Johann Christian Vollerdt (1708-1769) was a German painter.
Models of Bond Prices and LIBOR Rates.
The Heath-Jarrow-Morton methodology of term structure modelling presented in the previous section is based on the arbitrage-free dynamics of instantaneous, continuously compounded forward rates. The assumption that instantaneous rates exist is not always convenient, since it requires some degree of smoothness with respect to the {\bf tenor} (i.e. maturity) of bond prices and their volatilities. An alternative construction of an arbitrage-free family of bond prices, making no reference to the instantaneous, continuously compounded rates, is in some circumstances more suitable.
Let us comment briefly on the role of short-term interest rates and a savings account in bond price modelling. In the HJM model, in which the continuously compounded short rate \(r_{t}\) is well-defined, the savings account \(B\) represents the amount generated at time \(t\) by continuously reinvesting in the short rate. In particular, the absence of arbitrage between bonds and cash corresponds to nonnegativity of the short rate. In the more general setting considerd in this section, the absence of arbitrage between all bonds of different maturities implies the existence of a savings account which follows a process of finite variation. If, in addition, the absence of arbitrage between bonds and cash is assumed, the savings account is shown to be increasing. The sample paths of a savings account are not necessarily absolutely continuous with respect to the Lebesgue measure, in general, so that the short rate need not exist. In other words, term structure models in which the instantaneous rate of interest is not well-defined are also covered by the subsequent analysis.
Bond Price Models.
We assume that we are given a probability space \((\Omega ,{\bf F},\mathbb{P})\) equipped with a filtration \({\bf F}=\{{\cal F}_{t}\}_{t\in [0,T^{*}]}\) which satisfies the “usual conditions”. More specifically, we make the
following assumption.
Assumption F. The process \({\bf W}\) is a \(d\)-dimensional standard Brownian motion defined on a filtered probability space \((\Omega ,{\bf F},\mathbb{P})\). The filtration \({\bf F}\) is the usual \(\mathbb{P}\)-augmentation of the natural filtration of \({\bf W}\). We write \({\cal M}_{loc}(\mathbb{P})\) and \({\cal M}(\mathbb{P})\) to denote the class of all real-valued local martingales and the class of all real-valued martingales, respectively. \({\cal M}_{c}(\mathbb{P})\) also stands for the class of strictly positive \(\mathbb{P}\)-martingales with continuous sample paths. We denote by \({\cal V}\) (resp. by \({\cal A}\)) the class of all real-valued adapted (resp. predictable) processes of finite variation. We write \({\cal S}_{p}(\mathbb{P})\) to denote the class of all real-valued special semimartingales, i.e., those processes \(X\) which admit a decomposition \(X=X_{0}+M+A\), where \(M\in {\cal M}_{loc}(\mathbb{P})\) and \(A\in {\cal A}\). We also denote by \({\cal S}_{p}^{+}(\mathbb{P})\) the class of those special semimartingales \(X\in {\cal S}_{p}(\mathbb{P})\) which are strictly positive, and such that the process of left-hand limits, \(X_{t-}\), is also strictly positive. Notice that the class \({\cal S}_{p}(\mathbb{P})\) (as well as \({\cal S}_{p}^{+}(\mathbb{P})\)) is invariant with respect to an equivalent change of the underlying probability measure. More precisely, \({\cal S}_{p}(\mathbb{Q})={\cal S}_{p}(\mathbb{P})\) and \({\cal S}_{p}^{+}(\mathbb{Q})={\cal S}_{p}^{+}(\mathbb{P})\) if \(\mathbb{P}\) and \(\mathbb{Q}\) are mutually equivalent probability measures on \((\Omega ,{\cal F}_{T^{*}})\) such that the Radon-Nikodym derivative
\[L_{t}=\left .\frac{d\mathbb{Q}}{d\mathbb{P}}\right |_{{\cal F}_{t}}\mbox{ for all }t\in [0,T^{*}]\]
follows a locally bounded process. Since the filtration \({\bf F}\) is generated by a Brownian motion, the Radon-Nikodym derivative \(L\) will always follow a continuous exponential martingale, hence a locally bounded process. Therefore, we may and do write simply \({\cal S}_{p}\) and \({\cal S}_{p}^{+}\) in what follows.
Family of Bond Prices.
Let us fix a strictly positive horizon date \(T^{*}>0\), and let \(p(t,T)\) stand for the price at time \(t\leq T\) of a zero-coupon which matures at time \(T\leq T^{*}\). By a family of bond price, we mean an arbitrary family of strictly positive real-valued adapted processes \(p(t,T)\) for \(t\in [0,T]\) with \(p(T,T)=1\) for every \(T\in [0,T^{*}]\). We shall usually make the following assumptions.
(BP1). For any maturity date \(T\in [0,T^{*}]\), the bond price \(p(t,T)\) for \(t\in [0,T]\) belongs to the class \({\cal S}_{p}^{+}\).
(BP2). For any fixed \(T\in [0,T^{*}]\), the forward process
\[F_{p}(t,T,T^{*})\equiv\frac{p(t,T)}{p(t,T^{*})}\mbox{ for all }t\in [0,T]\]
follows a martingale under \(\mathbb{P}\), or equivalently
\begin{equation}{\label{museq141}}\tag{91}
p(t,T)=\mathbb{E}_{\scriptsize \mathbb{P}}\left [\left .\frac{p(t,T^{*})}{p(T,T^{*})}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T]
\end{equation}
since \(p(t,T^{*})\) is \({\cal F}_{t}\)-measurable and \(F_{p}(t,T,T^{*})=\mathbb{E}_{P}[F_{p}(T,T,T^{*})|{\cal F}_{t}]\) implies
\[\frac{p(t,T)}{p(t,T^{*})}=\mathbb{E}_{P}\left [\left .\frac{p(T,T)}{p(T,T^{*})}\right |{\cal F}_{t}\right ].\]
By virtue of (BP1) and (BP2), the process \(F_{p}(t,T,T^{*})\) follows, under \(\mathbb{P}\), a continuous and strictly positive martingale with respect to the filtration of a Brownian motion, so that \(F_{p}(t,T,T^{*})\) is in \({\cal M}_{c}^{+}(P)\). Consequently, for any fixed \(T\in [0,T^{*}]\) there exists a \(\mathbb{R}^{d}\)-valued predictable process \(\boldsymbol{\gamma}(t,T,T^{*})\), \(t\in [0,T]\), integrable with respect to the Brownian motion \({\bf W}\), and such that
\begin{equation}{\label{museq*66}}\tag{92}
F_{p}(t,T,T^{*})=F_{p}(0,T,T^{*})\cdot\exp\left (\int_{0}^{t}\boldsymbol{\gamma}(s,T,T^{*})\cdot d{\bf W}_{s}-\frac{1}{2}
\int_{0}^{t}\parallel\boldsymbol{\gamma}(t,T,T^{*})\parallel^{2}ds\right ).
\end{equation}
Put another way, for any fixed maturity \(T\in [0,T^{*}]\), the process \(F_{p}(t,T,T^{*})\) satifies
\begin{equation}{\label{museq*65}}\tag{93}
dF_{p}(t,T,T^{*})=F_{p}(t,T,T^{*})\boldsymbol{\gamma}(t,T,T^{*})\cdot d{\bf W}_{t}.
\end{equation}
Let us now consider two maturities \(T,U\in [0,T^{*}]\). We define the forward process \(F_{p}(t,T,U)\) by setting (ref. (\ref{museq1328}))
\begin{equation}{\label{museq142}}\tag{94}
F_{p}(t,T,U)\equiv\frac{F_{p}(t,T,T^{*})}{F_{p}(t,U,T^{*})}=\frac{p(t,T)}{p(t,U)}\mbox{ for all }t\in [0,\min\{T,U\}].
\end{equation}
Suppose first that \(U>T\), then the amount
\[f_{s}(t,T,U)=\frac{F_{p}(t,T,U)-1}{U-T}\]
is the {\bf add-on (annualized) forward rate} over the future time interval \([T,U]\) prevailing at time \(t\), and
\[f(t,T,U)=\frac{\ln F_{p}(t,T,U)}{U-T}\]
is the (continuously compounded) forward rate at time \(t\) over the interval (ref. (\ref{museq116})). On the other hand, if \(U<T\), then \(F_{p}(t,T,U)\) represents the value at time \(t\) of the forward price of a \(T\)-maturity bond for the forward contract that settles at time \(U\).
\begin{equation}{\label{musl1411}}\tag{95}\mbox{}\end{equation}
Proposition \ref{musl1411}. For any maturities \(T,U\in [0,T^{*}]\), the dynamics under \(P\) of the forward process are given by the following expression
\[dF_{p}(t,T,U)=F_{p}(t,T,U)\boldsymbol{\gamma}(t,T,U)\cdot (d{\bf W}_{t}-\boldsymbol{\gamma}(t,U,T^{*})dt),\]
where
\begin{equation}{\label{museq144}}\tag{96}
\boldsymbol{\gamma}(t,T,U)=\boldsymbol{\gamma}(t,T,T^{*})-\boldsymbol{\gamma}(t,U,T^{*})
\end{equation}
for every \(t\in [0,\min\{T,U\}]\).
Proof. Use the Ito’s formula. \(\blacksquare\)
Combining Proposition \ref{musl1411} with Girsanov’s theorem, we obtain
\begin{equation}{\label{museq145}}\tag{97}
dF_{p}(t,T,U)=F_{p}(t,T,U)\boldsymbol{\gamma}(t,T,U)\cdot d{\bf W}^{U}_{t},
\end{equation}
where for every \(t\in [0,U]\)
\[{\bf W}_{t}^{U}={\bf W}_{t}-\int_{0}^{t}\boldsymbol{\gamma}(s,U,T^{*})ds.\]
The process \({\bf W}^{U}\) is a standard Brownian motion on the filtered probability space \((\Omega .\{{\cal F}_{t}\}_{t\in [0,U]},\mathbb{P}_{U})\), where the probability measure \(\mathbb{P}_{U}\sim \mathbb{P}\) is defined on \((\Omega ,{\cal F}_{U})\)
by means of its Radon-Nikodym derivative with respect to the underlying probability measure \(\mathbb{P}\), i.e.,
\[\frac{d\mathbb{P}_{U}}{d\mathbb{P}}=\exp\left (\int_{0}^{U}\boldsymbol{\gamma}(s,U,T^{*})\cdot d{\bf W}_{s}-\frac{1}{2}
\int_{0}^{U}\parallel\boldsymbol{\gamma}(t,U,T^{*})\parallel^{2}ds\right )\mbox{ \(P\)-a.s.}.\]
It is apparent that the forward process \(F_{p}(t,T,U)\) follows an exponential local martingale under the “forward” probability measure \(\mathbb{P}_{U}\), since equality (\ref{museq145}) yields
\[F_{p}(t,T,U)=F_{p}(0,T,U)\cdot\exp\left (\int_{0}^{t}\boldsymbol{\gamma}(s,T,U)\cdot d{\bf W}_{s}^{U}-\frac{1}{2}
\int_{0}^{t}\parallel\boldsymbol{\gamma}(t,T,U)\parallel^{2}ds\right )\]
for \(t\in [0,\min\{T,U\}]\). Observe also that we \(\mathbb{P}_{T^{*}}=\mathbb{P}\) and \({\bf W}^{T^{*}}={\bf W}\). We thus recognize the underlying probability measure \(P\) as a forward martingale measure associated with the horizon date \(T^{*}\).
Spot and Forward Martingale Measures.
The concept of a forward martingale measure was introduced in terms of the behavior of relative bond prices. In the present context, we find it convenient to make use of the notion of a forward process. (ref. (\ref{museq1328}) and Proposition \ref{musp1322})
Definition. Let \(U\) be a fixed maturity date. A probability measure \(\mathbb{P}_{U}\sim\mathbb{P}\) on \((\Omega ,{\cal F}_{U})\) is called a forward martingale measure for the date \(U\) if, for any maturity \(T\in [0,T^{*}]\), the forward process \(F_{p}(t,T,U)\) for \(t\in [0,\min\{T,U\}]\) follows a local martingale under \(\mathbb{P}_{U}\). \(\sharp\)
It follows immediately from assumption (BP2) that the underlying probability measure \(\mathbb{P}\) is indeed a forward probability measure for the date \(T^{*}\) in the sense of above definition. Let us now introduce the notion of spot martingale measure within the present framework (ref. (\ref{bineq101})). Intuitively speaking, a spot measure is a forward measure associated with the initial date \(T=0\). Its formal definition relates to a very specific kind of discounting, however. It is also worth noting that neither a forward measure for the date \(T^{*}\), nor a spot measure, are uniquely defined, in general.
\begin{equation}{\label{musd1412}}\tag{98}\mbox{}\end{equation}
Definition \ref{musd1412}. A spot martingale measure for the setup (BP1) and (BP2) is any probability measure \(\mathbb{P}^{*}\sim \mathbb{P}\) on \((\Omega ,{\cal F}_{T^{*}})\) for which there exists a process \(B^{*}\in {\cal A}^{+}\) with \(B_{0}^{*}=1\), and such that for any maturity \(T\in [0,T^{*}]\), the bond price \(p(t,T)\) satisfies
\[p(t,T)=\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |{\cal F}_{t}\right ],\]
for all \(t\in [0,T]\). \(\sharp\)
Arbitrage-Free Properties.
We shall study two forms of absence of arbitrage. The first, weaker notion refers to a pure bond market. The second form assumes, in addition, that cash is also present. Note that by cash we mean here money which can be carried over at no cost, rather than a savings account yielding positive interest. We shall formulate now a sufficient condition for the absence of arbitrage between bonds with different maturities as well as between bonds and cash.
Definition. A family \(p(t,T)\) of bond prices is said to satisfy the weak no-arbitrage condition if and only if there exists a probability measure \(\mathbb{Q}\sim\mathbb{P}\) on \((\Omega ,{\cal F}_{T^{*}})\) such that for any maturity \(T<T^{*}\), the forward process \(F_{p}(t,T,T^{*})=p(t,T)/p(t,T^{*})\) belongs to \({\cal M}_{loc}(\mathbb{Q})\), We say that the family \(p(t,T)\) satisfies the no-arbitrage condition if, in addition, inequality \(p(T,U)\leq 1\) holds for any maturities \(T,U\in [0,T^{*}]\) such that \(T\leq U\). \(\sharp\)
Assumption (BP2) is manifestly sufficient for the family \(p(t,T)\) to satisfy a weak no-arbitrage condition, as we may take \(Q=P\). As mentioned, if a family \(p(t,T)\) satisfies a weak no-arbitrage condition, then it is possible to construct a model of the security market with the absence of arbitrage across bonds with different maturities (let us stress that the weak no-arbitrage condition makes no explicit reference to the presence of cash or a savings account). We shall now investigate the absence of arbitrage between all bonds and cash. Inequality \(p(T,U)\leq 1\), which is equivalent to \(F_{p}(T,U,T)\leq 1\), gives immediately
\begin{equation}{\label{museqa52}}\tag{99}
F_{p}(t,U,T)=\mathbb{E}_{P}[F_{p}(T,U,T)|{\cal F}_{t}]\leq 1
\end{equation}
for every \(t\in [0,T]\). Since almost all sample paths of the forward process \(F_{p}(t,U,T)\) are continuous functions, we may reformulate this condition in the following way. (by (\ref{museqa52}) and (\ref{museq142}))
(BP3). For any two maturities \(T\leq U\), the following inequality holds with probability one
\begin{equation}{\label{museq148}}\tag{100}
p(t,U)\leq p(t,T)\mbox{ for all }t\in [0,T].
\end{equation}
Suppose, on the contrary, that \(p(t,U)>p(t,T)\) for certain maturities \(U>T\). In such a case, by issuing at time \(t\) a bond of maturity \(U\), and purchasing a \(T\)-maturity bond, one could lock in a risk-free profit if, in addition, cash were present in the market. Indeed, to meet the liability at time \(U\) it would be enough to carry one unit of cash, received at time \(T\), over the period \([T,U]\) (at no cost). At the intuitive level, the following three conditions are equivalent.
- The bond price \(p(t,T)\) is a nonincreasing function of maturity \(T\).
- The forward process \(F_{p}(t,T,U)\) is never less than one.
- The bond price \(p(t,T)\) is never strictly greater than one (see Corollary \ref{musc1413}).
Not surprisingly, the absence of arbitrage between bonds and cash appears to be closely related to the question of the existence of an increasing savings account implied by the family \(p(t,T)\).
\begin{equation}{\label{musd1414}}\tag{101}\mbox{}\end{equation}
Definition \ref{musd1414}. A savings account implied by the family \(p(t,T)\) of bond prices is an arbitrary process \(B^{*}\) which belongs to \({\cal A}^{+}\) with \(B_{0}^{*}=1\), and such that there exists a probability measure \(\mathbb{P}^{*}\sim P\) on \((\Omega ,{\cal F}_{T^{*}})\) under which the relative bond price
\[Z(t,T)\equiv\frac{p(t,T)}{B_{t}^{*}}\mbox{ for all }t\in [0,T]\]
is a martingale for any maturity \(T\in [0,T^{*}]\). \(\sharp\)
It is clear that \(Z(t,T)\) is a \(\mathbb{P}^{*}\)-martingale if, for any maturity \(T\), the bond price \(p(t,T)\) satisfies
\begin{equation}{\label{museq*62}}\tag{102}
p(t,T)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
In particular, we have
\[p(0,T)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\frac{1}{B_{T}^{*}}\right ]\mbox{ for all }t\in [0,T^{*}],\]
so that an implied savings account \(B^{*}\) matches also the initial term structure \(p(0,T)\) for \(T\in [0,T^{*}]\). It is also clear that the probability measure \(\mathbb{P}^{*}\) of Definition \ref{musd1414} is a spot martingale measure for the family \(p(t,T)\) in the sense of Definition \ref{musd1412}.
Implied Savings Account.
Now we shall take up the issue of the existence of an implied savings account. We will show that, under assumptions (BP1)-(BP3), there exists an increasing process \(B^{*}\) which represents an implied savings account for the family \(p(t,T)\). To this end, we establish a preliminary result which deals with the behavior of the terminal discount factor $D_{t}=1/p(t,T^{*})$ for \(t\in [0,T^{*}]\). Note that the process \(D\) belongs to \({\cal S}_{p}^{+}\) since \(p(t,T^{*})\) does.
\begin{equation}{\label{musl1412}}\tag{103}\mbox{}\end{equation}
Proposition \ref{musl1412}. Under the assumptions (BP1)–(BP3), the terminal discount factor \(D\) follows a strictly positive semimartingale under the forward martingale measure \(\mathbb{P}\). (Recall that \(\mathbb{P}=\mathbb{P}_{T^{*}}\) is a forward martingale measure for the date \(T^{*}\)).
Proof. Combining (\ref{museq141}) with (\ref{museq148}), we obtain
\[p(t,U)=\mathbb{E}_{\scriptsize \mathbb{P}}\left [\left .\frac{p(t,T^{*})}{p(U,T^{*})}\right |
{\cal F}_{t}\right ]\leq \mathbb{E}_{\scriptsize \mathbb{P}}\left [\left .\frac{p(t,T^{*})}{p(T,T^{*})}\right |{\cal F}_{t}\right ]=p(t,T),\]
so that \(\mathbb{E}_{\scriptsize \mathbb{P}}[D_{U}|{\cal F}_{t}]\leq \mathbb{E}_{\scriptsize \mathbb{P}}[D_{T}|{\cal F}_{t}]\) for \(t\leq T\leq U\leq T^{*}\). Setting \(t=T\) in the last inequality, we find that \(\mathbb{E}_{\scriptsize \mathbb{P}}[D_{U}|{\cal F}_{T}]\leq \mathbb{E}_{P}[D_{T}|{\cal F}_{T}]=D_{T}\) for every \(T\leq U\leq T^{*}\), so that \(D\) is a \(\mathbb{P}\)-supermartingale. \(\blacksquare\)
To show the existence of an implied savings account, we make use of the following standard result of stochastic calculus.
\begin{equation}{\label{musp1411}}\tag{104}\mbox{}\end{equation}
Proposition \ref{musp1411}. Suppose that \(X\) belongs to class \({\cal S}_{p}^{+}\) with \(X_{0}=1\). There exists a unique pair \((M,A)\) of stochastic processes such that \(X=MA\), the process \(M\) belongs to \({\cal F}_{c}^{+}(P)\) with \(M_{0}=1\), and \(A\) belongs to \({\cal A}^{+}\) with \(A_{0}=1\). If, in addition, \(X\) is a supermartingale, then \(A\) is a decreasing process. \(\sharp\)
It is well-known that if a strictly positive special semimartingale \(X\) follows a supermartingale, then the process of left-hand limits \(X_{t-}\) is also strictly positive. Hence \(X\) belongs to the class \({\cal S}_{p}^{+}\). Assume that the process \(M\) in the decomposition above has continuous paths, this holds in this case since the underlying filtration is generated by a Brownian motion. Then the process \(A\) is easily seen to belong to \({\cal S}_{p}^{+}\).
Proposition. Under assumptions (BP1)–(BP3), there exists a predictable process \(\boldsymbol{\xi}\) integrable with respect to the Brownian motion \({\bf W}\), and such that the terminal discount factor \(D\) admits the unique decomposition
\begin{equation}{\label{museq149}}\tag{105}
D_{t}=D_{0}\tilde{A}_{t}\tilde{M}_{t}=D_{0}\tilde{A}_{t}\cdot\exp\left (\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}-\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\xi}_{s}\parallel^{2}ds\right )\mbox{ for all }t\in [0,T^{*}],
\end{equation}
where \(\tilde{M}\) is in \({\cal M}_{c,loc}^{+}(P)\) (i.e. the local martingales in \({\cal M}_{loc}^{+}\) with continuous sample paths) and \(\tilde{A}\) belongs to \({\cal A}^{+}\) with \(\tilde{A}_{0}=\tilde{M}_{0}=1\). If, in addition, assumptiom (BP3) is met, then \(\tilde{A}\) is a decreasing process.
Proof. All assertions are immediate conequences of Proposition \ref{musl1412} combined with Proposition \ref{musp1411}, and the representation theorem for strictly positive martingales with respect to a Brownian filtration. \(\blacksquare\)
For the further purposees, it is convenient to rewrite (\ref{museq149}) as follows
\begin{equation}{\label{museq1410}}\tag{106}
B_{t}^{*}\equiv\frac{1}{\tilde{A}_{t}}=\frac{p(t,T^{*}}{p(0,T^{*})}\cdot
\exp\left (\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}-
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\xi}_{s}\parallel^{2}ds\right )\mbox{ for all }t\in [0,T^{*}].
\end{equation}
To show the existence of an implied savings account, it is enough to check that the process \(B^{*}\) given by (\ref{museq1410}) satisfies Definition \ref{musd1414}. We formulate the next result under the assumptions (BP1)–(BP3). If assumption (BP3) is relaxed, all claims of Proposition \ref{musp1412} remain valid, except that \(B^{*}\) is not necessarily an increasing process.
\begin{equation}{\label{musp1412}}\tag{107}\mbox{}\end{equation}
Proposition \ref{musp1412}. Let the family satisfy the assumptions (BP1)–(BP3). Assume, in addition, that the process \(\tilde{M}\) defined by the multiplicative decomposition (\ref{museq149}) of the terminal discount factor \(D\) is a martingale, and not only a local martingale under \(\mathbb{P}\). Let \(B^{*}=1/\tilde{A}_{t}\) be an increasing predictable process uniquely determined by (\ref{museq149}). Then, we have the following properties.
(i) \(B^{*}\) represents a savings account implied by the family \(p(t,T)\).
(ii) \(B^{*}\) is associated with the spot martingale measure \(\mathbb{P}^{*}\), given by
\begin{equation}{\label{museq1411}}\tag{108}
\frac{d\mathbb{P}^{*}}{d\mathbb{P}}=B_{T^{*}}^{*}\cdot p(0,T^{*})\mbox{ \(\mathbb{P}\)-a.s.}.
\end{equation}
(iii) The relative price process \(B_{t}^{*}/p(t,T^{*})=1/Z(t,T^{*})\) follows a martingale under the forward martingale measure \(\mathbb{P}\) for the date \(T^{*}\).
Proof. Let \(\mathbb{P}^{*}\) be an arbitrary probability measure on \((\Omega ,{\cal F}_{T^{*}})\) equivalent to \(\mathbb{P}\). Then the Radon-Nikodym derivative of \(\mathbb{P}^{*}\) with respect to \(\mathbb{P}\), restricted to the \(\sigma\)-filed, equals
\begin{equation}{\label{museq1412}}\tag{109}
\left .\frac{d\mathbb{P}^{*}}{d\mathbb{P}}\right |_{{\cal F}_{t}}=\exp\left (\int_{0}^{t}\tilde{\boldsymbol{\xi}}_{s}\cdot d{\bf W}_{s}-\frac{1}{2}\int_{0}^{t}\parallel\tilde{\boldsymbol{\xi}}_{s}\parallel^{2}ds\right )\mbox{ \(\mathbb{P}\)-a.s.}
\end{equation}
for some predictable process \(\tilde{\boldsymbol{\xi}}\). We start by considering a zero-coupon bond of maturity \(T^{*}\). In view of (\ref{museq149}), we have that \(1/(\tilde{A}_{t}\cdot\tilde{M}_{t})=D_{0}/D_{t}\) implies \(B_{t}^{*}/\tilde{M}_{t}=p(t,T^{*})/p(0,T^{*})\). Then the relative bond price \(Z(t,T^{*})=p(t,T^{*})/B_{t}^{*}\) satisfies, under \(\mathbb{P}\),
\begin{equation}{\label{museq1413}}\tag{110}
Z(t,T^{*})=\frac{p(0,T^{*})}{\tilde{M}_{t}}=p(0,T^{*})\cdot\exp\left (-\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}+
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\xi}_{s}\parallel^{2}ds\right )\mbox{ for all }t\in [0,T^{*}].
\end{equation}
Consequently, under \(\mathbb{P}^{*}\), we have
\[Z(t,T^{*})=\frac{p(0,T^{*})}{\tilde{M}_{t}}=p(0,T^{*})\cdot\exp\left (-\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}^{*}-
\frac{1}{2}\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot (2\tilde{\boldsymbol{\xi}}_{s}-\boldsymbol{\xi}_{s})ds\right ),\]
where
\[{\bf W}_{t}^{*}={\bf W}_{t}-\int_{0}^{t}\tilde{\boldsymbol{\xi}}_{s}ds\mbox{ for all }t\in [0,T^{*}]\]
follows a Brownian motion under \(\mathbb{P}^{*}\). It is thus evident that the relative bond price \(Z(t,T^{*})\) is a local martingale under \(\mathbb{P}^{*}\) provided that \(\tilde{\boldsymbol{\xi}}=\boldsymbol{\xi}\). Under this assumption, we have
\begin{equation}{\label{museq1414}}\tag{111}
Z(t,T^{*})=p(0,T^{*})\cdot\exp\left (-\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}^{*}+
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\xi}_{s}\parallel^{2}ds\right ).
\end{equation}
We are in a position to define a candidate for a spot probability measure by setting \(\tilde{\boldsymbol{\xi}}=\boldsymbol{\xi}\) in (\ref{museq1412}). In view of Definition \ref{musd1414}, we have to check that for any maturity \(T<T^{*}\), the relative bond price \(Z(t,T^{*})\) follows a martingale under \(\mathbb{P}^{*}\). It is enough to show that for any maturity \(T<T^{*}\), we have (ref. (\ref{museq*62}))
\begin{equation}{\label{museq1415}}\tag{112}
p(t,T)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
From (\ref{museq1413}), we have
\begin{equation}{\label{museq*63}}\tag{113}
\exp\left (\int_{0}^{t}\boldsymbol{\xi}_{s}\cdot d{\bf W}_{s}-
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\xi}_{s}\parallel^{2}ds\right )=\frac{p(0,T^{*})}{Z(t,T^{*})}.
\end{equation}
Observe that equality \(\tilde{\boldsymbol{\xi}}=\boldsymbol{\xi}\), combined with (\ref{museq1412}) and (\ref{museq*63}), gives
\begin{equation}{\label{museq1416}}\tag{114}
\eta_{t}=\left .\frac{d\mathbb{P}^{*}}{d\mathbb{P}}\right |_{{\cal F}_{t}}=\frac{p(0,T^{*})}
{Z(t,T^{*})}=\frac{B_{t}^{*}\cdot p(0,T^{*})}{p(t,T^{*})}\mbox{ for all }t\in [0,T^{*}].
\end{equation}
Consequently, using the Bayes rule, (\ref{museq1416}) and (\ref{museq141}), we obtain
\[\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |
{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \mathbb{P}}\left [\left .\frac{B_{t}^{*}\cdot\eta_{T}}
{B_{T}^{*}\cdot\eta_{t}}\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \mathbb{P}}\left [\left .
\frac{p(t,T^{*})}{p(T,T^{*})}\right |{\cal F}_{t}\right ]=p(t,T).\]
This proves equation (\ref{museq1415}). We have thus shown that the process \(B^{*}=1/\tilde{A}\) satisfies all conditions of the definition of an implied savings account, and \(\mathbb{P}^{*}\) is the associated spot martingale measure. Equation (\ref{museq1411}) follows from (\ref{museq1416}) by taking \(\eta_{T^{*}}\). \(\blacksquare\)
Notice that the probability measure \(\mathbb{P}^{*}\) given by (\ref{museq1411}) is the spot martingale measure associated with the forward martingale measure \(\mathbb{P}\) for the date \(T^{*}\). More generally, the forward measure \(\mathbb{P}\) and the associated spot measure \(\mathbb{P}^{*}\) are related to each other through the formula
\[\frac{d\mathbb{P}^{*}}{d\mathbb{P}}=B_{T^{*}}^{*}\cdot p(0,T^{*})\mbox{ \(\mathbb{P}\)-a.s.}.\]
As mentioned earlier, the uniqueness of a spot and forward measures is not a universal property (see the statements just before Definition \ref{musd1412}). Summarizing, for any forward measure \(\mathbb{Q}\) for the date \(T^{*}\), the probability measure \(\mathbb{Q}^{*}\), which is defined on \((\Omega ,{\cal F}_{T^{*}})\) by the formula
\begin{equation}{\label{museq1418}}\tag{115}
\frac{d\mathbb{Q}^{*}}{d\mathbb{Q}}=B_{T^{*}}^{*}\cdot p(0,T^{*})\mbox{ \(\mathbb{Q}\)-a.s.}
\end{equation}
is a spot measure for the family \(p(t,T)\). Conversely, if \(\mathbb{Q}^{*}\) is a spot measure, then the probability measure \(\mathbb{Q}\) given by (\ref{museq1418}) is a forward measure for the date \(T^{*}\). We are in a position to examine the uniqueness of an implied savings account.
\begin{equation}{\label{musl1413}}\tag{116}\mbox{}\end{equation}
Proposition \ref{musl1413}. Let \(B^{*}\) and \(\widehat{B}\) be two processes from \({\cal A}^{+}\) such that for every \(T\in [0,T^{*}]\)
\[\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize widehat{\mathbb{P}}}\left [\left .\frac{\widehat{B_{t}}}{\widehat{B_{T}}}\right |{\cal F}_{t}\right ]\]
for every \(t\in [0,T]\), where \(\mathbb{P}^{*}\sim\widehat{\mathbb{P}}\) are two mutually equivalent probability measure on \((\Omega ,{\cal F}_{T^{*}})\). If \(B_{0}^{*}=\widehat{B}_{0}\), then \(B^{*}=\widehat{B}\). \(\sharp\)
The following result establishes the uniqueness of an implied savings account. The uniqueness of an implied savings account is in the sense of indistinguishability of stochastic processes. This last notion is invariant with respect to an equivalent change of a probability measure.
Proposition. Under assumptions (BP1) and (BP2), the uniqueness of an implied savings account holds.
Proof. Suppose that \(B^{*}\) and \(\widehat{B}\) are two arbitrary savings account implied by the family \(p(t,T)\) associated with the mutually equivalent probability measures \(\mathbb{P}^{*}\) and \(\widehat{\mathbb{P}}\), respectively. Since \(B^{*}\) and \(\widehat{B}\) are assumed to satisfy Definition \ref{musd1414}, we have
\[ $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{t}^{*}}{B_{T}^{*}}\right |{\cal F}_{t}
\right ]=p(t,T)=\mathbb{E}_{\widehat{P}}\left [\left .\frac{\widehat{B_{t}}}{\widehat{B_{T}}}\right |{\cal F}_{t}\right ]\]
for every \(t\in [0,T]\), and for any maturity date \(T\). Also, by the definition of an implied savings accounts, \(B^{*}\) and \(\widehat{B}\) are predictable processes of finite variation. Equality \(B^{*}=\widehat{B}\) is thus an immediate consequence of Proposition \ref{musl1413}. \(\blacksquare\)
\begin{equation}{\label{musc1413}}\tag{117}\mbox{}\end{equation}
Proposition \ref{musc1413}. Under assumptions (BP1) and (BP2), the following statements are equivalent.
(a) The bond price \(p(t,T)\) is a nonincreasing function of maturity date \(T\).
(b) The forward process \(F_{p}(t,T,U)\) for \(t\leq T\leq U\) is never strictly less than one.
(c) The bond price \(p(t,T)\) is never strictly greater than one.
(d) The implied savings account follows an increasing process.
Proof. Equivalence of (a), (b) and (d) is trivial. It is also obvious that (d) implies (c). It remains to check that (d) follows from (c). Let \(B^{*}\) be the unique savings account associated with the family \(p(t,T)\). Condition \(p(t,T)\leq 1\) implies easily that the process \(1/B^{*}_{t}\) follows a supermartingale under the spot measure \(\mathbb{P}^{*}\). Since it is a process of finite variation. its martingale part, being continuous by virtue of Assumption F, vanishes identically. Therefore, \(B^{*}\) is an increasing process. \(\blacksquare\)
Bond Price Volatility.
Here we assume that a family \(p(t,T)\) of bond prices satisfies (BP1)–(BP3).
Definition. An \(\mathbb{R}^{d}\)-valued adapted process \({\bf b}(t,T)\) is called a bond price volatility for maturity \(T\) when the bond price \(p(t,T)\) admits the representation
\begin{equation}{\label{museq1420}}\tag{118}
dp(t,T)=p(t,T)\cdot {\bf b}(t,T)\cdot d{\bf W}_{t}+dC_{t}^{T},
\end{equation}
where \(C^{T}\) is a predictable process of finite variation. \(\sharp\)
Under (BP1) and (BP2), the existence and uniqueness of the bond price volatility \({\bf b}(t,T)\) for any maturity \(T\) is a simple consequence of the canonical decomposition of the special semimartingale \(p(t,T)\in {\cal S}_{p}^{+}\), combined with the predictable representation theorem. Also, it is not hard to check that the bond price volatility is invariant with respect to an equivalent change of probability measure. More precisely, if (\ref{museq1420}) holds, then under any probability measure \(\tilde{\mathbb{P}}\sim\mathbb{P}\), we have
\[dp(t,T)=p(t,T)\cdot {\bf b}(t,T)\cdot d\tilde{{\bf W}}_{t}+d\tilde{C}_{t}^{T}\]
for some predictable process of finite variation \(\tilde{C}^{T}\), where \(\tilde{{\bf W}}\) follows a Brownian motion under \(\tilde{\mathbb{P}}\). Since we have assumed that assumptions (BP1)–(BP3) are satisfied, there exists a unique savings account \(B^{*}\) associated with a spot probability measure \(\mathbb{P}^{*}\). For any maturity \(T\), the relative bond price \(Z(t,T)=p(t,T)/B_{t}^{*}\) follows a local martingale under \(\mathbb{P}^{*}\) so that
\begin{equation}{\label{museq1421}}\tag{119}
Z(t,T)=p(0,T)\cdot\exp\left (\int_{0}^{t}{\bf b}(s,T)\cdot d{\bf W}_{s}^{*}-\frac{1}{2}\int_{0}^{t}\parallel {\bf b}(s,T)\parallel^{2}ds\right ).
\end{equation}
By comparing (\ref{museq1421}) with (\ref{museq1414}), we find that \({\bf b}(t,T)=-\boldsymbol{\xi}_{t}\), i.e., the volatility of a \(T^{*}\)-maturity bond is determined by the multiplicative decomposition (\ref{museq149}). Upon setting \(T=t\) in (\ref{museq1421}), we obtain the following representation for a savings account \(B^{*}\) in terms of bond price volatilities
\begin{equation}{\label{museq1422}}\tag{120}
B_{t}^{*}=p^{-1}(0,T)\cdot\exp\left (-\int_{0}^{t}{\bf b}(s,T)\cdot d{\bf W}_{s}^{*}+\frac{1}{2}
\int_{0}^{t}\parallel {\bf b}(s,T)\parallel^{2}ds\right )
\end{equation}
for every \(t\in [0,T^{*}]\).
Observe that for any maturities \(T,U\in [0,T^{*}]\), we have (ref. (\ref{museq144}))
\begin{equation}{\label{museq67}}\tag{121}
\boldsymbol{\gamma}(t,T,U)={\bf b}(t,T)-{\bf b}(t,U)\mbox{ for all }t\in [0,\min\{t,U\}],
\end{equation}
where \(\boldsymbol{\gamma}(t,T,U)\) is the volatility of the forward process \(F_{p}(t,T,U)\). Therefore, the forward volatilities \(\boldsymbol{\gamma}(t,T,U)\) are uniquely specified by the bond price volatilities \({\bf b}(t,T)\). Next we consider the converse implication; that is, whether the bond price volatilities are uniquely determined by the forward volatilities.
Forward Processes.
Here we no longer assume that we are given a family of bond prices. We make instead the following assumptions.
(FP1). For any \(T\in [0,T^{*})\), we are given an adapted \(\mathbb{R}^{d}\)-valued process \(\boldsymbol{\gamma}(t,T,T^{*})\) for
$t\in [0,T]$ satisfying
\[P\left\{\int_{0}^{T}\parallel\boldsymbol{\gamma}(s,T,T^{*})\parallel^{2}ds<\infty\right\}=1.\]
By convention, \(\boldsymbol{\gamma}(t,T^{*},T^{*})={\bf 0}\in\mathbb{R}^{d}\) for every \(t\in [0,T^{*}]\).
(FP2). We are given a deterministic function \(u(0,T)\) for \(T\in [0,T^{*}]\) with \(u(0,0)=1\), which represents an initial term structure of interest rates.
Definition. Given the assumptions (FP1) and (FP2), for any maturity \(T\in [0,T^{*}]\), we define the forward process \(F(t,T,T^{*})\) for \(t\in [0,T]\) by specifying its dynamics under \(\mathbb{P}\) (ref. (\ref{museq*65}))
\begin{equation}{\label{museq1429}}\tag{122}
dF(t,T,T^{*})=F(t,T,T^{*})\boldsymbol{\gamma}(t,T,T^{*})\cdot d{\bf W}_{t},
\end{equation}
and the initial condition
\[F(0,T,T^{*})=\frac{u(0,T)}{u(0,T^{*})}\mbox{ for all }T\in [0,T^{*}]. \sharp\]
For any \(T\leq T^{*}\), the unique solution of (\ref{museq1429}) is given by the standard exponential formula (ref. (\ref{museq*66}))
\begin{equation}{\label{museq1430}}\tag{123}
F(t,T,T^{*})=\frac{u(0,T)}{u(0,T^{*})}\cdot\exp\left (\int_{0}^{t}
\boldsymbol{\gamma}(s,T,T^{*})\cdot d{\bf W}_{s}-\frac{1}{2}
\int_{0}^{t}\parallel\boldsymbol{\gamma}(t,T,T^{*})\parallel^{2}ds\right )
\end{equation}
for \(t\in [0,T]\). We postulate that the process \(F(t,T,T^{*})\) has a financial interpretation as the ratio of bond prices; more exactly, we require that
\begin{equation}{\label{museq1431}}\tag{124}
F(t,T,T^{*})=\frac{p(t,T)}{p(t,T^{*})}\mbox{ for all }t\in [0,T],
\end{equation}
where bond prices \(p(t,T)\) remain unspecified. Indeed, the goal is to construct a family \(p(t,T)\) that would be consistent with the dynamics (\ref{museq1430}) of forward processes, and would match the initial term structure \(u(0,T)\), that is, \(p(0,T)=u(0,T)\) for any maturity \(T\leq T^{*}\). Note that here the bond price is not required a priori to be a semimartingale. Nevertheless, in some circumstances we will make reference to the volatility of a bond price, which is defined only for the bond price that follows a semimartingale. In view of assumptions (FP1) and (FP1) and (\ref{museq1431}), to find a family \(p(t,T)\), it is sufficient to specify the price of a \(T^{*}\)-maturity bond. When searching for a candidate for the process \(p(t,T^{*})\), we need to take into account the terminal condition \(p(T^{*},T^{*})=1\) and the initial condition \(p(0,T^{*})=u(0,T^{*})\). A family \(p(t,T)\) is then defined by setting \(p(t,T)\equiv F(t,T,T^{*})\cdot p(t,T^{*})\) for every \(t\leq T\leq T^{*}\). Such a family is easily seen to match a pre-specified initial term structure. The terminal condition \(p(T,T)=1\) is not necessarily satisfied, however, unless a judicious choice of the process \(p(t,T^{*})\) is made.Let us introduce a counterpart of condition (BP3). We find it convenient to introduce the family of processes \(F(t,T,U)\) by setting
\begin{equation}{\label{museq1432}}\tag{125}
F(t,T,U)\equiv\frac{F(t,T,T^{*})}{F(t,U,T^{*})}\mbox{ for all } t\in [0,\min\{T,U\}].
\end{equation}
(FP3). For any maturities \(T,U\in [0,T^{*}]\) such that \(T\leq U\), we have
\[F(t,T,U)\geq 1\mbox{ for all }t\in [0,T].\]
Notice that (FP3) implies that \(u(0,U)\leq u(0,T)\) for arbitrary maturities \(T\leq U\).
\begin{equation}{\label{musd1422}}\tag{126}\mbox{}\end{equation}
Definition \ref{musd1422}. We say that a family \(p(t,T)\) of bond prices is associated with (FP1) and (FP2) when the following conditions are satisfied.
- Processes \(F(t,T,T^{*})\) given by (\ref{museq1430}) coincide with processes \(F_{p}(t,T,T^{*})\), which are given by the formula
\[F_{p}(t,T,T^{*})\equiv\frac{p(t,T)}{p(t,T^{*})}\mbox{ for all }t\in [0,T^{*}].\] - Equality \(p(0,T)=u(0,T)\) is satisfied for every \(T\in [0,T^{*}]\). \(\sharp\)
To show that any family of forward processes \(F(t,T,T^{*})\) admits an associated family of bond prices, we will use the notion of a savings account implied by the assumptions (FP1) and (FP2). Formally, a savings account implied by (FP1) and (FP2) is any process which represents an implied savings account for some family of bond prices associated with (FP1) and (FP2).
It is clear that we may represent the volatility \(\boldsymbol{\gamma}(t,T,T^{*})\) as follows (ref. (\ref{museq67}))
\begin{equation}{\label{museq1434}}\tag{127}
\widehat{{\bf b}}(t,T)=\boldsymbol{\gamma}(t,T,T^{*})+\widehat{{\bf b}}(t,T^{*})\mbox{ for all }t\in [0,T]
\end{equation}
for some family of processes \(\widehat{{\bf b}}(t,T)\) for \(t\leq T\leq T^{*}\). Given a family of forward volatilities \(\boldsymbol{\gamma}(t,T,T^{*})\), in order to determine uniquely all processes \(\widehat{{\bf b}}(t,T)\), it sufficies to specify the process \(\widehat{{\bf b}}(t,T^{*})\). The bond price volatilities \({\bf b}(t,T)\) of any associated family \(p(t,T)\), if well-defined, necessarily satisfy relationship (\ref{museq1434}); that is, for any maturity \(T\leq T^{*}\), we have
\begin{equation}{\label{museq1435}}\tag{128}
{\bf b}(t,T)=\boldsymbol{\gamma}(t,T,T^{*})+{\bf b}(t,T^{*})\mbox{ for all }t\in [0,T]
\end{equation}
This does not mean that arbitrary processes \(\widehat{{\bf b}}(t,T)\) which satisfy (\ref{museq1434}) are indeed price volatilities of some family \(p(t,T)\) of bond prices associated with (FP1) and (FP2). On the other hand, It follows immediately from (\ref{museq1434}) and (\ref{museq1435}) that for an arbitrary choice of the process \(\widehat{{\bf b}}(t,T^{*})\), there exists a unique process \(\boldsymbol{\psi}\) such that the actual bond price volatility \({\bf b}(t,T)\) satisifes \({\bf b}(t,T)=\widehat{{\bf b}}(t,T)+ \boldsymbol{\psi}_{t}\) for all \(t\in [0,T]\) and for any maturity \(T\leq T^{*}\). Indeed, it is enough to set \(\boldsymbol{\psi}_{t}={\bf b}(t,T^{*})-\widehat{{\bf b}}(t.T^{*})\) for every \(t\in [0,T^{*}]\). We will assume from now on that the forward volatilites \(\boldsymbol{\gamma}(t,T,T^{*})\) are bounded. The goal is to find explicitly a family of bond prices associated with (FP1) and (FP2).
First, we take an arbitrary, bounded, adapted \(\mathbb{R}^{d}\)-valued process \(\widehat{{\bf b}}(t,T^{*})\), and we define the probability measure \(\mathbb{P}^{*}\sim\mathbb{P}\) on \((\Omega ,{\cal F}_{T^{*}})\) by setting
\begin{equation}{\label{museq1436}}\tag{129}
\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}=\exp\left (-\int_{0}^{T^{*}}\widehat{{\bf b}}(s,T^{*})\cdot d{\bf W}_{s}+\frac{1}{2}
\int_{0}^{T^{*}}\parallel\widehat{{\bf b}}(s,T^{*})\parallel^{2}ds\right )
\end{equation}
The process \(\widehat{{\bf W}}_{t}\) given by the formula
\[\widehat{{\bf W}}_{t}={\bf W}_{t}+\int_{0}^{t}\widehat{{\bf b}}(s,T^{*})ds\]
is a Brownian motion under \(\widehat{P}\). In the second step, we introduce a candidate for the savings account process \(\widehat{B}_{t}\) (ref. (\ref{museq1422}))
\begin{equation}{\label{museq1437}}\tag{130}
\widehat{B}_{t}=u^{-1}(0,t)\cdot\exp\left (-\int_{0}^{t}\widehat{{\bf b}}(s,t)\cdot d\widehat{{\bf W}}_{s}+\frac{1}{2}
\int_{0}^{t}\parallel\widehat{{\bf b}}(s,t)\parallel^{2}ds\right ),
\end{equation}
where \(\widehat{{\bf b}}(t,T)\) is defined by (\ref{museq1434}).
It is not known a priori whether the process \(\widehat{B}\) is of finite variation (or even if it follows a semimartingale). It appears that \(\widehat{B}\) is of finite variation if and only if it represents an implied savings account for a family \(p(t,T)\) of bond prices defined by formula (\ref{museq1438}) below. In the opposite case, neither the process \(\widehat{B}\) nor the bond prices are semimartingales.
We are in a position to introduce a family \(p(t,T)\) by setting
\begin{equation}{\label{museq1438}}\tag{131}
p(t,T)=u(0,T)\cdot\widehat{B}_{t}\cdot\exp\left (\int_{0}^{t}\widehat{{\bf b}}(s,T)\cdot d\widehat{{\bf W}}_{s}-\frac{1}{2}
\int_{0}^{t}\parallel\widehat{{\bf b}}(s,T)\parallel^{2}ds\right )\mbox{ for all }t\in [0,T],
\end{equation}
for any maturity \(T\in [0,T^{*}]\). We claim that \(p(t,T)\) is a family of bond prices associated with (FP1) and (FP2). To check this, we analyse the forward process \(F_{p}(t,T,T^{*})\) associated with the family \(p(t,T)\). It is clear that
\[F_{p}(t,T,T^{*})=\frac{u(0,T)}{u(0,T^{*})}\cdot\exp\left (
\int_{0}^{t}\boldsymbol{\gamma}(s,T,T^{*})\cdot d\widehat{{\bf W}}_{s}-
\frac{1}{2}\int_{0}^{t}\boldsymbol{\delta}_{s}(T,T^{*})ds\right ),\]
where
\[\boldsymbol{\delta}_{s}(T,T^{*})=\parallel\widehat{{\bf b}}(s,T)\parallel^{2}-\parallel\widehat{{\bf b}}(s,T^{*})\parallel^{2}\mbox{ for every }s\in [0,T].\]
Let us check that the condition (i) of Definition \ref{musd1422} is satisfied. To this end, notice that by making use of (\ref{museq1434}) and (\ref{museq1438}), we get, after simple manipulations,
\[F_{p}(t,T,T^{*})=\frac{u(0,T)}{u(0,T^{*})}\cdot\exp\left (
\int_{0}^{t}\boldsymbol{\gamma}(s,T,T^{*})\cdot d{\bf W}_{s}^{*}-
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\gamma}(s,T,T^{*})\parallel^{2}ds\right ).\]
Condition (ii) of Definition \ref{musd1422} is an immediate consequence of (\ref{museq1437}) and (\ref{museq1438}). Family \(p(t,T)\) of bond prices introduced above manifestly satisfies the weak no-arbitrage condition. Furthermore, if assumption (FP3) is met, family \(p(t,T)\) satisfies the no-arbitrage condition. Recall that we assume that the volatilities \(\boldsymbol{\gamma}(t,T,T^{*})\) of forward processes are bounded.
Proposition. For any bounded adapted process \(\widehat{{\bf b}}(t,T^{*})\), processes \(p(t,T)\) given by \((\ref{museq1436}), (\ref{museq1437})\) and \((\ref{museq1438})\) represent a family of bond prices associated with (FP1) and (FP2). This family satisfies the weak no-arbitrage condition (it satisfies the no-arbitrage condition if (FP3) holds). The process \(\widehat{B}\) given by \((\ref{museq1437})\) represents a savings account implied by the family \(p(t,T)\) if and only if it follows a predictable process of finite variation.
Proof. In view of previous considerations, only the last claim is not obvious. The “only if” clause follows directly from the definition of a savings account. The “if” clause is a consequence of results from the previous discussions. In fact, for any maturity \(T\), the relative process
\[\widehat{Z}(t,T)\equiv\frac{p(t,T)}{\widehat{B}_{t}}=u(0,t)\cdot\exp\left (
\int_{0}^{t}\widehat{{\bf b}}(s,T)\cdot d\widehat{{\bf W}}_{s}-\frac{1}{2}
\int_{0}^{t}\parallel\widehat{{\bf b}}(s,T)\parallel^{2}ds\right )\mbox{ for all }t\in [0,T]\]
is evident in \({\cal M}_{loc}(\widehat{P})\). If the volatility of a \(T^{*}\)-maturity bond equals \(\widehat{{\bf b}}(t,T^{*})\), then the process \(\widehat{{\bf b}}(t,T)\) given by (\ref{museq1434}) is the bond price volatility for maturity \(T\). To conclude, it is enough to compare (\ref{museq1437}) with (\ref{museq1422}). \(\blacksquare\)
Let us now examine the problem of uniqueness of a family of bond prices associated with a given collection of forward processes. Since any family \(p(t,T)\) of bond prices associated with the assumptions (FP1) and (FP2) satisfies
\[F(t,T,T^{*})=F_{p}(t,T,T^{*})=\frac{p(t,T)}{p(t,T^{*})}\mbox{ for all }t\in [0,T],\]
we have (see (\ref{museq1432}))
\begin{equation}{\label{museq1439}}\tag{132}
p(t,T)=\frac{p(t,T)}{p(t,t)}=\frac{F(t,T,T^{*})}{F(t,t,T^{*})}=F(t,T,t)\mbox{ for all }t\in [0,T].
\end{equation}
Therefore, a family of bond prices associated with a collection \(F(t,T,T^{*})\) of forward processes is uniquely determined; this implies
in turn the uniqueness of the savings account implied by the forward processes. Notice, however, that equality (\ref{museq1439}) is not very useful in practice since the dynamics of \(F(t,T,t)\) are not easily available.
