Johann Wilhelm Jankowski (1825-1870) was an Austrian painter.
Bond Market.
We assume that \({\bf W}=(W_{1},\cdots ,W_{d})\) is a standard \(d\)-dimensional Brownian motion and the filtration \(I\!\! F\) is the augmentation of the filtration generated by \({\bf W}\). The dynamics of the various processes are given as follows:
- Short rate dynamics: we consider
\begin{equation}{\label{ch5eq1}}\tag{7}
dr_{t}=a_{t}dt+{\bf b}_{t}\cdot d{\bf W}_{t}
\end{equation} - Bond price dynamics: we consider
\begin{equation}{\label{ch5eq2}}\tag{8}
dp(t,T)=p(t,T)[m(t,T)dt+{\bf n}(t,T)\cdot d{\bf W}_{t}]
\end{equation} - Forward rate dynamics: we consider
\begin{equation}{\label{ch5eq3}}\tag{9}
df(t,T)=\alpha (t,T)dt+\boldsymbol{\sigma}(t,T)\cdot d{\bf W}_{t}.
\end{equation}
We assume that in the above formulas the coefficients meet standard conditions required to guarantee the existence of the various processes, that is, existence of solutions of the various stochastic differential equations. Furthermore, we assume that the processes are smooth enough to allow differentiation and certain operations involving changing of order of integration and differentiation. We now give the relationships between the processes specified above.
\begin{equation}{\label{binch5p9}}\tag{10}\mbox{}\end{equation}
Proposition \ref{binch5p9}. We have the following properties.
(i) If \(p(t,T)\) satisfies \((\ref{ch5eq2})\), then for the forward rate dynamics, we have
\begin{equation}{\label{binch5eq24}}\tag{11}
df(t,T)=\alpha (t,T)dt+\boldsymbol{\sigma}(t,T)\cdot d{\bf W}_{t},
\end{equation}
where \(\alpha\) and \(\boldsymbol{\sigma}\) are given by
\[\alpha (t,T)={\bf n}_{T}(t,T)\cdot {\bf n}(t,T)-m_{T}(t,T)\mbox{ and }\boldsymbol{\sigma}(t,T)=-{\bf n}_{T}(t,T).\]
(ii) If \(f(t,T)\) satisfies \((\ref{ch5eq3})\), then the short rate satisfies
\[dr_{t}=a_{t}dt+{\bf b}_{t}\cdot d{\bf W}_{t},\]
where \(a\) and \(b\) are given by
\[a_{t}=f_{T}(t,t)+\alpha (t,t)\mbox{ and }{\bf b}_{t}=\boldsymbol{\sigma}(t,t).\]
(iii) If \(f(t,T)\) satisfies \((\ref{ch5eq3})\), then \(p(t,T)\) satisfies
\[dp(t,T)=p(t,T)\cdot\left [r_{t}+A(t,T)+\frac{1}{2}\parallel S(t,T)\parallel^{2}\right ]dt+p(t,T){\bf S}(t,T)\cdot d{\bf W}_{t},\]
where
\begin{equation}{\label{binch5eq12}}\tag{12}
A(t,T)=-\int_{t}^{T}\alpha (t,u)du\mbox{ and }{\bf S}(t,T)=-\int_{t}^{T}\boldsymbol{\sigma}(t,u)du.
\end{equation}
Proof. To prove (i), we only have to apply Ito’s formula to the defining equation for the forward rates. To prove (ii) we start by integrating the forward rate dynamics. This leads to
\begin{equation}{\label{bineq88}}\tag{13}
f(t,t)=r_{t}=f(0,t)+\int_{0}^{t}\alpha (u,t)du+\int_{0}^{t}\boldsymbol{\sigma}(u,T)\cdot d{\bf W}_{u}.
\end{equation}
Writing also \(\alpha\) and \(\boldsymbol{\sigma}\) in integrated form
\[\alpha (s,t)=\alpha (s,s)+\int_{s}^{t}\alpha_{T}(s,u)du\mbox{ and }
\boldsymbol{\sigma}(s,t)=\boldsymbol{\sigma}(s,s)+\int_{s}^{t}\boldsymbol{\sigma}_{T}(s,u)du,\]
and inserting this into (\ref{bineq88}) we find
\[r_{t}=f(0,t)+\int_{0}^{t}\alpha (u,u)du+\int_{0}^{t}\int_{s}^{t}
\alpha_{T}(s,u)duds+\int_{0}^{t}\boldsymbol{\sigma}(u,u)\cdot
d{\bf W}_{u}+\int_{0}^{t}\int_{s}^{t}\boldsymbol{\sigma}_{T}(s,u)dud{\bf W}_{s}.\]
After changing the order of integration we can identify terms to establish part (ii). This completes the proof. \(\blacksquare\)
The absence of arbitrage is guaranteed by the existence of an equivalent martingale measure \(\bar{\mathbb{P}}\). Recall that an equivalent martingale measure has to satisfy \(\bar{\mathbb{P}}\sim\mathbb{P}\) and the discounted price processes (with respect to a suitable numeraire) of the securities have to be local \(\bar{\mathbb{P}}\)-martingales. For the bond market, this implies that all zero-coupon bonds with maturities \(0\leq T\leq T^{*}\) have to be local martingale. More precisely, taking the risk-free bank account \(B_{t}\) as numeraire, we have the following definition.
\begin{equation}{\label{musd1211}}\tag{14}\mbox{}\end{equation}
Definition \ref{musd1211}. A measure \(\bar{\mathbb{P}}\sim \mathbb{P}\) defined on \((\Omega ,{\cal F}_{T^{*}}, \mathbb{P})\) is an equivalent martingale measure for the bond market, if for every fixed \(0\leq T\leq T^{*}\) the process \(p(t,T)/B_{t}\) for \(0\leq t\leq T\) is a local \(\bar{\mathbb{P}}\)-martingale. In this case, a family \(\{p(t,T)\}\) for \(t\leq T\leq T^{*}\) is called an arbitrage-free family of bond prices relative to \(r\). \(\sharp\)
Assume that there exists at least one equivalent martingale measure, say \(\bar{\mathbb{P}}\). Defining contingent claim as \({\cal F}_{T}\)-measurable random variables such that \(X/B_{t}\in L^{1}({\cal F}_{T},\bar{\mathbb{P}})\) with some \(0\leq T\leq T^{*}\), we can use the risk-neutral valuation principle to obtain the following result.
\begin{equation}{\label{binch5p1}}\tag{15}\mbox{}\end{equation}
Proposition \ref{binch5p1}. Consider a \(T\)-contingent claim \(X\). Then, the price process \(\Pi_{X}(t)\), for \(0\leq t\leq T\), of the contingent claim is given by
\[\Pi_{X}(t)=B_{t}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{X}{B_{T}}
\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .
X\cdot e^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ].\]
In particular, the price process of a zero-coupon bond with maturity \(T\) is given by
\begin{equation}{\label{bineq101}}\tag{16}
p(t,T)=B_{t}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{1}{B_{T}}
\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .e^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ].
\end{equation}
Proof. We just have to apply Theorem \ref{binp56}. In particular, we know \(X=p(T,T)=1\). \(\blacksquare\)
Expectations Hypotheses.
Suppose that equality (\ref{bineq101}) is satisfied under the actual probability measure, that is,
\begin{equation}{\label{museq124}}\tag{17}
p(t,T)=\mathbb{E}_{\mathbb{P}}\left [\left .e^{-\int_{t}^{T}r(s)ds}\right |{\cal F}_{t}\right ].
\end{equation}
Equality (\ref{museq124}) is traditionally referred to as the local expectations hypothesis, or a risk-neutral expectations hypothesis. The term “local expectations” refers to the fact that under (\ref{museq124}), the current bond price equals the expected
value, under the actual probability measure, of the bond price in the next (infinitesimal) period, discounted at the current short-term rate. This property can be made more explicit in a discrete-time setting. In our framework, given an arbitrage-free family of bond prices relative to a short-term rate \(r\), it is evident that (\ref{museq123}) holds under any martingale measure \(\bar{\mathbb{P}}\). This does not mean that the local expectations hypothesis, or any other traditional form of expectations hypothesis, is satisfied under the actual probability measure \(\mathbb{P}\).
The return-to-maturity expectations hypothesis assumes that the return from holding any discount bond to maturity is equal to the return expected from rolling over a series of a single-period bonds. Its continuous-time counterpart reads as follows
\[\frac{1}{p(t,T)}=\mathbb{E}_{\mathbb{P}}\left [\left .e^{\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T],\]
for every \(T\leq T^{*}\). Finally, the yield-to-maturity expectations hypothesis asserts that the yield from holding any bond is equal to the yield expected from rolling over a series of a single-period bonds. In a continuous-time framework, this means that for any maturity date \(T\leq T^{*}\), we have
\[p(t,T)=\exp\left (-\mathbb{E}_{\mathbb{P}}\left [\left .\int_{t}^{T}r_{u}du
\right |{\cal F}_{t}\right ]\right )\mbox{ for all }t\in [0,T].\]
The last formula may also be given the following equivalent form
\[y(t,T)=\frac{1}{T-t}\cdot \mathbb{E}_{\mathbb{P}}\left [\left .\int_{t}^{T}r_{u}du\right |{\cal F}_{t}\right ],\]
or finally
\begin{equation}{\label{museq125}}\tag{18}
f(t,T)=\mathbb{E}_{\mathbb{P}}[r_{T}|{\cal F}_{t}]\mbox{ for all }t\in [0,T].
\end{equation}
In view of (\ref{museq125}), under the yield-to-maturity expectations hypothesis, the forward interest rate is an unbiased estimate, under the actual probability \(\mathbb{P}\), of the future short-term interest rate \(r_{T}\). For this reason, the yield-to-maturity expectations hypothesis is also frequently referred to as the {\bf unbiased expectations hypothesis}. We will see in what follows that condition (\ref{museq125}) is always satisfied, not under the actual probability, however, but under the so-called forward martingale measure for the given date \(T\). Note that if the shot-term rate \(r\) is a deterministic function, then all expectations hypotheses coincide, and follow easily from the absence of arbitrage.
