With $P_{t}$ the price in current dollars of a dollar delivered $t$ time units
from now, we assume that $P$ is a decreasing function defined for
$t\in\mathbb{R}_{+}$ with $P_{0}=1$. The negative logarithmic derivative,
$-\stackrel{{\scriptstyle\bullet}}{{P}}_{t}/P_{t}$ defines the yield curve
function $Y_{t}$. An $n$ parameter linear yield curve model selects as
allowable yield curves $Y_{t}(r)=\sum_{i=1}^{n}r_{i}Y^{i}_{t}$ with the
functions $Y^{i}$ fixed and with $r$ varying over an open subset of
$\mathbb{R}^{n}$ on which $Y_{t}(r)\geq 0$ for all $t\in\mathbb{R}_{+}$. For
example, the flat yield curve model with $P_{t}(r)=e^{-rt}$ is a one parameter
linear model with $Y^{1}_{t}(r)=r>0$. We impose two natural economic
requirements on the model: (SPA) static prices allowed, i.e. it is always
possible that as time moves forward, relative prices do not change, and (NLA)
no local arbitrage, i.e. there does not exist a self-financing bundle of
futures such that the zero present value is a local minimum with respect to
small changes in the space of admissible yield curves. In that case the model
always contains one of four simple models. If we impose the additional
requirement (LRE) long rates exist, i.e. for every $r$
$Lim_{t\to\infty}Y_{t}(r)$ exists as a finite limit, then the number of simple
models is reduced to two.

###### Key words and phrases:

yield curve models, linearity, no arbitrage, static prices

2010 Mathematical Subject Classification 91B02, 91G10, 91G30, 97M30

###### Contents

  1. 1 Introduction
  2. 2 Yield Curve Models
    1. 2.1 Static Prices Allowed (SPA)
    2. 2.2 No Local Arbitrage (NLA)
    3. 2.3 Linearity (LIN)
    4. 2.4 Long Rates Exist (LRE)
    5. 2.5 Main Results
  3. 3 Arbitrage in Linear Models

##  1\. Introduction

Acknowledgement: We would like to express our thanks to our colleague Eli
Amzallag for helpful discussions about this work.

Notation: As usual, we use $\mathbb{R}$ and $\mathbb{C}$ for the real and
complex fields and $\mathbb{Z}$ for the ring of integers. We use
$\mathbb{R}_{+}$ to denote the set of non-negative reals. That is,
$\mathbb{R}_{+}=[0,\infty)$.

When the interest rate is $r$, dollars delivered $t$ years in the future are
discounted by the factor $e^{-rt}$. To put this another way the price $P_{t}$
of one dollar delivered $t$ years from now is $e^{-rt}$ in current dollars.
Here $t$ varies continuously in $\mathbb{R}_{+}$. Thus, the yield curve
$Y_{t}$, defined to be the negative logarithmic derivative of $P_{t}$, i.e.
$Y_{t}=-\stackrel{{\scriptstyle\bullet}}{{P}}_{t}/P_{t}$, is the constant $r$.
Even if we ignore risk, which induces a price spread among $t$ futures for
fixed $t$, the price curve $P_{t}$ may not exhibit this simple shape. To
describe the pattern of interest rates we then need the entire yield curve
$Y_{t}$ instead of the single number $r$.

Nonetheless, when we think of interest rates as unchanging, the flat yield
curve model with its single interest rate is a simple and useful tool adequate
for many theoretical purposes. However, when we begin to consider change in
interest rates, the flat model won’t do. This paper had its origin in our
discovery of its flaws.

We had, in fact, rediscovered an odd phenomenon well-known among financial
economists who consider the notion of duration for a bond.

Consider a finite set $D$ consisting of several future times. For each time
$t\in D$ allot $z_{t}>0$ current dollars for $t$ futures ($z_{t}$ is defined
to be 0 for $t\not\in D$). If the current interest rate is $r^{*}$, this
allocation buys a bond paying $x_{t}$ dollars $t$ years from now where

(1.1) $z_{t}\ =\ e^{-r^{*}t}x_{t}.$

The present value of the basket of futures is

(1.2) $V_{0}\ =\sum_{t}\ z_{t}\ =\sum_{t}\ e^{-r^{*}t}x_{t}.$

Following [5] (see also [2]) we define the duration $T$ of the basket of
futures as the weighted average of the payment times:

(1.3) $\begin{split}T\ =(\sum_{t}\ tz_{t})/(\sum_{t}\ z_{t})\hskip
28.45274pt\\\ =\ (\sum_{t}\ te^{-tr^{*}}x_{t})/(\sum_{t}\
e^{-tr^{*}}x_{t}).\end{split}$

Alternatively, we can define the duration as the unique time $T$ satisfying
the -equivalent- equations

(1.4) $\displaystyle\begin{split}0\ =&\sum_{t}\ (T-t)z_{t},\\\ 0\ =&\sum_{t}\
(T-t)e^{(T-t)r^{*}}x_{t}.\end{split}$

Suppose we finance this bond by a single sale of $V_{0}$ worth of time $T$
futures. That is, define

(1.5) $\tilde{x}_{s}\ =\left\\{\begin{array}[]{cr}x_{s}&s\neq T\\\ x_{T}\ -\
\sum_{t}\ e^{(T-t)r^{*}}x_{t}&s=T\end{array}\right.$

or, equivalently,

(1.6) $\tilde{z}_{s}\ =\left\\{\begin{array}[]{lr}z_{s}&s\neq T\\\
z_{T}-(\sum_{t}z_{t})&s=T,\end{array}\right.$

The investment $\tilde{x}_{t}$ is self-financing, i.e. its present value,
$\tilde{V}_{0}$ is zero.

Fisher and Weil in [1] point out that this sale immunizes the investor against
loss when interest rates change. To see this, suppose that a period $h$ as
passed, short enough that none of the payments are due at or before $h$, and
that the new interest rate is now $r$. The quantity $\tilde{x}_{t}$ is a
payment occurring $t-h$ in the future. So the new present value is:

(1.7) $\begin{split}\tilde{V}_{h}\ =\sum_{t}\ e^{-(t-h)r}\tilde{x}_{t}\
=\hskip 56.9055pt\\\ e^{-(T-h)r}[\sum_{t}\ e^{(T-t)r}x_{t}-\sum_{t}\
e^{(T-t)r^{*}}x_{t}].\end{split}$

By applying (1.4) we can write this as

(1.8) $\tilde{V}_{h}\ =e^{-(T-h)r}[\sum_{t}\
e^{(T-t)r^{*}}x_{t}[e^{(T-t)(r-r^{*})}\ -\ 1\ -\ (T-t)(r-r^{*})].$

Observe that $e^{a}-1-a=\int^{a}_{0}(e^{s}-1)ds$ is nonnegative for all $a$,
vanishing only when $a=0$. Thus, we obtain $\tilde{V}_{h}\geq 0$ vanishing
only when $r=r^{*}$.

As Ingersoll, Shelton and Weil remark, [3] page 635, this result is “too
good”. It is an arbitrage opportunity which exists at any interest rate. In
fact, (1.3) shows that $T$ can be defined using the current dollar quantities
$z_{t}$ in a way independent of the interest rate.

Arbitrage opportunities do happen. Noticing such an investment, a speculator
can reap the associated profit. However, the transactions involved cause
compensating price movements which tend to wipe out the original opportunity.
Here, however, as long as the yield curve remains flat and will continue to do
so the arbitrage possibilities described above remain as well. Such
speculations in the futures market would, instead, bend the yield curve.

A yield curve model selects as admissible certain special curves together with
their associated price curves. For example, for the flat yield curve model the
positive constants are the admissible yield curves. An arbitrage opportunity
for a model is an investment with no possibility of loss when the prices
curves are restricted to those admissible according to the model. Our first
condition on a yield curve model is that even local versions of such
arbitrages do not occur.

The second condition is that the model allows static prices. That is, starting
from an admissible price curve the possibility that relative prices among
futures do not change is not excluded from the menu of choices offered by the
family of admissible price curves. In the next section, we will examine in
some detail what this means.

Our third condition is linearity. This is a mathematical convenience rather
than an economic imperative. The flat yield curve model is described by
$Y_{t}(r)=r\cdot 1$ with a single positive parameter. An $n$ parameter linear
model is of the form
$Y_{t}(r_{1},\dots,r_{n})=r_{1}Y^{1}_{t}+\dots+r_{n}Y^{n}_{t}$ where
$(r_{1},\dots,r_{n})$ varies over some open subset of $\mathbb{R}^{n}$ and the
curves $Y^{1}_{t},\dots,Y^{n}_{t}$ are fixed for the model.

The infinitesimal or short rate for the yield curve $Y_{t}$ is $Y_{0}$. The
limit or long rate is $Lim_{t\rightarrow\infty}\ Y_{t}$ if the limit exists.
We say the model admits long rates if this limit exists as a finite real
number for any admissible yield curve.

Our main result is that an $n$ parameter linear model allows static prices,
admits no local arbitrage, contains the flat yield curves and admits long
rates if and only if it contains, in a natural way to be described later, one
of the following two models:

(1.9) $Y_{t}(r_{1},r_{2},r_{3})\ =r_{1}+r_{2}e^{-\rho t}+r_{3}e^{-2\rho t}$

where $\rho$ is a fixed positive constant, or

(1.10) $Y_{t}(r_{1},r_{2},r_{3},r_{4})\ =r_{1}+r_{2}e^{-\rho t}\cos\omega
t+r_{3}e^{-\rho t}\sin\omega t+r_{4}e^{-2\rho t}$

where $\rho$ and $\omega$ are fixed positive constants.

In each case $r_{1}$ is the long rate. If we further demand that the approach
to the long rate be eventually monotonic then the damped oscillation of (1.10)
is excluded and we are left with (1.9).

If we omit the long rate demand and allow the yield curves to be unbounded
then the constant $\rho$ in (1.9) can be negative, and we have two further
possible models:

(1.11) $Y_{t}(r_{1},r_{2})\ =r_{1}+r_{2}t,\hskip 170.71652pt$
$Y_{t}(r_{1},r_{2},r_{3},r_{4},r_{5})\
=r_{1}+r_{2}\cos\omega_{1}t+r_{3}\sin\omega_{1}t+r_{4}\cos\omega_{2}t+r_{4}\sin\omega_{2}t.$

where $\omega_{1},\omega_{2}$ are positive constants such that
$\omega_{1}/\omega_{2}$ is irrational.

A wide variety of yield curve models arise as diffusion models. Each of these
is an $n$ parameter linear model for some $n$. There is an additional dynamic
structure in that the change in prices over time is governed by a diffusion
stochastic process in the parameter space $\mathbb{R}^{n}$.

This is not how diffusion models are usually presented. Instead (e.g. see
Ingersoll, Shelton and Weil (1978) pages 637ff.) from diffusion equations for
general assets the yield curve at each parameter point is obtained by solving
an associated partial differential equation. The result is a linear yield
curve model with no local arbitrage.

These models do not usually satisfy the static price condition. This is not
serious because the static price condition can always be obtained by
introducing an additional translation parameter (see Section 1 below).
However, in most diffusion models the new parameter enters in a nonlinear
fashion. Thus, the new enlarged model no longer satisfies linearity.

It is the interplay between these three conditions which limits the range of
possible models. If we impose the additional condition that each yield curve
approaches a nonzero long rate limit in a monotone fashion then by rescaling
time we can assume $\rho$ in (1.9) is $1$. We thus obtain the unique simple
yield curve model, the three parameter linear model

(1.12) $Y_{t}(r_{1},r_{2},r_{3})\ =r_{1}+r_{2}e^{-t}+r_{3}e^{-2t},$

where $(r_{1},r_{2},r_{3})$ varies over the open set in $\mathbb{R}^{3}$ such
that

(1.13) $inf_{t\geq 0}\ \ Y_{t}(r_{1},r_{2},r_{3})\quad>\quad 0.$

##  2\. Yield Curve Models

In general, _price_ describes the exchange ratio between quantities of two
commodities in a market. If among many commodities we choose one, say corn, as
_numeraire_ then the price $p_{i}$ of commodity $i$ is an amount of corn for
which a unit quantity of $i$ exchanges. A basket of commodities is described
by a vector $x$ such that coordinate $x_{i}$ is the quantity of $i$ in the
basket. The cost or _value_ of such a basket, given by

(2.1) $V(x)=\sum_{i}\ p_{i}x_{i},$

is the quantity of corn for which the entire basket can be exchanged. In (2.1)
$x_{i}$ can be positive or negative. We interpret a negative $x_{i}$ as a
liability, a commitment to deliver quantity $|x_{i}|$ of good $i$. So $V(x)$
is the net value summed over the goods coming in and going out. To change
numeraire from corn to gold, observe that if $p_{g}$ is the corn price of gold
then $p_{i}/p_{g}$ is the gold price of commodity $i$. So we get the gold
value of basket $x$ by dividing $V(x)$ by $p_{g}$.

We assume that all prices are nonnegative. This is a consequence of the (often
tacit) assumption of _free disposal_. Similarly, the assumption that choice,
as well as disposal, is costless implies that the market evaluates a choice
between goods $A$ and $B$ as at least as valuable as good $A$ alone. For
mathematical convenience we will assume that prices are nonzero as well and so
are strictly positive.

We are also ignoring transaction costs and are assuming that there is a single
price to buy or to sell a commodity in the market.

The description of a commodity requires a specification of time and place of
delivery. We will ignore place but for futures markets the date of delivery is
exactly the central concern. Thus, the _price curve_ for dollar futures is
defined by:

> $P_{t}=$ the current dollar price of one
> dollar delivered $t$ time units from now.

The _present value_ of a finite bundle of payments $x=\\{x_{t}\\}$ is the
value in current dollars computed as in (2.1).

(2.2) $V(x)\ =\sum_{t}\ P_{t}x_{t}.$

As before $x_{t}$ is positive or negative according to whether the quantity
$|x_{t}|$ of dollars is incoming or outgoing $t$ time units from now.

The present value is a sum rather than an integral because $x$ consists of a
finite number of discrete payments rather than a continuous flow. For any set
$T$, we call $x:T\rightarrow\mathbb{R}$ a _finite function_ when the _support_
of $x$, denoted supp$(x)$, is a finite subset of $T$, where

(2.3) ${\rm supp}(x)\ =\\{\ t\ :\ x_{t}\neq 0\ \\}.$

So a bundle of payments $x$ is a finite function on $\mathbb{R}_{+}$. We call
$x$ a bundle of futures when $x_{0}=0$, i.e. $0\not\in{\rm supp}(x)$, so that
no current payments are involved.

Observe that we use continuous variables for time and later for interest rate
parameters. In contrast with our finite payment functions, the price functions
are assumed to be smooth. That is, continuous partial derivatives of all
orders exist.

The numeraire for our price function $P$ is current dollars and so $P_{0}=1$,
the current dollar price of one current dollar. Furthermore, the price
function is nonincreasing. This follows from the free disposal assumption and
the further assumption of _free storage._ To each consumer a dollar delivered
at an early date is at least as valuable as a later one because one could
always store the former and so obtain the latter at no additional cost. In
other words, if we regard the late dollar as good $A$ then the early dollar
represents a choice between good $A$, obtained by storage, and good $B$, the
consumption of the dollar during the intervening time. In particular, writing
$\stackrel{{\scriptstyle\bullet}}{{P}}_{t}$ for the derivative at time $t$, we
have that $\stackrel{{\scriptstyle\bullet}}{{P}}_{t}$ is nonpositive. We will
assume it is always negative. Thus, we have the following conditions on the
price curves, defined for all $t\geq 0$:

(2.4) $\begin{split}1\geq P_{t}>0\qquad\mbox{ with}\quad P_{0}=1,\\\
\stackrel{{\scriptstyle\bullet}}{{P}}_{t}<0.\hskip 85.35826pt\end{split}$

Associated with the price function there are several related curves. The _log-
price_ is defined by

(2.5) $L_{t}\ =\ -\ln P_{t},$

(where ln is the natural log) and its derivative, the _yield curve_ is defined
by

(2.6) $Y_{t}\ =\ \stackrel{{\scriptstyle\bullet}}{{L}}_{t}\ =\
-\stackrel{{\scriptstyle\bullet}}{{P}}_{t}/P_{t}.$

These satisfy the conditions

(2.7) $\displaystyle\begin{split}&L_{t}\geq 0\quad\mbox{with}\quad L_{0}\ =\
0,\\\ &Y_{t}>0.\end{split}$

From the initial conditions $P_{0}=1$, or equivalently $L_{0}=0$, we can
recover the price from the yield curve

(2.8) $\displaystyle\begin{split}&L_{t}\ =\int^{t}_{0}\ Y_{s}ds,\\\ &P_{t}\
=\exp(-L_{t}).\end{split}$

Observe that we are using the phrase “yield curve” for the marginal value of
the log-price. It is sometimes used instead to refer to the average value
$L_{t}/t$. The two concepts agree exactly when the yield curve is a constant.
In this, the _flat yield curve_ case, with $Y_{t}=r>0$, we have $L_{t}=rt$ and
$P_{t}=\exp(-rt)$. This is the classical price curve when “the interest rate”
is $r$.

A _yield curve model_ picks out from the set of all smooth, positive real
valued functions on $\mathbb{R}_{+}=[0,\infty)$ a subset, the _admissible
yield curves_ , with associated price curves given by (2.8). An $n$ _parameter
yield curve model_ associates to each admissible yield curve a unique vector
$r$ in some subset $O$ of $\mathbb{R}^{n}$. To be more precise, such a model
is a function $Y$ from $O$ to the vector space $\mathbb{R}^{\mathbb{R}_{+}}$
of real valued functions on $\mathbb{R}_{+}$, with special assumptions as
follows:

######  Definition 2.1.

Let $n$ be a positive integer and $O$ be a nonempty open subset of
$\mathbb{R}^{n}$. An _ $n$ parameter yield curve model defined on $O$_ is a
smooth, positive real valued function in $n+1$ real variables,
$Y_{t}(r_{1},\ldots,r_{n})$ defined for all $t\geq 0$ and
$r=(r_{1},\dots,r_{n})$ in $O$. Regarded as a function from $O$ to
$\mathbb{R}^{\mathbb{R}_{+}}$, we assume that $Y$ is injective. That is, if
for $r,r^{\prime}\in O$, $Y_{t}(r)=Y_{t}(r^{\prime})$ for all
$t\in\mathbb{R}_{+}$, then $r=r^{\prime}$.

For example, the _flat yield curve model_ is the one parameter model
$Y_{t}(r_{1})=r_{1}$ defined on $O=(0,\infty)$.

Changing prices is represented as continuous motion in the set of admissible
yield curves. For an $n$ parameter model defined on $O$, such change in time
is described by smooth motion of the parameter $r$ in the set $O$.

We now describe the conditions which we will require of a good model.

###  2.1. Static Prices Allowed (SPA)

If $Y_{t}$ is an admissible yield curve whose associated price curve $P_{t}$
represents prices in the current market, then we want the model to include the
possibility that over time relative prices do not change. We are not claiming
that this sort of stasis is at all likely, but we do not want it ruled out a
priori by the model.

At first glance this condition appears to be trivially satisfied. We are
fixing the set of admissible curves in a time independent way so if $Y_{t}$ is
admissible today, then it is admissible tomorrow. However, an unchanging yield
curve is usually not the same thing as unchanging prices. The confusion
between these two concepts comes from the two different roles played by time
in the futures market.

On the one hand, each instant on the time line is a _date_ , $d$, a moment on
the calendar. The date is the label attached to our commodities. For precision
we will use the notation

(2.9) $\$_{d}\ \equiv\ \mbox{ a dollar delivered at date }\ d.$

On the other hand, between two dates is an interval of time. Suppose that from
a date $d_{0}$ we move to a date $h$ time units later which we will denote
$d_{h}$. Then,

(2.10) $d_{h}\ =d_{0}+h\quad\mbox{and}\quad h\ =d_{h}-d_{0}.$

The variable $t$ in the price and yield curves is this interval length
measuring the wait from now, $d_{0}$, to a later date $d_{t}=d_{0}+t$. Thus,
if $P$ is the price function now at $d_{0}$, then $P_{t}$ is the price of
$\$_{d_{0}+t}$ in current dollars, i.e. in $\$_{d_{0}}$.

Suppose that when we move from the date $d_{0}$ to the date $d_{h}$ the price,
log-price and yield functions change from $P$, $L$, $Y$ to new ones $P^{h}$,
$L^{h}$, $Y^{h}$.

The numeraire has changed from $\$_{d_{0}}$ to $\$_{d_{h}}$. $P^{h}_{t}$ is
the price of $\$_{d_{h}+t}$ in the new numeraire $\$_{d_{h}}$. In the old
price system the commodity $\$_{d_{h}+t}=\$_{d_{0}+h+t}$ is priced at
$P_{t+h}$ valued in $d_{0}$ dollars, $\$_{d_{0}}$.

Similarly, suppose that we purchased at date $d_{0}$ a bundle of futures $x$
and that no payments have occurred in the interval from $d_{0}$ to $d_{h}$,
that is

(2.11) $x_{t}\ =0\quad\mbox{for}\quad 0\leq t\leq h$

or equivalently, $[0,h]\cap supp(x)\ =\emptyset.$

For each $t$, $x$ describes the quantity $x_{t}$ of commodity $\$_{d_{0}+t}$.
At date $d_{h}$ the asset remains the same but is described by the new finite
function $x^{h}$ with

(2.12) $x^{h}_{t-h}\ =x_{t},$

that is, the wait for each commodity has been reduced by $h$. The $d_{h}$
present value is now

(2.13) $V^{h}\ =\sum_{t}\ P^{h}_{t}x^{h}_{t}\ =\sum_{t}\ P^{h}_{t-h}x_{t}.$

Notice that $x_{t}=0$ for $0\leq t<h$ implies that although $P^{h}_{t-h}$ is
undefined for $t$ in this range we can still regard both sums with $t$ varying
in $\mathbb{R}_{+}$.

_Unchanging prices_ means that the relative prices of goods $\$_{d}$ with
$d\geq d_{h}$ are the same with respect to the price function $P^{h}$ as they
were with respect to $P$. That is, for all $t\geq 0$

(2.14) $\begin{split}P^{h}_{t}\ =P_{t+h}/P_{h},\hskip 85.35826pt\\\ L^{h}_{t}\
=L_{t+h}-L_{h},\hskip 85.35826pt\\\ Y^{h}_{t}\ =Y_{t+h}.\hskip
99.58464pt\end{split}$

The first equation describes unchanging relative prices. We are merely
changing the numeraire from $\$_{d_{0}}$ to $\$_{d_{h}}$. The second equation
follows by taking logs and the third, in turn, by taking the derivative with
respect to $t$ because the derivative of the constant $-L_{h}$ is zero. On the
other hand, by using (2.8) we can recover the first, price, equation from the
third, yield, equation in (2.14).

Thus, the SPA condition says that if $Y$ is an admissible curve, then for any
$h>0$ the $h$ translated curve $\Phi_{h}(Y)$ is also admissible where

(2.15) $\Phi_{h}(Y)_{t}\ \equiv\ Y_{t+h}\qquad t,h\geq 0.$

For an $n$ parameter model defined on $O$ the $h$ translate of the curve
$Y(r)$ for $r\in O$ must then correspond to a unique parameter value which we
denote $\varphi_{h}(r)$. That is,

(2.16) $\Phi_{h}(Y(r))\ =Y(\varphi_{h}(r))$

We assume the function $\varphi$ is smooth. Thus, we are led to

######  Definition 2.2.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined for
$(t,r)\in\mathbb{R}_{+}\times O\subset\mathbb{R}^{n+1}$, with associated log-
price and price curves: $L_{t}(r)\ =\int^{t}_{0}Y_{s}(r)ds$ and $P_{t}(r)\
=\exp[-L_{t}(r)]$. We say that the model allows static prices, or satisfies
condition SPA, if there is a smooth function $\varphi:\mathbb{R}_{+}\times
O\rightarrow O$ such that for all
$(t,h,r)\in\mathbb{R}_{+}\times\mathbb{R}_{+}\times O$

(2.17) $\begin{split}P_{t}(\varphi_{h}(r))\ =P_{t+h}(r)/P_{h}(r)\hskip
56.9055pt\\\ L_{t}(\varphi_{h}(r))\ =L_{t+h}(r)-L_{h}(r)\hskip 56.9055pt\\\
Y_{t}(\varphi_{h}(r))\ =Y_{t+h}(r).\hskip 71.13188pt\end{split}$

Remark. As with (2.14) the three equations in (2.17) are equivalent. By
Definition 2.1 the association from parameters to yield curves is assumed
injective and so the parameter value $\varphi_{h}(r)$ is uniquely determined
for each $(h,r)\in\mathbb{R}_{+}\times O$ by the yield curve equation in $t$
of (2.17).

From the remark above we obtain the Kolmogorov flow conditions

(2.18) $\begin{split}\varphi_{0}(r)\ =r;\hskip 85.35826pt\\\
\varphi_{h_{1}}(\varphi_{h_{2}}(r))\ =\varphi_{h_{1}+h_{2}}(r)\hskip
28.45274pt\end{split}$

for all $r\in O$ and $h_{1},h_{2}\in\mathbb{R}_{+}$. Observe that the two
sides of each equation have the same yield curve. We will call the function
$\varphi$ the static price flow on $O$.

Notice that if we define the short rate function
$Y_{0}:O\rightarrow(0,\infty)$ by setting $t=0$ in $Y_{t}(r)$ then we can
recover the full yield curve from the short rate function and the static price
flow $\varphi$ by using (2.17):

(2.19) $Y_{t}(r)\ =Y_{0}(\varphi_{t}(r))\hskip 56.9055pt$

for all $(t,r)\in\mathbb{R}_{+}\times O$.

Translation of a function by $h$ leaves the function unchanged for all
positive $h$ if and only if the function is constant. Thus, the flat yield
curve is the one case where the notions of unchanging yield curve and
unchanging prices agree. In particular, the one parameter flat yield curve
model allows static prices with $\varphi_{h}(r_{1})=r_{1}$ for all
$(h,r_{1})\in\mathbb{R}_{+}\times(0,\infty)$.

A smooth flow like $\varphi$ is the solution of a smooth system of ordinary
differential equations. For each $r\in O$ let
$\stackrel{{\scriptstyle\bullet}}{{\varphi}}_{h}(r)$ denote the derivative
$d\varphi_{h}(r)/dh$, i.e. the velocity vector of the path in $O$,
$\varphi_{h}(r)$, obtained by fixing $r$ and letting $h$ vary. The vector
field $\xi:O\rightarrow\mathbb{R}^{n}$ is obtained by taking the velocity
vector at $h=0$, that is,

(2.20) $\xi(r)\ =\ \stackrel{{\scriptstyle\bullet}}{{\varphi}}_{0}(r).$

Differentiating (2.18) with respect to $h_{1}$ and setting $h_{2}=h$, we see
that $\varphi_{h}(r)$ is the solution of the smooth initial value problem

(2.21) $\begin{split}\varphi_{0}(r)\ =r\hskip 56.9055pt\\\
\stackrel{{\scriptstyle\bullet}}{{\varphi}}_{h}(r)\ =\
\xi(\varphi_{h}(r)).\hskip 28.45274pt\end{split}$

Definition 2.2 includes the assumption that this solution path is defined for
all positive $h$. In o.d.e. terminology this says that the vector field $\xi$
is _complete_ in the positive direction.

With $O$ an open set, we can use this characterization of the flow to extend
$\varphi_{h}(r)$ to some negative values of $h$.

######  Lemma 2.3.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined on an open subset
$O$ of $\mathbb{R}^{n}$, with $P_{t}(r)$ and $L_{t}(r)$ the associated price
and log-price curves. Assume that the model satisfies SPA with
$\varphi:\mathbb{R}_{+}\times O\rightarrow O$ the static price flow. There
exists an open subset $\tilde{O}$ of $\mathbb{R}\times O$ which contains
$\mathbb{R}_{+}\times O$, and a smooth function
$\tilde{\varphi}:\tilde{O}\rightarrow O$ which extends $\varphi$, and for
$(t,-h,r)\in\mathbb{R}_{+}\times\tilde{O}$ such that $h,t-h\geq 0$

(2.22) $\begin{split}P_{t-h}(r)\
=P_{t}(\tilde{\varphi}_{-h}(r))/P_{h}(\tilde{\varphi}_{-h}(r))\\\ L_{t-h}(r)\
=L_{t}(\tilde{\varphi}_{-h}(r))-L_{h}(\tilde{\varphi}_{-h}(r))\\\ Y_{t-h}(r)\
=Y_{t}(\tilde{\varphi}_{-h}(r)).\hskip 56.9055pt\end{split}$

###### Proof.

The constructions and proof of the existence and uniqueness theorem for
o.d.e.’s yield the following results, see, for example, [4] Chapter IV
Sections 1 and 2. The solution flow for the vectorfield $\xi$ is a smooth
function $\tilde{\varphi}:\tilde{O}\rightarrow O$ with $\tilde{O}$ an open
subset of $\mathbb{R}\times O$. For each $r\in O$, $\\{h:(h,r)\in\tilde{O}\\}$
is an open interval containing $0$. It is the maximum open interval on which
the unique solution path of the initial value problem (2.21) is defined. This
interval thus includes $\mathbb{R}_{+}$ because $\xi$ is assumed positively
complete. Furthermore, by uniqueness, $\tilde{\varphi}$ extends $\varphi$.
Finally, the flow equation (2.18) for $\tilde{\varphi}$ holds in the sense
that if the left side is defined then the right side is as well.

In particular, if $(t,-h,r)$ is in $\mathbb{R}_{+}\times\tilde{O}$ with
$h,t-h\geq 0$, then $\varphi_{h}(\tilde{\varphi}_{-h}(r))=r$. Now let
$\tilde{r}=\tilde{\varphi}_{-h}(r)$, $\tilde{t}=t-h$ and apply equations
(2.17) to $(\tilde{t},h,\tilde{r})$. Since $\tilde{t}+h=t$ and
$\varphi_{h}(\tilde{r})=r,$ we get (2.22).

$\Box$

Thus, condition SPA imposes some structure upon an $n$ parameter model. On the
other hand, Joe Langsam pointed out that we can always obtain SPA by throwing
in another parameter.

(2.23) $\begin{split}\tilde{Y}_{t}(r_{1},\dots,r_{n},r_{n+1})\
=Y_{t+r_{n+1}}(r_{1},\dots,r_{n})\\\
\tilde{\varphi}(r_{1},\dots,r_{n},r_{n+1},h)\
=(r_{1},\dots,r_{n},r_{n+1}+h)\end{split}$

is an $n+1$ parameter model on $O\times\mathbb{R}_{+}$ with $\tilde{\varphi}$
the required static price flow.

###  2.2. No Local Arbitrage (NLA)

A bundle of futures $x$ is called a _self-financing investment_ if at current
prices the present value is zero. This means that the worth of the assets and
liabilities in the bundle exactly balance and no current cash is required for
$x$. Since we are ignoring transaction costs, anyone can obtain $x$ in the
current market.

Suppose that $h$ is small enough that no payments occur during the initial $h$
time interval, i.e. condition (2.11) holds. The bundle $x$, now labeled
$x^{h}$, is worth $V^{h}$ computed using the new price curve $P^{h}_{t}$, cf.
equations (2.12) and (2.13). For example, if relative prices are unchanged so
that $P^{h}$ is related to $P$ by equation (2.14) then the value $V^{h}$ is
still zero. We have merely changed the numeraire from $\$_{d_{0}}$ to
$\$_{d_{h}}$. In general, when prices have changed $V^{h}$ is pure profit when
positive, and loss when negative, measured in now current dollars
$\$_{d_{h}}$.

For a yield curve model a bundle of futures $x$ is an arbitrage for an
admissible price function $P$ if it has present value zero with respect to $P$
and if for any $h$ satisfying (2.11) and for any admissible price function
$P^{h}$ the value $V^{h}$ given by (2.13) is nonnegative. Thus, $x$ is a self-
financing investment at current prices and no admissible pattern of price
movements returns a loss on the investment. As described in the Introduction
the occurrence of such arbitrage possibilities is a serious flaw in a yield
curve model. We want to exclude even the local versions of such arbitrage.

######  Definition 2.4.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined on
$O\subset\mathbb{R}^{n}$ with associated price curve $P_{t}(r)\
=\exp[-\int^{t}_{0}Y_{s}(r)ds]$. Let $x$ be a bundle of futures, i.e. a finite
function $x:\mathbb{R}_{+}\rightarrow\mathbb{R}$ with $x_{0}=0$. $x$ is called
self-financing at $r\in O$ if

(2.24) $\sum_{t}\ P_{t}(r)x_{t}\ =0.$

$x$ is called a null investment at $r$ if for all $(h,r^{\prime})$ in some
neighborhood $[0,\epsilon)\times U$ of $(0,r)$ in $\mathbb{R}_{+}\times O$
with $[0,\epsilon]\cap{\rm supp}(x)=\emptyset$

(2.25) $\sum_{t}\ P_{t-h}(r^{\prime})x_{t}\ =0.$

$x$ is called a local arbitrage at $r$ if $x$ is self-financing at $r$ and if
for all $(h,r^{\prime})$ in some neighborhood $[0,\epsilon)\times U$ of
$(0,r)$ in $\mathbb{R}_{+}\times O$ with $[0,\epsilon]\cap{\rm
supp}(x)=\emptyset$

(2.26) $\sum_{t}\ P_{t-h}(r^{\prime})x_{t}\ \geq\ 0.$

We say that the model allows no local arbitrage, or satisfies condition NLA,
if the zero function, $x_{t}=0$ for all $t\in\mathbb{R}_{+}$, is the only
local arbitrage at any $r\in O$.

Thus, condition NLA has two aspects. First, it requires that $0$ is the only
null investment. This means that any two different bundles of futures can be
distinguished by some price movement near $r$ in the market. Notice that two
bundles of futures have the same value not only now with respect to $P_{t}(r)$
but at time $h$ from now at $P_{t}(r^{\prime})$, for all $(h,r^{\prime})$ near
$(0,r)$, exactly when the difference between the two bundles is null.
Secondly, the condition excludes entirely true arbitrage opportunities,
investments self-financing now for which no local motion leads to a loss and
some local motion leads to a profit.

When the model satisfies condition SPA the description of a local arbitrage
can be simplified.

######  Proposition 2.5.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined on the open set
$O\subset\mathbb{R}^{n}$. Assume that the model satisfies condition SPA with
$\varphi:\mathbb{R}_{+}\times O\rightarrow O$ the static price flow for the
model. Let $x$ be a self-financing investment at $r\in O$.

The bundle $x$ is a null investment at $r$ if and only if for all $r^{\prime}$
in some neighborhood of $r$ in $O$

(2.27) $\sum_{t}\ P_{t}(r^{\prime})x_{t}\ =0.$

The bundle $x$ is a local arbitrage at $r$ if and only if for all $r^{\prime}$
in some neighborhood of $r$ in $O$

(2.28) $\sum_{t}\ P_{t}(r^{\prime})x_{t}\ \geq\ 0.$

Proof. We obtain (2.27) and (2.28) from (2.25) and (2.26) respectively by
setting $h=0$. Thus, (2.27) and (2.28) are always necessary. To prove
sufficiency in the SPA case we apply Lemma 2.3 to get
$\tilde{\varphi}:\tilde{O}\rightarrow O$, the extension of the static price
flow $\varphi$. Now let $U_{0}$ be a neighborhood of $r$ in $O$ such that
$r^{\prime}\in U_{0}$ implies (2.27) or (2.28) holds. We can choose
$\epsilon>0$ and a neighborhood $U$ of $r$ such that for
$(h,r^{\prime})\in(-\epsilon,\epsilon)\times U$,
$\tilde{\varphi}_{-h}(r^{\prime})\in U_{0}$. Shrinking $\epsilon>0$ we can
assume that $x_{t}=0$ for $t\leq\epsilon$. Now for
$(h,r^{\prime})\in[0,\epsilon)\times U$ apply (2.27) or (2.28) with
$r^{\prime}$ replaced by $\tilde{\varphi}_{-h}(r^{\prime})$. Divide through by
the common factor $P_{h}(\tilde{\varphi}_{-h}(r^{\prime}))$ and apply (2.22)
to get (2.25) or (2.26) .

$\Box$

Let $x$ be a bundle of futures. For an $n$ parameter yield curve model on $O$
we define the _value function_ of $x$, $V:O\rightarrow\mathbb{R}$ by

(2.29) $V(r)\ =\sum_{t}\ P_{t}(r)x_{t}.$

If $O$ is an open subset of $\mathbb{R}^{n}$ and the yield curve model allows
static prices then Proposition 2.5 says that $x$ is a local arbitrage at $r$
exactly when $V(r)=0$ and the value function $V$ has a local minimum at $r$.
Recall that if $V$ has a local minimum at $r$ then the partial derivatives
$\partial V/\partial r_{i}$ ($i=1,\ldots,n$) vanish at $r$, i.e. $r$ is a
critical point for $V$, and the Hessian, the matrix of second partials
$(\partial^{2}V/\partial r_{i}\partial r_{j})$, is positive semidefinite at
$r$. Conversely, if $r$ is a critical point for $V$ and the Hessian matrix at
$r$ is positive definite then $V$ has a strict local minimum at $r$.

###  2.3. Linearity (LIN)

We call a yield curve model _linear_ when the set of admissible yield
functions is convex in the vector space of all smooth real functions on
$\mathbb{R}_{+}$. Recall that a subset $O$ of a real vector space is convex
when $r_{0},r_{1}$ in $O$ implies the segment between them is contained in
$O$, i.e. $(1-t)r_{0}+tr_{1}\in O$ for all $0\leq t\leq 1$. For an $n$
parameter model we want the parametrization itself to be a linear function.

######  Definition 2.6.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined on $O$. We say
that the model is linear, or satisfies condition LIN, if $O$ is a convex open
subset of $\mathbb{R}^{n}$ and for each $t$ in $\mathbb{R}_{+}$ the function
$r\mapsto Y_{t}(r)$ is the restriction to $O$ of a real-valued linear function
from $\mathbb{R}^{n}$ to $\mathbb{R}$.

Notice that if $F:O\rightarrow\mathbb{R}$ is the restriction to $O$, open in
$\mathbb{R}^{n}$, of a linear function, then $F^{i}=\partial F/\partial r_{i}$
is constant on $O$ and $F(r_{1},\ldots,r_{n})\ =\sum^{n}_{i=1}\ r_{i}F^{i}$.
So if $Y_{t}(r)$ is a linear $n$ parameter yield curve model on $O$ then
$Y^{i}_{t}=\partial Y_{t}(r)/\partial r_{i}$ is a smooth real valued function
of $t$ which is independent of $r$ in $O$. Furthermore, for
$r=(r_{1},\ldots,r_{n})$ in $O$:

(2.30) $Y_{t}(r_{1},\ldots,r_{n})\ =\sum^{n}_{i=1}\ r_{i}Y^{i}_{t}.$

######  Lemma 2.7.

For a set $T$ let $\mathbb{R}^{T}$ denote the vector space of real valued
functions defined on $T$. Let $\\{Y^{1},\ldots,Y^{n}\\}$ be a list of
functions in $\mathbb{R}^{T}$ for some positive integer $n$ and let $O$ be a
nonempty open subset of $\mathbb{R}^{n}$. For $r\in O$ define $Y(r)\
=\sum^{n}_{i=1}\ r_{i}Y^{i}$ in $\mathbb{R}^{T}$. The following conditions are
equivalent

  * i.

The list $\\{Y^{1},\ldots,Y^{n}\\}$ is linearly independent in
$\mathbb{R}^{T}$.

  * ii.

The function $r\mapsto Y(r)$ from $O$ to $\mathbb{R}^{T}$ is injective.

  * iii.

The set of vectors $\\{(Y^{1}_{t},\ldots,Y^{n}_{t}):t\in T\\}$ spans
$\mathbb{R}^{n}$.

###### Proof.

(i) $\Leftrightarrow$ (ii): If $Y(r)=Y(r^{\prime})$ then $\sum_{i}\
c_{i}Y^{i}\ =0$ where $c_{i}=r_{i}-r^{\prime}_{i}$ ($i=1,\dots,n)$.
Conversely, if $\sum_{i}\ c_{i}Y^{i}\ =0$ and $r\in O$ then because $O$ is
open there is an $\epsilon>0$ small enough that $r_{\epsilon}\ =r+\epsilon
c\in O$. Clearly, $Y(r)\ =Y(r_{\epsilon})$.

(i) $\Leftrightarrow$ (iii): The set of vectors
$\\{(Y^{1}_{t},\ldots,Y^{n}_{t})\\}$ does not span $\mathbb{R}^{n}$ if and
only if there is a nonzero vector $c$ in $\mathbb{R}^{n}$ perpendicular to all
of them, i.e. satisfying $\sum_{i}\ c_{i}Y^{i}_{t}\ =0$ for all $t\in T$.

$\Box$

Remark. With the same proof we can replace the real field $\mathbb{R}$ with
the complex field $\mathbb{C}$ in the above lemma.

######  Definition 2.8.

For a linearly independent list $\\{Y^{i}:i=1,\ldots,n\\}$ of smooth functions
in $\mathbb{R}^{\mathbb{R}_{+}}$ the _associated linear model_ is the linear
map $Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ given by (2.30),
i.e. $Y(r)\ =\sum_{i}\ r_{i}Y^{i}$.

The associated positive domain and strict positive domain are the subsets $O$
and $O_{+}$ defined by

$O(Y^{1},\ldots,Y^{n})\ =\\{r\in\mathbb{R}^{n}:Y^{i}_{t}(r)>0\;{\rm
for\;all\;}t\in\mathbb{R}_{+},i=1,\ldots,n\\}$ (2.31)
$O_{+}(Y^{1},\ldots,Y^{n})\
=\\{r\in\mathbb{R}^{n}:\inf_{t}Y^{i}_{t}(r)>0,i=1,\ldots,n\\}.$

So $r\in O_{+}(Y^{1},\ldots,Y^{n})$ exactly when for some $\delta>0$,
$Y^{i}_{t}(r)\geq\delta$ for all $t\in\mathbb{R}_{+},i=1,\dots,n$. It easily
follows that $O(Y^{1},\ldots,Y^{n})$ and $O_{+}(Y^{1},\ldots,Y^{n})$ are
convex subsets of $\mathbb{R}^{n}$.

######  Lemma 2.9.

If $\\{Y^{i}:i=1,\ldots,n\\}$ is a linearly independent list of bounded,
smooth functions in $\mathbb{R}^{\mathbb{R}_{+}}$, then
$O_{+}(Y^{1},\ldots,Y^{n})$ is open as well as convex.

###### Proof.

Assume $|Y^{i}_{t}|\leq M$ for $i=1,\ldots,n$ and $t\in\mathbb{R}_{+}$. If
$Y^{i}_{t}(r)\geq\delta$ for all $t$ with $\delta>0$ and
$|r^{\prime}_{i}-r_{i}|\ <\ \delta/2nM$ for $i=1,\ldots,n$, then
$Y^{i}_{t}(r^{\prime})\geq\delta/2$ for all $t$.

$\Box$

######  Proposition 2.10.

If $Y_{t}(r)$ is a linear $n$ parameter yield curve model defined on a convex
open set $O$, then the list $\\{Y^{1},\ldots,Y^{n}\\}$ of smooth functions in
$\mathbb{R}^{\mathbb{R}_{+}}$ defined by

(2.32) $Y^{i}_{t}\ =\partial Y_{t}(r)/\partial r_{i}\;\;\;(i=1,\ldots,n)$

is linearly independent.

Conversely, if $\\{Y^{1},\ldots,Y^{n}\\}$ is a linearly independent list of
smooth functions in $\mathbb{R}^{\mathbb{R}_{+}}$ then the associated linear
model defines a linear yield curve model on any nonempty, convex, open set $O$
contained in the positive domain $O(Y^{1},\ldots,Y^{n})$.

###### Proof.

By Lemma 3.8 the map $Y:O\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ is injective
iff $\\{Y^{1},\ldots,Y^{n}\\}$ is linearly independent. The results then
follow from Definition 1.

$\Box$

Remark. It follows that for any linear yield curve model the function
$Y:O\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ extends uniquely as a linear map
from $\mathbb{R}^{n}$ to ${\bf R}^{\mathbb{R}_{+}}$. Thus, we use the formula
(2.30) to define $Y(r)$ for all $r$ in $\mathbb{R}^{n}$. In particular, a
linearly independent list $\\{Y^{1},\ldots,Y^{n}\\}$ defines a linear yield
curve model for some $O$ exactly when the interior of $O(Y^{1},\ldots Y^{n})$
in $\mathbb{R}^{n}$ is nonempty.

To combine conditions SPA and LIN we recall that for an $n\times n$ matrix
$A=(A^{i}_{j})$ the smooth matrix function defined for $h\in\mathbb{R}$:

(2.33) $E_{h}\ =\exp[hA]\ =\sum^{\infty}_{N=0}\ (hA)^{N}/N!$

is the fundamental solution for the linear system associated with $A$. That
is, differentiating with respect to $h$:

$E_{0}\ =I,\;\;{\rm the\;identity\;matrix}$
$\stackrel{{\scriptstyle\bullet}}{{E}}_{h}\ =E_{h}A\ =AE_{h},\;{\rm i.e.}$
(2.34) $\stackrel{{\scriptstyle\bullet}}{{E^{i}_{jh}}}\ =\sum^{n}_{k=1}\
E^{i}_{kh}A^{k}_{j}\ =\sum^{n}_{k=1}\ A^{i}_{k}E^{k}_{jh}.$

For $r\in\mathbb{R}^{n}$, define $\psi_{h}(r)\ =rE_{h}$ so that
$\psi_{h}(r)_{j}\ =\sum^{n}_{i=1}\ r_{i}E^{i}_{jh}$. Then $\psi_{h}(r)$ is the
solution of the linear initial value problem on $\mathbb{R}^{n}$.

$\psi_{0}(r)\ =r$ (2.35) $\stackrel{{\scriptstyle\bullet}}{{\psi}}_{h}(r)\
=\psi_{h}(r)A$

That is, regarded as a function from $\mathbb{R}\times\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$, $\psi$ is the linear solution flow for the linear
vectorfield defined by $\xi(r)\ =rA$.

######  Proposition 2.11.

Suppose that $\varphi:\mathbb{R}_{+}\times O\rightarrow O$ is the static price
flow of a linear $n$ parameter yield curve which is defined on the open convex
set $O$ and which satisfies condition SPA. For each $h\geq 0$, $\varphi_{h}$
is the restriction to $O$ of some linear automorphism of $\mathbb{R}^{n}$. In
fact, there is an $n\times n$ matrix $A$ such that for
$(h,r)\in\mathbb{R}_{+}\times O$, $\varphi_{h}(r)=rE_{h}$ where $E$ is the
fundamental solution for the matrix differential equation associated with $A$.

###### Proof.

By hypothesis the map $Y:O\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ associating
to $r$ the function $Y(r)$ is the restriction of a linear map
$\tilde{Y}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$, and
$\tilde{Y}$ is injective because its restriction to $O$ is (apply Lemma 2.7).
For $h\geq 0$ the translation map
$\Phi_{h}:\mathbb{R}^{\mathbb{R}_{+}}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$
defined by (2.15) is clearly linear. Because $\Phi_{h}\circ Y\
=Y\circ\varphi_{h}$ on $O$ by (2.16) we can define the linear map
$\tilde{Y}^{-1}\circ\Phi_{h}\circ\tilde{Y}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
which restricts to $\varphi_{h}$ on $O$.

Because $\varphi_{h}$ is linear in the $r$ variables for each $h$, the same is
true of the derivative, the vectorfield $\xi:O\rightarrow\mathbb{R}^{n}$
defined by (2.20). That is, there exists an $n\times n$ matrix $A$ such that

(2.36) $\xi(r)\ =rA.$

Comparing (2.35) with (2.21), we see that uniqueness for solutions of smooth
differential equations implies that $\varphi_{h}$ is the restriction to $O$ of
$\psi_{h}.$

$\Box$

######  Corollary 2.12.

Let $Y_{t}(r)$ be a linear $n$ parameter yield curve model defined on $O$
which satisfies condition SPA. Let $A$ be the $n\times n$ matrix whose
fundamental solution $E$ defines the static price flow as described in
Proposition 2.11. There is a vector $(Y^{1}_{0},\ldots,Y^{n}_{0})$ in
$\mathbb{R}^{n}$ such that the short rate function $Y_{0}(r)$ is given by
$\sum^{n}_{i=1}\ r_{i}Y^{i}_{0}$ and for all $(t,r)\in\mathbb{R}_{+}\times O$,
and the basic yield curves satisfy:

$Y^{i}_{t}\ =\sum^{n}_{k=1}\ E^{i}_{kt}Y^{k}_{0}$ (2.37)
$\stackrel{{\scriptstyle\bullet}}{{Y}}_{t}^{i}\ =\sum^{n}_{k=1}\
A^{i}_{k}Y^{k}_{t}.$

###### Proof.

For the short rate apply (2.30) with $t=0$. For the first equation of (2.37)
apply (2.19) with $\varphi(r)\ =rE_{t}$. Then use the second equation in
(2.34) to compute $\stackrel{{\scriptstyle\bullet}}{{Y}}^{i}_{t}$.

$\Box$

###  2.4. Long Rates Exist (LRE)

For any yield curve $Y_{t}$ the _short rate_ is the positive number $Y_{0}$.
We define the _long rate_ to be $Lim_{t\rightarrow\infty}\ Y_{t}$ if this
limit exists as a finite real number. If the long rate exists then $Y$ is
bounded on $\mathbb{R}_{+}$. A bounded function may however fail to have a
limit $t$ as $t$ approaches infinity. The function may instead oscillate.

######  Definition 2.13.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model defined on $O$. We say
that the model has long rates, or satisfies condition LRE, if for each $r\in
O$

(2.38) $Y_{\infty}(r)\ \equiv\ Lim_{t\rightarrow\infty}\ \ Y_{t}(r)$

exists and is finite.

Remark. If the model satisfies LRE then for each $r\in O$, $Y(r)$ is a bounded
smooth function in $\mathbb{R}^{\mathbb{R}_{+}}.$

As usual, for the linear case there are special results.

######  Proposition 2.14.

Let $Y_{t}(r)\ =\sum_{i}\ r_{i}Y^{i}_{t}$ be a linear $n$ parameter yield
curve model. The model has long rates if and only if for $i=1,\ldots,n$, the
limit $Y^{i}_{\infty}\ =Lim_{t\rightarrow\infty}\ Y^{i}_{t}\ $ exists. In that
case the function $Y_{\infty}(r)$ is well defined by (2.38) for all
$r\in\mathbb{R}^{n}$ and

(2.39) $Y_{\infty}(r)\ =\sum^{n}_{i=1}\ r_{i}Y^{i}_{\infty}.$

###### Proof.

Those $Y$ in $\mathbb{R}^{\mathbb{R}_{+}}$ for which the limit $Y_{\infty}$
exists constitute a linear subspace. This subspace contains the image of the
linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{\mathbb{R}_{+}}$ given by the
model when it contains the $Y(r)$’s as $r$ varies over some basis for
$\mathbb{R}^{n}$. So containing the $Y^{i}$’s is sufficient as is containing
the $Y(r)$’s for $r$ varying over some open set in $\mathbb{R}^{n}$. Equation
(2.39) is then clear from the linearity of taking limits.

$\Box$

###  2.5. Main Results

In comparing two yield curve models we say that the first is included in the
second if every yield curve admissible with respect to the first model is
admissible for the second. In comparing two linear models we have special
results.

######  Definition 2.15.

Assume that $\\{Y^{1},\ldots,Y^{n}\\}$ and
$\\{\tilde{Y}^{1},\ldots,\tilde{Y}^{m}\\}$ are linearly independent lists of
smooth functions in $\mathbb{R}^{\mathbb{R}_{+}}$ with associated linear
models $Y_{t}(r)=\sum^{n}_{i=1}\ r_{i}Y^{i}_{t}$ and
$\tilde{Y}_{t}(s)=\sum^{m}_{j=1}\ s_{j}\tilde{Y}^{j}_{t}$. We say that the
$\tilde{Y}_{t}(s)$ model includes the $Y_{t}(r)$ model when the following
equivalent conditions hold.

  1. (1)

There exists a nonempty open subset $O_{1}\subset\mathbb{R}^{n}$ such that for
each $r\in O_{1}$ there exists $s\in\mathbb{R}^{m}$ such that $Y_{t}(r)\
=\tilde{Y}_{t}(s)$ for all $t\in\mathbb{R}_{+}$.

  2. (2)

There is an injective linear mapping
$S:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ such that for all
$r\in\mathbb{R}^{n}$, $Y(r)=\tilde{Y}(S(r))$.

In that case, $S$ maps $O(Y^{1},\ldots,Y^{n})$ into
$O(\tilde{Y}^{1},\ldots,\tilde{Y}^{m})$ and $O_{+}(Y^{1},\ldots,Y^{n})$ into
$O_{+}(\tilde{Y}^{1},\ldots,\tilde{Y}^{m})$.

Proof of equivalence. (1) $\Leftrightarrow$ (2): By assumption the linear maps
$Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ and
$\tilde{Y}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ are
injective. Assumption (1) says that on the nonempty open set $O_{1}$ the map
$\tilde{Y}^{-1}\circ Y$ is well-defined. As $O_{1}$ spans $\mathbb{R}^{n}$ the
image of $Y$ must lie in the image of $\tilde{Y}$ by linearity. Hence,
$\tilde{Y}^{-1}\circ Y:O_{1}\rightarrow\mathbb{R}^{m}$ extends to define the
linear map $S$.

The converse is obvious.

Since $Y(r)=\tilde{Y}(S(r))$ it is clear that $r$ is in $O$ or $O_{+}$ if and
only if $S(r)$ is.

$\Box$

We are now ready to state our main results.

######  Theorem 2.16.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model which is linear and
allows static prices, i.e. $Y_{t}(r)$ satisfies SPA and $LIN$. If the model
allows no local arbitrage, i.e. satisfies condition NLA, then it includes one
of the following four linear models:

  1. (1)

With $\rho\neq 0$ fixed, let

(2.40) $Y_{t}(r_{1},r_{2})\ =r_{1}e^{\rho t}+r_{2}e^{2\rho t}.$

  2. (2)

With $\rho\neq 0$ and $\omega>0$ fixed, let

(2.41) $Y_{t}(r_{1},r_{2},r_{3})\ =r_{1}e^{\rho t}\cos\omega t+r_{2}e^{\rho
t}\sin\omega t+r_{3}e^{2\rho t}.$

  3. (3)

Let

(2.42) $Y_{t}(r_{1},r_{2})\ =r_{1}+r_{2}t.$

  4. (4)

With $\omega>0$ fixed, let

(2.43) $Y_{t}(r_{1},r_{2},r_{3})\ =r_{1}\cos\omega t+r_{2}\sin\omega t.$

Conversely, if the yield curve model includes either model (1), (2) or (3)
then it satisfies NLA.

The proof of all this occupies the next section. We will also describe the
oddities associated with model (4). Because model (3) contains unbounded
functions and model (4) has undamped oscillation we regard them as anomalous.
Also in (1) and (2), $\rho>0$ leads to unbounded functions.

There are some positivity problems in the above linear models. In (1) if
$\rho$ is negative, the set $O_{+}(e^{\rho t},e^{2\rho t})$ is empty, although
$\\{(r_{1},r_{2}):r_{1},r_{2}>0\\}$ is contained in $O(e^{\rho t},e^{2\rho
t})$. In (2) we can write $(r_{1},r_{2})$ in polar coordinates as
$(a\cos\phi,a\sin\phi)$ and then model (2) can be written as $e^{\rho
t}[a\cos(\omega t-\phi)+r_{3}e^{\rho t}]$. It follows that if $\rho$ is
negative, then $O(e^{\rho t}\cos\omega t,e^{\rho t}\sin\omega t,e^{2\rho t})$
is empty. Similarly, in (4) $O(\cos\omega t,\sin\omega t)$ is empty. These
problems are eliminated if we also demand that the model includes the flat
rate model. That is,

(2.44) $\displaystyle\begin{split}&O_{+}(1,e^{\rho t},e^{2\rho t}),\\\
&O_{+}(1,e^{\rho t}\cos\omega t,e^{\rho t}\sin\omega t,e^{2\rho t}),\\\
&O_{+}(1,t)\quad\text{and}\quad O_{+}(1,\cos\omega t,\sin\omega t)\end{split}$

are nonempty for all $\rho$ and $\omega$.

The two simple yield curve models in the title come from the following:

######  Corollary 2.17.

Let $Y_{t}(r)$ be an $n$ parameter yield curve model which is linear, allows
static prices and has long rates, i.e. $Y_{t}(r)$ satisfies SPA, LIN and LRE.
The model contains the flat yield curve model and allows no local arbitrage,
i.e. satisfies NLA, if and only if it includes one of the following two yield
curve models.

  1. (1)

Exponential Model (3 parameters): With $\rho>0$ fixed, let

(2.45) $Y_{t}(r_{1},r_{2},r_{3})\ =r_{1}+r_{2}e^{-\rho t}+r_{3}e^{-2\rho t},$

defined on the open convex set $O_{+}(1,e^{-\rho t},e^{-2\rho t})$. The short
rate is $r_{1}+r_{2}+r_{3}$ and the long rate is $r_{1}$.

  2. (2)

Exponential-Oscillation Model (4 parameters): With $\rho,\omega>0$ fixed, let

$Y_{t}(r_{1},r_{2},r_{3},r_{4})\ =r_{1}+r_{2}e^{-\rho t}\cos\omega t+$ (2.46)
$r_{3}e^{-\rho t}\sin\omega t+r_{4}e^{-2\rho t},$

defined on the open convex set $O_{+}(1,e^{-\rho t}\cos\omega t,e^{-\rho
t}\sin\omega t,e^{-2\rho t})$ The short rate is $r_{1}+r_{2}+r_{4}$ and the
long rate is $r_{1}$.

By choosing time units property, i.e. by a constant rescaling, we can assume
$\rho=1$ in the models.

###### Proof.

In each case the $r_{1}$ coordinate comes from the assumption that the flat
yield curve model is included. Theorem 2.16 says that any model which includes
one of these satisfies NLA because (1) and (2) here contain (1) and (2) of the
theorem, respectively. Conversely, condition LRE excludes unbounded growth,
i.e. model (3) of the theorem and $\rho>0$ in models (1)and (2). It also
excludes the undamped oscillation of model (4). So the theorem implies either
model (1) or (2) here is included.

The rescaling in the last paragraph is the replacement of the time variable
$t$ by $\rho t$. $\hfill\Box$

##  3\. Arbitrage in Linear Models

Linear models occur as subspaces of the real vector space $\mathbb{R}^{T}$ for
some set $T$. As $\mathbb{R}$ is a subspace of $\mathbb{C}$, we can regard
$\mathbb{R}^{T}$ as a real subspace of the complex vector space
$\mathbb{C}^{T}$. For $g\in\mathbb{C}^{T}$ we can write
$g=f_{1}+\mathbf{i}f_{2}$, with $f_{1},f_{2}\in\mathbb{R}^{T}$ and we write
$f_{1}=Re(g),f_{2}=Im(g)$ with the conjugate $\bar{g}=f_{1}-\mathbf{i}f_{2}$.

For $V$ a subspace of $\mathbb{R}^{T}$ we define $V_{\mathbb{C}}$ to be the
complex subspace of $\mathbb{C}^{T}$ given by:

(3.1) $V_{\mathbb{C}}\ =\ \\{f_{1}+\mathbf{i}f_{2}:f_{1},f_{2}\in V\\}.$

So, for example, $\mathbb{C}^{T}=(\mathbb{R}^{T})_{\mathbb{C}}$.

######  Lemma 3.1.

Let $W$ be a complex subspace of $\mathbb{C}^{T}$. There exists a real
subspace $V$ of $\mathbb{R}^{T}$ such that $W=V_{\mathbb{C}}$ if and only if
$W$ satisfies the following equivalent conditions.

  * (i)

If $g\in W$, then $\bar{g}\in W$.

  * (ii)

If $g\in W$, then $Re(g)\in W$.

  * (iii)

If $g\in W$, then $Re(g),Im(g)\in W$.

In that case,

(3.2) $V\ =\ W\cap\mathbb{R}^{T}\ =\ \\{Re(g):g\in W\\}\ =\
\\{Re(g),Im(g):g\in W\\},$

and for any index set $I$, if $\\{g_{i}:i\in I\\}$ spans $W$, then
$\\{Re(g_{i}),Im(g_{i}):i\in I\\}$ spans $V$.

Proof: (i) $\Rightarrow$ (ii): $Re(g)=\frac{1}{2}(g+\bar{g})$.

(ii) $\Rightarrow$ (iii): $Im(g)=-Re(\mathbf{i}g)$.

(iii) $\Rightarrow$ (i): $\bar{g}=Re(g)-\mathbf{i}Im(g).$

Clearly, $V_{\mathbb{C}}$ satisfies (iii). On the other hand, given (i)-(iii)
it is clear that $W=V_{\mathbb{C}}$ for $V$ given by (3.2).

Finally, $g\ =\ \sum_{i}\ a_{i}g_{i}$ implies
$Re(g)=\sum_{i}Re(a_{i})Re(g_{i})-Im(a_{i})Im(g_{i})$ (with $a_{i}=0$ except
for finitely many $i\in I$).

$\Box$

Throughout this section we consider linear models of the form
$Y_{t}(r_{1},\ldots,r_{n})\ =\sum^{n}_{i=1}r_{i}Y^{i}_{t}$, which we regard as
a linear mapping $Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ with
$\\{Y^{1}_{t}(r),\dots,Y^{n}_{t}(r)\\}$ a linearly independent list of smooth
functions in $\mathbb{R}^{\mathbb{R}_{+}}.$ For a yield curve model, we
restrict to some nonempty convex open set $O\subset\mathbb{R}^{n}$ such that
$r\in O$ implies $Y^{i}_{t}(r)>0$ for all $t\geq 0$ and $i=1,\dots,n$. We can
regard the log-price as the linear function
$L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ given by

$L_{t}(r_{1},\ldots,r_{n})\ =\sum^{n}_{i=1}\ r_{i}L^{i}_{t}$ (3.3) $L^{i}_{t}\
=\int^{t}_{0}\ Y^{i}_{s}\ ds.$

So, in particular, $L_{0}(r)=0$ for all $r\in\mathbb{R}^{n}$. The price-
function extends as well to a nonlinear function
$P:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ by

(3.4) $P_{t}(r_{1},\ldots,r_{n})\ =\exp[-\sum^{n}_{i=1}\ r_{i}L^{i}_{t}].$

For $r\in O$ the price and log-price curves satisfy (2.4) and (2.7).

Because of the linear structure of the models we can describe investment
procedures which might lead to an arbitrage possibility at every $r\in O$.

Let $z_{t}$ be a finite function on $(0,\infty)$. That is, it is a real-valued
function on $\mathbb{R}_{+}$ whose support is a finite subset excluding $0$.
Since the function $z_{t}$, once chosen, will be fixed during some extended
arguments it will be useful to introduce the notation:

(3.5) $\langle f\rangle\ =\sum_{t}\ z_{t}f(t)$

for any function $f$ in $\mathbb{R}^{\mathbb{R}_{+}}$ or in
$\mathbb{C}^{\mathbb{R}_{+}}$.

Think of $z_{t}$ as the positive or negative amount of current dollars which
we allocate to futures $t$ time units from now. If the current price curve is
$P_{t}(r^{*})$ then the quantity $x_{t}$ of time $t$ futures with current
worth $z_{t}$ is given by

(3.6) $x_{t}\ =z_{t}/P_{t}(r^{*}).$

So the present value of the bundle of futures $x_{t}$ is given by

(3.7) $\sum\ x_{t}P_{t}(r^{*})\ =\langle 1\rangle.$

For any $r\in\mathbb{R}^{n}$ the value function, defined by (2.29), is at $r$
given by

(3.8) $V(r)\ =\langle P(r-r^{*})\rangle$

because for each $t$ $P_{t}(r)=P_{t}(r^{*})P_{t}(r-r^{*})$ by (3.4). Taking
the partial derivative with respect to $r_{i}$ we see that

(3.9) $\frac{\partial V}{\partial r_{i}}|_{r=r^{*}}\ =-\langle L^{i}\rangle$

and the Hessian matrix is given by

(3.10) $\frac{\partial V}{\partial r_{i}\partial r_{j}}|_{r=r^{*}}\ =\langle
L^{i}L^{j}\rangle.$

Thus, when $r=r^{*}$ the investment $x_{t}$ is self-financing provided
$\langle 1\rangle=0$. It is then a critical point for the value function when
$\langle L^{i}\rangle=0$ for $i=1,\ldots,n$. It is a strict local minimum when
the $n\times n$ Hessian matrix $\langle L^{i}L^{j}\rangle$ is positive
definite. By Proposition 2.5, if the linear yield curve model satisfies SPA,
then for $r^{*}\in O$ such a local minimum is a local arbitrage for the model.

We can reverse this analysis as well. If $x_{t}$ is a bundle of futures with
current prices given by $P_{t}(r^{*})$ we can use (3.6) to define the finite
function $z_{t}$.

The following _No Arbitrage Lemma_ does not require the SPA condition. We
postpone the proof.

######  Lemma 3.2.

For the $n$ parameter linear yield curve model whose log-price function $L$ is
given by (3.3) let $L(\mathbb{R}^{n})$ be the linear subspace of
$\mathbb{R}^{\mathbb{R}_{+}}$ which is the image of $L$. Assume that for some
positive integer $p$ there exist functions $F^{1},\ldots,F^{p}$ in
$L(\mathbb{R}^{n})$ which satisfy the following conditions:

  1. (1)

The function $t\mapsto(F^{1}_{t},\ldots,F^{p}_{t})$ is an injective function
from $\mathbb{R}_{+}$ to $R^{p}$, i.e. if $t_{1}\neq t_{2}$ then for some
$j=1,\ldots,p$, $F^{j}_{t_{1}}\neq F^{j}_{t_{2}}$.

  2. (2)

The function $\sum^{p}_{j=1}\ (F^{j})^{2}$, the sum of the squares, is in the
subspace $L(\mathbb{R}^{n})$.

The model satisfies condition NLA. In fact, if $z_{t}$ is a finite function on
$(0,\infty)$ not identically zero but satisfying $\langle 1\rangle=0$ and
$\langle L^{i}\rangle=0$ for $i=1,\ldots,n$, then for any $r^{*}$ in
$\mathbb{R}^{n}$, the value function defined by using
$x_{t}=z_{t}/P_{t}(r^{*})$ has a critical point at $r=r^{*}$ which is not a
local minimum.

Now we assume that the linear model satisfies condition SPA. By Corollary 2.12
the yield functions are solutions of linear ordinary differential equations
with constant coefficients. For such functions it is useful to introduce some
special notation.

In the additive group $\mathbb{Z}\times\mathbb{C}$ the set of _exponents_
$\mathcal{E}$ consists of the elements whose integer coordinate is
nonnegative. So an exponent $q$ is a pair $(m,\lambda)$ with $m$ a nonnegative
integer and $\lambda$ a complex number. For $q=(m,\lambda)$ with
$\lambda=a+ib$ we define the conjugate $\overline{q}=(m,\overline{\lambda})$,
the real part $\rho(q)=(m,a)$ and the imaginary part $\omega(q)=b$. Define

(3.11) $q^{\prime}\ =(\max(0,m-1),\lambda)\quad\mbox{for}\quad q\
=(m,\lambda).$

so that $q^{\prime}=q$ when $q=(0,\lambda)$.

For the exponent $q=(m,\lambda)$ define the function
$f^{q}:\mathbb{R}\rightarrow\mathbb{C}$ by

$f^{q}(t)=t^{m}\exp(\lambda t)$ (3.12) $\
=f^{\rho(q)}(t)[\cos(\omega(q)t)+i\sin(\omega(q)t)].$

Clearly $f^{\overline{q}}\ =\overline{f^{q}}$ and $f^{q_{1}+q_{2}}\
=f^{q_{1}}\cdot f^{q_{2}}$ for $q,q_{1},q_{2}\in\mathcal{E}$. In addition, the
derivative satisfies

(3.13) $\stackrel{{\scriptstyle\bullet}}{{f^{q}}}\ =\lambda
f^{q}+mf^{q^{\prime}}\;\;\;(q=(m,\lambda)).$

A _closed set_ of exponents $Q$ is a finite subset of $\mathcal{E}$ such that
$q\in Q$ implies $\overline{q}$, $q^{\prime}\in Q$. For a closed set $Q$ and a
complex number $\lambda$ we define the _multiplicity_ of $\lambda$ in $Q$ to
be

(3.14) $\min\\{m\geq 0:(m,\lambda)\not\in Q\\}.$

######  Lemma 3.3.

(a) Let $Q$ be any finite subset of $\mathcal{E}$ and $T$ be any nonempty open
subset of $\mathbb{R}$. The set of restrictions to $T$ of the functions
$\\{f^{q}:q\in Q\\}$ is linearly independent in the complex vector space
$\mathbb{C}^{T}$.

(b) Let $V$ be a finite dimensional subspace of $\mathbb{R}^{\mathbb{R}_{+}}$
consisting of differentiable functions. If $V$ is spanned by $\\{{\rm
Re}(f^{q}),{\rm Im}(f^{q}):q\in Q\\}$ for some closed set of exponents $Q$,
then $V$ is closed under differentiation, i.e. $f\in V$ implies
$\stackrel{{\scriptstyle\bullet}}{{f}}\in V$. Conversely, if $V$ is closed
under differentiation, then $Q\ =\\{q\in\mathcal{E}:{\rm Re}(f^{q}),{\rm
Im}(f^{q})\in V\\}$ is a closed set of exponents such that $\\{{\rm
Re}(f^{q}),{\rm Im}(f^{q}):q\in Q\\}$ spans $V$. In particular, if
$\\{Y^{1},\ldots,Y^{n}\\}$ is a list of differentiable functions such that

(3.15) $\stackrel{{\scriptstyle\bullet}}{{Y^{i}_{t}}}\ =\sum^{n}_{k=1}\
A^{i}_{k}Y^{k}_{t}\;\;\;(i=1,\ldots,n),$

then the vector space $V$ spanned by $\\{Y^{1},\ldots,Y^{n}\\}$ is spanned by
the real and imaginary parts of $f^{q}$ for $q$ varying in such a closed set
of exponents.

###### Proof.

(a) Denote by $D$ the derivative operator on the subspace of differentiable
functions in $\mathbb{C}^{\mathbb{R}_{+}}$, i.e.
$Df=\stackrel{{\scriptstyle\bullet}}{{f}}$. Suppose that $\sum\ c_{q}f^{q}$ is
identically zero on the open set $T$. Let $q_{0}=(m_{0},\lambda_{0})$ be an
exponent such that $c_{q_{0}}\neq 0$. For each $\lambda\neq\lambda_{0}$ apply
the operator $(D-\lambda)$ $m$ times where $m$ is the multiplicity of
$\lambda$ in $Q$, and $D-\lambda_{0}$ $m_{0}-1$ times. The only term remaining
in the relation is $c_{q_{0}}$ times a nonzero number times
$f^{(0,\lambda_{0})}$. This term is not identically zero on $T$ unless
$c_{q_{0}}=0$ after all.

(b) The first statement is obvious by (3.13). For the converse, assume $V$ is
closed under differentiation so that the complex subspace
$V_{\mathbb{C}}=\\{g\in\mathbb{C}^{\mathbb{R}_{+}}:{\rm Re}(g),{\rm Im}(g)\in
V\\}$ and $Q\ =\\{q\in\mathcal{E}:f^{q}\in V_{\mathbb{C}}\\}$. Since $V$ and
hence $V_{\mathbb{C}}$ are finite dimensional $Q$ is a finite subset of
$\mathcal{E}$ by (a). If $q\in Q$, then $\overline{q}\in Q$ because
$V_{\mathbb{C}}$ is closed under conjugation. By (3.13) $q^{\prime}\in Q$ when
$m>0$. Since $q^{\prime}=q$ when $m=0$, it follows that $Q$ is closed.

The differentiation operator $D$ is a linear map of $V_{\mathbb{C}}$ to itself
and so we can choose a basis $\\{Y^{1},\ldots,Y^{n}\\}$ so that $D$ is
represented by a matrix in Jordan canonical form or, to be more precise, so
that each $k\times k$ Jordan bloc is of the form

(3.16) $\left(\begin{array}[]{ccccc}\lambda&k-1&0&\ldots&0\\\
0&\lambda&k-2&\ldots&0\\\ 0&\ldots&\ldots&\lambda&1\\\
0&\ldots&\ldots&\ldots&\lambda\end{array}\right).$

So we see from (3.13) that each $Y^{i}$ can be chosen a function $f^{q}$ with
$q=(m,\lambda)$ and $\lambda$ an eigenvalue of the matrix. Consequently, the
$f^{q}$’s span $V_{\mathbb{C}}$ and so the Re$(f^{q})$’s and Im$(f^{q})$’s
span $V$ by Lemma 3.1.

$\Box$

Remark. Regard $q\in\mathbb{C}^{n}$ and $t\in\mathbb{R}^{n}$ as column vectors
i.e. $n\times 1$ matrices, and define

(3.17) $F^{q}(t)\ =\exp[q^{T}t]\ =\exp[\sum^{n}_{i=1}\ q_{i}t_{i}].$

For any finite subset $Q$ of $\mathbb{C}^{n}$ and any nonempty open subset $O$
of $\mathbb{R}^{n}$ the set of restrictions to $O$ of the functions
$\\{F^{q}:q\in Q\\}$ is linearly independent in the complex vector space
$\mathbb{C}^{O}$. The proof proceeds as in (a) using partial derivative
operators.

######  Corollary 3.4.

For the $n$ parameter linear yield curve model whose yield function is
$Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$, let
$Y(\mathbb{R}^{n})$ denote the linear subspace of
$\mathbb{R}^{\mathbb{R}_{+}}$ which is the image of $Y$. If the model
satisfies condition SPA, then there is a closed set of exponents $Q$ such that

$\\{f^{q}:q\in Q\;{\rm with}\;\omega(q)=0\\}\cup$ (3.18) $\\{{\rm
Re}(f^{q}),{\rm Im}(f^{q}):q\in Q\;{\rm with}\;\omega(q)>0\\}$

is a basis for $Y(\mathbb{R}^{n})$.

###### Proof.

The yield curve function is given by (2.30) with $\\{Y^{1},\ldots,Y^{n}\\}$
linearly independent by Proposition 2.10. So letting $V=Y(\mathbb{R}^{n})$ we
see that $V$ is an $n$ dimensional subspace of functions to which part (b) of
the lemma applies because Corollary 2.12 implies (3.15). It follows that there
is a closed set of exponents $Q$ such that the functions listed in (3.18) span
$Y(\mathbb{R}^{n})$. The functions $\\{f^{q}:q\in Q\\}$ span the complex space
$V_{\mathbb{C}}$ and so by part (a) of the lemma they form a basis for
$V_{\mathbb{C}}$. If $\omega(q)=0$ then $f^{q}$ is a real function and if
$\omega(q)>0$ then Re$(f^{q})\ =(f^{q}+f^{\overline{q}})/2$ while Im$(f^{q})\
=(f^{q}-f^{\overline{q}})/2i$. So the real functions listed in (3.18) form a
linearly independent set as well.

$\Box$

The set $Q$ of exponents in Corollary 3.4 is given by:

(3.19) $Q\ =\\{q\in\mathcal{E}:{\rm Re}(f^{q}),{\rm Im}(f^{q})\in
Y(\mathbb{R}^{n})\\}.$

We call this set the exponent set for the linear yield curve model $Y_{t}(r)$,
defined when the model satisfies SPA.

Now for any exponent $q$ define the exponent $\tilde{q}$ by

(3.20) $\tilde{q}\ =\left\\{\begin{array}[]{cll}q&{\rm if}\;q=(m,\lambda)&{\rm
with}\;\lambda\neq 0\\\ (m+1,0)&{\rm if}\;q=(m,0)&\end{array}\right.$

In particular, $\tilde{0}=(1,0)$. For any set of exponents $Q$ let $\tilde{Q}\
=$
$\\{\tilde{q}:q\in Q\\}$. Clearly, $0\not\in\tilde{Q}$.

For $q$ any nonzero exponent, i.e. $q\neq(0,0)$, define

(3.21) $l^{q}(t)\ =f^{q}(t)-f^{q}(0),$

so that $l^{q}=f^{q}$ if $m>0$ and $l^{q}=f^{q}-1$ if $m=0$.

######  Lemma 3.5.

Assume that $Y:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ and
$L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\mathbb{R}_{+}}$ are the yield and
log-price functions for a linear yield curve model satisfying SPA. Let $Q$ be
the exponent set for the yield curve model.

$\\{l^{q}:q\in\tilde{Q}\;{\rm with}\;\omega(q)=0\\}\cup$ (3.22) $\\{{\rm
Re}(l^{q}),{\rm Im}(l^{q}):q\in\tilde{Q}\;{\rm with}\;\omega(q)>0\\}$

is a basis for $L(\mathbb{R}^{n})$.

###### Proof.

From (3.13) we have, for $q=(m,\lambda)$:

(3.23) $l^{q}(t)\
=\lambda\int^{t}_{0}f^{q}(s)ds+m\int^{t}_{0}f^{q^{\prime}}(s)ds.$

Using induction on $m$, it easily follows that $\\{l^{q}:q\in\tilde{Q}\\}$ is
a basis for the complex vector space $V_{\mathbb{C}}$ where
$V=L(\mathbb{R}^{n})$. So the list in (3.22) is a basis for the real vector
space $L(\mathbb{R}^{n})$ just as in Corollary 3.4.

$\Box$

We will call $\tilde{Q}$ the adjusted exponent set for the yield curve model
when $Q$ is the exponent set.

We are now ready to state our second technical result, the Arbitrage
Everywhere Lemma.

######  Lemma 3.6.

For a linear $n$ parameter yield curve model defined on $O$ and which
satisfies SPA, let $\tilde{Q}$ be the adjusted exponent set. Assume that for
all $q\in\tilde{Q}$, $q+\overline{q}\not\in\tilde{Q}\cup\\{0\\}$. The model
admits strict local arbitrage at every parameter value $r\in O$. In fact,
there is a finite function $z_{t}$ whose support is a subset of $(0,\infty)$
consisting of at most $\frac{1}{2}(n+1)(n+2)$ times, so that for any $r^{*}$
in $\mathbb{R}^{n}$ the value function defined using $x_{t}\
=z_{t}/P_{t}(r^{*})$ has a strict local minimum at $r=r^{*}$.

$\Box$

Again we postpone the proof, first showing how Lemmas 3.2 and 3.6 yield the
main result, stated in the previous section.

Proof of Theorem 2.16: Since the model is linear and satisfies SPA we can
apply the contrapositive of Lemma 3.6. When the model satisfies NLA there must
exist $q\in\tilde{Q}$ such that $q+\overline{q}\in\tilde{Q}\cup\\{0\\}$.
Suppose $q=(m,\lambda)$ so that
$q+\overline{q}=(2m,\lambda+\overline{\lambda})$. Let $q_{0}=(0,\lambda)$. We
number the four possibilities following the statement of the theorem.

(1) $\lambda=\rho+i0$ with $\rho\neq 0$: By (3.20) $q=(m,\rho)$ and
$q+\overline{q}=(2m,2\rho)$ are in $Q$ which is closed. Hence,
$q_{0}=(0,\rho)$ and $q_{0}+\overline{q}_{0}=(0,2\rho)$ are in $Q$.
$f^{q_{0}}(t)=\exp(\rho t)$ and $f^{q_{0}+\overline{q}_{0}}(t)\ =\exp(2\rho
t)$ are in $Y(\mathbb{R}^{n})$.

(2) $\lambda=\rho+i\omega$ with $\rho\neq 0$ and $\omega\neq 0$: Replacing $q$
by $\overline{q}$ if necessary we may assume $\omega>0$. Again $q$ and
$q+\overline{q}=(2m,2\rho)$ are in $Q$, and again $q_{0}$ and
$q_{0}+\overline{q}_{0}=(0,2\rho)$ are in $Q$. Hence, Re$(f^{q_{0}}(t))\
=\exp(\rho t)\cos\omega t$, Im$(f^{q_{0}}(t))\ =\exp(\rho t)\sin\omega t$ and
$f^{q_{0}+\overline{q}_{0}}(t)\ =\exp(2\rho t)$ are in $Y(\mathbb{R}^{n})$.

(4) $\lambda=0+i\omega$ with $\omega\neq 0$: Again we can assume $\omega>0$
and as before $q$ and $q_{0}=(0,i\omega)$ are in $Q$. Hence,
Re$(f^{q_{0}}(t))\ =\cos\omega t$ and Im$(f^{q_{0}}(t))\ =\sin\omega t$ are in
$Y(\mathbb{R}^{n})$.

(3) $\lambda=0+i0$: Since $q=(m,0)\in\tilde{Q}$, $m\geq 1$. Furthermore,
$q+\overline{q}=(2m,0)\in\tilde{Q}$. By (3.20), $(m-1,0)$ and $(2m-1,0)$ are
in $Q$. Because $Q$ is closed $(0,0)$ and $(1,0)$ are in $Q$. Hence, $1\
=t^{0}\exp(0t)$ and $t\ =t^{1}\exp(0t)$ are in $Y(\mathbb{R}^{n})$.

To show that models which contain (1), (2) and (3) satisfy the NLA condition
we apply Lemma 3.2 using the basis in each case given by Lemma 3.5.

(1) With $q=(0,\rho),$ let $F=l^{q}$ so that $F(t)=\exp(\rho t)-1$.
$(F(t))^{2}=(\exp(2\rho t)-1)-2(\exp(\rho
t)-1)=l^{q+\overline{q}}(t)-2l^{q}(t)$. Hence, $F$, $(F)^{2}\in
L(\mathbb{R}^{n})$. Furthermore, $F:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$
is injective.

(2) With $q=(0,\rho+i\omega),$ let $F_{1}(t)={\rm Re}(l^{q}(t))\ =\exp(\rho
t)[\cos(\omega t)]-1$ and $F_{2}(t)={\rm Im}(l^{q}(t))\ =\exp(\rho
t)[\sin(\omega t)]$. $(F_{1}(t))^{2}+(F_{2}(t))^{2}\
=l^{q+\overline{q}}(t)-2F_{1}(t)$. Hence, $F_{1}$, $F_{2}$ and
$(F_{1})^{2}+(F_{2})^{2}$ are in $L(\mathbb{R}^{n})$. Since
$l^{q+\overline{q}}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is injective it
follows that $(F^{1},F^{2}):\mathbb{R}_{+}\rightarrow\mathbb{R}^{2}$ is
injective.

(3) With $(0,0)$ and $q=(1,0)$ in $Q$, $q$ and $q^{\prime}=(2,0)$ are in
$\tilde{Q}$. Let $F(t)\ =l^{q}(t)=t$ so that $(F(t))^{2}\
=t^{2}=l^{q^{\prime}}(t)$. Hence, $F$ and $(F)^{2}$ are in
$L(\mathbb{R}^{n})$. Clearly, $F:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is
injective.

(4) With $q=(0,i\omega)$, let $F_{1}(t)={\rm Re}(l^{q}(t))=\cos(\omega t)-1$
and $F_{2}(t)={\rm Im}(l^{q}(t))=\sin\omega t$.
$(F_{1}(t))^{2}+(F_{2}(t))^{2}=-2F_{1}(t)$. Hence, $F_{1}$, $F_{2}$ and
$(F_{1})^{2}+(F_{2})^{2}$ are in $L(\mathbb{R}^{n})$. However,
$(F_{1},F_{2}):\mathbb{R}_{+}\rightarrow\mathbb{R}^{2}$ is not injective
because $F_{1}$ and $F_{2}$ are $2\pi/\omega$ periodic.

Thus, Lemma 3.3 applies to cases (1), (2) and (3) but not to case (4).

$\Box$

Remark. If $\omega_{1},\omega_{2}>0$ and the ratio $\omega_{1}/\omega_{2}$ is
irrational then with $q_{\alpha}=(0,i\omega_{\alpha})$ $(\alpha=1,2)$ we
define $F_{1}(t)+iF_{2}(t)\ =l^{q_{1}}(t)$ and $F_{3}(t)+iF_{4}(t)\
=l^{q_{2}}(t)$. $\sum^{4}_{k=1}\ (F_{k}(t))^{2}\ =-2F_{1}(t)-2F_{3}(t)$.
Furthermore, $(l^{q_{1}},l^{q_{2}}):\mathbb{R}_{+}\rightarrow{\mathbb{C}}^{2}$
is injective. So Lemma 3.2 shows that any linear yield curve model which
satisfies SPA and which contains the four parameter linear model

(3.24) $Y(r_{1},r_{2},r_{3},r_{4})\ =\
r_{1}\cos\omega_{1}t+r_{2}\sin\omega_{1}t+r_{3}\cos\omega_{2}t+r_{4}\sin\omega_{2}t$

satisfies NLA.

Proof of Lemma 3.6: Define
$Q^{2}=\\{q_{1}+q_{2}:q_{1},q_{2}\in\tilde{Q}\cup\\{0\\}\\}$ and
$Q^{\\#}=\\{q+\overline{q}:q\in\tilde{Q}\\}$. Clearly, $Q^{\\#}$ consists of
real exponents only. $Q^{2}$ and $\tilde{Q}\cup\\{0\\}$ are closed because $Q$
is closed. $Q^{2}$ contains both $Q^{\\#}$ and $\tilde{Q}\cup\\{0\\}$. The
hypothesis of the lemma says exactly that $Q^{\\#}$ and $\tilde{Q}\cup\\{0\\}$
are disjoint, i.e.

(3.25) $\tilde{Q}\cup\\{0\\}\subset Q^{2}\backslash Q^{\\#}.$

Notice first that the assumptions imply that

(3.26) $\begin{split}(m,\mathbf{i}\omega)\not\in Q\quad\text{with}\ \
\omega\not=0,\\\ (m,0)\not\in Q\quad\text{with}\ \ m\geq 1.\end{split}$

If $q=(m,\mathbf{i}\omega)\in Q$ then $q_{0}=(0,\mathbf{i}\omega)\in\tilde{Q}$
and $(0,0)=q_{0}+\overline{q}_{0}\in\tilde{Q}\cup\\{0\\}$. If $q=(m,0)\in Q$
with $m\geq 1$, then $(0,0),(1,0)\in Q$ and so $(1,0),(2,0)\in\tilde{Q}$. With
$q_{0}=(1,0)\in\tilde{Q}$,
$(2,0)=q_{0}+\overline{q}_{0}\in\tilde{Q}\cup\\{0\\}$.

By Lemma 3.3a, the set $\\{f^{q}:q\in Q^{2}\\}$ restricts to a linearly
independent set in the complex vector space ${\mathbb{C}}^{T}$ for $T$ any
nonempty open set in $\mathbb{R}$. By Corollary 3.4 the exponent set $Q$ for
the $n$ parameter model has $n$ elements. So $Q^{2}$ has at most
$\frac{1}{2}(n+1)(n+2)$ elements. By Lemma 2.7 and the Remark thereafter, the
set $\\{f^{q}(t):t\in T\\}$ spans the vector space ${\mathbb{C}}^{(Q^{2})}$.
Thus, if we select for each $q\in Q^{\\#}$ a positive number $C_{q}$, then we
can choose a finite function $z_{t}$ whose support consists of at most
$\frac{1}{2}(n+1)(n+2)$ times in $T$, which we assume is contained in
$(0,\infty)$, such that

(3.27) $\begin{array}[]{ll}\langle f^{q}(t)\rangle=0&q\in Q^{2}\backslash
Q^{\\#}\\\ \langle f^{q}(t)\rangle=C_{q}&q\in Q^{\\#}.\end{array}$

Here we are using the notation of (3.5) $\langle
f(t)\rangle=\sum_{t}z_{t}f(t)$. It will be convenient to write $\langle
f(t)\rangle$ rather than $\langle f\rangle$ in these arguments.

A priori the numbers $z_{t}$ may be complex. But $f^{q}(t)$ is a real function
for $q\in Q^{\\#}$ and $\overline{f^{q}(t)}=f^{\overline{q}}(t)$. So
$Q^{2}-Q^{\\#}$ closed under conjugation implies that the conjugate vector
$\overline{z}_{t}$ would yield equations (3.27) as well. Hence, so would
$\\{{\rm Re}(z_{t})=\frac{1}{2}(z_{t}+\overline{z}_{t})\\}$. Thus, we can
choose $z_{t}$ to be a real vector.

Now let $\tilde{Q}_{0}=\\{q\in\tilde{Q}:\omega(q)=0\\}$ and
$\tilde{Q}_{+}=\\{q\in\tilde{Q}:\omega(q)>0\\}$. From (3.25) and (3.27) we can
take real and imaginary parts to get:

(3.28) $\begin{split}\langle 1\rangle=0,\qquad\langle
f^{q}(t)\rangle=0\;\;\;q\in\tilde{Q}_{0}\hskip 85.35826pt\\\ \langle
f^{\rho(q)}(t)\cos(\omega(q)t)\rangle=\langle
f^{\rho(q)}(t)\sin(\omega(q)t)\rangle=0\;\;\;\;q\in\tilde{Q}_{+}.\end{split}$

Furthermore, we will now prove that

$\langle(f^{q}(t))^{2}\rangle=C_{2q}\;\;\;\;q\in\tilde{Q}_{0}.$
$\langle[f^{\rho(q)}(t)\cos(\omega(q)t)]^{2}\rangle=\langle[f^{\rho(q)}(t)\sin(\omega(q)t)]^{2}\rangle=\frac{1}{2}C_{q+\overline{q},}$
(3.29)
$\langle[f^{\rho(q)}(t)\cos(\omega(q)t)][f^{\rho(q)}(t)\sin(\omega(q)t)]\rangle=0\;\;\;q\in\tilde{Q}_{+}.$

For $q\in\tilde{Q}_{0}$, $2q=q+\overline{q}\in Q^{\\#}$ and
$(f^{q}(t))^{2}=f^{2q}(t)$. So the first equation follows from (3.27) for
$2q$.

For $q\in\tilde{Q}_{+}$, $q+\overline{q}=2\rho(q)\in Q^{\\#}$ and so
$\sin^{2}+\cos^{2}=1$ implies

$\langle[f^{\rho(q)}(t)\cos(\omega(q)t)]^{2}\rangle+\langle[f^{\rho(q)}\sin(\omega(q)t)]^{2}\rangle$
(3.30) $=\langle f^{q+\overline{q}}(t)\rangle=C_{q+\overline{q}}.$

On the other hand, since $q+q$ is not real it is in $Q^{2}\backslash Q^{\\#}$
and so $\langle f^{2q}(t)\rangle=0$. Taking real and imaginary points we have

(3.31) $\begin{split}0=\langle f^{2\rho(q)}(t)\cos(2\omega(q)t)\rangle=\hskip
142.26378pt\\\
\langle[f^{\rho(q)}(t)\cos(\omega(q)t)]^{2}\rangle-\langle[f^{\rho(q)}(t)\sin(\omega(q)t)]^{2}\rangle.\hskip
85.35826pt\\\
\;\;0=\langle[f^{2\rho(q)}(t)\sin(2\omega(q)(t)]\rangle=2\langle[f^{\rho(q)}(t)\cos(\omega(q)t)][f^{\rho(q)}(t)\sin(\omega(q)t)]\rangle.\end{split}$

Together with (3.30) these imply the rest of (3.29).

By Lemma 3.5 we can use as the basis $\\{L^{i}:i=1,\ldots,n\\}$ for
$L(\mathbb{R}^{n})$ the set $\\{l^{q}:q\in\tilde{Q}_{0}\\}\cup\\{{\rm
Re}(l^{q}),{\rm Im}(l^{q}):q\in\tilde{Q}_{+}\\}$ listed in any order. Recall
from (3.21) that $l^{q}(t)=f^{q}(t)-f^{q}(0)$. Thus, up to an added constant,
the basis $\\{L^{i}:i=1,\ldots,n\\}$ consists of the set
$\\{f^{q}:q\in\tilde{Q}_{0}\\}\cup\\{{\rm Re}(f^{q}),{\rm
Im}(f^{q}):q\in\tilde{Q}_{+}\\}$ listed in the same order. Thus (3.28) implies
$\langle 1\rangle=0$ and $\langle L^{i}\rangle=0$ for $i=1,\ldots,n$. We show
that for some choice of positive constants $C_{q}$ the Hessian matrix $\langle
L^{i}L^{j}\rangle$ is positive definite. Again in computing the Hessian we may
replace each $l^{q}$ by the corresponding $f^{q}$.

As an example, consider the case $Q=\\{(0,0)\\}$, which is the flat yield
curve example. Here $\tilde{Q}=\\{(1,0)\\},Q^{\\#}=\\{(2,0)\\}$ and $Q^{2}$ is
the disjoint union of $Q^{\\#}$ and $\tilde{Q}\cup\\{0\\}$. $L(\mathbb{R})$ is
one dimensional with basis $l^{(1.0)}(t)=t$. Any positive choice for
$C_{(2,0)}$ yields a positive definite $1\times 1$ Hessian.

Returning to the general case, we see that the equations (3.29) imply that the
diagonal entries are positive and that many of the off-diagonal entries are
zero. However, they need not all be zero and this is why we have to be a bit
careful about the choice of $C_{q}$’s.

To illustrate this consider the case

(3.32) $Q=\tilde{Q}=\\{(0,-1),(0,-3),(0,-5)\\}.$

Write $C_{a}$ for $C_{(0,-a)}$ $(a=2,6,10)$. The Hessian matrix in this case
is

(3.33) $B=\left(\begin{array}[]{ccc}C_{2}&0&C_{6}\\\ 0&C_{6}&0\\\
C_{6}&0&C_{10}\end{array}\right).$

Now if we simply chose $C_{a}=1$ for $a=2,6,10$ then $B$ would not be positive
definite. The remainder of the proof is exemplified here by the idea that if
$C_{2}>0$ and $C_{6}>0$ are already chosen then $B$ is always positive
definite provided that $C_{10}$ is chosen large enough.

We complete the proof that $C_{q}$’s can always be chosen so that the Hessian
is positive definite by using induction on the number of points in $Q$.

Begin by choosing a set of extreme points $Q^{*}$ of $Q$ as follows. Among the
nonnegative numbers $\\{|{\rm Re}(\lambda)|:(m,\lambda)\in Q\\}$ choose the
largest and denote it by $a^{*}$. If $a^{*}=0$, then (3.26) implies that
$Q=\\{(0,0)\\}$, the flat yield curve case we considered above, and so we may
assume that $a^{*}>0$.

At least one of the sets $\\{(m,\lambda)\in Q:{\rm Re}(\lambda)=a^{*}\\}$ and
$\\{(m,\lambda)\in Q:{\rm Re}(\lambda)=-a^{*}\\}$ is nonempty. We will suppose
that the first one is nonempty and let
$m^{*}=\max\\{m:(m,\lambda)\in\tilde{Q}$ and ${\rm Re}(\lambda)=a^{*}\\}$. Now
let $Q^{*}=\\{(m^{*},\lambda)\in\tilde{Q}:{\rm Re}(\lambda)=a^{*}\\}$. Since
$a^{*}>0$, $Q^{*}\subset Q$. Notice that $q^{*}\in Q^{*}$ implies
$q^{*}+\overline{q}^{*}=(2m^{*},2a^{*})$ which we will denote by $\rho^{*}$.

Define $Q^{\prime}=Q\setminus Q^{*}$ and notice that $Q^{\prime}$ is a smaller
closed subset still satisfying the hypothesis of the Theorem, i.e.
$\tilde{Q}^{\prime}\cup\\{0\\}\subset Q^{\prime 2}\setminus Q^{\prime\\#}$.
Furthermore, $\rho^{*}\not\in Q^{\prime\\#}$. In fact, we have

(3.34) $\rho^{*}\not\in\\{{\rm
Re}(q_{1}+q_{2}):q_{1}\in\tilde{Q}^{\prime}\;\;{\rm
and}\;\;q_{2}\in\tilde{Q}\\}.$

For suppose $q_{1}=(m_{1},\lambda_{1})$ and $q_{2}=(m_{2},\lambda_{2})$. By
construction either ${\rm Re}(\lambda_{1})<a^{*}$ or $m_{1}<m^{*}$.
Furthermore, ${\rm Re}(\lambda_{2})\leq a^{*}$ and if ${\rm
Re}(\lambda_{2})=a^{*}$, then $m_{2}\leq m^{*}$. Hence, ${\rm
Re}(\lambda_{1}+\lambda_{2})\leq 2a^{*}$ and if ${\rm
Re}(\lambda_{1}+\lambda_{2})=2a^{*}$, then $m_{1}+m_{2}<2m^{*}$.

Now arrange the list $\\{L^{i}:i=1,\ldots,n\\}$ so that the functions $l^{q}$
with $q\in Q^{*}$ occur at the end. The Hessian matrix $B$ has the bloc form.

(3.35) $B=\left(\begin{array}[]{cc}B^{\prime}&U\\\
U^{T}&B^{*}\end{array}\right)$

where $B^{\prime}$ is the matrix associated with $Q^{\prime}$. By inductive
hypothesis we can choose $C_{q}>0$ for $q\in Q^{\prime\\#}$ so that
$B^{\prime}$ is positive definite.

Now some of the $C_{q}$’s or $\frac{1}{2}C_{q}$’s may also appear in $U$ and
$U^{T}$ for $q\in\tilde{Q}^{\prime}$. This is what happened in the above
example. However, (3.34) implies that $q_{1}+q_{2}$ in $Q^{2}$ can equal
$\rho^{*}$ only for $q_{1},q_{2}$ in $\tilde{Q}^{*}$. Thus, if the $C_{q}$’s
have been chosen for $q\in Q^{\prime 2}$ so that $B^{\prime}$ is positive
definite, then $U$ is also determined.

Furthermore, $B^{*}$ is a positive diagonal matrix. By (3.29) every diagonal
entry is either $C_{\rho^{*}}$ or $\frac{1}{2}C_{\rho^{*}}$. If $Q^{*}$
consists of the real exponent $(m^{*},a^{*})$ alone then $B^{*}$ is the
$1\times 1$ matrix $(C_{\rho^{*}})$. Otherwise the off-diagonal entries are
all zero because if $q_{1},q_{2}\in\tilde{Q}^{*}$ is not a conjugate pair then
$q_{1}+q_{2}$ is not real and so is not in $Q^{\\#}$. By (3.27), $\langle
f^{q_{1}+q_{2}}\rangle$ and its conjugate are zero.

In detail, suppose that
$q_{1}=(m^{*},a^{*}+\mathbf{i}\omega_{1}),q_{2}=(m^{*},a^{*}+\mathbf{i}\omega_{2})$
so that
$q_{1}+q_{2}=(2m^{*},2a^{*}+\mathbf{i}(\omega_{1}+\omega_{2})),q_{1}+\overline{q}_{2}=(2m^{*},2a^{*}+\mathbf{i}(\omega_{1}-\omega_{2}))$.
It follows that $\langle
f^{2\rho^{*}}(t)\cos(\omega_{1}+\omega_{2})t\rangle=0=\langle
f^{2\rho^{*}}(t)\cos(\omega_{1}-\omega_{2})t\rangle$ and $\langle
f^{2\rho^{*}}(t)\sin(\omega_{1}+\omega_{2})t\rangle=0=\langle
f^{2\rho^{*}}(t)\sin(\omega_{1}-\omega_{2})t\rangle$. Now apply the sum
formulae for sine and cosine.

It follows that $B^{*}=C_{\rho^{*}}\Delta$ where $\Delta$ is a diagonal matrix
with diagonal entries all $1$ or $\frac{1}{2}$.

In the special case where $Q=Q^{*}$, $B=B^{*}$ and any positive choice of
$C_{\rho^{*}}$ will do. This includes the initial step for the induction.

In general $B$ is positive definite provided $C_{\rho^{*}}$ is now chosen
large enough. This comes from the following lemma whose proof completes the
proof of Lemma 3.6.

######  Lemma 3.7.

Let $B^{\prime}$ be a $k\times k$ positive definite symmetric matrix, $D$ be a
$n-k\times n-k$ positive definite symmetric matrix and $U$ be a $k\times n-k$
matrix. If $C>0$ is chosen sufficiently large then

$B=\left(\begin{array}[]{cc}B^{\prime}&U\\\ U^{T}&CD\end{array}\right)$

is positive definite.

###### Proof.

Write $z$ in ${\mathbb{R}}^{n}$ as $z=(x,y)$ where $x\in{\mathbb{R}}^{k}$ and
$y\in{\mathbb{R}}^{n-k}$. Then

$z^{T}Bz=x^{T}B^{\prime}x+2x^{T}Uy+Cy^{T}Dy.$

We want to choose $C>0$ large enough that $z\neq 0$ implies $z^{T}Bz>0$.

Because $B^{\prime}$ and $D$ are positive definite we can find $\epsilon>0$
and $M>0$ such that

$x^{T}B^{\prime}x\geq\epsilon\parallel x\parallel^{2}$
$y^{T}Dy\geq\epsilon\parallel y\parallel^{2}$ $|2x^{T}Uy|\leq M\parallel
x\parallel\parallel y\parallel.$

Now if $\epsilon M^{-1}\parallel x\parallel>\parallel y\parallel$ then

$z^{T}Bz\geq x^{T}B^{\prime}x-|2x^{T}Uy|$ $\;\;\;\;\;>\epsilon\parallel
x\parallel^{2}-M\parallel x\parallel(\epsilon M^{-1}\parallel x\parallel)=0.$

Similarly, $C\epsilon M^{-1}\parallel y\parallel>\parallel x\parallel$ implies

$z^{T}Bz>Cy^{T}Dy-|2x^{T}Uy|$ $\;\;\;\;\;>C\epsilon\parallel
y\parallel^{2}-M(C\epsilon M^{-1}\parallel y\parallel)\parallel y\parallel=0.$

If we choose $C>(M/\epsilon)^{2}$ then for any $z\neq 0$ at least one of the
two inequalities will hold, i.e. if $\parallel y\parallel\;\geq\epsilon
M^{-1}\parallel x\parallel$ and $\parallel y\parallel\;>0$ then $C\epsilon
M^{-1}\parallel y\parallel\;>M\epsilon^{-1}\parallel y\parallel\ \geq\
\parallel x\parallel$.

$\Box$

Proof of Lemma 3.2: For $j=1,\ldots,p$ there exist real constants $u^{j}_{i}$
($i=1,\ldots,n)$ such that $F^{j}=\sum_{i}u_{i}^{j}L^{i}$ and $u_{i}^{p+1}$
($i=1,\ldots,n$) such that $\sum_{j}(F^{j})^{2}=\sum_{i}u_{i}^{p+1}L^{i}$. For
$s\in{\mathbb{R}}^{p}$ and $S\in{\mathbb{R}}$ define $r(s,S)$ by
$r(s,S)_{i}=-\sum_{j}s_{j}u^{j}_{i}-Su^{p+1}_{i}$ ($i=1,\ldots,n)$.

Suppose we are given a finite function $z_{t}$ on $\mathbb{R}_{+}$ such that
$z_{0}=0$ and $\langle 1\rangle=\langle L^{i}\rangle=0$ for $i=1,\ldots,n$.
Clearly,

$\langle 1\rangle=0$ $\langle F^{j}\rangle=0\;\;\;(j=1,\ldots,p)$ (3.36)
$\langle\sum^{p}_{j=1}(F^{j})^{2}\rangle=0.$

Now given $r^{*}$ we define the basket of futures $x_{t}$ by (3.6), i.e.
$x_{t}=z_{t}/P_{t}(r^{*})$. Then from (3.8) we have for $r=r^{*}+r(s,S)$ that
$V(r)$ is given by

$V(s,S)\equiv\langle\exp[\sum^{p}_{j=1}s_{j}F^{j}+S\sum^{p}_{j=1}(F^{j})^{2}]\rangle$
(3.37) $=\langle\exp[s^{T}F+SF^{T}F]\rangle.$

For the last expression we are regarding $s$ and each $F_{t}$ as column
vectors in ${\mathbb{R}}^{p}$. By (3.7) and (3.9), $x_{t}$ is a self-financing
basket of futures with respect to the original price curve $P_{t}(r^{*})$ and
$r=r^{*}$ is a critical point for the value function $V(r)$. In particular, at
the origin $(s,S)=(0,0)$ $V(0,0)$ equals zero. For $s$ a nonzero vector in
${\mathbb{R}}^{p}$ define

(3.38) $\epsilon=\parallel s\parallel=(s^{T}s)^{1/2}\;\;{\rm
and}\;\;\theta=s/\parallel s\parallel.$

Thus, $\theta$ is a variable on the unit sphere in ${\mathbb{R}}^{p}$.

For $\epsilon\neq 0$ we define $M=S/\epsilon^{2}$. So we have the change of
variables:

(3.39) $s=\epsilon\theta\;\;\;{\rm and}\;\;\;S=M\epsilon^{2}.$

We prove there exist $(\theta_{+},M_{+})$ and $(\theta_{-},M_{-})$ such that
for $\epsilon>0$ small enough

$V(\epsilon\theta_{+},\epsilon^{2}M_{+})>0$ (3.40)
$V(\epsilon\theta_{-},\epsilon^{2}M_{-})<0.$

It follows that the origin is not a local minimum (or local maximum) point for
$V(s,S)$ and so $r^{*}$ is not for $V(r)$.

Notice first that $V(s,S)$ does not vanish identically on any neighborhood $O$
of the origin. For if $\sum z_{t}\exp[s^{T}F_{t}+SF^{T}_{t}F_{t}]=0$ as a
function on $O$ then by the Remark after Lemma 3.3 it would have to be true
that for some $t_{1}\neq t_{2}$ both in the support of $\\{z_{t}\\}$.
$F^{j}_{t_{1}}=F^{j}_{t_{2}}$ ($j=1,\ldots,p$). This contradicts the
assumption of condition (1) in the statement of Lemma 3.2. That is, by that
assumption
$(F^{1}_{t_{1}},\dots,F^{p}_{t_{1}})\not=(F^{1}_{t_{2}},\dots,F^{p}_{t_{2}})$
for for $t_{1}\not=t_{2}$ in the support of $z$ and so the functions
$\exp[s^{T}F_{t}+SF^{T}_{t}F_{t}]$ of $(s,S)\in O$ are linearly independent
and we would not have $\sum z_{t}\exp[s^{T}F_{t}+SF^{T}_{t}F_{t}]=0$ on $O$.

Now we use the series expansion for the exponential and the binomial theorem.

$V(s,S)=\sum^{\infty}_{n=1}\frac{1}{n!}\langle(s^{T}F+SF^{T}F)^{n}\rangle$
$=\sum^{\infty}_{n=1}\sum^{n}_{k=0}\frac{1}{k!(n-k)!}\langle(s^{T}F)^{n-k}S^{k}(F^{T}F)^{k}\rangle$
(3.41)
$=\sum^{\infty}_{n=1}\sum^{n}_{k=0}\frac{M^{k}}{k!(n-k)!}\epsilon^{n+k}\langle(\theta^{T}F)^{n-k}(F^{T}F)^{k}\rangle$

Observe that $\langle 1\rangle=0$ implies we can begin the series with the
$n=1$ term. Having used the polar coordinate change (3.40) we similarly write,
with $\parallel F\parallel=(F^{T}F)^{1/2}$,

(3.42) $F=\parallel F\parallel\omega,\;\;\;{\rm i.e.}\;\;\;\omega=F/\parallel
F\parallel\;\;(\parallel F\parallel\;\neq 0),$

and change variables in the sum letting $N=n+k$:

$V(s,S)=\sum^{\infty}_{n=1}\sum^{n}_{k=0}\frac{M^{k}}{k!(n-k)!}\epsilon^{n+k}\langle\parallel
F\parallel^{n+k}(\theta^{T}\omega)^{n-k}\rangle$ (3.43)
$=\sum^{\infty}_{N=1}\epsilon^{N}\sum^{[N/2]}_{k=0}\frac{M^{k}}{k!(N-2k)!}\langle\parallel
F\parallel^{N}(\theta^{T}\omega)^{N-2k}\rangle,$

where $[N/2]$ is the greatest integer less than or equal to $N/2$.

Because this series does not vanish identically there is a smallest positive
value $N=N^{*}$ such that

(3.44)
$V_{N^{*}}(\theta,M)\equiv\sum^{[N^{*}/2]}_{k=0}\frac{M^{k}}{k!(N^{*}-2k)!}\langle\parallel
F\parallel^{N^{*}}(\theta^{T}\omega)^{N^{*}-2k}\rangle$

is not identically zero. Furthermore, absolute convergence of the series
implies for any $(\theta,M)$ such that $V_{N^{*}}(\theta,M)\neq 0$ the sign of
$V(\epsilon\theta,\epsilon^{2}M)$ is the same as the sign of
$V_{N^{*}}(\theta,M)$ provided $\epsilon>0$ is sufficiently small. Thus, it
suffices to find $(\theta_{+},M_{+})$ and $(\theta_{-},M_{-})$ such that

$V_{N^{*}}(\theta_{+},M_{+})>0$ (3.45) $V_{N^{*}}(\theta_{-},M_{-})<0.$

Then (3.40) follows for $\epsilon>0$ sufficiently small.

We proceed by averaging the $\theta$ variable over the unit sphere in
${\mathbb{R}}^{p}$. There are two cases:

Case i: $\int_{S^{p-1}}V_{N^{*}}(\theta,M)d\theta$ vanishes for all $M$.

In this case, there is some $M_{0}$ such that $V_{N^{*}}(\theta,M_{0})$ is not
identically zero as a function of $\theta$ by choice of $N^{*}$. Since the
average in $\theta$ is $0$ at $M_{0}$, there must exist values $\theta_{+}$
and $\theta_{-}$ satisfying (3.45) with $M_{+}=M_{0}=M_{-}$.

Case ii: $\int_{S^{p-1}}V_{N^{*}}(\theta,M)d\theta$ does not vanish
identically.

In this case we compute the average. The results depend on whether $p=1$ or
$p>1$.

When $p=1$, $\theta$ and $\omega$ are each $\pm 1$. For each $\omega$, the
function $\theta^{T}\omega$ takes the two values $\pm 1$, each weighted $1/2$.
So the average of $(\theta^{T}\omega)^{N^{*}-2k}$ is $0$ when $N^{*}$ is odd
and is $1$ when $N^{*}$ is even.

Thus, when $p=1$, we have

(3.46)
$\int\;V_{N^{*}}(\theta,M)d\theta=\left\\{\begin{array}[]{ll}0&N^{*}\;\;\;{\rm
odd}\\\ P_{R}(M)\cdot\langle\parallel F\parallel^{N^{*}}\rangle&N^{*}=2R\\\
\end{array}\right.$

where

(3.47) $P_{R}(M)\ \equiv\ \sum^{R}_{k=0}\ \frac{M^{k}}{k!(2R-2k)!}.$

When $p>1$, we let $\alpha$ denote the angle between $\omega$, fixed on
$S^{p-1}$, and $\theta$. Thus,

(3.48) $\theta=(\cos\alpha)\omega+(\sin\alpha)\psi$

where $\psi$ varies on the $p-2$ sphere equatorial with respect to the pole at
$\omega$. We normalize so that

(3.49) $1\ =\int_{S^{p-1}}1d\theta\ =\int_{S^{p-2}}\int^{\pi}_{0}d\alpha
d\psi\ =\pi\int_{S^{p-2}}d\psi.$

It follows that

$\int_{S^{p-1}}(\theta^{T}\omega)^{N^{*}-2k}d\theta\
=\int_{S^{p-2}}\int^{\pi}_{0}(\cos\alpha)^{N^{*}-2k}d\alpha d\psi$ (3.50) $\
=\frac{1}{\pi}\int^{\pi}_{0}(\cos\alpha)^{N^{*}-2k}d\alpha\ =\ \frac{1}{\pi}\
\left\\{\begin{array}[]{cl}0&N^{*}\;\;{\rm odd}\\\
\frac{(N^{*}-2k-1)!!}{(N^{*}-2k)!!}&N^{*}\;\;{\rm even}\end{array}\right..$

The formula for $\int^{\pi}_{0}(\cos\alpha)^{R}d\alpha$ is proved by using
induction on $R$ and the formula

(3.51)
$\frac{d^{2}}{d\alpha^{2}}(\cos\alpha)^{R}=R(R-1)(\cos\alpha)^{R-2}-R^{2}(\cos\alpha)^{R}.$

The semi-factorial $R!!$ for a positive integer $R$ is $R\cdot(R-2)\ldots$
descending to $1$ or $2$ . In particular, $(2R-2k-1)!!/(2R-2k)!\
=1/(2R-2k)!!$.

Thus, when $p>1$, we have

(3.52) $\int V_{N^{*}}(\theta,M)d\theta\
=\left\\{\begin{array}[]{ll}0&N^{*}\;\;{\rm odd}\\\
\tilde{P}_{R}(M)\langle\parallel F\parallel^{N^{*}}\rangle/\pi&N^{*}\
=2R\end{array}\right..$

where

(3.53) $\tilde{P}_{R}(M)\ \equiv\ \sum^{R}_{k=0}\
\frac{M^{k}}{k![(2R-2k)!!]^{2}}.$

Thus, for any positive integer $p$, if $N^{*}$ is odd or $\langle\parallel
F\parallel^{N^{*}}\rangle=0$, then we return to case $i$.

When $N^{*}=2R$ and $\langle\parallel F\parallel^{N^{*}}\rangle\neq 0$ we
complete the proof by using Lemma 7 below. When $p=1$ it says that we can
choose $M_{+}$ and $M_{-}$ so that

(3.54) $P_{R}(M_{+})\langle\parallel F\parallel^{N^{*}}\rangle>0\;{\rm
and}\;P_{R}(M_{-})\langle\parallel F\parallel^{N^{*}}\rangle<0.$

When the average of a function of $\theta$ is nonzero there will certainly
exist values having the same sign as the average. So by (3.46) we can choose
$\theta_{+}$ and $\theta_{-}$ to satisfy (3.45). When $p>1$ we use the same
argument, replacing $P_{R}$ by $\tilde{P}_{R}$ in (3.54) and replacing (3.46)
by (3.52). So the proof of the following lemma completes the proof of Lemma
3.2.

######  Lemma 3.8.

For every positive integer $R$ define the polynomials of degree $R$:

$P_{R}(M)=\sum^{R}_{k=0}\frac{M^{k}}{k!(2R-2k)!}$
$\tilde{P}_{R}(M)=\sum^{R}_{k=0}\frac{M^{k}}{k![(2R-2k)!!]^{2}}.$

If $M\geq 0$ then $P_{R}(M)$, $\tilde{P}_{R}(M)>0$ and for every $R$ there
exist negative numbers $M$, $\tilde{M}$ such that $P_{R}(M)<0$ and
$\tilde{P}_{R}(\tilde{M})<0$.

###### Proof.

Because the coefficients are positive it is clear that $P_{R}(M)\geq
P_{R}(0)>0$ when $M>0$. Furthermore if $R$ is odd then ${\rm
Lim}_{M\rightarrow-\infty}P_{R}(M)=-\infty$ and so $P_{R}(M)<0$ for $M<0$ with
$|M|$ sufficiently large. We complete the proof for $P_{R}$ by finding a range
of negative values $M$ where $P_{R}(M)$ has the same sign as the $R-1$ term of
the sum. When $R$ is even this term is negative.

Let $T_{k}$ denote the $k^{th}$ term and $S_{k}$ the $k^{th}$ partial sum:

(3.55) $T_{k}=\frac{M^{k}}{k!(2R-2k)!}$

(3.56) $S_{k}=\sum^{k}_{j=0}T_{j}=S_{k-1}+T_{k}.$

Define the absolute ratio $(k=1,\ldots,R)$:

(3.57) $\frac{|T_{k-1}|}{|T_{k}|}=\frac{k}{(2R-2k+2)(2R-2k+1)|M|}=\mu_{k}.$

Notice that $\mu_{k}$ increases monotonically with $k$ provided $|M|$ is
fixed. If $|M|>(R-1)/6$, then $\mu_{R-1}<1/2$ (direct computation) implies
$\mu_{k}<1/2$ for all $k=1,\ldots,R-1$.

By induction on $k$ we then prove that with $M$ negative and $|M|>(R-1)/6$:

(3.58) $\frac{1}{2}<\frac{S_{k}}{T_{k}}\leq 1\;\;\;\;k=0,\ldots,R-1.$

Because $S_{0}=T_{0}\neq 0$ the initial step is clear. Inductively,

(3.59)
$\frac{S_{k}}{T_{k}}=\frac{S_{k-1}+T_{k}}{T_{k}}=1+\frac{S_{k-1}}{T_{k}}.$

Because $M$ is negative the terms of the series alternate in sign and so
$T_{k-1}/T_{k}=-\mu_{k}$. Thus,

(3.60) $\frac{S_{k}}{T_{k}}=1-\mu_{k}\frac{S_{k-1}}{T_{k-1}}.$

Now $\mu_{k}>0$ and $S_{k-1}/T_{k-1}>0$ (in fact $>1/2$) imply
$S_{k}/T_{k}<1$. Also, $\mu_{k}<1/2$ and $S_{k-1}/T_{k-1}\leq 1$ imply
$S_{k}/T_{k}>1/2$.

Now choose $M$ negative and so that $(R-1)/6<|M|<R/4$. $S_{R}$ satisfies

(3.61) $S_{R}=S_{R-1}+T_{R}=S_{R-1}-\frac{1}{\mu_{R}}T_{R-1}$

and so

(3.62) $\frac{S_{R}}{T_{R-1}}=\frac{S_{R-1}}{T_{R-1}}-\frac{1}{\mu_{R}}.$

By (3.59) $S_{R-1}/T_{R-1}>1/2$. On the other hand $|M|<R/4$ implies by direct
computation again that $\mu_{R}>2$ and so $1/\mu_{R}<1/2$. So with $M$
negative in the range $(R-1)/6<|M|<R/4$

(3.63) $\frac{S_{R}}{T_{R-1}}>0.$

That is, $P_{R}(M)=S_{R}$ has the same sign as $T_{R-1}$. This sign is
negative when $R$ is even.

For $\tilde{P}_{R}(M)$ the proof follows exactly the same pattern. Equation
(3.57) is replaced by

(3.64) $\frac{|T_{k-1}|}{|T_{k}|}=\frac{k}{(2R-2k+2)^{2}|M|}\equiv\mu_{k}.$

The required range for $\tilde{M}$ negative such that
$\tilde{P}_{R}(\tilde{M})$ has the same sign as its $R-1$ term as above is
$(R-1)/8<|\tilde{M}|<R/8$.

$\Box$

## References

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