Johann Heinrich Ludolf Steinike (1825-1909) was a German landscape painter.
Models of the Short-Term Rate.
In a continuous-time framework, it is customary to model the short-term rate of interest by means of an Ito process. We thus assume that the dynamics of \(r\) are given in a differential form
\begin{equation}{\label{binch5eq4}}\tag{19}
dr_{t}=\mu (r_{t},t)dt+\boldsymbol{\sigma}(r_{t},t)\cdot
d{\bf W}_{t}\equiv \mu_{t}+\boldsymbol{\sigma}_{t}\cdot d{\bf W}_{t}\mbox{ with }r_{0}>0,
\end{equation}
where \(\mu\) and \(\boldsymbol{\sigma}\) are adapted stochastic processes with values in \(\mathbb{R}\) and \(\mathbb{R}^{d}\), respectively, and \({\bf W}\) a \(d\)-dimensional standard Brownian motion. Recall that (\ref{binch5eq4}) is a shorthand form of the following integral representation
\[r_{t}=r_{0}+\int_{0}^{t}\mu_{s}ds+\int_{0}^{t}\boldsymbol{\sigma}_{u}\cdot d{\bf W}_{u}\mbox{ for all }t\in [0,T].\]
It is implicitly assumed that \(\mu\) and \(\boldsymbol{\sigma}\) satisfy the suitable integrability conditions, so that the process \(r\) is well-defined. In financial interpretation, the underlying probability measure \(\mathbb{P}\) is believed to reflect a subjective assessment of the “market” upon the future behavior of the short-term interest rate. In other words, the underlying probability \(\mathbb{P}\) is the actual probability, as opposed to a martingale probability measure for the bond market, which we are going to construct .
If we model under the objective probability measure \(\mathbb{P}\) and assume that a locally risk-free asset \(B\) (the money market) exists, we face the question whether in an arbitrage-free market bond prices, quite naturally viewed as derivatives with the short rate as underlying, are uniquely determined. In contrast to the equity market setting, with a risky asset and a risk-free asset available for trading, the short rate \(r\) is not the price of a traded asset, and hence we only set up portfolios consisting of putting money in the bank account. We thus face an incomplete market situation, and the best we can hope for is to find consistency requirements for bonds of different maturity. Given a single “benchmark” bond, we should then be able to price all other bonds relative to this given bond.
On the other hand, if we assume that the short rate is modelled under an equivalent martingale measure, we can immediately price all contingent claims via the risk-neutral valuation formula in Proposition \ref{binch5p1}. The drawback in this case is the question of calibrating model (we do not observe the parameters of the process under an equivalent martingale measure, but rather under the objective probability measure).
It is well-known that any probability measure \(\bar{\mathbb{P}}\) equivalent to \(\mathbb{P}\) on \((\Omega ,{\cal F}_{T^{*}})\) is given by the Radon-Nikodym derivative (a Girsanov density)
\[\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=L_{T^{*}}=\exp\left [-\int_{0}^{T^{*}}
\boldsymbol{\lambda}_{u}\cdot d{\bf W}_{u}-\frac{1}{2}\int_{0}^{T^{*}}
\parallel\boldsymbol{\lambda}_{u}\parallel^{2}du\right ]\mbox{ \(\mathbb{P}\)-a.s.},\]
for some predictable \(\mathbb{R}^{d}\)-valued process \(\boldsymbol{\lambda}\), where
\[L_{t}=\exp\left [-\int_{0}^{t}\boldsymbol{\lambda}_{u}\cdot d{\bf W}_{u}-\frac{1}{2}\int_{0}^{t}\parallel \boldsymbol{\lambda}_{u}\parallel^{2}du\right ].\]
We will use the notation \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\) to emphasize the dependence of the equivalent martingale measure on \(\boldsymbol{\lambda}\). By Girsanov’s theorem \ref{binp66}, we know that
\begin{equation}{\label{museq128}}\tag{20}
{\bf W}_{t}^{(\boldsymbol{\lambda})}={\bf W}_{t}+\int_{0}^{t}\boldsymbol{\lambda}_{u}du\mbox{ for all }t\in [0,T^{*}]
\end{equation}
follows a \(d\)-dimensional standard \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\)-Brownian motion. So the \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\)-dynamics of \(r\) are given by
\[dr_{t}=(\mu_{t}-\boldsymbol{\lambda}_{t}\cdot\boldsymbol{\sigma}_{t})dt+\boldsymbol{\sigma}_{t}\cdot
d{\bf W}_{t}^{(\boldsymbol{\lambda})}\]
(also Proposition \ref{musp1221}). Let us mention that the natural filtrations of the Brownian motions \({\bf W}\) and \({\bf W}^{(\boldsymbol{\lambda})}\) do not coincide, in general.
\begin{equation}{\label{musp1221}}\tag{21}\mbox{}\end{equation}
Proposition \ref{musp1221}. Assume the short-term interest rate \(r\) follows an Ito process under the actual probability measure \(\mathbb{P}\), as specified by (\ref{binch5eq4}). Let \(\{p(t,T)\}\) be an arbitrage-free family of bond prices relative to \(r\). For any martingale measure \(\bar{\mathbb{P}}=\bar{\mathbb{P}}(\boldsymbol{\lambda})\) taken as in Definition \ref{musd1211}, the following holds
(i) The process \(r\) satisfies under \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\)
\[dr_{t}=(\mu_{t}-\boldsymbol{\sigma}_{t}\cdot\boldsymbol{\lambda}_{t})dt+\boldsymbol{\sigma}_{t}\cdot
d{\bf W}_{t}^{(\boldsymbol{\lambda})}.\]
(ii) There exists an adapted \(\mathbb{R}^{d}\)-valued process \(c^{\scriptsize (\boldsymbol{\lambda})}(t,T)\) satisfying
\begin{equation}{\label{museq129}}\tag{22}
dp(t,T)=p(t,T)\left (r_{t}dt+c^{\scriptsize (\boldsymbol{\lambda})}(t,T)\cdot d{\bf W}^{(\boldsymbol{\lambda})}_{t}\right ).
\end{equation}
Consequently, for any fixed maturity \(T\in (0,T^{*}]\), we have
\[p(t,T)=p(0,T)\cdot B_{t}\cdot\exp\left (\int_{0}^{t}c^{\scriptsize (\boldsymbol{\lambda})}(u,T)\cdot
d{\bf W}_{u}^{(\boldsymbol{\lambda})}-\frac{1}{2}\int_{0}^{t}\parallel
c^{\scriptsize (\boldsymbol{\lambda})}(u,T)\parallel^{2}du\right ).\]
Proof. To prove part (i), it is enough to combine (\ref{binch5eq4}) with (\ref{museq128}). Let \(Z^{*}(t,T)=p(t,T)/B_{t}\). To prove part (ii), it is sufficient to observe that the process \(M_{t}=Z^{*}(t,T)\cdot L_{t}\) follows a (local) martingale under \(P\). In view of Theorem \ref{binp68}, we have
\begin{equation}{\label{museq17}}\tag{23}
M_{t}=Z^{*}(t,T)\cdot L_{t}=M_{0}+\int_{0}^{t}\boldsymbol{\gamma}_{u}
\cdot d{\bf W}_{u}=Z^{*}(0,T)+\int_{0}^{t}\boldsymbol{\gamma}_{u}\cdot d{\bf W}_{u}\mbox{ for all }t\in [0,T]
\end{equation}
for some adapted process \(\boldsymbol{\gamma}\). We now have
\begin{equation}{\label{museq12}}\tag{24}
dL_{t}^{-1}=L_{t}^{-1}\boldsymbol{\lambda}_{t}\cdot d{\bf W}_{t}^{(\boldsymbol{\lambda})}=L_{t}^{-1}
\boldsymbol{\lambda}_{t}\cdot (d{\bf W}_{t}+\boldsymbol{\lambda}_{t}dt).
\end{equation}
Applying the product rule (\ref{bineq48}), we obtain
\begin{align*}
dZ^{*}(t,T) & =d(M_{t}\cdot L_{t}^{-1})\\
& =L_{t}^{-1}dM_{t}+M_{t}dL_{t}^{-1}+d\langle M,L^{-1}\rangle\mathbb{E}_{t}\\
& =L_{t}^{-1}\boldsymbol{\gamma}_{t}\cdot d{\bf W}_{t}+M_{t}L_{t}^{-1}\boldsymbol{\lambda}_{t}\cdot
d{\bf W}_{t}^{(\boldsymbol{\lambda})}+L_{t}^{-1}\boldsymbol{\gamma}_{t}\boldsymbol{\lambda}_{t}dt
\mbox{ (by (\ref{museq17}) and (\ref{museq12}))}\\
& =L_{t}^{-1}\boldsymbol{\gamma}_{t}\cdot (d{\bf W}_{t}^{(\boldsymbol{\lambda})}-\boldsymbol{\lambda}_{t})+
M_{t}L_{t}^{-1}\boldsymbol{\lambda}_{t}\cdot d{\bf W}_{t}^{(\boldsymbol{\lambda})}+L_{t}^{-1}
\boldsymbol{\gamma}_{t}\boldsymbol{\lambda}_{t}dt\\
& =L_{t}^{-1}(\boldsymbol{\gamma}_{t}+M_{t}\boldsymbol{\lambda}_{t})d{\bf W}_{t}^{(\boldsymbol{\lambda})}
\end{align*}
Since \(B_{t}=\exp (\int_{0}^{t}r_{u}du)\), applying the product rule again, we have
\begin{align*}
dp(t,T) & =d(B_{t}\cdot Z^{*}(t,T))\\
& =Z^{*}(t,T)dB_{t}+B_{t}dZ^{*}(t,T)+d\langle B,Z^{*}\rangle\mathbb{E}_{t}\\
& =Z^{*}(t,T)B_{t}r_{t}dt+B_{t}L_{t}^{-1}(\boldsymbol{\gamma}_{t}+M_{t}\boldsymbol{\lambda}_{t})
d{\bf W}_{t}^{(\boldsymbol{\lambda})}\\
& =p(t,T)r_{t}dt+p(t,T)(M_{t}^{-1}\boldsymbol{\gamma}_{t}+\boldsymbol{\lambda}_{t})d{\bf W}_{t}^{(\boldsymbol{\lambda})}.
\end{align*}
This completes the proof. \(\blacksquare\)
Proposition. Let \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\) and \(\bar{\mathbb{P}}(\bar{\boldsymbol{\lambda}})\) be two probability measures equivalent to the underlying probability measure \(\mathbb{P}\). Assume that the bond price \(p(t,T)\) is given by \((\ref{bineq101})\) with \(\bar{\mathbb{P}}=\bar{\mathbb{P}}(\boldsymbol{\lambda})\). Then, for every \(t\in [0,T^{*}]\)
\[p(t,T)=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}(\bar{\boldsymbol{\lambda}})}\left [\left .
\exp\left (-\int_{t}^{T}r_{u}du\right )\cdot\exp\left (\int_{t}^{T^{*}}(\bar{\boldsymbol{\lambda}}_{u}-\boldsymbol{\lambda}_{u})
\cdot d{\bf W}_{u}^{(\bar{\boldsymbol{\lambda}})}-\frac{1}{2}\int_{t}^{T^{*}}\parallel\boldsymbol{\lambda}_{u}-
\bar{\boldsymbol{\lambda}}_{u}\parallel^{2}du\right )\right |{\cal F}_{t}\right ].\]
Proof. We write the Radon-Nikodym derivative as
\[\frac{d\bar{\mathbb{P}}(\boldsymbol{\lambda})}{d\mathbb{P}}=L_{t}^{(\boldsymbol{\lambda})}.\]
Then, we have
\[\frac{d\bar{\mathbb{P}}(\boldsymbol{\lambda})}{d\bar{\mathbb{P}}(\bar{\boldsymbol{\lambda}})}=\frac
{d\bar{\mathbb{P}}(\boldsymbol{\lambda})/d\mathbb{P}}{d\bar{\mathbb{P}}(\bar{\boldsymbol{\lambda}})/d\mathbb{P}}=\frac
{L_{t}^{(\boldsymbol{\lambda})}}{L_{t}^{(\bar{\boldsymbol{\lambda}})}}=
\exp\left (-\int_{0}^{t}(\boldsymbol{\lambda}_{u}-\bar{\boldsymbol{\lambda}}_{u})d{\bf W}_{u}-\frac{1}{2}\int_{0}^{t}
(\parallel\boldsymbol{\lambda}_{u}\parallel^{2}-\parallel\bar{\boldsymbol{\lambda}}_{u}\parallel^{2})du\right ).\]
Since
\[d{\bf W}_{t}=d{\bf W}_{t}^{(\bar{\boldsymbol{\lambda}})}-\bar{\boldsymbol{\lambda}}_{t}dt\]
and
\[\parallel\boldsymbol{\lambda}_{u}-\bar{\boldsymbol{\lambda}}_{u}\parallel^{2}=\parallel\boldsymbol{\lambda}_{u}\parallel^{2}-
2\boldsymbol{\lambda}_{u}\cdot\bar{\boldsymbol{\lambda}}_{u}+\parallel\bar{\boldsymbol{\lambda}}_{u}\parallel^{2},\]
we have
\[\frac{L_{t}^{(\boldsymbol{\lambda})}}{L_{t}^{(\bar{\boldsymbol{\lambda}})}}=\exp\left (-\int_{0}^{t}
\boldsymbol{\gamma}_{u}\cdot d{\bf W}_{u}^{(\bar{\boldsymbol{\lambda}})}-\frac{1}{2}\int_{0}^{t}
\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\equiv U_{t}^{(\boldsymbol{\gamma})},\]
where \(\boldsymbol{\gamma}_{t}=\boldsymbol{\lambda}_{t}-\bar{\boldsymbol{\lambda}}_{t}\). Applying the Bayes formula in Proposition \ref{binch4p1}, we get
\[p(t,T)=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}(\boldsymbol{\lambda})}\left [\left .
e^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}(\bar{\boldsymbol{\lambda}})}
\left [\left .\frac{U_{T^{*}}^{(\boldsymbol{\lambda})}}
{U_{t}^{(\boldsymbol{\lambda})}}\cdot e^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ].\]
This yields the asserted equality upon simplification. \(\blacksquare\)
Suppose that the short-term interest rate \(r\) satisfies (\ref{binch5eq4}) under a probability \(\mathbb{P}\). Let \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\) be an arbitrary probability measure equivalent to \(\mathbb{P}\). Then we may define a bond price \(p(t,T)\) by setting
\begin{equation}{\label{museq1211}}\tag{25}
p(t,T)=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}(\boldsymbol{\lambda})}\left [\left .e^{-\int_{t}^{T}r_{u}du}\right |
{\cal F}_{t}^{\scriptsize {\bf W}^{(\boldsymbol{\lambda})}}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
It follows from (\ref{museq129}) that the bond price \(p(t,T)\) satisfies, under the actual probability measure \(P\),
\[dp(t,T)=p(t,T)\left ((r_{t}-\boldsymbol{\lambda}_{t}\cdot c^{\scriptsize (\boldsymbol{\lambda})}(t,T))dt+c^{\scriptsize (\boldsymbol{\lambda})}(t,T)\cdot d{\bf W}_{t}\right ).\]
This means that the instantaneous returns from holding the bond differ, in general, from the short-term interest rate \(r\). In financial literature, the additional term is commonly referred to as the risk premium or the market price of risk. Summarizing, we have a certain degree of freedom: if the short-term rate \(r\) is given by (\ref{binch5eq4}), then any probability measure \(\bar{\mathbb{P}}(\boldsymbol{\lambda})\) equivalent to \(\mathbb{P}\) can formally be used to construct an arbitrage-free family of bond prices through formula (\ref{museq1211}). Notice, however, that if the actual probability measure \(\mathbb{P}\) is used to define the bond price through (\ref{museq1211}), the market prices for risk vanish.
Suppose that the dynamics of \(r\) under the actual probability measure \(\mathbb{P}\) satisfy
\begin{equation}{\label{museq1234}}\tag{26}
dr_{t}=\mu (r_{t},t)dt+\sigma (r_{t},t)dW_{t}
\end{equation}
for some sufficiently regular functions \(\mu\) and \(\sigma\), where the Brownian motion \(W\) is assumed to be one-dimensional. Assume, in addition, that the risk premium process equals \(\lambda_{t}=\lambda (r_{t},t)\) for some function \(\lambda (r,t)\). In financial interpretation, the last condition means that the excess rate of return of a given zero-coupon bond depends only on the current short-term rate and the price volatility of this bond. Using (\ref{museq1234}) and Girsanov’s theorem, we conclude that under the martingale measure \(\bar{\mathbb{P}}=\bar{\mathbb{P}}(\lambda )\), the process \(r\) satisfies
\begin{equation}{\label{museq1235}}\tag{27}
dr_{t}=(\mu (r_{t},t)-\lambda (r_{t},t)\cdot\sigma (r_{t},t))dt+\sigma (r_{t},t)d\bar{W}_{t}\equiv
\mu^{\lambda}(r_{t},t)dt+\sigma (r_{t},t)d\bar{W}_{t}.
\end{equation}
since \(dW_{t}=d\bar{W}_{t}-\lambda dt\). Let us stress once again that it is essential to assume that the functions \(\mu ,\sigma\) and \(\lambda\) are sufficiently regular (for instance, locally Lipschitz with respect to the first variable, and satisfying the linear growth condition), so that the SDE (\ref{museq1235}), with initial condition \(r_{0}>0\), admits a unique global strong solution. Under such assumptions, the process \(r\) is known to follow, under the martingale measure \(\bar{\mathbb{P}}\), a strong Markov process with continuous sample paths. The arbitrage price \(\Pi_{t}(X)\) of any attainable contingent claim \(X\), which is of the form \(X=g(r_{T})\) for some function \(g:\mathbb{R}\rightarrow \mathbb{R}\), is given by the risk-neutral valuation formula
\[\Pi_{t}(X)=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .g(r_{T})\cdot
e^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ]\equiv v(r_{t},t),\]
where \(v:\mathbb{R}\times [0,T^{*}]\rightarrow \mathbb{R}\). It follows, from the result known as the Feynman-Kac formula, that under mild technical assumption, if a security pays dividends continuously at rate \(h(r_{t},t)\) and yields a terminal payoff \(G_{T}=g(r_{T})\) at time \(T\), then the valuations function \(v\) solves the following PDE
\begin{equation}{\label{museq9}}\tag{28}
\frac{\partial v}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}(r,t)\cdot\frac
{\partial^{2}v}{\partial r^{2}}(r,t)+\mu^{\lambda}(r,t)\cdot\frac{\partial v}{\partial r}(r,t)-r\cdot v(r,t)+h(r,t)=0
\end{equation}
or
\[\frac{\partial v}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}(r,t)\cdot\frac
{\partial^{2}v}{\partial r^{2}}(r,t)+(\mu (r,t)-\lambda (r,t)\cdot
\sigma (r,t))\cdot\frac{\partial v}{\partial r}(r,t)-r\cdot v(r,t)+h(r,t)=0\]
subject to the terminal condition \(v(r,T)=g(r)\). Existence of a closed-form solution of this equation for the most typical derivative securities (in particular, for a zero-coupon bond and an option on such a bond) is a desirable property of a term structure model of diffusion type. Otherwise, the efficiency of numerical procedures used to solve the fundamental PDE becomes an important practical issue.
Suppose now that the zero-coupon bond price process \(p(t,T)\) is determined by the assessment, at time \(t\), of the segment \(\{r_{\tau}:t\leq\tau\leq T\}\) of the short rate process over the term of the bond. So we assume
\begin{equation}{\label{museq6}}\tag{29}
p(t,T)=v(t,r_{t})\equiv F(t,r_{t};T)
\end{equation}
with a sufficiently smooth function. Since we know that the value of a zero-coupon is one unit at maturity (also \(h(r,t)=0\)), we have the terminal condition \(F(T,r;T)=1\). Thus we have the following result.
Proposition (Term Structure Equation). If there exists an equivalent martingale measure of type \(\bar{\mathbb{P}}(\lambda )\) for the bond market (implying that the no-arbitrage condition holds) and the zero-coupon bond price processes \(p(t,T)\) are given by a sufficiently smooth function \(v\) as in \((\ref{museq6})\), then \(F\) will satisfy the partial differential equation
\begin{equation}{\label{binch5eq5}}\tag{30}
\frac{\partial F}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}(r,t)\cdot\frac
{\partial^{2}F}{\partial r^{2}}(r,t)+(\mu (r,t)-\lambda (r,t)\cdot
\sigma (r,t))\cdot\frac{\partial F}{\partial r}(r,t)-r\cdot F(r,t)=0
\end{equation}
with terminal condition \(F(T,r;T)=1\). \(\sharp\)
We emphasize again that the existence of \(\bar{\mathbb{P}}(\lambda)\) implies that the bond market is free of arbitrage, but that the reverse implication is not true in general. Comparison of the above PDE (\ref{binch5eq5}) with the Black-Scholes (\ref{bineq8}) shows that we have the additional appearance of the term \(\lambda\). The important fact to notice is that \(\lambda\) is not determined within the model, so in order to solve the equation we have to specify \(\lambda\) a priori. This reflects the fact that the model is incomplete. Specification of \(\lambda\) is a statistical problem, and is done by calibrating the model to “benchmark” bond prices.
We investigate the role of the function \(\lambda\) a little further. Under the assumption on the bond prices, an application of Ito’s formula to the process \(F(t,r_{t};T)\) (recall \(r_{t}\) has dynamics given in (\ref{museq1234})) yields
\[dF(t,r_{t};T)=F(t,r_{t};T)\cdot [m(t,T)dt+n(t,T)dW_{t}]\]
with the functions \(m(t,T)=m(t,r_{t};T)\) and \(n(t,T)=n(t,r_{t};T)\) given by
\[m=\frac{F_{t}+\mu F_{r}+\frac{1}{2}\sigma^{2}\cdot F_{rr}}{F}\mbox{ and }n=\frac{\sigma F_{r}}{F}.\]
We are now in the framework of (\ref{ch5eq2}) with the functions \(m\) and \(n^{2}\) being the mean and variance, respectively, of the instantaneous rate of return at time \(t\) on a bond with marurity date \(T\) given that the current spot rate is \(r_{t}=r\). Rewritting (\ref{binch5eq5}), \(\lambda\) satisfies
\[\lambda =\frac{F_{t}+\mu F_{r}+\frac{1}{2}\sigma^{2}F_{rr}-rF}{\sigma F_{r}}\]
or in terms of \(m\) and \(n\) above,
\[\lambda (t,r)=\frac{m(t,r;T)-r}{n(t,r;T)}\mbox{ for }T\geq t.\]
Observe that in the numerator \(m(t,r:T)-r\) we have the risk premium for the \(T\)-bond, i.e., the rate of return over the risk-free rate commanded by a \(T\)-bond. In the denominator, we have the volatility of the \(T\)-bond, and thus the quotient can be interpreted as the risk premium per unit of volatility of the \(T\)-bond. The ratio is often called the market price of risk, reflecting the fact that it specifies the expected instantaneous rate of return on a bond per additional unit of risk.
Single-Factor Models.
We survey the most widely accepted single-factor models of the short-term rate. It is assumed throughout that the dynamics of \(r\) are specified under the martingale probability measure \(\bar{\mathbb{P}}\) (i.e., the risk premium vanishes identically). The underlying Browinian \(\bar{W}\) is assumed to be one-dimensional. The models are based on a single source of uncertainty, i.e., they belong to the class of single-factor models.
By referring (\ref{museq9}), we are clearly in need of efficient methods of solving this partial differential equations. Fortunately, there is a class of models, exhibiting an affine term structure (ATS), which allows for simplication.
Definition. If bond prices are given as
\[p(t,T)=\exp [A(t,T)-r_{t}\cdot B(t,T)]\mbox{ for }0\leq t\leq T\]
with \(A(t,T)\) and \(B(t,T)\) are deterministic functions, we say that the model possesses an affine term structure. \(\sharp\)
Assuming that we have such a model in which both \(\mu\) and \(\sigma^{2}\) are affine in \(r\), say \(\mu (t,r_{t})=\alpha (t)-r_{t} \cdot\beta (t)\) and \(\sigma (t,r_{t})=\sqrt{\gamma (t)+r_{t}\cdot\delta (t)}\), we find that \(A\) and \(B\) are given as solutions of ordinary differential equations
\[A_{t}(t,T)-\alpha (t)B(t,T)+\frac{\gamma (t)}{2}\cdot B^{2}(t,T)=0\mbox{ with }A(T,T)=0,\]
\[1+B_{t}(t,T)+\beta (t)\cdot B(t,T)-\frac{\delta (t)}{2}\cdot B^{2}(t,T)=0,\mbox{ with }B(T,T)=0.\]
The equation for \(B\) is a Riccati equation, which can be solved analytically. Using the solution for \(B\) we get \(A\) by integrating. Examples of short rate models exhibiting an affine term structure are given below:
- Vasicek model: \(dr_{t}=(\alpha -\beta r_{t})dt+\gamma d\bar{W}_{t}\);
- Cox-Ingersoll-Ross (CIR) model: \(dr_{t}=(\alpha -\beta r_{t})dt+\delta\sqrt{r_{t}}d\bar{W}_{t}\);
- Ho-Lee model: \(dr_{t}=\alpha (t)dt+\gamma d\bar{W}_{t}\);
- Hull-White (extended Vasicek) model: \(dr_{t}=(\alpha (t)-r_{t}\cdot\beta (t))dt+\gamma (t)d\bar{W}_{t}\);
- Hull-White (extended CIR) model: \(dr_{t}=(\alpha (t)-r_{t}\cdot\beta (t))dt+\delta (t)\sqrt{r_{t}}d\bar{W}_{t}\).
Time-homogeneous Models.
Vasicek’s model. The model analysed by Vasicek \cite{vas} is one of the earliest models of term structure. The diffusion process proposed by Vasieck is a mean-reverting version of the Ornstein-Uhlenbeck process. The short-term interest rate \(r\) is defined as the unique strong solution of the SDE
\begin{equation}{\label{museq1213}}\tag{31}
dr_{t}=(a-br_{t})dt+\sigma d\bar{W}_{t},
\end{equation}
where \(a,b\) and \(\sigma\) are strictly positive constants. It is well-known that the solution of (\ref{museq1213}) is a Markov process with continuous sample paths and a Gausian distribution with the mean \(a/b\) and the variance \(\sigma^{2}/2b\). It is evident that Vasieck’s model, as any Gaussian model, allows for negative values of (nominal) interest rates. This property is manifestly incompatible with no-arbitrage in the presence of cash in the economy. Let us consider any security whose payoff depends on the short-term rate \(r\) as the only state variable. More specifically, we assume that this security is of European style, pays dividends continuously at rate \(h(r_{t},t)\), and yields a terminal payoff \(G_{T}=g(r_{T})\) at time \(T\). Using the well-known relationship between diffusion processes and the PDEs, one can show that the price process \(G_{t}\) of such a security admits the representation \(G_{t}=v(r_{t},t)\) solves the following PDE by referring (\ref{museq9})
\[\frac{\partial v}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}\cdot\frac{\partial^{2}v}{\partial r^{2}}(r,t)+(a-br)
\cdot\frac{\partial v}{\partial r}(r,t)-r\cdot v(r,t)+h(r,t)=0\]
subject to the terminal condition \(v(r,T)=g(r)\). Solving this equation with \(h=0\) and \(g(r)=1\), Vasicek showed that the price of zero-coupon bond is
\begin{equation}{\label{museq1214}}\tag{32}
p(t,T)=v(r_{t},t,T)=e^{m(t,T)-n(t,T)\cdot r_{t}},
\end{equation}
where
\begin{equation}{\label{museq1215}}\tag{33}
n(t,T)=\frac{1}{b}\left (1-e^{-b(T-t)}\right )\mbox{ and }
m(t,T)=\frac{\sigma^{2}}{2}\int_{t}^{T}n^{2}(u,T)du-a\int_{t}^{T}n(u,T)du.
\end{equation}
To establish this result, it is enough to assume that the bond price is given by (\ref{museq1214}) with the functions \(m\) and \(n\) satisfying \(m(T,T)=n(T,T)=0\), and to make use of the fundamental PDE. By separating terms which do not depend on \(r\), and those that are linear in \(r\), we arrive at the following system of differential equations
\begin{equation}{\label{museq1217}}\tag{34}
n_{t}(t,T)=b\cdot n(t,T)-1\mbox{ with }n(T,T)=0
\end{equation}
and
\begin{equation}{\label{museq1218}}\tag{35}
m_{t}(t,T)=a\cdot n(t,T)-\frac{1}{2}\sigma^{2}\cdot n^{2}(t,T)\mbox{ with }m(T,T)=0,
\end{equation}
which in turn yields easily the expressions above. One may check that we have
\begin{equation}{\label{museq1219}}\tag{36}
dp(t,T)=p(t,T)(r_{t}dt+\sigma\cdot n(t,T)d\bar{W}_{t})\equiv p(t,T)(r_{t}dt+b(t,T)d\bar{W}_{t}).
\end{equation}
If the bond price admits representation (\ref{museq1214}), then obviously
\[y(t,T)=\frac{n(t,T)\cdot r_{t}-m(t,T)}{T-t},\]
and thus the bond’s yield, \(y(t,T)\), is an affine function of the short-term rate \(r_{t}\). For this reason, models of the short-term rate in which the bond price satisfies (\ref{museq1214}) for some functions \(m\) and \(n\) are termed affine models of the term structure.
Jamshidian \cite{jam} obtained closed-form solutions for the prices of a European option written on a zero-coupon and on a coupon-bearing bond for Vasicek’s model. He showed that the arbitrage price at time \(t\) of a call option on a \(U\)-maturity zero-coupon bond, with strike price \(K\) and expiry \(T\leq U\), equals
\[C_{t}=p(t,T)\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .(\xi\cdot\eta-K)^{+}\right |{\cal F}_{t}\right ],\]
where \(\eta =p(t,U)/p(t,T)\), and \(\mathbb{P}^{*}\) stands for some probability measure equivalent to \(\bar{\mathbb{P}}\). The random variable \(\xi\) is independent of the \(\sigma\)-field \({\cal F}_{t}\) under \(\mathbb{P}^{*}\), and has under \(\mathbb{P}^{*}\) a lognormal law such that the variance \(Var_{\mathbb{P}^{*}}(\ln\xi )\) equals \(v_{U}(t,T)\), where
\[v^{2}_{U}(t,T)=\int_{t}^{T}|b(s,T)-b(s,U)|^{2}ds,\]
or explicitly
\[v^{2}_{U}(t,T)=\frac{\sigma^{2}}{2b^{3}}\cdot\left (1-e^{-2b(T-t)}\right )\cdot\left (1-e^{-b(U-t)}\right )^{2}.\]
The bond option price valuation formula established in Jamshidian \cite{jam} reads as follows
\[C_{t}=p(t,U)\cdot N\left (h_{1}(t,T)\right )-K\cdot p(t,T)\cdot N\left (h_{2}(t,T)\right ),\]
where for every \(t\leq T\leq U\)
\[h_{1,2}(t,T)=\frac{\ln\left (\frac{p(t,U)}{p(t,T)}\right )-\ln K\pm\frac{1}{2}\cdot v_{U}^{2}(t,T)}{v_{U}(t,T)}.\]
It is important to observe that the coefficient \(a\) does not enter the bond option valuation formula. This suggests that the actual value of the risk premium has no impact whatsoever on the bond option rice; the only relevant qantities are in fact the bond price volatilities \(b(t,T)\) and \(b(t,U)\) (ref. (\ref{museq1219})). To account for the risk premium, it is enough to make an equivalent change of the probability measure in (\ref{museq1219}). Since the volatility of the bond price is invariant with respect to such a transformation of the underlying probability measure, the bond price is independent of the risk premium, provided that the bond price volatility is deterministic. The determination of the risk premium may thus appear irrelevant, if we concentrate on the valuation of derivatives. This is not the case, however, if the aim is to model the actual behavior of bond prices.
Cox-Ingersoll-Ross model (CIR model). The general equilibrium approach to term structure modelling developed by Cox et al. \cite{cox} leads to the following modification of the mean-reverting diffusion of Vasicek, known as the square-root process
\[dr_{t}=(a-br_{t})dt+\sigma\cdot\sqrt{r_{t}}d\bar{W}_{t},\]
where \(a,b\) and \(\sigma\) are strictly positive constants. Due to the presence of the square-root in the diffusion coefficient, the CIR diffusion takes only positive values; it can reach zero, but it never becomes negative. In a way similar to the previous case, the price process \(G_{t}=v(r_{t},t)\) of any standard European interest rate derivative, which settles at time \(T\), can be found by solving the valuation PDE by referring (\ref{museq9})
\[\frac{\partial v}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}\cdot r\cdot\frac{\partial^{2}v}{\partial t^{2}}(r,t)+(a-br)
\cdot\frac{\partial v}{\partial t}(r,t)-r\cdot v(r,t)+h(r,t)=0\]
subject to the terminal condition \(v(r,T)g(r)\). Cox et al. \cite{cox} found closed-form solutions for the price of a zero-coupon bond. If we assume that the bond price \(p(t,T)\) satisfies (\ref{museq1214}), then using the valuation PDE above, we obtain
\[m(t,T)=\frac{2a}{\sigma^{2}}\cdot\ln\left (\frac{\gamma\cdot
e^{b\cdot\tau/2}}{\gamma\cdot\cosh (\gamma\tau )+\frac{1}{2}\cdot b\cdot\sinh (\gamma\tau )}\right )\mbox{ and }
n(t,T)=\frac{\sinh (\gamma\tau )}{\gamma\cdot\cosh (\gamma\tau )+\frac{1}{2}\cdot b\cdot\sinh (\gamma\tau )},\]
where \(\gamma =T-t\) and \(2\gamma =\sqrt{b^{2}+2\sigma^{2}}\). Closed-form expressions for the price of an option on a zero-coupon and an option on a coupon-bearing bond in the CIR framework were also derived in Cox el al. \cite{cox} and in Longstaff \cite{lon93}.
Longstaff’s model. Longstaff \cite{lon} modified the CIR model by postulating the following dynamics for the short-term rate
\[dr_{t}=a(b-c\sqrt{r_{t}})dt+\sigma\cdot\sqrt{r_{t}}d\bar{W}_{t},\]
referred to as the {\bf double square-root process}. Longstaff derived a closed-form expression for the price of a zero-coupon bond
\[p(t,T)=v(r_{t},t;T)=e^{m(t,T)-n(t,T)-q(t,T)\cdot\sqrt{r_{t}}}\]
for some explcitly known functions \(m,n\) and \(q\), which are not reproduced here.
Time-inhomogeneous Models
Hull-White model. Note that both Vasieck’s and the CIR models are special cases of the following means-reverting diffusion process
\[dr_{t}=a(b-c\cdot r_{t})dt+\sigma\cdot r_{t}^{\beta}d\bar{W}_{t},\]
where \(0\leq\beta\leq 1\) is a constant. These models of the short-term rate are thus built upon a certain diffusion process with constant (i.e. time-independent) coefficients. In practical applications, it is more reasonable to expect that in some situations, the market’s expectations about future interest rates involve time-dependent coefficients. Also, it would be plausible if a model fitted not merely the initial value of the short-term rate, but rather the whole initial yield curve. This desirable property of a bond price model motivated Hull and White \cite{hul} to propose an essential modification of the models above. The fairly general interest rate model they proposed extends both Vasieck’s model and CIR model in such a way that the model is able to fit exactly any initial yield curve. In some circumstances, it leads also to a closed-form solution for the price of a European bond option. In its most general form, the Hull-White methodology assumes that
\begin{equation}{\label{museq1229}}\tag{37}
dr_{t}=(a(t)-b(t)\cdot r_{t})dt+\sigma (t)\cdot r_{t}^{\beta}d\bar{W}_{t}
\end{equation}
for some constant \(\beta\geq 0\), where \(\bar{W}\) is a one-dimensional Brownian motion, and \(a,b,\sigma :\mathbb{R}_{+}\rightarrow \mathbb{R}\) are locally bounded functions. By setting \(\beta =0\) in (\ref{museq1229}), we obtain the
generalized Vasieck model, in which the dynamics of \(r\) are
\[dr_{t}=(a(t)-b(t)\cdot r_{t})dt+\sigma (t)d\bar{W}_{t}.\]
To explicitly solve this equation, let us denote \(l(t)=\int_{0}^{t}b(u)du\). Then we have
\[d\left (e^{l(t)}\cdot r_{t}\right )=e^{l(t)}\cdot\left (a(t)dt+\sigma (t)d\bar{W}_{t}\right ),\]
so that
\[r_{t}=e^{-l(t)}\cdot\left (r_{0}+\int_{0}^{t}e^{l(u)}\cdot a(u)du+\int_{0}^{t}e^{l(u)}\cdot\sigma (u)dW_{u}^{*}\right ).\]
It is not suprising that closed-form solutions for the bond and bond option prices are not hard to derive in this setting. On the other hand, if we put \(\beta =1/2\), then we obtain the generalized CIR model
\[dr_{t}=(a(t)-b(t)\cdot r_{t})dt+\sigma (t)\cdot\sqrt{r_{t}}d\bar{W}_{t}.\]
In this case, however, the cloed-form expressions for the bond and bond option prices are not easily available.
The most important feature of the Hull-White approach is the possibility of the exact fit of the initial term structure and, in some circumstances, also of the term structure of forward rate volatilities. This can be done by means of a judicious choice of the functions \(a,b\) and \(\sigma\). Since the details of the fitting procedure depend on the particular model (i.e. on the choice of \(\beta\)), let us illustrate this point by restricting the attention to the generalized Vasieck model. We start by assuming that the bond price \(p(t,T)\) can be represented in the following way
\begin{equation}{\label{museq1230}}\tag{38}
p(t,T)=p(r_{t},t,T)=e^{m(t,T)-n(t,T)\cdot r_{t}}
\end{equation}
for some functions \(m\) and with \(m(T,T)=0\) and \(n(T,T)=0\). Plugging (\ref{museq1230}) into the fundamental PDE for the zero-coupon bond, which is
\[\frac{\partial v}{\partial t}(r,t)+\frac{1}{2}\sigma^{2}(t)\cdot\frac{\partial^{2}v}{\partial r^{2}}(r,t)+(a(t)-b(t)\cdot r)\cdot
\frac{\partial v}{\partial r}(r,t)-r\cdot v(r,t)=0,\]
we obtain
\[m_{t}(t,T)-a(t)\cdot n(t,T)+\frac{1}{2}\sigma^{2}(t)\cdot n^{2}(t,T)-r\cdot (1+n_{t}(t,T)-b(t)\cdot n(t,T))=0.\]
Since the last equation holds for every \(t,T\) and \(r\), we deduce that \(m\) and \(n\) satisfy the following system of differential equations (ref. (\ref{museq1217}) and (\ref{museq1218}))
\[n_{t}(t,T)=b(t)\cdot n(t,T)-1\mbox{ with }n(T,T)=0\]
and
\begin{equation}{\label{museq1231}}\tag{39}
m_{t}(t,T)=a(t)\cdot n(t,T)-\frac{1}{2}\sigma^{2}(t)\cdot n^{2}(t,T)\mbox{ with }m(T,T)=0.
\end{equation}
Suppose that an initial term structure \(u(0,T)\) is exogenously given. We adopt the convention to denote by \(u(0,T)\) the initial term structure, which is given (it can, for instance, be inferred from the market data), as opposed to the initial structure \(p(0,T)\), which is implied by a particular stochastic model of the term structure. Assume also that the forward rate volatility is not prespecified. In this case, we may and do assume that \(b(t)=b\) and \(\sigma (t)=\sigma\) are given constants; only the function \(a\) is thus unknown. Since \(b\) and \(\sigma\) are constants, \(n\) is given by (\ref{museq1215}). Furthermore, in view of (\ref{museq1231}), \(m\) equals
\[m(t,T)=\int_{0}^{t} a(u)\cdot n(u,T)du-\frac{1}{2}\int_{0}^{t}\sigma^{2}(u)\cdot n^{2}(u,T)du.\]
Since \(m(T,T)=0\), we have \(m(T,T)-m(t,T)=-m(t,T)\). Then we obtain
\begin{equation}{\label{museq1232}}\tag{40}
m(t,T)=\frac{1}{2}\int_{t}^{T}\sigma^{2}(u)\cdot n^{2}(u,T)du-\int_{t}^{T}a(u)\cdot n(u,T)du.
\end{equation}
Since the forward rates implied by the model equal (ref. (\ref{museq115}))
\[f(0,T)=-\frac{\partial\ln p(0,T)}{\partial T}=n_{T}(o,T)\cdot r_{0}-m_{T}(0,T),\]
easy calculations involving (\ref{museq1215}) and (\ref{museq1232}) show that
\[\hat{f}(0,T)\equiv -\frac{\partial\ln u(0,T)}{\partial T}=e^{-bT}\cdot
r_{0}+\int_{0}^{T} e^{-b(T-s)}\cdot a(s)ds-\frac{\sigma^{2}}{2b^{2}}\cdot (1-e^{-bT})^{2}.\]
Put another way, \(\hat{f}(0,T)=g(T)-h(T)\), where \(g'(T)=-bg(T)+a(T)\) with \(g(0)=r_{0}\) and \(h(T)=\sigma^{2}\cdot (1-e^{-bT})^{2}/(2b^{2})\). consequently, we obtain
\[a(T)=g'(T)+bg(T)=\hat{f}_{T}(0,T)+h'(T)+b\cdot (\hat{f}(0,T)+h(T)),\]
and the function \(a\) is indeed uniquely determined. This terminates the fitting procedure. Though, at least theoretically, this procedure can be extended to fit the volatility structure, is should be stressed that the possibility of an exact match with the historical data is only one of several desirable properties of a model of the term structure. If the forward rate volatilities are also fitted, the Hull-White approach becomes close to the Heath-Jarrow-Morton methodology. In the matching procedure, the exact knowledge of the first derivative of the initial term structure \(u(0,T)\) with respect to \(T\) is assumed. In practice, the yield curve is known only at a finite number of points, corresponding to maturities of traded bonds, and the accuracy of data is also largely limited. Therefore, the actual shape of the yirld curve is known only approximately. Furthermore, in fitting additional initial data, we would typically need to use also the higher derivatives of the initial yield curve.
Parameter Estimation.
In the sequel, we discuss the parameter estimations. Using the martingale modelling approach, the model has been specified under a fixed martingale measure \(\bar{\mathbb{P}}\). This means in particular that all parameters are \(\bar{\mathbb{P}}\)-parameters. The observations, on the other hand, have been made under the objective probability measure \(\mathbb{P}\). Thus we cannot use the historical data to estimate parameters directly.
If we had specified the model under the objective probability measure \(\mathbb{P}\), we could use historical data to estimate the parameters. In order to compute bond prices we would have to specify an equivalent martingale measure \(\bar{\mathbb{P}}\). Now this leads to the problem of estimating the market price of risk (or more exactly the Girsanov density), but this is more or less equivalent to estimating the \(\bar{\mathbb{P}}\)-parameters. So due to the incompleteness of the market implied by only specifying the short rate, the problem boils down to using the current price information from the market in order to specify \(\bar{\mathbb{P}}\). This is done by calibrating the model to data or fitting the yield curve.
Given a concrete model involving a parameter vector \(\theta\), we write
\[dr_{t}=a(t,r_{t};\theta )dt+b(t,r_{t};\theta )d\bar{W}_{t}.\]
If we solve the term strcture equation for all times \(T\), we obtain theoretical bond price \(p(t,T;\theta )\). In the market, we observe today’s price \(p(0,T)\) for all values of \(T\), giving us the empirical term structure \(\{\hat{p}(0,T), T\geq 0\}\). We choose the parameter \(\theta\) in such a way that the theoretical curve \(\{p(t,T;\theta ),T\geq 0\}\) fits the empirical curve \(\{\hat{p}(0,T),T\geq 0\}\) as closely as possible in the sense specified by some objective function. This gives us the estimated parameter \(\hat{\theta}\). Using the procedure, we specified a martingale measure out of the class \(\{\bar{\mathbb{P}}_{\theta}\}\) by the requirement that the \(\bar{\mathbb{P}}\)-dynamics for \(r\) given by
\[dr_{t}=a(t,r_{t};\hat{\theta})dt+b(t,r_{t};\hat{\theta})d\bar{W}_{t}.\]
For practical purpose, we would like to have
\[\hat{p}(0,T)=p(0,T;\hat{\theta})\mbox{ for all }T\geq 0.\]
This involves an infinite-dimensional system of equations. So if we work with a model containing only a finite-dimensional parameter vector (such as the Vasicek model) there is no hope of obtaining a complete fit between observed and theoretical prices. This means that we are not able to price the simplest contingent claims in this model, let alone more complicated derivative structures.
The above procedure is a parametric statistical problem. We specify functions \(a,b\) depending on a parameter \(\theta\) with value in a parameter space \(\Theta\), and try to find the best parameter \(\hat{\theta}\) according to some objective function. Since this restricts the structure of possible functions \(a,b\), nonparametric approaches to estimating the functions \(a,b\) have been proposed.
Let us outline the calibration procedure for the Hull-White (extended Vasicek) model. The specification of the functions \(a(t,r)\) and \(b(t,r)\) as affine (deterministic) functions of \(r\) leads to a parametric problem in an infinite-dimensional parameter space. We assume that at the current time \(t=0\) we know
(i) the zero-coupon bond prices \(p(0,T)\) for \(0\leq T\leq T^{*}\);
(ii) the volatility at time zero of all bonds with maturities \(0\leq T\leq T^{*}\);
(iii) the current spot rate \(r(0)\);
(iv) the current pull-back level \(\alpha (0)\);
(v) the volatility function of the short rate \(\gamma (t)\) for \(t\in [0,T^{*}]\).
The task is to specify the functions \(\alpha (t)\) and \(\beta (t)\). Using the affine term structure of the model, we find that the volatility function of the bonds is given as \(-\gamma (0)B(0,T)p(0,T)\), and together with (i), (ii) and (v) this gives us \(B(0,T)\) for all \(0\leq T\leq T^{*}\). Since \(B(0,T)=-\int_{0}^{T}e^{-K(s)}ds\), we can recover \(\beta (t)\) from the definition of \(K(t)=\int_{0}^{t}\beta (s)ds\). We also use the function \(B(0,T)\) to find the function \(A(0,T)\) (use the affine term structure). Differentiating \(A(0,T)\) three times leads to an ordinary differential equation for \(\alpha (t)\), for which we can specify an unique solution, since we are given an initial value \(\alpha (0)\).
Multi-factor Models.
Since the short-term rate was assumed to follow a (one-dimensional) Markov process, it could also be identified with a unique state variable of the model. A more general approach to term structure modelling incorporates multi-factor models; that is, those term structure models which involve several sources of uncertainty (typically represented by a multidimensional Brownian motion).
In most two-factor models, the term structure is inferred from the evolution of the short-term interest rate and some other economic variable (for instance, the long-term interest rate, the spread between the short-term and long-term rates, the yields on a fixed number of bonds, etc.). From the theoretical point of view, a general multi-factor model is based on the specification of a multidimensional Markov process \({\bf X}\). Assume that \({\bf X}=(X_{1},\cdots ,X_{d})\) is a multidimensional diffusion process, defined as a unique strong solution of the SDE
\[d{\bf X}_{t}=\boldsymbol{\mu}({\bf X}_{t})+\boldsymbol{\sigma}({\bf X}_{t})\cdot d{\bf W}_{t},\]
where \({\bf W}\) is a \(d\)-dimensional Brownian motion, and the coefficients \(\boldsymbol{\mu}\) and \(\boldsymbol{\sigma}\) take values in \(\mathbb{R}^{d}\) and \(\mathbb{R}^{d}\times \mathbb{R}^{d}\), respectively. Processes \(X_{1},\cdots ,X_{d}\) are termed state variables; their economic interpretation, if any, is not always apparent, however. Notice that we have assumed that the number of factors coincides with the number of state variables. Generally speaking, the latter may be greater than the former. The short-term rate is now defined by setting \(r_{t}=g({\bf X}_{t})\) for some \(g:\mathbb{R}^{d}\rightarrow \mathbb{R}\). A special class of such models is obtained by postulating that \({\bf X}\) solves a linear SDE with time-dependent coefficients
\[d{\bf X}_{t}=({\bf a}(t)+{\bf b}(t)\cdot {\bf X}_{t})dt+\boldsymbol{\sigma}(t)\cdot d{\bf W}_{t},\]
where \({\bf a}:[0,T^{*}]\rightarrow \mathbb{R}^{d}\) and \({\bf b},\boldsymbol{\sigma}:[0,T^{*}]\rightarrow \mathbb{R}^{d}\times \mathbb{R}^{d}\) are bounded functions. Furthermore, it is common to set either \(r_{t}=\frac{1}{2}\parallel {\bf X}_{t}\parallel^{2}\) or \(r_{t}=\boldsymbol{\gamma}\cdot {\bf X}_{t}\) for some \(\boldsymbol{\gamma}\in \mathbb{R}^{d}\).
Consol Yield Model.
At the intuitive level, the consol yield can be defined as the yield on a bond that has a constant continuous coupon and infinite maturity. To make this concept precise, we need to consider an economy with an infinite horizon date, thst is, we set \(T^{*}=+\infty\). A consol (or consolidated bond) is a special kind of coupon-bearing bond with no final maturity date. An investor purchasing a consol is entitled to receive coupons from the issuer forever. In a continuous-time framework, it is convenient to view a consol as a risk-free security which continuously pays dividends at a constant rate, \(k\) say. In the framework of term structure models based on a (nonegative) short-term interest rate \(r\), the price of a consol at time \(0\) equals
\[B(0,\infty ,k)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\int_{0}^{\infty}k\cdot
e^{-\int_{0}^{s}r_{u}du}ds\right ]=\int_{0}^{\infty}k\cdot p(0,s)ds.\]
The last result can be extended to the case of arbitrary \(t\in \mathbb{R}_{+}\), yielding following equality for the price of a consol at time \(t\)
\begin{equation}{\label{museq1237}}\tag{41}
B(t,\infty ,k)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\int_{t}^{\infty}k\cdot e^{-\int_{t}^{s}r_{u}du}ds\right ]=\int_{t}^{\infty}k\cdot p(t,s)ds.
\end{equation}
It is clear that we may and do assume, without loss of generality, that the rate \(k\) equals \(1\). For brevity, we denote \(D_{t}=B(t,\infty ,1)\) with \(k=1\) in what follows. A substitution of \(r_{s}\) by a \({\cal F}_{t}\)-measurable random variable \(\eta\) in the last formula gives
\[\int_{t}^{\infty} e^{-\int_{t}^{s}\eta du}ds=\int_{t}^{\infty}e^{-\eta\cdot (s-t)}ds=\int_{0}^{\infty}e^{-\eta x}dx=\frac{1}{\eta}.\]
For this reason, the yield on a consol, called also the {\bf consol rate}, is simply defined as the reciprocal of its price, \(y_{t}=D_{t}^{-1}\). The consol rate can be seen as a proxy of a long-term rate of interest. To extend the short-term rate model, we may take into account the natural interdependence between both rates. In this way, we arrive at a two-factor diffusion-type model, in which the short-term rate \(r\) and the consol yield \(y\) are intertwined. Since \(D_{t}=y_{t}^{-1}\), we may work directly with the price \(D\) of the consol equally well. Formally, such a model assumes that the two-dimensional process \((r_{t},D_{t})\) solves the following pair of SDEs
\begin{equation}{\label{museq1238}}\tag{42}
dr_{t}=\mu (r_{t},D_{t})dt+\sigma (r_{t},D_{t})\cdot d{\bf W}_{t}^{*}\mbox{ and }
dD_{t}=\nu (r_{t},D_{t})dt+\gamma (r_{t},D_{t})\cdot d{\bf W}_{t}^{*},
\end{equation}
where \({\bf W}^{*}\) follows a two-dimensional standard Brownian motion under the martingale measure \(\mathbb{P}^{*}\). Notice that the drift coefficient \(\nu\) in the dynamics of \(D\) cannot be chosen arbitrarily. Indeed, an application of Ito’s formula to (\ref{museq1237}) gives the following SDE governing a consol’s price dynamics
\begin{equation}{\label{museq1240}}\tag{43}
dD_{t}=(r_{t}D_{t}-k)dt+\gamma (r_{t},D_{t})\cdot d{\bf W}_{t}^{*}
\end{equation}
under the martingale measure \(\mathbb{P}^{*}\). Equations (\ref{museq1238}) and (\ref{museq1240}) should still be considered simultaneously with equality (\ref{museq1237}), which links the price \(D\) to the short-term rate \(r\). since otherwise the term structure model would be inconsistent. Therefore, a reasonable approach is not to specify all coefficients in (\ref{museq1238}) and (\ref{museq1240}), but rather to leave a certain degree of freedom, for instance in the choice of the diffusion coefficient \(\gamma\). It appears that the consol price, and thus also the consol rate, is necessarily of the form \(D_{t}=g(r_{t})\) for some function \(g\). The function \(g=g(x)\) in question is shown to be a unique solution of the following ordinary differential equation
\[g'(x)\mu (x,g(x))-xg(x)+\frac{1}{2}g”(x)\sigma^{2}(x,g(x))+1=0.\]
Concluding, the model can be reduced to a model with a single state variable \(r\), in which the dynamics of the short-term rate are given by formula (\ref{museq1238}) with \(D_{t}=g(r_{t})\).
