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\begin{center}
\vspace*{0.25in} %% 11pt 
\noindent{\LARGE \bf 
The Potential Approach to Bond and Currency Pricing} 

\vspace{0.3in} 
\noindent{\Large Markus Leippold$^a$\footnote{Corresponding author. Tel.: +41 71 220 3066; fax: +41 71 311 2690; e-mail: \\markus.leippold@unisg.ch.}; Liuren Wu$^b$\footnote{Tel.: +1 212 636 6117; e-mail: liwu@mary.fordham.edu.}}\\
\vspace{0.1in}
\noindent{\small  \it $^a$ Swiss Institute of 
Banking and Finance, 
University of St. Gallen, Merkurstr. 1, 9000 St.Gallen, 
Switzerland} \\
 
\noindent{\small \it $^b$Graduate School of Business, 
Fordham University; 113 West 60th Street, New York, NY 10023, USA} 

\end{center}

\vspace{0.4in} 
\noindent{\large First draft: June, 1998} 

\noindent{\large This draft: \today} 
%\newline {\large Very Preliminary: For Discussion Only} 
\vspace{0.30in}\\ 
\noindent{\large \bf Abstract} 

\medskip 
\noindent In this paper, we begin the modeling of bond and currency prices 
from the modeling of the state-price density satisfying basic properties 
of a potential. We provide extensive examples and show their implications on bond and currency pricing. Most classic short rate models are special cases of this general approach. We also investigate the connection to the Heath, Jarrow, and Morton model. One advantage of the potential approach resides in its ease in simultaneously modeling the yield curves of 
many countries and their exchange rates. The properties of exchange rates under each example are derived and we illustrate their possibility in explaining the forward premium puzzle. 



\vspace{.4in} 
\bigskip 
\medskip 
\noindent{\it JEL Classification Codes: } G12, G13, G15, E43, C60. 

\medskip 
\paper{\noindent{\it Keywords: } potential; 
interest rates; foreign exchange; 
international term structure; 
state-price density; resolvent; 
Markov process; Heath-Jarrow-Morton; forward premium puzzle.} 

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\section{Introduction} 


The pricing kernel, or the state-price density, which 
relates future cash flows to today's price, 
is the fundamental building 
block of modern asset pricing theory. 
In abstract, the state-price density 
process can be regarded as a positive supermartingale, 
or, under some regularity conditions, a {\it potential}. 
The theory of Markov processes 
provides a rich framework for the generation of examples of 
potentials. 
In this paper, we begin the modeling of bond and currency prices 
from the modeling of the state-price density satisfying basic properties 
of a potential. 
We provide extensive examples on the potential modeling of 
the state-price densities and their implications on bond and 
currency pricing. We show that most 
classic interest rate models 
are special cases of this general approach. 
We also investigate the connection between the potential approach 
and the Heath et al. (1992) approach (henceforth: HJM) widely used 
in the finance area. 
One advantage of the potential approach resides in its 
great ease in modeling the yield curves of 
many countries at the same time, 
together with the exchange rates 
between them. We derive 
the properties of exchange rates under each example and 
illustrate their possibility in explaining the forward 
premium puzzle. 




The bulk of literature on bond pricing focuses on two approaches. 
One stream of literature, the {\it short rate models}, 
specifies the instantaneous interest rate process 
directly and then comes up with expressions 
for the prices of zero-coupon bond 
and other interest rate derivatives. 
Cox et al. (1985), 
Beaglehole and Tenney (1991), 
Black et al. (1990), 
Brennan and Schwartz (1979), 
Duffie and Kan (1994), 
Fong and Vasicek (1991), 
Hull and White (1990), 
Longstaff and Schwartz (1991), 
Richard (1978), 
Schaefer and Schwartz (1987), 
Vasicek (1977), 
are just some of the many papers which study different 
models for the spot rate process and explore the consequences 
of this model choice. All these models are based either on a general 
(e.g. Cox et al. (1985)) or partial equilibrium 
(e.g. Hull and White (1990)) framework. 

Another approach to interest rate modeling, 
the {\it forward rate models}, began with the paper 
of Ho and Lee (1986) and was thoroughly analyzed in the continuous 
time setting by Babbs (1990) and Heath et al. (1992) (see Jamshidian (1988) and Sommer (1996) for the continuum limit of the 
discrete-time Ho and Lee model and Amin and Jarrow (1991) for an extension of the Heath et al. model to an international economy). 
The idea of this approach is to model the forward rate process 
(or, equivalently, to model the movement of the yield curve) 
directly. 

\noindent Only recently, a less developed approach 
has emerged in the bond-pricing literature. 
These models are based on the direct specification 
of the state-price process $\xi_t$. 
Refer to the bond price equation (\ref{bondprice}), the 
positivity of bond prices implies that the state-price 
density process $\xi_t$ is a {\it positive 
supermartingale}. If additionally we assume the 
economically reasonable condition $P(0,t) \rightarrow 0$ 
as $t\rightarrow \infty$, then the state-price density $\xi_t$ 
is what is known as a {\it potential}.\footnote{A 
positive supermartingale tending to $0$ in expectation 
is called a 
potential because of the very close links with the Markov process 
concept of a potential. 
See for example, Bhattacharya and Waymire (1990).} 
Therefore, this approach has 
also been referred to as the {\it potential approach}. 

To our knowledge, the earliest published reference to 
the state-price density approach appears 
to be Constantinides (1992), where it is 
used to generate a fairly general squared-Gaussian model. 
Backus et al. (1998a,b,c) illustrate the 
role of pricing kernel in bond and currency pricing in 
a discrete-time setup. 
Rogers (1997) formalizes the potential 
approach and illustrates the application of a {\it resolvent} 
representation in modeling the potential ( a special case of Rogers' approach is presented in Flesaker and Hughston (1996)). 
In this paper, 
we follow the procedure of Rogers (1997) and apply 
the resolvent representation to bond and currency pricing. 

Given a state price density $\xi_t$, the price of a zero-coupon bond is given by 
\begin{equation} 
P(t,T)=E_t\left[\xi_T\right]/\xi_t, 
\label{bondprice} 
\end{equation} 
where the expectation is taken under the objective probability measure. 
Using the above equation as the starting point has several advantages compared 
to the traditional approaches of term structure modeling. 
First, it offers great flexibility for the construction of 
interest rate models with strictly positive interest rates. 
Requiring nonnegative interest rate at the cost of analytical 
complexity may not be appropriate for some classes of instruments; 
however, for the calculation of long term contracts and most 
structured products there is a substantial increase in accuracy if 
a positive interest rate model is used (see e.g. Rogers (1995,1996)). 

Secondly, state-price models based on (\ref{bondprice}) offer great 
simplifications 
when modeling international term structures. The exchange rate 
between two countries equals the ratio of their state-price 
densities. This observation was made by Sa\'{a}-Requejo (1993) and 
Backus et al. (1998b) in a discrete-time version and by 
Ahn (1997) in the continuous-time framework. 
Starting from the state price density, 
we can therefore simultaneously 
model the term structures of interest rates in any two countries and 
the exchange rate between them. 
Due to this very ease of the potential approach in modeling 
the international term structures, 
we are enabled to explore implications on the {\it forward premium puzzle} 
in great detail and clarity under different specifications. 
We are able to point directions in model specification to account 
for the forward premium puzzle. 

Lastly, since the state-price density 
is more closely related to the equilibrium of an economy, 
the correct specification of the state-price density also 
serves as a benchmark for future equilibrium modeling 
of the economy under the sense of reverse engineering. 


The paper is structured as follows. 
In the next section, we describe the resolvent procedure 
for the specification of the potentials and show how interest 
rates, bond prices, and exchange rates can be derived from the 
state-price density. 
Section \ref{Examples} analyzes several examples and explores their 
correspondence to the classic interest rate models. In section 
\ref{sec:HJM1}, we analyze the connection between the state-price density 
approach and the HJM approach. 
Since the key to the construction of 
interest rate models within the HJM 
framework is the specification of the forward rate volatility, we will 
elaborate on the connection of the forward rate volatility and the 
volatility of the pricing kernel process. Section \ref{sec:Puzzle} 
extends the pricing kernel model from a one-country economy to a 
multi-country economy. The flexibility of the 
state-price density approach allows us to construct international 
term structure models in very convenient ways 
and provides us with insights in model specifications to 
account for the forward premium 
anomaly of currency prices. 
Section \ref{conclusion} concludes. 


\setcounter{exmpl}{0} 
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\section{The Resolvent Representation} 

Let $(\Omega,{\cal F}, \P)$ be a probability space equipped with a filtration 
$F = ({\cal F}_t)_{t \leq 0}$ satisfying the usual conditions of 
right-continuity and completeness. 
The fundamental building block of modern asset pricing theory is 
the {\it state-price density}, $\xi_t$, 
or the {\it pricing kernel}, $\xi_{t,T}=\xi_T/\xi_t$, 
which relates future cash flows, denoted as $K_T$, to today's price, 
denoted as $p_t$: 
\begin{equation} 
p_t=E\left[\frac{\xi_T}{\xi_t} K_T |{\cal F}_t \right] 
=E_t\left[\frac{\xi_T}{\xi_t} K_T\right], 
\label{price} 
\end{equation} 
where $E[\cdot|{\cal F}_t]=E_t[\cdot]$ 
is the conditional expectation operator defined on $\left(\Omega, {\cal F}, 
\P\right)$. 
The existence and uniqueness of a positive state-price density is guaranteed 
by 
the assumption of an arbitrage-free and complete market. We refer to Duffie 
(1996) for details. 

For any Markov process $\{X\}_{t \geq 0}$ with {\it resolvent} 
$\left(R_{\lambda}\right)_{\lambda>0}$, 
we can take $\alpha>0$ and a positive 
function $g$ on the state space $x\in {\cal X}$ and define 
the state-price density $\xi_t$ as 
\begin{equation} 
\xi_t = e^{-\alpha t} R_{\alpha} g(X_t). 
\label{kernel} 
\end{equation} 
The resulting state price density $\xi_t$ is a 
potential.\footnote{See 
Appendix (\ref{resolvent}) for details and proof.} 
Different choices of $g$ and $\alpha$ give a wide range of possible 
potentials, even within the context of a fixed Markov process. 
Using this framework, the spot rate can be given as 
\begin{equation} 
r(t) = \left. \frac{\partial E\left[\xi_T|{\cal F}_t\right]} 
{\partial T} \right|_{T=t} 
= \frac{g(X_t)}{R_{\alpha} g(X_t)} \, \, . 
\label{interest} 
\end{equation} 
The bond price corresponds to 
\begin{equation} 
P(t,T)=\frac{E \left[e^{-\alpha \tau} R_{\alpha} g(X_T)|{\cal F}_t\right] } 
{R_{\alpha} g(X_t)}, 
\label{bond} 
\end{equation} 
where $\tau=T-t$ is the maturity of the bond. 

Since the resolvent of a Markov process is hard to specify in a usable 
closed 
form, we can use the {\it resolvent operator}, 
\[ 
R_{\lambda}=(\lambda-{\cal L})^{-1} 
\] 
where ${\cal L}$ is the infinitesimal generator 
of the Markov process. If we pick some positive function 
$f: {\cal X} \rightarrow (0,\infty)$ and then define $g$ via 
\[ 
g=(\lambda-{\cal L})f , 
\] 
we have $R_{\lambda} g=f$, and, provided $g$ is everywhere non-negative, 
we have the state price density, as in (\ref{kernel}), now given by 
\[ 
\xi_t = e^{-\alpha t}R_{\alpha}g(X_t)=e^{-\alpha t}f(X_t). 
\] 

From (\ref{interest}), 
we derive the instantaneous interest rate as 
\[ 
r(t) = \frac{g(X_t)}{R_{\alpha} g(X_t)} = 
\frac{(\alpha -{\cal L}) f(X_t)}{f(X_t)} \, , 
\] 
and from (\ref{bond}) we obtain the bond price as 
\[ 
P(t,T)=\frac{E_t\left[ e^{-\alpha \tau}f(X_T)\right]}{f(X_t)}. 
\] 

\subsection{The Markov process} 

Before choosing any functional forms for the $f$ function, we need 
to 
specify the Markov process. Of interest are the following class of 
$d$% 
-dimensional stationary Markov processes: 
\begin{equation} 
dX=\kappa \left( \theta -X\right) dt+ \sqrt{V(X^{\eta })} dW 
\label{ito1} 
\end{equation} 
where $\kappa $ is a $d\times d$ matrix capturing the speed of 
mean-reversion. The $d\times 1$ vector $\theta $ captures the long-run mean 
of $X$. The instantaneous variance matrix $V(X^{\eta})$ 
is assumed to be diagonal with $V_{ii}=a_i+b_i X_i^{\eta}$. 
The infinitesimal generator of $X$ is given by 
\begin{equation} 
\left( {\cal L}f\right) \left( X\right) =\left( \theta -X\right) 
^{\prime 
}\kappa ^{\prime }\left( \nabla f\right) +\frac{1}{2} 
\tr 
\left[V(X^{\eta}) \cdot \left( Hf\right) \right] , 
\label{generator} 
\end{equation} 
where $\nabla f$ denotes the vector gradient, $Hf$ the Hessian 
matrix of $f$, and ``$\cdot $'' the dot-product. 
To simplify, we can re-scale the Markov process, 
with no 
loss of generality, such that $a_i=0$ and $b_i=1$, and therefore 
$V_{ii}=X_i^{\eta}$. 

The following three commonly used examples for the Markov process 
fit into 
the specification of (\ref{ito1}), with $\eta$ equal to $0$, 
$1$, and $2$, respectively. 

\begin{example} 
\label{ex:ou} 
({\it An Ornstein-Uhlenbeck process with mean-reversion:}) 
\begin{equation} 
dX=\kappa \left( \theta -X\right) dt+dW. \label{ou1} 
\end{equation} 
This is one of the most simple cases for a stationary process, 
corresponding to (\ref{ito1}) with $\eta =0$. 
With this process, $X$ is multi-variate normal with 
\begin{eqnarray} 
\mu _{t} &=&E_{t}\left[ X_{T}\right] =\left( I-e^{-\kappa \tau 
}\right) 
\theta +e^{-\kappa \tau }X_{t}; \nonumber \\ 
V_{t} &=&E_{t}\left[ (X_{T}-\mu _{t})^{2}\right] =\int_{t}^{T}e^{- 
\kappa 
(T-s)}\left( e^{-\kappa (T-s)}\right) ^{\prime }ds; 
\label{ou2} 
\end{eqnarray} 
where $\mu _{t}$ denotes the conditional mean and $V_{t}$ the 
conditional 
variance. The 
characteristic function is 
\begin{equation} 
\phi _{t}\left( X_{T},s\right) =\exp \left( is^{\prime }\mu - 
\frac{1}{2}% 
s^{\prime }V_{t}s\right) . \label{ou3} 
\end{equation} 
Refer to Stuart and Ord (1987) for the derivation. 
\end{example} 

\begin{example} 
\label{ex:bes} 
({\it A mean-reverting Bessel process:}) 
\begin{equation} 
dX=\kappa \left( \theta -X\right) dt+\sqrt{X}dW, \label{bessel1} 
\end{equation} 
corresponding to (\ref{ito1}) with $\eta =1$. With such a 
Bessel 
process, $X$ is a linear combination of non-central chi-square 
variates. For 
any affine transformations of $X$: $y=b^{\prime }x$, we have the 
following 
characteristic function 
\begin{equation} 
\phi \left( y,\tau ;s\right) =\left| \left( I-isC\right) ^{\bullet 
\left( -A\right) }\right| \exp 
\left( isb^{\prime }D\left( I-isC\right) ^{-1}x\right) , 
\label{bessel3} 
\end{equation} 
where ``$\bullet $'' denotes ``dot power operator,'' and $A$, $C$, 
and $D$ 
are diagonal matrices with 
$A_{ii} = 2\hat{\kappa}_{i}\theta _{i}$ , 
$C_{ii}= \left(b_{i}/2\hat{\kappa}_{i}\right)( 1-e^{-% 
\hat{\kappa}_{i}\tau })$, 
$D_{ii} =\exp \left( -\hat{\kappa}_i\tau \right)$, and 
$\hat{\kappa}_i =(\kappa'b)_i/b_i$. 
The conditional mean $\mu \left( y_{\tau }\right) $ and variance 
$V\left( 
y_{\tau }\right) $ are 
\begin{eqnarray} 
\mu \left( y_{\tau }\right) &=&b^{\prime }\left( I-e^{-\tau 
\hat{\kappa}% 
}\right) \theta +b^{\prime }e^{-\tau \hat{\kappa}}x; \nonumber \\ 
V\left( y_{\tau }\right) &=&\sum_{i=1}^{d}\theta 
_{i}\frac{b_{i}^{2}}{2\hat{\kappa}_{i}}\left( 1-e^{-\hat{\kappa}_{i}\tau 
}\right) 
^{2}+2\sum_{i=1}^{d}x_{i}\frac{b_{i}^{2} 
}{2\hat{\kappa}_{i}}% 
\left( 1-e^{-\hat{\kappa}_{i}\tau }\right) e^{-\hat{\kappa}_{i}\tau 
}. 
\label{bessel2} 
\end{eqnarray} 
The unconditional moments are 
\begin{eqnarray*} 
\mu \left( y\right) &=&b^{\prime }\theta ;\quad V\left( y\right) 
=\sum_{i=1}^{d}\theta _{i}\frac{b_{i}^{2} 
}{2\hat{\kappa}_{i}}. 
\end{eqnarray*} 
Refer to Appendix \ref{bessel} for the derivation. 
\end{example} 

\begin{example} 
({\it A geometric Brownian motion process:}) 
\begin{equation} 
dX=-\kappa Xdt+XdW, \label{geometric1} 
\end{equation} 
corresponding to (\ref{ito1}) with $\eta =2$ and $\theta =0$. When 
$\kappa $ 
is diagonal such that the elements of $X$ are independent from each 
other, $X_T$ are conditionally log normal: $\ln X_T\sim N(\mu_t, V_t)$ 
with 
\begin{eqnarray} 
\mu _{t} &=&e^{-(\frac{1}{2}+\kappa )\tau }X_{t}; \nonumber \\ 
V_{t} &=&\int_{t}^{T}e^{-(\frac{1}{2}+\kappa )s}\left( e^{- 
(\frac{1}{2}% 
+\kappa )s}\right) ^{\prime }ds. \label{geometric3} 
\end{eqnarray} 
\end{example} 

\subsection{The $f$ function} 

One requirement for $f$ is that it has to be a positive function: 
$f:{\cal X}% 
\rightarrow (0,\infty )$. The most common example is that of an 
exponential 
form 
\begin{equation} 
f\left( x\right) = 
\exp\left( b_{0}+b_{1}^{\prime }x+x^{\prime }Bx\right) 
\label{f-function} 
\end{equation} 
where $b_{0}$ is a scalar, $b_{1}$ is a $d\times 1$ 
vector, and $B$ is a $d\times d$ matrix. 
We assume $B$ is a symmetric matrix with no loss of generality. 
%\footnote{% 
%Note that for any matrix $B$, 
%$x^{\prime }Bx=\frac{1}{2}x^{\prime }\left( 
%B+B^{\prime }\right) x$ since the asymmetric part 
%$\frac{1}{2}\left( 
%B-B^{\prime }\right) $ of $B$ gives the contribution 
%$\frac{1}{2}x^{\prime 
%}\left( B-B^{\prime }\right) x=0$, and thus we may without any 
%loss of 
%generality assume that $B$ is symmetric.} 
The gradient vector and the 
Hessian matrix of $f$ are, respectively, 
\begin{eqnarray*} 
\nabla f &=&f\left( x\right) B_1;\\ 
Hf &=&f\left( x\right) \left[ 2B+B_{1}B_{1}^{\prime }\right] ; 
\end{eqnarray*} 
with 
\[ 
B_{1} =b_{1}+2Bx, 
\] 
being a $d\times 1$ vector. 
Applying the resolvent operator, we can obtain the $g$ function: 
\begin{eqnarray} 
g(X_{t}) &=&\left( \alpha -{\cal L}\right) f(X_{t}) \nonumber \\ 
&=&f(X_t)\left[\alpha -(\theta -X_{t})^{\prime }\kappa ^{\prime} 
B_{1} 
-\frac{1}{2}\tr\left[ (X_{t}^{\eta }X_{t}^{\eta \prime })\cdot 
\left( 2B+B_{1}B_{1}^{\prime }\right)\right] \right] . \label{gx} 
\end{eqnarray} 

\subsection{The state price density and asset pricing} 

Given the Markov process in (\ref{ito1}) and the $f$ function in 
(\ref 
{f-function}), we can obtain the state price density $\xi _{t}$ as, 
\[ 
\xi _{t}=e^{-\alpha t}R_{\alpha }g(X_{t})=e^{-\alpha t}f(X_{t}), 
\] 
from which we can price contingent claims. Specifically, the 
instantaneous 
rate can be expressed as
\begin{eqnarray} 
r\left( t\right) &=&\frac{\left( \alpha -{\cal L}\right) f\left( 
X_{t}\right) }{f\left( X_{t}\right) } \nonumber\\ 
&=& 
\alpha -(\theta -X_{t})^{\prime }\kappa ^{\prime} 
B_{1} 
-\frac{1}{2}\tr\left[ (X_{t}^{\eta }X_{t}^{\eta \prime })\cdot 
\left( 2B+B_{1}B_{1}^{\prime }\right)\right] . 
\label{interest1} 
\end{eqnarray} 
The 
zero-coupon bond prices, 
\[ 
P(t,T)=\frac{E_{t}[\xi _{T}]}{\xi _{t}}=\frac{E_{t}\left[ e^{- 
\alpha \tau 
}f(X_{T})\right] }{f(X_{t})}. 
\] 
contain expectation operation on the $f$ function that needs to be 
worked 
out before one can obtain an analytical form. 
In general, any contingent claim $p_t$ with time-$T$ payoff 
$K_T=K(X_T)$ can be priced under such a framework 
as 
\[ 
p_t=E_t\left[\frac{\xi_T}{\xi_t} K(X_T)\right] 
=\frac{E_t\left[e^{-\alpha \tau} f(X_T) K(X_T)\right]}{f(X_t)}. 
\] 
Obviously, a combination of different specifications of the 
$f$ function and the Markov process $X$ 
can generate a wide range of asset pricing models. 
As we will show later, all diffusion-based interest rate models 
can de derived from this potential framework. 



\setcounter{exmpl}{0} 
\setcounter{equation}{0} 
\section{Bond Pricing: Examples} 
\label{Examples} 

In this section we provide extensive 
examples where the bond prices 
can be worked out in reasonably simple forms. 
We also derive many traditional 
models as our special cases. 


\subsection{Exponential quadratic $f$ functions with 
Ornstein-Uhlenbeck 
process for $X_t$} 
\label{ou-expquad} 

The exponential quadratic $f$ function can be rewritten as 
\begin{equation} 
f(x)=\exp \left[ \frac{1}{2}(x-c)^{\prime }Q(x-c)+\gamma \right] , 
\label{f-function2} 
\end{equation} 
where $Q=2B$, $c=-\frac{1}{2}B^{-1}b_1$, and 
$\gamma=b_0-\frac{1}{4}b_1'B^{-1}b_1$. 
The pricing kernel is therefore 
\[ 
\xi _{t,T}=e^{-\alpha \tau}\frac{f(X_T)}{f(X_t)}= 
\exp \left[ -\alpha \tau +\frac{1}{2}(X_{T}-c)^{\prime 
}Q(X_{T}-c)-\frac{1}{2}(X_{t}-c)^{\prime }Q(X_{t}-c)\right] . 
\] 
Note that $\gamma $ drops out of the pricing kernel. As a result, 
we can 
assume $\gamma =0$ by setting $b_0=\frac{1}{4}b_1'B^{-1}b_1$ 
with no loss of generality. Further, we assume that $Q$ is diagonal 
for simplicity. 
The $g(x)$ function is of the form 
\[ 
g(X_{t}) 
=f(X_{t})\left[ \alpha +(X_{t}-\theta )^{\prime }\kappa ^{\prime 
}Q\left( 
X_{t}-c\right) 
-\frac{1}{2}\tr\left( Q\right) -\frac{1}{2}\left( X_{t}- 
c\right) 
^{\prime }Q^{2}\left( X_{t}-c\right) \right] . 
\] 
The instantaneous interest rate can thus be simplified to 
\[ 
r\left( t\right) 
=\frac{1}{2}\left( X_{t}-\widetilde{c}\right) ^{\prime 
}\widetilde{Q}% 
\left( X_{t}-\widetilde{c}\right) +\widetilde{\gamma }, 
\] 
with 
\begin{eqnarray*} 
\widetilde{Q} &=&\kappa ^{\prime }Q+Q^{\prime }\kappa -Q^{2}; \\ 
\widetilde{c} &=&\widetilde{Q}^{-1}\left( \kappa ^{\prime 
}Qc+Q\kappa \theta 
-Q^{2}c\right) ; \\ 
\widetilde{\gamma } &=&\alpha -\frac{1}{2}\tr\left( Q\right) 
+\theta ^{\prime 
}\kappa ^{\prime }Qc-\frac{1}{2}c^{\prime }Q^{2}c- 
\frac{1}{2}\widetilde{c}% 
^{\prime }\widetilde{Q}\widetilde{c}. 
\end{eqnarray*} 
The interest rate is therefore 
a quadratic function of the normally distributed 
variates $X_{t}$. The properties of quadratic functions of normal variates 
are well-documented by, for example, Holmquist (1996) and Searle (1971). 


The expectation of an exponential quadratic $f(X_{T})$ function, such as 
(\ref 
{f-function2}) (with $\gamma =0$), of normal variates can be 
shown to equal 
\begin{equation} 
E_{t}[f(X_{T})]=|I-QV_{t}|^{-1/2}\exp \left[ \frac{1}{2}(\mu _{t}- 
c)^{\prime 
}(I-QV_{t})^{-1}Q(\mu _{t}-c)\right] . \label{quad1} 
\end{equation} 
Refer to Appendix \ref{efx} for the derivation. 
The bond prices are therefore given by 
\begin{equation} 
P(t,T)=|I-QV_{t}|^{-1/2}e^{\left[ -\alpha \tau +\frac{1}{2}(\mu 
_{t}-c)^{\prime }(I-QV_{t})^{-1}Q(\mu _{t}-c)-\frac{1}{2}(X_{t}- 
c)^{\prime 
}Q(X_{t}-c)\right] } , \label{ou-expquad-b} 
\end{equation} 
which is exponential-quadratic in $X_{t}$. Constantinides (1992) 
developed a model similar to this example. 
Ahn (1998) developed a general equilibrium that sustains such a 
model. As shown in the following 
examples, this class of models fit in the affine class of Duffie and 
Kan (1996) only under very specific conditions. Ahn (1998) shows how 
this general model can be reduced to parameterized Cox et al. (1985) model under special conditions. 


\begin{example} 
({\it Affine cases:}) 
We can show that 
when $\kappa$ and $Q$ are both scalars and $\theta =c$ 
this class of models (exponential-quadratic $f$ functions with 
Gaussian $X_{t}$ process) can be reduced to an affine structure. 
The instantaneous interest rate becomes 
\[ 
r\left( t\right) =\left( \alpha -\frac{1}{2}Q\right) +\left( \kappa 
Q-\frac{1}{2}Q^{2}\right) (X_{t}-\theta)'(X_t-\theta), 
\] 
which follows a square-root process 
\[ 
dr=\hat{\kappa}\left( \hat{\theta}-r\right) 
dt+\sqrt{\hat{\alpha}+\beta r}d \hat{W} 
\] 
where $d \hat{W}$ is a one-dimensional Brownian motion 
and where 
\begin{eqnarray*} 
\hat{\kappa} &=&2\kappa ; \\ 
\hat{\theta} &=&\alpha-\frac{1}{2}Q+d(2 \kappa Q-Q^2)/(2\kappa);\\ 
\hat{\alpha} &=& -(2\kappa Q-Q^2)(2\alpha-Q) ;\\ 
\beta &=&4 \kappa Q-2Q^{2}, 
\end{eqnarray*} 
and $d$ is the dimension of $X_t$. 
The pricing kernel is 
\begin{eqnarray*} 
\xi _{t,T} 
=\exp \left[ -\alpha \tau +\left( 2\kappa -Q\right) ^{-1}\left( 
r_{T}-r_{t}\right) \right] , 
\end{eqnarray*} 
which is exponential-affine in $r$. As stated in Duffie and Kan 
(1996), an 
affine interest rate with an exponential-affine pricing kernel 
generates an 
affine model of bond pricing. That is, the bond prices $P(t,T)$ will be 
exponential 
affine functions of the instantaneous interest rate $r(t)$: 
\begin{equation} 
P(t,T) =\exp \left( -a_{\tau }-b_{\tau }r(t)\right) , 
\label{eq:affinepb} 
\end{equation} 
with 
\begin{eqnarray*} 
a_{\tau } &=&\frac{1}{2}\log \left( 1-QV_{t}\right) +\alpha \tau 
-\left[ 1-(1-QV_{t})^{-1}e^{-2\kappa \tau }\right] \left( 2\kappa - 
Q\right) 
^{-1}\left( \alpha -\frac{1}{2}Q\right) ; \\ 
b_{\tau } &=&\left[ 1-(1-QV_{t})^{-1}e^{-2\kappa \tau }\right] 
\left( 2\kappa -Q\right) ^{-1}. 
\end{eqnarray*} 
Note that $V_t=\int_t^T e^{-2\kappa (T-s)}ds$ is reduced to a scalar 
now that $\kappa$ is a scalar. 
\end{example} 

\begin{example} 
({\it Beaglehole and Tenney's (1991) univariate quadratic model}): 
\\ 
When $d=1$ and $\theta =\widetilde{\gamma}=0$, we have the interest rate 
\[ 
r\left( t\right) =\frac{1}{2}\widetilde{Q}\left( X_{t}- 
\widetilde{c}\right) 
^{2}, 
\] 
with 
\[ 
\widetilde{Q}=\left( 2\kappa -Q\right) Q;\quad \mbox{ and } 
\widetilde{c}=\frac{\left( 
\kappa -Q\right) c}{2\kappa -Q}. 
\] 
Apply Ito's lemma, we have the following stochastic process 
\begin{eqnarray*} 
dr 
&=&\left( \alpha -\beta \sqrt{r}-\gamma r\right) dt+\sigma 
\sqrt{r}dW, 
\end{eqnarray*} 
where $\alpha =\frac{1}{2}\widetilde{Q}$; $\beta =\kappa 
\widetilde{c}\sqrt{2% 
\widetilde{Q}}$; $\gamma =2\kappa $; and $\sigma 
=\sqrt{2\widetilde{Q}}$. 
This is exactly the model proposed by Beaglehole and Tenney (1991). 
\end{example} 

\begin{example} 
({\it Longstaff (1989) Double Square Root Model}): 
When $d=1$ and $\kappa =0$, but $\kappa \theta =\mu \neq 0$, 
$Q=iq$, and $% 
\tilde{\gamma}=0$, we have 
\begin{eqnarray*} 
dX_{t} &=&\mu dt+dW; \\ 
r\left( t\right) &=&\frac{1}{2}q^{2}\left( X_{t}- 
\widetilde{c}\right) ^{2}, 
\end{eqnarray*} 
where $\widetilde{c}=c-\mu /Q$. 
The interest rate is then given by 
\begin{eqnarray*} 
dr &=&\left( \alpha +\beta \sqrt{r}\right) dt+\sigma \sqrt{r}dW 
\end{eqnarray*} 
where $\alpha =\frac{1}{2}q^{2}$, $\beta =\sqrt{2}\mu q$, and 
$\sigma =\sqrt{% 
2}q$. This is just the double-square-root model developed by 
Longstaff 
(1989). 
\end{example} 



\subsection{Exponential-linear $f$ functions with affine $X_t$: 
Affine models} 


With an exponential-linear $f$ function of the form 
\[ 
f(x)=\exp (a+b^{\prime }x), 
\] 
we have the pricing kernel of the form, 
\[ 
\xi _{t,T}=\exp \left[ -\alpha \tau +b^{\prime }\left( X_{T}- 
X_{t}\right) 
\right] , 
\] 
which is also exponential linear in $X$. The constant term $a$ 
drops out of 
the pricing kernel and thus can be set to zero with no loss of 
generality. 
The $g\left( x\right) $ function is then 
\begin{equation} 
g(X_{t})=f\left( X_{t}\right) \left[ \alpha -(\theta - 
X_{t})^{\prime }\kappa 
^{\prime }b-\frac{1}{2} b^{\prime }V(X_{t}^{\eta}) b\right] , 
\nonumber 
\end{equation} 
and the instantaneous interest rate is 
\[ 
r\left( t\right) =\alpha -(\theta -X_{t})^{\prime }\kappa ^{\prime 
}b-\frac{1}{2} b^{\prime }V(X_{t}^{\eta })b, 
\] 
which is a linear function of $X_{t}$ as long as the instantaneous 
variance matrix 
$V(X_t^{\eta})$ of the Markov process is affine in $X_t$, that is, 
as long as $\eta =0$ or $1$. 
Recall that $V(X_{t}^{\eta })$ is re-scaled to a diagonal matrix with 
$V_{ii}=X_i^{\eta}$. 
This affine interest rate, together with the exponential-affine 
pricing 
kernel, will generate exponential-affine bond prices, as stated in 
Duffie and Kan (1996). 
Specifically, the bond prices are given by 
\[ 
P(t,T)=\phi _{t}(X_{T},b)\exp (-\alpha \tau -b^{\prime }X_{t}). 
\] 
where $\phi _{t}(X_{T},b)=E_{t}[e^{b^{\prime }X_{T}}]$ denotes the 
moment 
generating function of $X_{T}$ conditional on time $t$ information 
and with 
moment generating parameter $b$. Note, however, that when $V(X^{\eta})$ 
is not affine in $X$, 
for example, when $\eta=2$, the interest rate $r(t)$ may 
no longer be Markovian. 
We will confine ourselves to the affine structure. 

\begin{example} 
When $X_t$ follows a Bessel process, that is, when 
$\eta=1$, referring to the characteristic function of 
$y=b^{\prime }x$ in (\ref{bessel3}), 
we have 
\[ 
\phi _{t}(X_{T},b)=\left| \left( I-C\right) ^{\bullet \left( - 
A\right) 
}\right| \exp \left( b'D\left( I-C\right) ^{-1}X_{t}\right) . 
\] 
The bond price is therefore 
\begin{equation} 
P(t,T)=\left| \left( I-C\right) ^{\bullet \left( -A\right) } 
\right|\exp 
\left[ -\alpha \tau -b'\left(I -D\left( I-C\right) ^{- 
1}\right)X_{t}\right] , 
\label{eq:expaffbp} 
\end{equation} 
which is exponential affine in $X_{t}$. The continuously compounded 
yields 
\[ 
y(t,T)=\frac{-\log P(t,T)}{\tau }=-\log \left| \left( I-C\right) 
^{\bullet 
\left( -A\right) }\right| +\alpha +\frac{1}{\tau }b'\left(I-D\left( 
I-C\right) ^{-1}\right)X_{t}, 
\] 
are thus affine functions of the Markov process $X_{t}$. This 
corresponds to 
a generalized multi-factor Cox et al. (1985) model as 
studied by, 
among others, Dai and Singleton (1997) and Backus et al. (1998d). 
\end{example} 

\begin{example} 
When $\eta =0$ and thus $X_{t}$ follows an 
Ornstein-Uhlenbeck process, 
referring to (\ref{ou3}), we have the moment generating function, 
\begin{eqnarray*} 
\phi _{t}(X_{T},b) 
&=&\exp \left( b^{\prime }\left( I-e^{-\kappa \tau }\right) \theta 
+b^{\prime }e^{-\kappa \tau }X_{t}+\frac{1}{2}b^{\prime 
}V_{t}b\right) . 
\end{eqnarray*} 
with $V_t=\int_t^T e^{-\kappa (T-s)}\left(e^{-\kappa (T-s)}\right) ds$. 
The bond price is then 
\begin{equation} 
P(t,T)=\exp \left( -\alpha \tau +b^{\prime }\left( I-e^{-\kappa 
\tau 
}\right) \theta +\frac{1}{2}b^{\prime }V_{t}b-b^{\prime }\left( 
I-e^{-\kappa \tau }\right) X_{t}\right) , 
\label{eq:bpexpaff2} 
\end{equation} 
which is also exponential-affine in $X_{t}$. 
\end{example} 

%More words: Summarize the potential pricing results. 

The extensive examples provided in this section illustrate the 
flexibility of the potential approach in bond pricing. By varying the 
specifications of the $f$ function and the Markov process $X$, we can 
generate a wide variety of bond pricing models which virtually incorporate 
all traditional interest rate models as special cases or examples. 
In the next section, we derive the links between this potential 
approach and the widely used HJM approach in term structure modeling. 


\section{Potential versus HJM Approach} 
\label{sec:HJM1} 

In this section, we connect the potential approach to the 
well-know Heath-Jarrow-Morton (HJM) framework. 
These two approaches are in sharp contrast in that 
the HJM approach makes use of the information 
contained in the current forward curve and intends to avoid 
specifying the market price of risk, which is incorporated in the 
forward curve, while the potential approach directly specifies the 
pricing kernel, and thus the market price of risk. These two 
approaches have pros and cons of their own. For example, 
the HJM framework is better suited for fixed-income 
derivatives pricing since the only unobservable input is 
the volatility structure of the forward rates, which can also be 
directly estimated from the forward rate data. 
However, the potential approach works best for modeling 
term structures of different countries and the exchange rates between 
them at the same time. It also provides more insight to the 
underlying economy since the pricing kernel is more closely related to the 
equilibrium of the economy and the underlying macroeconomic 
fundamentals 
such as preference, inflation, 
and monetary policy. 
In any case, it will be interesting to see the links between these 
two frameworks. 

\subsection{Forward rates and the pricing kernel} 
The state price density relates to 
the instantaneous interest rate by 
\[ 
\xi_t = \exp\left( -\int_0^t r(s) ds\right) \cdot Z_t. 
\] 
Since we have 
\begin{equation} 
P(t,T) = \frac{E_t\left( \xi_T)\right)}{\xi_t} = E_t^{*} \left( 
\exp\left(- \int_t^T r(s)\, ds \right)\right) 
\label{eq:bpequation}, 
\end{equation} 
we can interpret the variable $Z_t$ as the {\it Radon-Nikod\'{y}m 
derivative}, which takes us from the objective measure $\P$ to the 
risk-neutral measure $\Pr \sim \P $ defined as 
\[ 
Z_t = \frac{d\Pr}{d\P} = {\cal E}\left( - \int_0^t 
\gamma(s) \cdot dW(s) \right), 
\] 
where 
${\cal E}$ denotes the {\it Dol\'{e}ans exponential} 
\[ 
{\cal E}\left( U_t \right) = \exp \left( U_t - \frac{1}{2} \langle 
U \rangle_t \right) 
\] 
and $\gamma(t)$ is an ${\cal F}_t$-adapted process. We can now 
rewrite the expression for the pricing kernel as 
\[ 
\xi_t = P(0,t) {\cal E}\left( - \int_0^t \gamma(s)\cdot dW(s) 
\right) 
,\] 
or in differential notation 
\[ 
\frac{d\xi_t}{\xi_t} = - r(t)dt - \gamma(t)\cdot dW .\\ 
\] 

\noindent Let $B(t)$ be the money market account defined as $B(t) = \int_0^t 
r(s)\,ds$. Since the discounted price process of the zero bond $P(t,T)/B(t)$ 
is a martingale under the risk-neutral measure $\Pr$, by the abstract version 
of Bayes formula the expression $Z(t)P(t,T)/B(t)$ is a 
martingale under the objective measure $\P$. From the martingale 
representation theorem, there exists an adapted process 
$\gamma(t,T)$ such that 
\[ 
Z(t)P(t,T) = B(t)P(0,T){\cal E} \left( - \int_0^t \gamma(s,T)\cdot 
dW(s) \right) 
.\] 
Hence, application of Ito's Lemma yields the bond price dynamics under the 
objective measure $\P$ as 
\[ 
\frac{d P(t,T)}{P(t,T)} = \left( r(t) - v(t,T) \cdot 
\gamma(t)\right) dt + v(t,T)\cdot dW(t) 
,\] 
where for the bond price volatility we have \\ 
\parbox{12cm}{ 
\begin{eqnarray*} 
v(t,T) &=& \gamma(t) - \gamma(t,T)\\ 
&=& - \int_t^T \frac{\gamma(t,T)}{\partial T}\Big |_{T = s} \, ds\\ 
&=& - \int_t^T \sigma(t,s) \,ds 
\end{eqnarray*}}\hfill 
\parbox{8mm}{\begin{eqnarray}\label{eq:bondvola}\end{eqnarray}}\\ 
where $\sigma(t,s)$, as will be clear right away, 
is the instantaneous volatility of the forward rate 
$f(t,s)$. 
Equation (\ref{eq:bpequation}) allows us to express the forward 
rate in terms of the pricing kernel. Using the stochastic version 
of Fubini's Theorem,\footnote{See Baxter (1997) for a derivation of 
the stochastic version of the Fubini Theorem.} the representation 
of the forward rate $f(t,T)$ is 
\begin{equation} 
f(t,T) = \frac{1}{E_t \left( \xi_T \right)} E_t \left( 
\frac{\partial \xi_T}{\partial T} \right) 
\label{eq:forward1}. 
\end{equation} 
Obviously, the forward rate and short rate dynamics obey 
\begin{eqnarray*} 
df(t,T) &=& \left(\sigma(t,T) \cdot \left( \int_t^T \sigma(t,s) - 
\gamma(t)\right) \right) dt + \sigma(t,T)\cdot dW\\ 
dr(t) &=& \left( \frac{\partial f(t,T)}{\partial T} \Big |_{T=t} - 
\sigma(t,t) \cdot \gamma(t) \right) dt + \sigma(t,t) \cdot dW 
.\end{eqnarray*} 
Using the expression for the bond price volatility in 
(\ref{eq:bondvola}), the forward rate dynamics simplifies to 
\begin{equation} 
df(t,T) = \frac{\partial \gamma(t,T)}{\partial T} \cdot \gamma(t,T)\, 
dt + \frac{\partial \gamma(t,T)}{\partial T} \cdot dW 
\label{eq:forward2}.\\ 
\end{equation} 

It is interesting to compare the forward dynamics in 
(\ref{eq:forward2}) with the forward rate dynamics under the risk-neutral 
measure derived in the HJM framework. 
Under the HJM framework, 
the arbitrage-free drift of the forward rate under the risk-neutral measure, 
is {\it completely} determined by the forward rate volatility $\sigma(t,T)$ 
and its integrals. 
In the state-price model, the drift is fully 
determined by the adapted process $\gamma(t,T)$ and its derivative 
with respect to time-of-maturity. The interpretation of the term $\gamma(t,T)$ 
becomes obvious when we compare equation 
(\ref{eq:forward2}) with the forward rate representation in equation 
(\ref{eq:forward1}). Application of Ito's Lemma gives us 
\[ 
\frac{d\left[ E_t \left( \xi_T \right) \right]}{E_t \left( \xi_T 
\right)} = -\gamma(t,T)\cdot dW. 
\] 
Therefore, 
$-\gamma(t,T)$ is equal to the diffusion term of the 
process of the pricing kernel's time-$t$ expectation value for time 
$T$ and $\gamma(t,t)=\gamma(t)$ is often labeled as market price of risk. 

\subsection{The Volatility Structure of Forward Rates} 
The volatility of the forward rate plays a crucial role in the HJM 
framework: Not only does it determine the arbitrage-free drift of the 
forward rate under the risk-neutral measure, but also through the choice 
of a specific volatility function we generate a particular interest 
rate model, which can be used to price interest rate contingent 
claims. Since starting from the HJM framework it is not obvious 
which forward rate volatility structure gives rise to the interest 
rate models presented in section \ref{Examples}, we do the reverse: starting 
from the potential approach we derive the 
volatility structure of the forward rate 
which would yield the same interest rate 
model within the HJM framework. In what follows we delineate the 
recipe on how the volatility structure can be derived. 


By Ito's Lemma, the forward rate process can be 
expressed as a function of the state variable and the bond price, i.e. 
\[ 
df(t,T) = - \frac{\partial^2 \log(P(t,T))}{\partial T \partial X} 
dX_t - \frac{1}{2} \frac{\partial^3 \log(P(t,T))}{\partial T 
\partial X^2} \langle dX \rangle_t - \frac{\partial^2 
\log(P(t,T))}{\partial T \partial t} dt 
.\] 
If the short rate can be identified as the state variable, i.e. 
$r(t) = X_t$, it is obvious that the volatility structure of the 
forward rates is 
\begin{equation} 
\frac{\partial \gamma(t,T)}{\partial T} = - \frac{\partial^2 
\log(P(t,T))}{\partial T \partial r} 
\cdot \frac{\partial 
\gamma(t,T)}{\partial T}\Big |_{T=t} 
\label{eq:volastructure} 
.\end{equation} 
Otherwise, in our setting we would have 
\[ 
\frac{\partial \gamma(t,T)}{\partial T} = - 
\frac{\partial^2 \log(P(t,T))}{\partial T 
\partial X} \cdot X_t^{\eta} 
\label{eq:volastructure2} 
. 
\] 
Assuming an affine term structure model of the form 
\[ 
P(t,T) = A(t,T)\exp\left( - B(t,T)\cdot X_t \right) 
,\] 
for some functions $A(t,T)$ and $B(t,T)$, then equation 
(\ref{eq:volastructure}) can be further simplified to 
\begin{equation} 
\frac{\partial \gamma(t,T)}{\partial T} = 
\frac{\partial B(t,T)}{\partial T} \cdot \frac{\partial 
\gamma(t,T)}{\partial T}\Big |_{T=t} . 
\label{eq:affvolastructure} 
\end{equation} 
Equipped with these results, we are now able to derive the 
volatility structures for the calculated examples in Section 3, 
which would make a HJM framework equivalent of the {\it potential} 
approach considered. 

\subsection{Examples} 

\subsubsection{Exponential 
quadratic function $f$ and OU-process $X_t$} 
In this case, the bond price is (see equation 
\ref{ou-expquad-b}) 
\[ 
P(t,T)=|I-QV_{t}|^{-1/2}e^{\left[ -\alpha \tau +\frac{1}{2}(\mu 
_{t}-c)^{\prime }(I-QV_{t})^{-1}Q(\mu _{t}-c)-\frac{1}{2}(X_{t}- 
c)^{\prime 
}Q(X_{t}-c)\right] } . 
\] 
If we make the restriction that $\kappa$ and $Q$ are scalars as 
well as $\theta = c$ we have shown that the bond prices given in 
(\ref{ou-expquad-b}) can be reduced to an affine structure. The 
bond price equation then simplifies to (see equation 
\ref{eq:affinepb}) 
\[ 
P(t,T) =\exp \left( -a_{\tau }-b_{\tau }r(t)\right) , 
\] 
The dynamics of the pricing kernel can be stated in differential 
notation as 
\beq 
\frac{d\xi_t}{\xi_t} 
&=& \left( \frac{\hat{\kappa}\left( \hat{\theta}- r(t) 
\right)}{\hat{\kappa} -Q} - \alpha \right ) dt + \left ( 
\frac{\sqrt{\hat{\alpha}+\beta r(t) }}{ \hat{\kappa} -Q } \right ) 
\, dW . 
\eeq 
Thus, since $Q = \frac{1}{2} ( \hat{\kappa} - 
\sqrt{\hat{\kappa}^2 - 2\beta} ) $, the risk premium is 
% I made some change here. I think the sign should be minus because 
% $Q$ has be smaller than \hat{\kappa}. The reason is not self-obvious, but 
has to 
%be true. 
\beq 
\gamma(t) &=& - \frac{\sqrt{\beta\left ( \hat{\kappa} - 4\alpha 
- \sqrt{\hat{\kappa}^2-2\beta} + 4 r(t)\right )} 
}{\sqrt{\hat{\kappa}^2-2\beta} + \hat{\kappa}}. 
\eeq 
In the Cox et al. (1985) model where $\hat{\alpha}=0$, 
the risk premium simplifies 
to 
\begin{equation} 
\gamma(t) = - \frac{2 \sqrt{\beta r(t)}}{ \hat{\kappa} + 
\sqrt{\hat{\kappa}^2-2\beta}} 
=\lambda \sqrt{r(t)} ,\label{cirrp}\\ 
\end{equation} 
where 
\[ 
\lambda=-\frac{2\sqrt{\beta}}{\hat{\kappa} + 
\sqrt{\hat{\kappa}^2-2\beta}} 
\] 
In the original Cox et al. (1985) model the risk premium is given $\lambda 
\sqrt{r(t)}$, where $\lambda$ is an exogenously given constant; however, 
the $\lambda$ in equation (\ref{cirrp}) is determined by the parameters of the 
interest rate process under the objective measure. We have therefore derived 
a parameterized version of the Cox et al. (1985) model.\\ 


From the bond price equation, the volatility structure 
of the forward rate can be easily derived as 
\beq 
\sigma(t,T) = \frac 
{4 e^{\hat{\kappa}(T-t)} \hat{\kappa}^2 
} 
{ 
\left( 
\hat{\kappa} \left( 1 + e^{\hat{\kappa}(T-t)} \right ) 
+\left( e^{\hat{\kappa}(T-t)}-1\right) \sqrt{\hat{\kappa}^2 - 
2\beta} 
\right ) 
} \sqrt{\hat{\alpha}+\beta r(t)}.\\ 
\eeq 
Again, if we set $\hat{\alpha} = 0$, we obtain the volatility 
structure of the forward rates for the Cox et al. (1985) model.\\ 

\noindent Next, we want to derive the volatility structure of the 
Beaglehole and Tenney (1991) model. The pricing kernel given in 
their model is 
\[ 
\xi_t = \exp \left ( - \alpha t + \frac{1}{2} Q \left ( X_t - c 
\right )^2 \right ) 
,\] 
which can be written in differential form as 
\begin{eqnarray*} 
\frac{d\xi_t}{\xi_t} 
&=& \left ( - \alpha - X_t(X_t -c)Q\kappa + \frac{1}{2} \left ( Q 
+Q^2 (X_t - c)^2 \right ) \right ) dt + (X_t - c) Q \, dW \\ 
&=& - r(t) dt - \gamma(t) \,dW . 
\end{eqnarray*} 
The market price of risk becomes 
\begin{eqnarray*} 
\gamma(t) &=& \frac{c Q \kappa}{2\kappa - Q} + \sqrt{\frac{2 
Q}{2\kappa - Q}}\sqrt{r(t)} 
\end{eqnarray*} 
which is proportional to the square-root of the interest rate $r(t)$. 

The bond price can be derived as 
\[ 
P(t,T) = 
{\frac{ \displaystyle {\exp \left ( 
{- \left( T - t \right) \alpha - 
{\frac{Q {{\left( c - X_t \right) }^2}}{2}} + 
{\frac{Q \kappa 
{{\left( c - 
{e^{-\kappa \left( T - t \right) }} X_t 
\right) }^2}}{\left( 
{e^{-2 \kappa \left( T - t \right) }} -1 
\right) Q + 2\kappa }}}\right ) }} 
{\displaystyle {\sqrt{{ 
\frac{\left( 
{e^{- 2\kappa \left( T - t \right) }} -1 
\right) Q + 2\kappa }{2 \kappa }}}}}}. 
\] 
The volatility structure of the forward rates is then 
\begin{eqnarray*} 
\sigma(t,T) &=& \frac{ 
2 e^{( t + T ) \kappa} Q \kappa^2 
\left[ c\left( 
\left( e^{2t \kappa} + e^{2 T \kappa} \right) Q - 
2 e^{2 T\kappa} \kappa \right) + 
2e^{( t+T )\kappa }\left( 2\kappa - Q \right) X_t \right ] } 
{\left[ \left( e^{2t\kappa} - 
e^{2T\kappa} \right) Q + 
2 e^{2T\kappa}\kappa \right]^2} \\ 
&=& A+B\, X_t , 
\end{eqnarray*} 
which is a linear function of the Markov process $X_t$ and thus 
is proportional to the square-root of the interest rate $r(t)$. 
Note that the volatility of the short rate is $\sigma(t,t) = 
\sqrt{2\widetilde{Q}r(t)}$, where $\widetilde{Q} = (2\kappa -Q)Q$. 


\subsubsection{Exponential-linear $f$ 
and affine $X_t$} 

In the case of an exponential-linear $f$ function with affine 
$X_t$, the pricing kernel dynamics is given as 
\begin{eqnarray*} 
\frac{d\xi_t}{\xi_t} &=& 
\left( -\alpha + \left( \theta - X_t\right ) ' \kappa ' b + 
\frac{1}{2} (b^2)' X_t^{2\eta} 
\right ) dt 
+ b' X_t^{\eta}dW . 
\end{eqnarray*} 
The market price of risk is thus 
\[ 
\gamma(t) = - b\cdot X_t^{\eta}. 
\] 


When $X_t$ follows a Bessel process, the bond price becomes 
exponential affine in $X_t$, i.e. we have (see equation 
\ref{eq:expaffbp}) 
\[ 
P(t,T)=\left| \left( I-C\right) ^{\bullet \left( -A\right) } 
\right|\exp 
\left[ -\alpha \tau -b'\left(I -D\left( I-C\right) ^{- 
1}\right)X_{t}\right] . 
\] 
The market price of risk is $\gamma(t) = -b \cdot \sqrt{X_t}.$ 
Using the bond price equation, the volatility structure of the 
forward rates can be derived as 
\begin{eqnarray*} 
\sigma(t,T) = \left[\frac{\partial \left( -D(I-C)^ 
{-1}\right )}{\partial T} \right] b \cdot \sqrt{X_t} 
\end{eqnarray*} 


When $X_{t}$ follows an Ornstein-Uhlenbeck process, the bond price 
is, as shown in equation (\ref{eq:bpexpaff2}), 
\[ 
P(t,T)=\exp \left( -\alpha \tau +b^{\prime }\left( I-e^{-\kappa 
\tau}\right) \theta +\frac{1}{2}b^{\prime }V_{t}b+b^{\prime }\left( 
I-e^{-\kappa \tau }\right) X_{t}\right) , 
\] 
which is also exponential-affine in $X_{t}$. 
The pricing kernel follows 
\[ 
\frac{d\xi_t}{\xi_t} = 
\left ( -\alpha + \left ( \theta - X_t\right ) ' \kappa 'b+ 
\frac{1}{2}b'b \right ) dt + b' dW 
\] 
and the market price of risk equals $\gamma(t) = - b$ 
and is therefore constant. Using the bond price equation, the 
volatility structure of the forward rates can be derived as 
\begin{eqnarray*} 
\sigma(t,T) &=& e^{-\kappa' (T-t)} \kappa'b 
= e^{-\kappa' (T-t)} \sigma(t) 
,\end{eqnarray*} 
where $\sigma(t) = \kappa'b$ is the volatility structure of the short 
interest rate. This gives rise to a Gaussian interest rate model, 
where the possibility of generating negative interest rates is 
positive. 



Under the HJM framework, the volatility structure of the forward rates 
are exogenously specified while drift is derived from the 
volatility structure as a result of no-arbitrage. Market prices of risk 
are incorporated in the current forward curve. 
The potential approach, on the other hand, 
directly 
specifies the state price and thus the market price of risk. 
Although we can, as we just did, derive the volatility structure of the 
forward rate from the state price specification, 
except under very special cases such as the Gaussian interest rate model, 
the derived volatility structures are often functions of the state price 
density and are intertwined with the market price of risk $\gamma(t)$. 
That is, the volatility structures are generally not exogenous, as is 
the case in a HJM framework. 

%??? You may check its validity of this above comment 

As illustrated in previous sections, 
the potential approach provides great flexibility in generating 
a wide variety of term structure models, yet its advantage is even more 
pronounced in simultaneously 
modeling international term structures and the exchange rate 
between them. In the next section, we will fully exploit this advantage and 
extend the analysis to exchange rate market. 




\section{Accounting For Forward Premium Puzzle} 
\label{sec:Puzzle} 

In this section, we will extend the analysis above to a 
multi-currency economy. We will focus on the most puzzling 
feature of currencies: high interest rate 
currencies tend to appreciate while one might guess, instead, that 
investors would demand higher interest rates on currencies 
expected to fall in value. 
This departure from uncovered interest parity, which we term 
the {\it forward premium puzzle}, 
has been documented in numerous studies and has spawned a second generation 
of papers attempting to account for it. 
One of the 
most influential studies is by Fama (1984). He attributes the 
behavior of forward and spot exchange rates to the time-varying 
risk premia which have to possess certain properties. 
Traditional asset pricing models have been notably unsuccessful in 
producing risk premiums with the desired property. 
The potential approach provides a 
consistent framework for the valuation of interest rates as well as 
foreign exchange rate contingent claims. 
We examine, 
from the perspective of the potential approach, model 
specifications that have ``potentials'' to explain the puzzle. 

\subsection{The potential approach of currency pricing} 

One of the biggest advantage of the 
potential approach lies in its great ease in simultaneously 
modeling international term 
structures and the exchange rates between them. Specifically, 
if we consider several countries at once and assume that they are 
governed by the same vector of Markov process $X_{t}$, 
if further we assume that at 
time $t$, one unit of country $j$'s currency is worth $S_{t}^{ij}$ 
units of country $i$'s currency, then under certain technical assumptions, 
we have that the development of the exchange rate $S_{t}^{ij}$ is governed 
by the ratio of the pricing kernels of the two countries: 
\begin{equation} 
\frac{S_{T}^{ij}}{S_{t}^{ij}}=\frac{\xi _{t,T}^{j}}{\xi 
_{t,T}^{i}} . 
\label{exchange} 
\end{equation} 
This observation was made by Sa\'{a}-Requejo (1993) and 
Backus et al. (1998b)) in a discrete-time version and by 
Ahn (1997) in the continuous-time framework. 
Starting from the state price density, 
we can therefore simultaneously 
model the term structures of interest rates in any two countries and 
the exchange rate between them. 
Under the resolvent representation, we have 
\begin{equation} 
\frac{S_{T}^{ij}}{S_{t}^{ij}} 
=e^{(\alpha ^{i}-\alpha ^{j})\tau 
}\frac{f^{j}(X_{T})/f^{j}(X_{t})}{% 
f^{i}(X_{T})/f^{i}(X_{t})}, 
\label{fexchange} 
\end{equation} 
where $\tau=T-t$. By assuming the same vector of Markov process but 
different 
$f$ functions for different countries, we implicitly assume that 
the world economies share the same vector of shocks but the impacts and 
repercussions of these 
shocks are different to different countries. 

The time-$t$ forward price of the currency with maturity 
$\tau=T-t$ can also be written as the ratio 
of the conditional expectations of two state-price densities involved: 
\begin{equation} 
F^{ij}(t,T)=\frac{E_t\left[\xi^j_T\right]}{E_t\left[\xi^i_T\right]} 
=e^{(\alpha ^{i}-\alpha ^{j})T} 
\frac{E_t\left[f^j(X_T)\right]}{E_t\left[f^i(X_T)\right]}. 
\label{forward} 
\end{equation} 

In general, we can, as we do, set $\alpha^i=\alpha^j$ by further assuming 
that the long-run mean of exchange 
rate depreciation rates are zero (i.e., 
no currency consistently beats the other 
in the long run). 



\subsection{The forward premium puzzle} 

Let $s_t=\ln S^{ij}_t$ and $f^{\tau}_t=\ln F^{ij}(t,t+\tau)$. 
Then $s_{t+1}-s_t$ 
captures the continuously compounded depreciation rate over time 
interval $[t,t+\tau]$, and $f^{\tau}_t-s_t$ 
captures the forward premium. By covered interest rate parity, 
\[ 
f^{\tau}_t-s_t =r^i_{t,\tau}-r^j_{t,\tau} 
\] 
the forward premium equals the difference between the yields 
on the two countries' zero-coupon bonds with maturity $\tau$. 
Consider the following regression, 
\begin{equation} 
s_{t+\tau}-s_t =\alpha+ \beta (f^{\tau}_t-s_t)+ \varepsilon_{t+\tau}. 
\label{regression1} 
\end{equation} 
The expectation hypothesis implies $\alpha=0$ and a regression slope 
$\beta = 1$, yet most studies estimate $\beta$ to be negative. 
See for example Bilson(1981), Cumby and Obstfeld (1984), Fama (1984), 
Hansen and Hodrick (1983), Hodrick and Srivastava (1986) and Hsieh 
(1984).\footnote{Recently, there are some preliminary 
evidence that the estimate 
of $\beta$ is closer to unity when the regression is on US dollar prices of 
currencies of emerging markets or countries with strong capital controls.} 
They find not only that the expectations hypothesis provides a poor 
approximation to the data, but that its predictions of future currency 
movements are in the wrong direction. In principal, equation 
(\ref{regression1}) can be used to construct profitable investment 
strategies. Bekaert and Hodrick 
(1992) show that while such strategies are not riskless, they have 
positive and statistically significant average excess returns. 


Following Fama (1984), we decompose the forward premium 
into two parts: the expected exchange 
rate depreciation $E_t[\delta_{t+\tau}]$ 
and the expected forward risk premium $E_t[p_{t+\tau}]$: 
\ban 
f^{\tau}_t-s_t &=&E_t[s_{T}-s_t]+E_t[f^{\tau}_t-s_{T}]\\ 
&\equiv& E_t[\delta_{t,T}]+E_t[p_{t,T}], 
\ean 
where $T=t+\tau$. 
From the linear projection theorem, 
the slope coefficient $\beta$ of the regression can be written 
as 
\[ 
\beta=\frac{\cov(\delta,\delta+p)}{\var(\delta+p)} 
=\frac{\cov(\delta,p)+\var(\delta)}{\var(\delta+p)} 
\] 
Note that the expectation hypothesis $\beta=1$ if and only if 
the forward risk premium $p_{t}$ is constant over time: $\var(p)=0$. 
To account for the actual negative estimate for $\beta$, Fama (1984) 
notes that we need (1) {\it time-varying} forward risk premium, 
(2) {\it negative correlation} between the risk premium ($p$) 
and the depreciation rate ($\delta$), and (3) greater variance of 
the risk premium ($p$) than the depreciation ($\delta$). 
We label this negative estimate as a puzzle because 
most asset pricing models so far 
have been notoriously unsuccessful in 
producing a risk premium that satisfies these properties, 
particularly the negative correlation between 
the risk premium and the depreciation rate. 


Representing currency spot price $S^{ij}$ and forward price $F^{ij}$ 
in terms of the state-price density as 
in (\ref{exchange}) and (\ref{forward}), 
we can rewrite the slope coefficient as 
\begin{equation} 
\beta = \frac{\cov \left[ \ln \frac{ \xi^j_{t,T}} {\xi^i_{t,T} }, 
\ln \frac{E_t \left[\xi^j_{t,T}\right] } 
{E_t\left[\xi^i_{t,T}\right] } 
\right ]} 
{ \var \left [ 
\ln \frac{E_t [\xi^j_{t,T} ]} 
{E_t\left[\xi^i_{t,T}\right] } 
\right ] 
}. 
\label{eq:bias} 
\end{equation} 
Clearly, the slope coefficient $\beta$ is {\it completely} 
determined once we have chosen a specific form for the pricing 
kernels of the two countries. To account for the forward 
premium puzzle, we need to specify pricing kernels that satisfy 
the required properties. In particular, the pricing kernels need to 
generate negative covariance between 
the depreciation rate and the forward premium. 

Under the resolvent representation of currency pricing in (\ref{fexchange}) 
and (\ref{forward}), assuming zero long-run mean for exchange rate 
depreciation rate ($\alpha^i=\alpha^j$), we can write the depreciation rate 
$\delta_{t,T}$, 
the risk premium $p_{t,T}$ and 
the forward premium $f^{\tau}_t-s_t$ as 
\ban 
\delta_{t,T}&=&\ln \frac{S^{ij}_{T}}{S^{ij}_t} 
=\ln f^j(X_{T})/f^j(X_t) -\ln f^i(X_{T})/f^i(X_t);\\ 
p_{t,T}&=&\ln \frac{F^{ij}(t,T)}{S^{ij}_{T}} 
=\ln E_t[f^j(X_{T})]/f^j(X_{T}) -\ln E_t[f^i(X_{T})]/f^i(X_{T});\\ 
f^{\tau}_t-s_t 
&=&\ln \frac{F^{ij}(t,T)}{S^{ij}_{t}} 
=\ln E_t[f^j(X_{T})]/f^j(X_{t}) -\ln E_t[f^i(X_{t})]/f^i(X_{t}). 
\ean 
As illustrated in Section \ref{Examples}, 
different combinations of the $X$ Markov process and the $f$ function 
can generate a wide range of term structure models of interest rates. 
Similarly, we can also obtain a wide variety of specifications 
for the exchange rates and forward premia, some of which 
hold great potentials in explaining the forward premium puzzle. 

%llp 
\subsection{Potential models of currency pricing: examples} 

With the properties of the foreign exchange rate data in mind, we 
will examine, in the following, whether (and which category of) 
potential models 
can explain the forward premium anomaly. Specifically, 
we investigate what combinations of the Markov process for $X$ and the 
$f$ function can generate exchange rates and forward premia that 
satisfy the Fama condition. 


\subsection{Exponential quadratic models} 

When the $f$ function is exponential quadratic of the form, 
\[ 
f^i(x)=\frac{1}{2}(x-c_i)'Q_i(x-c_i), 
\] 
the state-price density for country $i$ can be given as 
\[ 
\xi^i_t=\exp \left [ -\alpha_i t +\frac{1}{2}(X_{t}-c_i)^{\prime 
}Q_i(X_{t}-c_i) \right ] . 
\] 
We also assume that the state variable vector $X$ follows an 
Ornstein-Uhlenbeck process as specified in 
{\it Example \ref{ex:ou}}: 
\[ 
dX_t={\kappa} \left( \theta -X_t\right) dt+d W(t). 
\] 
Even within this same category, different choices of $Q_i$, 
$Q_j$ and $c_i$, $c_j$ yield a wide variety of currency pricing models. 
We consider two specific examples. 

\begin{example} 
The first example is based on the assumption that the diagonal 
matrix $Q$ is equal for both countries, i.e. $Q_i = Q_j = Q$ and 
$c_i \not= c_j$, the foreign exchange rate becomes 
\begin{equation} 
S^{ij}_t = \exp\left[ 
\frac{1}{2} \left ( c_j - c_i \right )' Q\left( c_j + c_i\right) 
+ \left ( c_i - c_j \right) 'Q X_t 
\right ] , 
\label{eq:expex1} 
\end{equation} 
which implies that 
the exchange rate depreciation rate $\delta$ follows an 
Ornstein-Uhlenbeck process with mean reversion, 
and that all the 
interest rates are squared-Gaussian. 
These features should give a tractable class 
of international term structure models. 

From (\ref{eq:expex1}), we have the depreciation rate 
\[ 
\delta_{t,T} = 
\left ( c_i - c_j \right )'Q \left ( X_T 
- X_t \right ) =D'(X_T-X_t), 
\] 
where $D'=(c_i-c_j)'Q$. 
The risk premium $p_{t,T}$ can be obtained by taking the 
expectation, 
\begin{eqnarray*} 
p_{t,T} &=& \log \frac{E_t\left[\xi^j_T\right] \xi^i_T} 
{E_t\left[\xi^i_T\right] \xi^j_T} \\ 
&=&\frac{1}{2} (c_i - c_j )' Q \left [ \left ( I - Q V_{t} \right 
)^{-1} 
\left ( 2 \mu_{t} - c_i -c_j \right ) 
- \left ( 2 X_{T} - c_i -c_j\right ) \right ]\\ 
&=&D'\bar Q e^{-\kappa\tau}X_t -D'X_T +constant, 
\end{eqnarray*} 
with $\bar Q=(I-Q V_{t})^{-1}$. Refer to 
{\it Example \ref{ex:ou}} for $\mu_{t}$ and $V_t$. 

Let $V$ denote the unconditional variance of $X$, we have 
\begin{eqnarray*} 
Var\left[\delta_{t,T}\right]&=&2D'V\left(I-e^{-\kappa \tau}\right)D;\\ 
Var\left[p_{t,T}\right] &=&D'\bar{Q}Ve^{-2\kappa \tau}\bar Q D+D'VD 
-2D'\bar{Q}Ve^{-2\kappa \tau} D ;\\ 
Cov\left[\delta_{t,T}, p_{t,T}\right] &=& 
-D'V\left(I-e^{-\kappa \tau}\right)D 
-DVe^{-\kappa \tau}\left(I-e^{-\kappa \tau}\right)\bar{Q}D. 
\end{eqnarray*} 
We can see that the covariance of the depreciation rate and 
the forward risk premium is negative. 
However, the relative magnitude of the variances of the depreciation rate 
and the forward risk premium depends on the parameters values. 
Presumably, the exponential quadratic model can, 
from a theoretical viewpoint, account for the forward premium 
anomaly found in empirical studies. Again, from a practical 
viewpoint, the exponential quadratic model presented in this 
section is particularly attractive because it entails some 
attractive features with respect to the distribution of the foreign 
exchange rates and the interest rates. 
\end{example} 

\begin{example} 
The second example is based on the assumption $Q_i \not= Q_j$, but 
$c_i = c_j =c$. Then the exchange rate becomes 
\begin{equation} 
S^{ij}_t 
= \exp\left[ 
\frac{1}{2} \left ( X_t - c \right ) ' Q_{ji} \left ( X_t - c\right) 
\right ], 
\label{expex2} 
\end{equation} 
with $Q_{ji} = Q_{j}-Q_{i}$. 
The depreciation rate $\delta_{t,T}$ is given as 
\[ 
\delta_{t,T}= 
\frac{1}{2} \left ( X_T - X_t \right ) Q_{ij} \left ( X_T + X_t 
- 2c \right ), 
\] 
which is a quadratic function of normal variates. 
The forward risk premium $p_{t,T}$ can be derived as 
\begin{eqnarray*} 
p_{t,T} &=& \log\left ( \frac{\sqrt{|I - Q_iV_{t} | } }{\sqrt{|I 
- Q_j V_{t}|} } \right ) 
+ \frac{1}{2} \left ( \mu_{t} - c \right )' \hat{Q}_{ji} 
\left ( \mu_{t} - c \right ) 
\\ 
&&- \frac{1}{2} \left ( X_T - c \right )'Q_{ji} \left ( X_T - c 
\right ) , 
\end{eqnarray*} 
with $\hat{Q}_{ji} = \left ( I - Q_j V_{t,T} \right )^{-1} 
Q_j - \left ( I - Q_i V_{t,T} \right )^{-1} Q_i$. 
Through messy but straightforward manipulations, 
we have the variances and covariance of the forward risk premium and the 
depreciation rate as 
\begin{eqnarray*} 
\var (p_{t,T}) &=& \frac{1}{2} 
tr( \hat{Q}_{ji}V e^{-2\kappa\tau} (\hat{Q}_{ji} -2 
Q_{ji}) V e^{-2\kappa\tau} + ( Q_{ji} V )^2) \\ 
&& + (\theta - c)' \left [ ( \hat{Q}_{ji} - 2 Q_{ji}) V e ^{- 
2\kappa \tau} \hat{Q}_{ji} + Q_{ji} V Q_{ji} \right ] 
(\theta - c);\\ 
Var (\delta_{t,T})&=& 
tr \left[ ( Q_{ji} V)^2 (I - e^{-2\kappa\tau}) \right ] 
+ 2 (\theta - c)' Q_{ji} V(I -e^{- \kappa\tau} ) Q_{ji} (\theta 
- c);\\ 
Cov\left[\delta_{t,T}, p_{t,T}\right] &=& 
-\frac{1}{2}tr\left[Q_{ji}^2 V^2 \left(I-e^{-2\kappa \tau}\right)\right] 
-(\theta-c)'Q_{ji}V\left(I-e^{-\kappa \tau}\right)Q_{ji}(\theta-c)\\ 
&& 
-\frac{1}{2}tr\left[\hat{Q}_{ji}Ve^{-2\kappa \tau} 
\left(I-e^{-2\kappa \tau}\right)Q_{ji}V\right] \\ 
&&-(\theta-c)'\hat{Q}_{ji}Ve^{-\kappa\tau} 
\left(I-e^{-\kappa \tau}\right)Q_{ji}(\theta-c). 
\end{eqnarray*} 
Again, the requirement for negative covariance between 
$\delta_{t,T}$ and $p_{t,T}$ is easily fulfilled, even for 
one-factor models; however, the relative 
magnitude of variances depends on the parameter values. 
\end{example} 


\subsection{Exponential Linear Models} 
When the $f$ function is exponential linear, 
the state-price density for country $i$ is 
\[ 
\xi^i_t = \exp \left ( \alpha_i t + b_i'X_t \right ). 
\] 
The exchange rate then follows 
\[ 
S^{ij}_t = \exp \left ( \left ( \alpha_j - \alpha_i \right ) t 
+ \left ( b_j - b_i \right )'X_t \right ) 
.\] 

\noindent Again, we will discuss two cases. 

\begin{example} 
We assume that $X_t$ follows an Ornstein-Uhlenbeck process as 
specified in {\it Example \ref{ex:ou}}. 
The depreciation rate is given as 
\begin{eqnarray*} 
\delta_{t,T} = \left ( b_j - b_i \right )' \left ( X_T - X_t\right 
), 
\end{eqnarray*} 
The variance of $\delta_{t,T}$ can easily be derived as 
\begin{eqnarray*} 
Var(\delta_{t,T}) = 2 \left ( b_j - b_i \right )' V \left ( I - 
e^{-\kappa \tau } \right )\left ( b_j - b_i \right ). 
\end{eqnarray*} 
The forward risk premium $p_{t,T}$ can be derived as 
\begin{eqnarray*} 
p_{t,T} = \frac{1}{2}\left ( b_j - b_i \right )' V \left ( b_j + 
b_i \right ) 
+ 
\left ( b_j - b_i \right )'\left ( e^{-\kappa \tau } X_t - X_T 
\right ) 
\end{eqnarray*} 
and variance of the forward risk premium is 
\begin{eqnarray*} 
Var(p_{t,T}) = \left ( b_j - b_i \right )' V\left ( I + e^{- 2 
\kappa \tau } \right ) \left ( b_j - b_i \right ). 
\end{eqnarray*} 
The risk premium is time varying, as required by Fama condition. 
However, the covariance between the depreciation rates and 
the forward premium is positive in such a set up. 
Since 
\begin{eqnarray*} 
\delta_{t,T} + p_{t,T} = 
\frac{1}{2}\left ( b_j - b_i \right )' V \left ( b_j + b_i \right ) 
- 
\left ( b_j - b_i \right )'\left ( I - e^{-\kappa \tau } \right ) 
X_t 
,\end{eqnarray*} 
the covariance term of the regression slope in equation 
(\ref{regression1}) comes down to 
\begin{eqnarray*} 
Cov\left [ \delta_{t,T}, \delta_{t,T} + p_{t,T}\right ] &=& 
\left ( b_j - b_i \right )' \left (I - e^{-\kappa \tau} \right ) 
V \left ( I - e^{-\kappa \tau}\right ) \left ( b_j - b_i \right ), 
\end{eqnarray*} 
which is positive. Therefore, the linear exponential model with the 
state variable $X_t$ following an Ornstein-Uhlenbeck process fails 
to generate a negative regression slope. 
\end{example} 


\begin{example} 
In this example we assume that $X_t$ follows the square-root 
process specified in {\it Example \ref{ex:bes}}. Then we obtain a 
generalized multi-factor Cox et al. (1985) model. 
The depreciation rate is given as 
\begin{eqnarray*} 
\delta_{t,T} = \left ( b_j - b_i \right )' \left ( X_T - X_t\right 
) 
,\end{eqnarray*} 
with variance 
\begin{eqnarray*} 
Var(\delta_{t,T}) = 2 \left ( b_j - b_i \right )' V \left ( I - 
e^{-\kappa \tau } \right )\left ( b_j - b_i \right ), 
\end{eqnarray*} 
where $V$ is the unconditional variance-covariance matrix of $X_t$. 
The forward risk premium is 
\begin{eqnarray*} 
p_{t,T} &=&\ln E_{t}\left[ f^{j}(X_{T})\right] /f^{j}\left( X_{T}\right) 
-\ln E_{t}\left[ f^{i}(X_{T})\right] /f^{i}\left( X_{T}\right) \\ 
&=&\ln \frac{\left| \left( I-C^{j}\right) ^{-A^{j}}\right| }{\left| \left( 
I-C^{i}\right) ^{-A^{i}}\right| }+\left[ b_{j}^{\prime }D_{j}\left( 
I-C_{j}\right) ^{-1}-b_{i}^{\prime }D_{i}\left( I-C_{i}\right) ^{-1}\right] 
X_{t}-\left( b_{j}-b_{i}\right) ^{\prime }X_{T}. 
\end{eqnarray*} 
The forward premium $f^{\tau}-s_t=\delta_{t,T}+p_{t,T}$ is 
\begin{eqnarray*} 
f_{t}^{\tau }-s_{t} &=&\ln E_{t}\left[ f^{j}(X_{T})\right] /f^{j}\left( 
X_{t}\right) -\ln E_{t}\left[ f^{i}(X_{T})\right] /f^{i}\left( X_{t}\right) 
\\ 
&=&\ln \frac{\left| \left( I-C^{j}\right) ^{-A^{j}}\right| }{\left| \left( 
I-C^{i}\right) ^{-A^{i}}\right| }+\left[ b_{j}^{\prime }D_{j}\left( 
I-C_{j}\right) ^{-1}-b_{i}^{\prime }D_{i}\left( I-C_{i}\right) ^{-1}-\left( 
b_{j}-b_{i}\right) ^{\prime }\right] X_{t}. 
\end{eqnarray*} 
The covariance term in the regression is therefore 
\ban 
&&\cov\left(\delta _{t,T},\delta_{t,T}+p_{t,T}\right) \\ 
&&=\left[ b_{j}^{\prime 
}D_{j}\left( I-C_{j}\right) ^{-1}-b_{i}^{\prime }D_{i}\left( I-C_{i}\right) 
^{-1}-\left( b_{j}-b_{i}\right) ^{\prime }\right] V\left( I-e^{-\kappa \tau 
}\right) \left( b_{j}-b_{i}\right) . 
\ean 
In theory, 
whether we can obatin a negative covariance or not depends on the arcane 
factor $\left[ b_{j}^{\prime }D_{j}\left( I-C_{j}\right) ^{-1}-b_{i}^{\prime 
}D_{i}\left( I-C_{i}\right) ^{-1}-\left( b_{j}-b_{i}\right) ^{\prime }\right] 
$, the sign of which is not clear. In practice, however, 
Backus et al. (1998b) find extreme difficulty 
in trying to account for the 
forward premium puzzle with this class of models. 
\end{example} 


While both the exponential quadratic $f$ function and the exponential 
linear $f$ function can generate affine structures for bond prices, 
the exponential-linear category, which has been widely used for bond 
pricing, 
has a harder time explaining the forward premium anomaly. 
Backus et al. (1998b), in a discrete-time set-up, find that 
in affine models with exponential linear pricing kernels, the forward 
premium anomaly either requires that the state variables 
have asymmetric effects on state prices in different currencies or that 
we abandon the requirement that interest rates be strictly positive. 
The exponential quadratic class of models are better suited for currency 
pricing for their increased degrees of freedom 
in specifying country-specific parameters. Since both countries share the 
same 
underlying Markov process, the only country-specific parameters in 
exponential-linear models is $b_i$ in the pricing kernel, a 
vector with dimension equal to the number of state variables ($d$). However, 
in the exponential quadratic models, country-specific parameters 
include both $c_i$, a $d$-dimension vector and $Q$, a $d\times d$ matrix, 
and thus the degree of freedom is vastly increased. The advantage becomes 
obvious when we model international term structures although 
the two categories may generate similar results when one focuses 
on only one country. 

\setcounter{exmpl}{0} 
\setcounter{equation}{0} 
\section{Concluding Remarks} 
\label{conclusion} 
The potential approach provides a general framework for modeling 
interest rates and, in particular, exchange rates. 
Through comprehensive examples, we illustrate their correspondence, 
as special cases, to the classical interest rate models. We also show the 
relationship between the potential formulation of interest rate 
models and the HJM framework. 
The great ease of the potential approach in modeling 
international term structures enables us to 
explore the potential specifications of the state-price density 
to account for the forward premium puzzle in the foreign exchange market. 
At least two dimensions can be worked on 
simultaneously in the future research: 
On the one hand, 
since the direct specification of the state-price density imposes a 
closer link to the equilibrium economy, potential models in this paper 
should provide a benchmark for building practical equilibrium models 
in the future. 
On the other hand, more comprehensive empirical work needs 
to be done to calibrate these models to the international term structure 
of interest rates and exchange rates and to test which specification 
is best to reconcile the anomalies observed in interest rate and 
exchange rate markets. 



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% A P P E N D I X 

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\appendix 
\pagebreak \parskip=\bigskipamount 
\setcounter{exmpl}{0} 
\setcounter{equation}{0} 
\section{Derivations and Proofs} 

\subsection{Resolvent representation of a potential} 

\label{resolvent} Let $f\in C({\cal X})$ be an arbitrary continuous 
function 
on ${\cal X}$. Consider the semigroup $\{T_t\}$ acting on $C({\cal 
X})$ by $% 
T_tf(x)=E_xf(X_t), x \in {\cal X}, t\geq 0$. For $\lambda >0$, 
write 
\[ 
R_{\lambda}f(x) =\int^{\infty}_0 e^{-\lambda s} T_s f(x) ds\, 
\,\,\, x \in 
{\cal X} ; 
\] 
$(R_{\lambda})_{\lambda>0}$ is called the {\it resolvent} (Laplace 
transform) of the semigroup $\{T_t\}$. A basic property of the 
resolvent is 
that $\lambda R_{\lambda}$ behaves like the identity operator on 
$C({\cal X}% 
) $ for $\lambda$ large in the sense that $\parallel f-\lambda 
R_{\lambda} 
f\parallel \rightarrow 0$ as $\lambda \rightarrow \infty$; i.e., 
$\lambda 
R_{\lambda} f \rightarrow f$ uniformly on ${\cal X}$ as $\lambda 
\rightarrow 
\infty$. With this property of the resolvent in mind, consider the 
process $% 
(Y_t)$ defined by 
\[ 
Y_t = e^{-\lambda t} R_{\lambda} f(X_t), \,\,\,\, t \geq 0, 
\] 
where $f\in C({\cal X})$ is fixed but arbitrary {\it nonnegative} 
function 
on ${\cal X}$. Then $(Y_t)$ is a supermartingale with respect to 
${\cal F}_t 
=\sigma\{X_s : s \leq t\}, t \geq 0$, since $E|Y_t|<\infty$, $Y_t$ 
is ${\cal % 
F}_t$-measurable, one has $T_tf(x) \geq 0 (x\in {\cal X}, t \geq 
0)$, and 
\begin{eqnarray*} 
E\left[Y_{t+h}|{\cal F}_t\right] &=& e^{-\lambda (t+h)}E\left[ 
R_{\lambda} 
f(X_{t+h})|{\cal F}_t\right] =e^{-\lambda(t+h)}T_h R_{\lambda} 
f(X_t) \\ 
&=& e^{-\lambda t} \int ^{\infty}_0 e^{-\lambda (s+h)} T_{s+h} 
f(X_t) ds 
=e^{-\lambda t} \int^{\infty}_h e^{-\lambda s} T_s f(X_t) ds \\ 
&\leq & e^{-\lambda t} \int^{\infty}_0 e^{-\lambda s} T_sf(X_t)ds 
=Y_t \,\, . 
\end{eqnarray*} 

\subsection{Exponential-quadratic functions of normal variates} 

\label{efx} This section derives the expectation of the 
exponential-quadratic functions of normal variates. Specifically, 
when $% 
X_{T} $ is normally distributed with mean $\mu _{t}$ and variance- 
covariance 
matrix $V_{t}$, we want to derive the expectation of a general 
exponential-quadratic function of $X_{T}$, 
\[ 
f(X_{T})=\exp \left[ {\frac{1}{2}(X_{T}-c)^{\prime }Q(X_{T}- 
c)}\right] 
=e^{q}, 
\] 
where $Q$ is assumed to be symmetric and positive-definite and $q$ 
is 
defined as $\frac{1}{2}(X_{T}-c)^{\prime }Q(X_{T}-c)$. First, we 
expand the $% 
q$ term as 
\begin{eqnarray*} 
q &=&\frac{1}{2}z^{\prime }Az+D^{\prime }z+{\cal C} 
\end{eqnarray*} 
where $\Sigma $ is defined such that $\Sigma \Sigma ^{\prime 
}=V_{t}$, $% 
z=\Sigma ^{-1}(X_{T}-\mu _{t})$ is a vector of standardized normal 
variates, 
$A=V_{t}Q$ is a $\left( d\times d\right) $symmetric and positive 
definite 
matrix, $D^{\prime }=(\mu _{t}-c)^{\prime }Q\Sigma $ is a $\left( 
1\times 
d\right) $ row of coefficients, and ${\cal C}=\frac{1}{2}(\mu 
_{t}-c)^{\prime }Q(\mu _{t}-c)$ is a constant. 

Then we can write the expectation of $f\left( X_{T}\right) $ in the 
integral 
form: 
\begin{equation} 
E\left[ f\left( X_{T}\right) \right] =\left( 2\pi \right) ^{- 
\frac{1}{2}% 
d}\int_{-\infty }^{+\infty }\cdots \int_{-\infty }^{+\infty }\exp 
\left[ 
\frac{1}{2}z^{\prime }Az+D^{\prime }z+{\cal C-}\frac{1}{2}z^{\prime 
}z\right] \prod dz_{i} \label{efx1} 
\end{equation} 
We perform an orthogonal transformation on $z$ such that 
\begin{equation} 
z=Gx \label{ortho2} 
\end{equation} 
where $G$ is an orthogonal matrix: $GG^{\prime }=I$. Substitute 
(\ref{ortho2}% 
) into (\ref{efx1}), we have 
\begin{equation} 
E\left[ f\left( X_{T}\right) \right] =\left( 2\pi \right) ^{- 
\frac{1}{2}% 
d}\int_{-\infty }^{+\infty }\cdots \int_{-\infty }^{+\infty }\exp 
\left[ -% 
\frac{1}{2}x^{\prime }\left( I-C\right) x+u^{\prime }x+{\cal 
C}\right] \prod 
dx_{i}, \label{efx2} 
\end{equation} 
where the Jacobian is unity since $G$ is orthogonal and $u^{\prime 
}=D^{\prime }G=(\mu _{t}-c)^{\prime }Q\Sigma G$. Since $C=G^{\prime 
}AG$ is 
a diagonal matrix, the right hand side of (\ref{efx2}), apart from 
the 
constant $\left( \exp {\cal C}\right) $ term, factorizes into $d$ 
single 
integrals of type 
\[ 
I_{j}=\left( 2\pi \right) ^{-\frac{1}{2}}\int_{-\infty }^{+\infty 
}\exp 
\left[ -\frac{1}{2}\left( 1-c_{ii}\right) 
x_{j}^{2}+u_{j}x_{j}\right] 
dx_{i}, 
\] 
which can be obtained directly from the moment generating function 
of the 
univariate normal distribution as 
\begin{equation} 
I_{j}=\left( 1-c_{jj}\right) ^{-\frac{1}{2}}\exp \left( 
\frac{1}{2}\frac{% 
u_{j}^{2}}{1-c_{jj}}\right) . \label{ij} 
\end{equation} 
Applying (\ref{ij}) to each term in (\ref{efx2}) gives 
\ban 
E\left[ f\left( X_{T}\right) \right] &=&\prod_{j=1}^{d}\left( 1- 
c_{jj}\right) 
^{-\frac{1}{2}}\exp \left[ 
\frac{1}{2}\sum_{j=1}^{d}\frac{u_{j}^{2}}{1-c_{jj}% 
}+{\cal C}\right] \\ 
& =&\left| I-QV_{t}\right| ^{- 
\frac{1}{2}% 
}\exp \left[ \frac{1}{2}(\mu _{t}-c)^{\prime }\left( I- 
QV_{t}\right) 
^{-1}Q\left( \mu _{t}-c\right) \right] . \label{efx4} 
\ean 

\subsection{Bessel processes} 

\label{bessel} 

In general, a multi-dimensional $\left( d\right) $ Bessel process 
can be 
written as 
\[ 
dX=\kappa \left( \theta -X\right) dt+\Sigma \sqrt{X}dW 
\] 
where $\kappa $ is a $\left( d\times d\right) $ matrix and $\Sigma 
$ is a 
diagonal positive definite matrix (In the context, we scale $\Sigma 
=I$). 

The properties of one-dimensional bessel process is 
well-known, see, for example, 
Cox et al. (1985). 
$X$ has a conditional non-central chi-square distribution with 
the characteristic function given by 
\ban 
\phi \left( x,\tau ;s\right) &=&E_0[\exp(isX_{\tau}]\\ 
&=&\left| 1-\frac{is\sigma ^{2}}{2\kappa 
}\left( 
1-e^{-\kappa \tau }\right) \right| ^{-\frac{2\kappa \theta }{\sigma 
^{2}}% 
}\exp \left( \frac{isxe^{-\kappa \tau }}{1-\frac{is\sigma 
^{2}}{2\kappa }% 
\left( 1-e^{-\kappa \tau }\right) }\right) \\ 
& =&\left| 1-2it\right| ^{- 
\frac{v}{2}}\exp \left( 
\frac{it\delta }{1-2it}\right) , 
\ean 
with the shape parameter $v=\frac{4\kappa \theta }{\sigma ^{2}}$, 
the noncentrality parameter $\delta =\frac{4\kappa 
xe^{-\kappa 
\tau }}{\sigma ^{2}\left( 1-e^{-\kappa \tau }\right) }$, 
and 
$t=\frac{s\sigma 
^{2}}{4\kappa }% 
\left( 1-e^{-\kappa \tau }\right) $. 
With $\tau \rightarrow \infty $, we have the 
unconditional 
characteristic function 
\[ 
\phi \left( x,\infty ;s\right) =\left| 1-\frac{is\sigma 
^{2}}{2\kappa }% 
\right| ^{-\frac{2\kappa \theta }{\sigma ^{2}}}=\left( 1-isb\right) 
^{-a}, 
\] 
where $a=\frac{2\kappa \theta }{\sigma ^{2}}$ and $b=\frac{\sigma 
^{2}}{% 
2\kappa }$, which is the characteristic function of a gamma 
distribution 
with mean $ab=\theta $ and variance $ab^{2}=\frac{\theta \sigma 
^{2}}{% 
2\kappa }$. 


For a $d$-dimensional Bessel process $x$, the characteristic function 
of its linear combination can be obtained through orthogonalization: 
\[ 
y=b^{\prime }x=c^{\prime }z 
\] 
such that the elements are $z$ are independent from each other 
with 
\[ 
dz=\hat{\kappa}\cdot \left( \theta -z\right) dt+\Sigma \sqrt{z}dW, 
\] 
where $\hat{\kappa}$ is a diagonal matrix with $i$-th element being 
$\left(\kappa ^{\prime }b\right) _{i}/b_{i}$. Now the problem is reduced 
to that of an affine function of independent Bessel processes. We have 
\begin{eqnarray*} 
\phi \left( y,\tau ;s\right) &=&\prod_{i=1}^{d}\phi \left( 
x_{i},\tau 
;sb_{i}\right) \nonumber \\ 
&=&\prod_{i=1}^{d}\left| 1-\frac{isb_{i}\sigma 
_{i}^{2}}{2\hat{\kappa}_{i}}% 
\left( 1-e^{-\hat{\kappa}_{i}\tau }\right) \right| ^{- 
\frac{2\hat{\kappa}% 
_{i}\theta _{i}}{\sigma _{i}^{2}}}\exp \left( \sum_{i=1}^{d}\frac{% 
isb_{i}x_{i}e^{-\hat{\kappa}_{i}\tau }}{1-\frac{isb_{i}\sigma 
_{i}^{2}}{2% 
\hat{\kappa}_{i}}\left( 1-e^{-\hat{\kappa}_{i}\tau }\right) 
}\right) \\ 
&=&\left| \left( I-isC\right) ^{\bullet \left( -A\right) }\right| 
\exp 
\left( isb^{\prime }D\left( I-isC\right) ^{-1}x\right) 
\end{eqnarray*} 
where ``$\bullet $'' denotes ``dot power operator,'' and $A$, $C$, 
and $D$ are diagonal matrices with 
\begin{eqnarray*} 
A_{ii} &=& \frac{2\hat{\kappa}_{i}\theta _{i}}{\sigma 
_{i}^{2}} ; 
\\ 
C_{ii} &=& \frac{isb_{i}\sigma 
_{i}^{2}}{2\hat{\kappa}_{i}}\left( 1-e^{-% 
\hat{\kappa}_{i}\tau }\right) ; \\ 
D_{ii} &=&\exp \left( -\hat{\kappa}_i\tau \right) . 
\end{eqnarray*} 
The conditional mean $\mu \left( y_{\tau }\right) $ and variance 
$V\left( 
y_{\tau }\right) $ are thus 
\begin{eqnarray*} 
\mu \left( y_{\tau }\right) &=&b^{\prime }\left( I-e^{-\tau 
\hat{\kappa}% 
}\right) \theta +b^{\prime }e^{-\tau \hat{\kappa}}x; \\ 
V\left( y_{\tau }\right) &=&\sum_{i=1}^{d}\theta 
_{i}\frac{b_{i}^{2}\sigma 
_{i}^{2}}{2\hat{\kappa}_{i}}\left( 1-e^{-\hat{\kappa}_{i}\tau 
}\right) 
^{2}+2\sum_{i=1}^{d}x_{i}\frac{b_{i}^{2}\sigma 
_{i}^{2}}{2\hat{\kappa}_{i}}% 
\left( 1-e^{-\hat{\kappa}_{i}\tau }\right) e^{-\hat{\kappa}_{i}\tau 
}. 
\end{eqnarray*} 
The unconditional moments are 
\[ 
\mu \left( y\right) =b^{\prime }\theta ;\quad V\left( y\right) 
=\sum_{i=1}^{d}\theta _{i}\frac{b_{i}^{2}\sigma 
_{i}^{2}}{2\hat{\kappa}_{i}}. 
\] 
The variance of the orthogonalized vector $z(t)$ is 
\[ 
V\left( z\right) = \diag\left[ \frac{b_{i}^{3}\sigma 
_{i}^{2}}{2\left( 
\kappa ^{\prime }b\right) _{i}}\theta _{i}\right] . 
\] 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\pagebreak 
\parskip=0.0in 
\parindent=0.0in 
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\setlength{\baselineskip}{12pt} 

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%llll 

\end{document} 










% Figures 
************************************************************* 
\newpage 
\noindent{\large \bf Figure 1 \\ 
Mean Yield Curve for the Quadratic Model} 

\vspace{0.1in} 
\hbox{\centerline{\psfig{figure=mkfig1.ps,height=3in,width=\textwid 
th}}} 

% Figures 
************************************************************* 
%\newpage 
\noindent{\large \bf Figure 2 \\ 
Mean Yield Curve for the 
Exponential Quadratic Model} 

\vspace{0.1in} 
\hbox{\centerline{\psfig{figure=mkfig2.ps,height=3in,width=\textwid 
th}}} 

% Figures 
************************************************************* 
\newpage 
\noindent{\large \bf Figure 3 \\ 
Mean Yield Curve for the 2-Factor 
Quadratic Model} 

\vspace{0.1in} 
\hbox{\centerline{\psfig{figure=mkfig3.ps,height=3in,width=\textwid 
th}}} 

% Figures 
************************************************************* 
%\newpage 
\noindent{\large \bf Figure 4 \\ 
Mean Yield Curve for the 
2-Factor Exponential Quadratic Model} 

\vspace{0.1in} 
\hbox{\centerline{\psfig{figure=mkfig4.ps,height=3in,width=\textwid 
th}}} 


\end{document} 


\end{document} 



Specifically, 
llp 
\ed 



