturities
of the bonds in the hedging portfolio cannot be chosen freely, and the
maturities also vary with running time t. In contrast, we can prespecify
arbitrary maturities (as long as they are distinct) and these maturities are
allowed to stay ?xed as t varies. For practical purposes this is extremely
important, since in real life we only have access to a ?nite set of maturities
for traded bonds.
Corollary 5.7 Assume that the forward rate volatilities have the form
_
?
j
(t, T) = q
j?1
(T ?t), j = 1, , m,
?
i
(t, T) = ?
i
, i = 1, , n,
(69)
where ?
1
, , ?
n
are constants and q
j?1
(s) is a polynomial of degree j ?1
with a non-vanishing leading term. Then the market is complete. Fur-
thermore, the maturities can be chosen arbitrarily.
35
Proof. Follows immediately from Theorem 5.6.
The next result and its proof explain Shirakawa’s idea of using the
Vandermonde matrix to construct the “basic bonds”.
Corollary 5.8 Let m = 0 and ?
i
(t, T) = ?
i
?(T ?t) where ? is a strictly
positive function and ?
i
are distinct non-zero constants. Then the market
is complete.
Proof. One can always choose a number a > 0 and a monotone sequence
of u
k
such that
_
u
k
0
?(s)ds = ka, k = 1, . . . n.
Take maturities T
k
= t + u
k
. Since D
i
(t, T
k
) = ka?
i
we have, putting
?
i
= e
a?
i
, that
det A(t, T
1
, . . . , T
n
) = det (?
k
i
?1) ,= 0.
Indeed, the linear dependence condition can be written as
f(?
i
) :=
n
k=1
?
k
?
k
i
?
n
k=1
?
k
= 0, i = 1, ..., n,
with a nontrivial vector ?; this is impossible: since also f(1) = 0 the
coe?cients of the polynomial f(?) of degree n must be equal to zero.
Remark 5.9 There is an important practical as well as mathematical
di?erence between the situation when the maturities of bonds in hedging
portfolios depend or do not depend on the current time t. In the for-
mer case the portfolio contains only instantaneously a ?nite set of bonds
(“basic bonds” at t) but when t varies, then the union of these sets of
securities may happen to be in?nite and even non-countable, and hence
one can not apply the classical theory of stochastic integration. As can be
seen from Corollary 5.8, the system of “basic bonds” constructed in [28]
depends unfortunately on t. We note again that in our results above we
may, in fact, chose maturities which stay ?xed during the entire trading
period.
36
6 A?ne term structures
As soon as one moves from abstract theory to practical applications, and
in particular to algorithms which have to be executed in real time on a
computer, the need emerges of easily manageable analytical formulas. In
the case of interest rate derivatives one is particularly fortunate if the
models possess a so-called a?ne term structure.
In this section the starting point is that we take as given the dynam-
ics of the short rate. The notations are a bit di?erent from those of the
others sections and we omit certain somewhat boring mathematical de-
tails, such as e.g. technical conditions ensuring that solutions to certain
equations below actually exist and have desirable properties (integrabil-
ity etc.).
De?nition 6.1 An interest rate model is said to have an a?ne term
structure if bond prices can be described as
p(t, T) = F(t, r
t
, T), (70)
where
log F(t, r, T) = A(t, T) ?B(t, T)r, (71)
and where A and B are deterministic functions. We sometimes use the
notation
F(t, r, T) = F
T
(t, r).
A model exhibiting an a?ne term structure occurs naturally only
in a Markovian environment and so the starting point in this section is
that we consider the dynamics of the short rate of interest given a priori
as a Markov process. Furthermore, we choose to specify the r-dynamics
directly under the martingale measure Q.
Assumption 6.2 We assume that under Q all bounded discounted price
processes are martingales, the short rate is assumed to be the solution of
a stochastic di?erential equation of the form
dr
t
= a(t, r
t
)dt + b(t, r
t
)d
˜
W
t
+
_
E
q(t, r
t
, x)µ(dt, dx), (72)
where a(t, r), b(t, r), and q(t, r, x) are given deterministic functions. The
process
˜
W is Q-Wiener and µ has a predictable Q-intensity
?(?, t, dx) = ?(t, r
t?
, dx), (73)
where ?(t, r, dx) is a deterministic measure for each t and r.
37
The main problem here is that of ?nding su?cient conditions on a, b, q,
and ? for the existence of an a?ne term structure. We start by presenting
the fundamental partial di?erential-di?erence equation in this context
concerning the pricing of simple claims in a general Markovian setting.
Proposition 6.3 Suppose that the short rate is given by (72) and con-
sider, for a ?xed T, any bounded (discounted) contingent claim X, to be
paid at T, of the form
X = ?(r
T
). (74)
Then the arbitrage-free price process ?(t; X) of this asset is given by
?(t; X) = F(t, r
t
), (75)
where F is a (su?ciently regular) function which is the a solution of the
Cauchy problem
_
¸
_
¸
_
?F
?t
(t, r) +/F(t, r) ?rF(t, r) = 0,
F(T, r) = ?(r),
(76)
with
/F(t, r) = a(t, r)
?F
?r
(t, r) +
1
2
b
2
(t, r)
?
2
F
?r
2
(t, r)
+
_
E
¦F(t, r + q(t, r, x)) ?F(t, r)¦ ?(t, r, dx).
(77)
Proof. By the Itˆ o formula we have the following representation:
F(t, r
t
) exp
_
?
_
t
0
r
s
ds
_
= F
0
+
_
t
0
?F(s, r
s
)
?r
exp
_
?
_
s
0
r
u
du
_
b(s, r
s
)d
˜
W
s
+
_
t
0
_
E
¦F(s, r
s?
+ q(s, r
s?
, x)) ?F(s, r
s?
)¦ exp
_
?
_
s
0
r
u
du
_
˜ µ(ds, dx)
where F
0
= F(0, r
0
) and ˜ µ(ds, dx) = µ(ds, dx) ? ?(s, r
s?
, dx)ds. The
right-hand side of this representation de?nes a local martingale. By As-
sumption 6.2 it is in fact a true martingale so, using the boundary con-
dition, we see that it is the discounted price process of the contingent
claim X and (75) follows.
Notice that, in general,
?(t; X) = E
Q
_
?(r
T
) exp
_
?
_
T
t
r
s
ds
_
[T
t
_
, (78)
38
and, because of the Markovian setting, we have in fact (75) with
F(t, r) = E
Q
_
?(r
T
) exp
_
?
_
T
t
r
s
ds
_
[r
t
= r
_
. (79)
Since / is easily seen to be the in?nitesimal operator of r, the relation
(76) is nothing but the Kolmogorov backward equation.
Corollary 6.4 Given the short rate dynamics (72) – (73), bond prices
are given by (70), where
_
¸
_
¸
_
?F
T
?t
(t, r) +/F
T
(t, r) ?rF
T
(t, r) = 0.
F
T
(T, r) = 1.
(80)
We now turn to the existence of the a?ne term structure. The as-
sertion below is an extension of a result by Du?e ([13], see also [8]).
Proposition 6.5 Suppose that the r-dynamics under Q is given by (72)
and the model parameters a, b, q, and ? have the following structure
a(t, r) = ?
1
(t) +?
2
(t)r,
b(t, r) =
_
?
1
(t) +?
2
(t)r,
q(t, r, x) = q(t, x),
?(t, r, dx) = l
1
(t, dx) +l
2
(t, dx)r,
(81)
Suppose that the functions A(., T) and B(., T) on [0, T] solve the following
system of ODE’s
?B
?t
(t, T) +?
2
(t)B(t, T) ?
1
2
?
2
(t)B
2
(t, T) + ?
2
(t, B(t, T)) = ?1,
B(T, T) = 0,
(82)
?A
?t
(t, T) +?
1
(t)B(t, T) +
1
2
?
1
(t)B
2
(t, T) + ?
1
(t, B(t, T)) = 0,
A(T, T) = 0,
(83)
where
?
i
(t, y) =
_
E
_
1 ?e
?yq(t,x)
_
l
i
(t, dx), i = 1, 2. (84)
Then the model has an a?ne term structure of the form (70) – (71).
39
We see that, for ?xed T, the system (82) – (83) has a nice recursive
structure (and, in particular, when l
2
= 0 (82) is a Riccati-type equation
for B). Given a solution to (82), we can then easily determine A by a
simple integration.
Notice also that one needs some caution to ensure ? to be positive.
This is the case if l
1
? 0 and l
2
= 0 or l
1
? 0, l
2
? 0, and the process r
is positive (recall that, in principle, we admit negative values of interest
rates).
Remark 6.6 Notice that the class of models satisfying (81) general-
izes some well-known term structure models such as Vasi?cek [29], Cox–
Ingersoll–Ross [9], Ho–Lee [20], Hull–White [21].
Remark 6.7 If a model possesses an a?ne term structure, then, always
under the assumption 6.2 and that of Proposition 6.5, the forward rate
dynamics are easily obtained as
df(t, T) = ¦B
t,T
(t, T)r
t
?A
t,T
(t, T) +B
T
(t, T)a(t, r)¦dt
+B
T
(t, T)b(t, r)d
˜
W
t
+
_
E
B
T
(t, T)q(t, x)µ(dt, dx) (85)
where B
T
(t, T) is the partial derivative with respect to T and B
t,T
the
partial with respect to t and T.
Remark 6.8 We close this section by pointing out that, if a model pos-
sesses an a?ne term structure, then (see Remark 6.7) the forward rate
dynamics satisfy (85) from which it is immediately seen that assump-
tion 5.1, in particular the decomposition property (51), is satis?ed with
c
t
(x) = ?q(t, x) and ?
0
(t, T) = B
T
(t, T). This implies that for the a?ne
term structure models the hedging problem can be approached by means
of a Laplace transform inversion (see section 5.1), which makes this prob-
lem considerably easier compared to the general setting of section 4.2.
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