Johann Gottfried Steffan (1815-1905) was a Swiss painter.
Models of Instantaneous Forward Rates.
Heath-Jarrow-Morton Methodology.
As pointed out in Remark \ref{binch5r1}, modelling the term structure with only one explanatory variable lead to various undesirable properties of the model. Various authors have proposed models with more than one state variable, e.g., the short rate and a long rate and/or intermediate rates. The Heath-Jarrow-Morton method is at the far end of this spectrum. They propose to use the entire forward rate curve as their (infinite-dimensional) state variable. More precisely, for any fixed \(T\leq T^{*}\), the dynamics of instantaneous, continuously compounded forward rates \(f(t,T)\) are exogenously given by
\begin{equation}{\label{binch5eq11}}\tag{44}
df(t,T)=\alpha (t,T)dt+\boldsymbol{\sigma}(t,T)d{\bf W}_{t},
\end{equation}
where \({\bf W}\) is a \(d\)-dimensional Brownian motion with respect to the underlying objective probability measure \(P\) and \(\alpha (t,T)\) is a \(\mathbb{R}\)-valued process and \(\boldsymbol{\sigma}(t,T)\) is a \(\mathbb{R}^{d}\)-valued process. For any fixed maturity \(T\), the initial condition of the stochastic differential equation (\ref{binch5eq11}) is determined by the current value of the empirical (observed) forward rate for the future date \(T\) which prevails at time \(0\). Observe that we have defined an infinite-dimensional stochastic system, and that by construction we obtain a perfect fit to the observed term structure (thus avoiding the problem of inverting the yield curve).
The exogenous specification of the family of forward rates \(\{f(t,T),T>0\}\) is equivalent to a specification of the entire family of bond prices \(\{p(t,T),T>0\}\) from Proposition \ref{binch5p8}. Furthermore, by Proposition \ref{binch5p9}, we obtain the dynamics of the bond price processes as
\begin{equation}{\label{binch5eq20}}\tag{45}
dp(t,T)=p(t,T)\cdot [m(t,T)dt+{\bf S}(t,T)d{\bf W}_{t}],
\end{equation}
where
\begin{equation}{\label{binch5eq200}}\tag{46}
m(t,T)=r_{t}+A(t,T)+\frac{1}{2}\cdot\parallel {\bf S}(t,T)\parallel^{2},
\end{equation}
\begin{equation}{\label{binch5eq17}}\tag{47}
A(t,T)=-\int_{t}^{T}\alpha (t,s)ds\mbox{ and }
{\bf S}(t,T)=-\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds
\end{equation}
(compare with (\ref{binch5eq12})).
Absence of Arbitrage.
We now explore what conditions we must impose on the coefficients in order to ensure the existence of an equivalent martingale measure with respect to a suitable numeraire. Then, we could conclude that the bond market is free of arbitrage. In contrast to the previous market models, we now have a continum of bonds with different maturities available for trade. Therefore a portfolio could include an infinite number of financial securities. We shall, however, in accordance with practical restrictions only consider portfolios involving an arbitrary, but finite, number of financial securities. For any collection of maturities \(0<T_{1}<T_{2}<\cdots <T_{k}=T^{*}\), we consider bond trading strategies \(\boldsymbol{\phi}\), where \(\boldsymbol{\phi}\) is a predictable, locally bounded \(\mathbb{R}^{k}\)-valued stochastic process which satisfies \(\phi_{i}(t)=0\) for every \(t\in (T_{i},T^{*}]\) and \(i=1,\cdots ,k\). This condition is necessary since the \(i\)th bond no longer exists after its maturity \(T_{i}\) and hence the holdings in this bond must be equal to zero.
A bond trading strategy \(\boldsymbol{\phi}\) is said to be self-financing when the wealth process \(V\), given by
\[V_{\boldsymbol{\phi}}(t)=\sum_{i=1}^{k}\phi_{i}(t)\cdot p(t,T_{i}),\]
satisfies
\[V_{\boldsymbol{\phi}}(t)=V_{\boldsymbol{\phi}}(0)+\sum_{i=1}^{k}\int_{0}^{t}\phi_{i}(s)dp(s,T_{i})\]
for every \(t\in [0,T^{*}]\). To ensure the arbitrage-free properties of the bond market model, we need to examine the existence of a martingale measure for a suitable choice of a numeriare; in the present setup, we can take either the savings account \(B\) or the bond price
$p(t,T^{*})$.
Using the savings account as the numeraire.
We use the money market account \(B\) by assuming that there exists a measurable version \(f(t,t)\) in \([0,T^{*}]\), given by
\[B_{t}=\exp\left (\int_{0}^{t}f(s,s)ds\right )=\exp\left (\int_{0}^{t}r(s)ds\right ).\]
So we allow investments in a saving account too. We must find an equivalent martingale measure such that
\[Z(t,T)=\frac{p(t,T)}{B_{t}}\]
is a martingale for every \(0\leq T\leq T^{*}\). We will call such a measure risk-neutral martingale measure to emphasize the dependence on numeraire (or the {\bf spot maringale measure} for the HJM model).
\begin{equation}{\label{binch5p11}}\tag{48}\mbox{}\end{equation}
Theorem \ref{binch5p11}. Assume that the family of forward rates is given by (\ref{binch5eq11}). Then, there exists a risk-neutral martingale measure if and only if there exists an adapted process \(\boldsymbol{\lambda}(t)\) with the properties given below.
(a) We have \(\int_{0}^{T}\parallel\boldsymbol{\lambda}(t)\parallel^{2}dt<\infty\) a.s. and \(\mathbb{E}[L(T)]=1\) with
\begin{equation}{\label{binch5eq14}}\tag{49}
L(t)=\exp\left (-\int_{0}^{T}(\boldsymbol{\lambda}(s))^{t}d{\bf W}(s)-\frac{1}{2}\int_{0}^{t}\parallel \boldsymbol{\lambda}(s)
\parallel^{2}ds\right ).
\end{equation}
(b} For all \(0\leq T\leq T^{*}\) and for all \(t\leq T\), we have
\begin{equation}{\label{binch5eq15}}\tag{50}
\alpha (t,T)=\boldsymbol{\sigma}(t,T)\cdot\left (\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\lambda}(t)\right ).
\end{equation}
Proof. Since we are working in a Brownian framework, we know that any equivalent measure \(\bar{\mathbb{P}}\sim\mathbb{P}\) is given via a Girsanov density (\ref{binch5eq14}). Using the product rule (\ref{bineq48}) and Girsanov’s theorem \ref{binp66}, we find the \(\bar{\mathbb{P}}\)-dynamics of \(Z(t,T)\) as
\begin{equation}{\label{binch5eq16}}\tag{51}
dZ=Z\left (A+\frac{1}{2}\parallel {\bf S}\parallel^{2}-{\bf S}\boldsymbol{\lambda}\right )dt+Z{\bf S}\cdot d\bar{{\bf W}},
\end{equation}
where \(\bar{{\bf W}}\) is a \(\bar{\mathbb{P}}\)-Brownian motion. In order for \(Z\) to be a local \(\bar{\mathbb{P}}\)-martingale, the drift coefficient in (\ref{binch5eq16}) has to be zero, so we obtain
\[A(t,T)+\frac{1}{2}\parallel {\bf S}(t,T)\parallel^{2}={\bf S}(t,T)\boldsymbol{\lambda}(t,T).\]
Taking the derivative with respect to \(T\) from the above formula and (\ref{binch5eq17}), we get
\[-\alpha (t,T)-\boldsymbol{\sigma}(t,T){\bf S}(t,T)=-\boldsymbol{\sigma}(t,T)\boldsymbol{\lambda}(t),\]
which after rearranging is (\ref{binch5eq15}). \(\blacksquare\)
It is possible to interpret \(\boldsymbol{\lambda}\) as a risk premium, which has to be exogenously specified to allow the choice of a particular risk-neutral martingale measure. In view of (\ref{binch5eq15}), this leads to a restriction on drift and volatility coefficients in the specification of the forward rate of dynamics (\ref{binch5eq11}). The particular choice \(\boldsymbol{\lambda}={\bf 0}\) means that we assume the model directly under a risk-neutral martingale measure \(\bar{\mathbb{P}}\). In that case, the relations between the various infinitesimal characteristics for the forward rate are known as the “Heath-Jarrow-Morton” drift condition.
\begin{equation}{\label{binch5p14}}\tag{52}\mbox{}\end{equation}
Theorem \ref{binch5p14} (Heath-Jarrow-Morton Model). Assume that \(\bar{\mathbb{P}}\) is a risk-neutral martingale measure for the bond market and that the forward rate dynamics under \(\bar{\mathbb{P}}\) are given by
\begin{equation}{\label{binch5eq19}}\tag{53}
df_{0}(t,T)=\alpha_{0}(t,T)dt+\boldsymbol{\sigma}(t,T)d\bar{{\bf W}}(t)
\end{equation}
with \(\bar{{\bf W}}\) a \(\bar{\mathbb{P}}\)-Brownian motion. Then, we have the Heath-Jarrow-Morton drift condition
\begin{equation}{\label{binch5eq26}}\tag{54}
\alpha_{0}(t,T)=\boldsymbol{\sigma}(t,T)\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds=-\boldsymbol{\sigma}(t,T)
{\bf S}(t,T)\mbox{ \(Q\)-a.s. for }0\leq t\leq T\leq T^{*},
\end{equation}
and bond price dynamics under \(\bar{\mathbb{P}}\) are given by
\begin{equation}{\label{eq1}}\tag{55}dp_{0}(t,T)=p_{0}(t,T)\cdot (r_{t}dt+{\bf S}(t,T)d\bar{{\bf W}}(t))\end{equation}
with \({\bf S}\) as in (\ref{binch5eq12}).
Proof. Just use \(\boldsymbol{\lambda}={\bf 0}\) in Theorem \ref{binch5p11}. In this case, \(d\bar{{\bf W}}(t)=d{\bf W}_{t}\) by Girsanov’s theorem. The forward rate dynamics under \(\bar{\mathbb{P}}\) in (\ref{binch5eq19}) can also be regarded as the forward rate dynamics under \(P\) written as \(df_{0}(t,T)=\alpha_{0}(t,T)dt+\boldsymbol{\sigma}(t,T)d{\bf W}_{t}\). Since
\[\frac{d}{dT}\left (\frac{1}{2}\cdot\parallel {\bf S}(t,T)\parallel^{2}\right )={\bf S}(t,T)\cdot\frac{d}{dT}{\bf S}(t,T)=
-\boldsymbol{\sigma}(t,T)\cdot {\bf S}(t,T),\]
it says
\[\int_{t}^{T}\boldsymbol{\sigma}(t,s)\cdot {\bf S}(t,s)ds=-\frac{1}{2}\cdot\parallel {\bf S}(t,T)\parallel^{2}.\]
Then, we have
\[A(t,T)=-\int_{t}^{T}\alpha (t,s)ds=\int_{t}^{T}\boldsymbol{\sigma}(t,s)\cdot {\bf S}(t,s)ds=-\frac{1}{2}
\cdot\parallel {\bf S}(t,T)\parallel^{2}.\]
The formula (\ref{eq1}) follows from the fact that \(d\bar{{\bf W}}(t)=d{\bf W}_{t}\) and part (iii) of Proposition \ref{binch5p9}. \(\blacksquare\)
As already mentioned, calibration of a Heath-Jarrow-Morton (HJM) model is done by definition: today’s forward rates are used as initial values for the stochastic differential equation (\ref{binch5eq19}) and so perfectly fitted. Even if we specify the forward rate dynamics under a risk-neutral martingale measure \(\bar{\mathbb{P}}\), Theorem \ref{binch5p14} tells us that we may freely specify the volatility structure. The drift parameters are then uniquely determined. Notice also that, since \(\mathbb{P}\) and \(\bar{\mathbb{P}}\) are equivalent, the volatility process is the same under \(\mathbb{P}\) as under \(\bar{\mathbb{P}}\).
Proposition. For any fixed maturity \(T\leq T^{*}\), the dynamics of the bond price \(p(t,T)\) under the risk-neutral martingale measure \(\bar{\mathbb{P}}\) (the spot martingale measure) are
\begin{equation}{\label{museq1314}}\tag{56}
dp(t,T)=p(t,T)(r_{t}dt+{\bf S}(t,T)\cdot d\bar{{\bf W}}_{t}),
\end{equation}
and the forward rate \(f(t,T)\) satisfies
\begin{equation}{\label{museq1315}}\tag{57}
df(t,T)=-\boldsymbol{\sigma}(t,T)\cdot {\bf S}(t,T)+\boldsymbol{\sigma}(t,T)\cdot d\bar{{\bf W}}_{t}.
\end{equation}
Finally, the short-term interest \(r_{t}=f(t,t)\) is given by the expression
\begin{equation}{\label{museq1316}}\tag{58}
r_{t}=f(0,t)-\int_{0}^{t}\boldsymbol{\sigma}(s,t){\bf S}(s,t)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,t)d\bar{{\bf W}}(s).
\end{equation}
Proof. Integrating the forward rate dynamics (\ref{binch5eq19}) (specified under a risk-neutral martingale measure), we get
\[f(t,T)=f(0,T)+\int_{0}^{t}\alpha (s,T)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,T)d\bar{{\bf W}}(s).\]
By definition \(r_{t}=f(t,t)\), we then have, from (\ref{binch5eq26}),
\[r_{t}=f(0,t)-\int_{0}^{t}\boldsymbol{\sigma}(s,t){\bf S}(s,t)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,t)d\bar{{\bf W}}(s).\]
This completes the proof. \(\blacksquare\)
It follows from (\ref{museq1316}) that the expectation of the future short-term rate under the risk-neutral martingale measure \(\bar{\mathbb{P}}\) (the spot martingale measure) does not equal the current value \(f(0,T)\) of the instantaneous forward rate; that is \(\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[r_{t}]\neq f(0,T)\) in general. We shall see soon that \(f(0,T)\) equals the expectation of \(r_{T}\) under the forward risk-neutral martingale measure for the date \(T\). Therefore under a spot martingale measure, the unbiased forward rate form of the expectation \(\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[r_{t}]=f(0,T)\), is not true.
In view of (\ref{museq1314}), the relative bond price \(Z(t,T)=p(t,T)/B_{t}\) satisfies
\[dZ(t,T)=Z(t,T){\bf S}(t,T)\cdot d\bar{{\bf W}}_{t},\]
and thus
\begin{equation}{\label{museq23}}\tag{59}
Z(t,T)=p(0,T)\cdot\exp\left (\int_{0}^{t}{\bf S}(s,T)\cdot d\bar{{\bf W}}_{s}-\frac{1}{2}\int_{0}^{t}\parallel {\bf S}(s,T)\parallel^{2}ds\right ),
\end{equation}
or equivalently
\[\ln p(t,T)=\ln p(0,T)+\int_{0}^{t}\left (t_{s}-\frac{1}{2}\parallel
{\bf S}(s,T)\parallel^{2}\right )ds+\int_{0}^{t}{\bf S}(s,T)\cdot d\bar{{\bf W}}_{s}.\]
Next we use the bond price as the numeraire.
Using the bond price as the numeriare.
For many valuation problems in the bond market it is more suitable to use the bond price process \(p(t,T^{*})\) as numeraire. We then have to find an equivalent probability measure \(\bar{\mathbb{P}}^{*}\) such that the auxiliary process
\[Z^{*}(t,T)=\frac{p(t,T)}{p(t,T^{*})}\mbox{ for }t\in [0,T]\]
is a martingale under \(\bar{\mathbb{P}}^{*}\) for all \(T\leq T^{*}\). We will call such a measure forward risk-neutral martingale measure for the date \(T^{*}\). In this setting, a saving account is not used and the existence of a martingale measure \(\bar{\mathbb{P}}^{*}\) guarantees that there are no arbitrage opportunities between bonds of different maturities.
The bond price dynamics under the original probability measure \(\mathbb{P}\) are given in (\ref{binch5eq20}). Now applying the product rule (\ref{bineq48}) to the quotient \(p(t,T)/p(t,T^{*})\) we find
\begin{equation}{\label{binch5eq21}}\tag{60}
dZ^{*}(t,T)=Z^{*}(t,T)\cdot [\bar{m}(t,T)dt+({\bf S}(t,T)-{\bf S}(t,T^{*}))d{\bf W}_{t}]
\end{equation}
with
\begin{equation}{\label{binch5eq201}}\tag{61}
\tilde{m}(t,T)=m(t,T)-m(t,T^{*})-{\bf S}(t,T^{*})\cdot [{\bf S}(t,T)-{\bf S}(t,T^{*})].
\end{equation}
Again any equivalent martingale measure \(\mathbb{Q}^{*}\) is given via a Girsanov density \(L(t)\) defined by a function \(\boldsymbol{\gamma}(t)\) as in Theorem \ref{binch5p11}. Since \(d{\bf W}_{t}=d\bar{{\bf W}}(t)-\boldsymbol{\gamma}(t)dt\), we get, from (\ref{binch5eq21}),
\[dZ^{*}(t,T)=Z^{*}(t,T)\cdot [(\tilde{m}(t,T)-({\bf S}(t,T)-{\bf S}(t,T^{*}))\cdot\boldsymbol{\gamma}(t))dt+
({\bf S}(t,T)-{\bf S}(t,T^{*}))d\bar{{\bf W}}(t)\]
Now for \(Z^{*}(t,T)\) to be a \(\mathbb{Q}^{*}\)-martingale the drift coefficient has to be zero, i.e.,
\[\tilde{m}(t,T)-({\bf S}(t,T)-{\bf S}(t,T^{*}))\cdot\boldsymbol{\gamma}(t)=0.\]
Replacing \(\tilde{m}\) with its definiton in (\ref{binch5eq200}) and (\ref{binch5eq201}), we get
\[A(t,T)-A(t,T^{*})+\frac{1}{2}\left (\parallel {\bf S}(t,T)\parallel^{2}-\parallel {\bf S}(t,T^{*})\parallel^{2}\right )=
[{\bf S}(t,T^{*})+\boldsymbol{\gamma}(t)][{\bf S}(t,T)-{\bf S}(t,T^{*})].\]
Written in terms of the coefficients of the forward rate dynamics using (\ref{binch5eq17}), this identity simplifies to
\[\int_{T}^{T^{*}}\alpha (t,s)ds+\frac{1}{2}\left |\!\left |\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds\right |\!\right |^{2}=
\boldsymbol{\gamma}(t)\cdot\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds.\]
Taking the derivative with respect to \(T\), we obtain
\[\alpha (t,T)+\boldsymbol{\sigma}(t,T)\cdot\int_{T}^{T^{*}}
\boldsymbol{\sigma}(t,s)ds=\boldsymbol{\gamma}(t)\cdot\boldsymbol{\sigma}(t,T),\]
that is,
\[\alpha (t,T)-\boldsymbol{\sigma}(t,T){\bf S}(T,T^{*})=\boldsymbol{\gamma}(t)\boldsymbol{\sigma}(t,T),\]
We have thus proved the following result.
Theorem. Assume that the family of forward rates is given by (\ref{ch5eq3}). Then, there exists a forward risk-neutral martingale measure if and only if there exists an adapted process \(\boldsymbol{\gamma}(t)\) with the properties of Theorem \ref{binch5p11} such that
\begin{equation}{\label{museq1312}}\tag{62}
\alpha (t,T)=\boldsymbol{\sigma}(t,T)[{\bf S}(T,T^{*})+\boldsymbol{\gamma}(t)]
=\boldsymbol{\sigma}(t,T)\cdot\left (-\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\gamma}(t)\right )
\mbox{ for }0\leq t\leq T\leq T^{*}. \sharp
\end{equation}
We can check that, under mild technical assumptions, the no-arbitrage conditions (\ref{binch5eq15}) and (\ref{museq1312}) are equivalent.
Proposition. Conditions (\ref{binch5eq15}) and (\ref{museq1312}) are equivalent.
Proof. Let us sketch the proof. Suppose first that condition (\ref{museq1312}) holds. Let us define
\[\boldsymbol{\lambda}(t)=-\int_{t}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\gamma}(t)\mbox{ for all }t\in [0,T^{*}],\]
Now we obtain
\begin{align*}
\boldsymbol{\sigma}(t,T)\cdot\left (\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\lambda}(t)\right ) & =
\boldsymbol{\sigma}(t,T)\cdot\left (\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds-\int_{t}^{T^{*}}
\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\gamma}(t)\right )\\
& =\boldsymbol{\sigma}(t,T)\cdot\left (-\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\gamma}(t)\right )=\alpha (t,T),
\end{align*}
which is the right-hand side of (\ref{binch5eq15}) by referring to (\ref{museq1312}). Conversely, given a process \(\boldsymbol{\lambda}\) which satisfies (\ref{binch5eq15}), we define the process \(\boldsymbol{\gamma}\) by setting
\[\boldsymbol{\gamma}(t)=\int_{t}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\lambda}(t)\mbox{ for all }t\in [0,T^{*}].\]
Then we have
\begin{align*}
\boldsymbol{\sigma}(t,T)\cdot\left (-\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\gamma}(t)\right ) & =
\boldsymbol{\sigma}(t,T)\cdot\left (-\int_{T}^{T^{*}}\boldsymbol{\sigma}(t,s)ds+\int_{t}^{T^{*}}
\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\lambda}(t)\right )\\
& =\boldsymbol{\sigma}(t,T)\cdot\left (\int_{t}^{T}\boldsymbol{\sigma}(t,s)ds+\boldsymbol{\lambda}(t)\right )=\alpha (t,T),
\end{align*}
which is the right-hand side of (\ref{museq1312}) by referring to (\ref{binch5eq15}). \(\blacksquare\)
We assume from now that the no-arbitrage condition (\ref{museq1312}) (or equivalently, (\ref{binch5eq15})) is satisfied. It is not essential to assume that the martingale measure for the bond market is unique, so long as we are not concerned with the completeness of the model. Recall that if a market is arbitrage-free, any attainable claim admits a unique arbitrage price anyway (it is uniquely determined by the replicating strategy), whether a market model is complete or not. For instance, in a Gaussian HJM setting, several claims, such as options on bond or a stock, are attainable with replicating strategy given by closed-form expressions. One may argue that from the practical viewpoint, the possibility of explicit replication of most typical claims is a more important feature of a model than its theoretical completeness. It is interesting to note that under some regularity assumptions, the short-term interest rate process specified by the HJM model of instantaneous forward rates follows a continuous semimartingale, or even a diffusion process.
Proposition. Suppose that the coefficients \(\alpha (t,T)\) and \(\boldsymbol{\sigma}(t,T)\) and the initial forward curve \(f(0,T)\) are differentiable with respect to maturity \(T\) with bounded partial derivatives \(\alpha_{T}(t,T),\boldsymbol{\sigma}_{T}(t,T)\) and \(f_{T}(0,T)\). Then the short-term rate \(r\) follows a continuous semimartingale under \(\mathbb{P}\). More specifically, for any \(t\in [0,T^{*}]\) we have
\begin{equation}{\label{museq1322}}\tag{63}
r_{t}=r_{0}+\int_{0}^{t}\zeta_{s}ds+\int_{0}^{t}\boldsymbol{\sigma}(s,s)\cdot d{\bf W}_{s},
\end{equation}
where \(\zeta\) stands for the following process
\[\zeta_{t}=\alpha (t,T)+f_{T}(0,t)+\int_{0}^{t}\alpha_{T}(s,t)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,t)\cdot d{\bf W}_{s}.\]
Proof. Observe first that \(r\) satisfies
\[r_{t}=f(t,t)=f(0,t)+\int_{0}^{t}\alpha (s,t)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,t)\cdot d{\bf W}_{s}.\]
Applying the stochastic Fubini theorem to the Ito integral, we obtain
\begin{align*}
\int_{0}^{t}\boldsymbol{\sigma}(s,t)\cdot d{\bf W}_{s} & =\int_{0}^{t}\boldsymbol{\sigma}(s,s)\cdot d{\bf W}_{s}+
\int_{0}^{t}(\boldsymbol{\sigma}(s,t)-\boldsymbol{\sigma}(s,s))\cdot d{\bf W}_{s}\\
& =\int_{0}^{t}\boldsymbol{\sigma}(s,s)\cdot d{\bf W}_{s}+\int_{0}^{t}\int_{s}^{t}\boldsymbol{\sigma}_{T}(s,u)du\cdot
d{\bf W}_{s}\\
& =\int_{0}^{t}\boldsymbol{\sigma}(s,s)\cdot d{\bf W}_{s}+
\int_{0}^{t}\int_{0}^{u}\boldsymbol{\sigma}_{T}(s,u)\cdot d{\bf W}_{s}du.
\end{align*}
Furthermore,
\[\int_{0}^{t}\alpha (s,t)ds=\int_{0}^{t}\alpha (s,s)ds+\int_{0}^{t}\int_{0}^{s}\alpha_{T}(u,s)duds,\]
and finally
\[f(0,t)=r_{0}+\int_{0}^{t}f_{T}(o,s)ds.\]
Combining these formulas, we obtain (\ref{museq1322}). \(\blacksquare\)
Forward Measure Approach.
We assume that the price \(p(t,T)\) of a zero-coupon bond of maturity \(T\leq T^{*}\) follows an Ito process under the martingale measure \(\mathbb{P}^{*}\) (ref. (\ref{museq1314}))
\begin{equation}{\label{museq1323}}\tag{64}
dp(t,T)=p(t,T)(r_{t}dt+{\bf b}(t,T)\cdot d{\bf W}^{*}_{t}),
\end{equation}
with \(p(T,T)=1\), wher \({\bf W}^{*}\) denotes the \(d\)-dimensional standard Brownian motion defined on a filtered probability space \((\Omega ,{\cal F},\mathbb{P}^{*})\), and \(r_{t}\) stands for the instantaneous, continuously compounded rate of interest. In other words, we take for granted the existence of an arbitrage-free family \(p(t,T)\) of bond prices (ref. Definition \ref{musd1211}) associated with a certain process \(r\) which models the short-term interest rate. Moreover, it is implicitly assumed that we have already constructed an arbitrage-free model of a market in which all bonds of different maturities, as well as a certain number of other assets (called stocks), are primary traded securities. The dynamics of a stock price \(S^{i}\) for \(i=1,\cdots ,m\), under the martingale measure \(\mathbb{P}^{*}\), are given by the following expression
\[dS^{i}_{t}=S^{i}_{t}(r_{t}dt+\boldsymbol{\sigma}^{i}_{t}\cdot d{\bf W}^{*}_{t})\mbox{ for }S_{0}^{i}>0,\]
where \(\boldsymbol{\sigma}^{i}\) represents the volatility of the stock price \(S^{i}\). The bond price volatility \({\bf b}(t,T)\) and the stock price volatility \(\boldsymbol{\sigma}^{i}_{t}\) are assumed to be \(\mathbb{R}^{d}\)-valued, bounded, adapted processes. Generally speaking, we assume that the prices of all primary securities follow strictly positive processes with continuous sample paths. We denote by \(\Pi_{t}(X)\) the arbitrage price at time \(t\) of an attainable contingent claim \(X\) which settles at time \(T\). Therefore
\begin{equation}{\label{museq1325}}\tag{65}
\Pi_{t}(X)=B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T],
\end{equation}
by virtue of the risk-neutral valuation formula, where \(B\) represents the savings account. Recall that the price \(p(t,T)\) of a zero-coupon bond which matures at time \(T\) admits the following representation (ref. (\ref{bineq101}))
\begin{equation}{\label{museq1326}}\tag{66}
p(t,T)=B_{t}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{1}{B_{T}}\right |{\cal F}_{t}
\right ]\mbox{ for all }t\in [0,T],
\end{equation}
for any maturity \(0\leq T\leq T^{*}\). Suppose now that we wish to price a European call option with expiry data \(T\), which is written on a zero-coupon bond of maturity \(U>T\). The option’s payoff at expiry equals
\[C_{T}=(p(T,U)-K)^{+},\]
so that the option price \(C_{t}\) at any date \(t\leq T\) is
\[C_{t}=B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{(p(T,U)-K)^{+}}{B_{T}}\right |{\cal F}_{t}\right ].\]
To find the option’s price using the last equality, we need to know that joint (conditional) probability distribution of \({\cal F}_{T}\)-measurable random variables \(B_{T}\) and \(p(T,U)\). The technique which was developed to circumvent this step is based on an equivalent change of probability measure.
Forward Price.
Recall that a forward constract is an agreement, established at the date \(t<T\), to pay or receive on settlement date \(T\) a preassigned payoff, say \(X\), at an agreed forward price. Let us emphasize that there is no cash flow at the contract’s initiation and the contract is not marked to market. We may and do assume, without loss of generality, that a forward contract is settled by cash on date \(T\). Therefore, a forward contract written at time \(t\) with the underlying contingent claim \(X\) and prescribed settlement date \(T>t\) may be summarized by the following two basic rules:
- A cash amount \(X\) will be received at time \(T\), and a pre-assigned amount \(F_{X}(t,T)\) of cash will be paid at time \(T\);
- The amount \(F_{X}(t,T)\) should be pre-determined at time \(t\) (according to the information available at this time) in such a way that the arbitrage price of the forward contract at time \(t\) is zero.
In fact, since nothing is paid up front, it is natural to admit that a forward contract is worthless at its initiation. We adopt the following formal definition of a forward contract.
Definition. Let us fix \(0\leq t\leq T\leq T^{*}\). A forward contract written at time \(t\) on a time \(T\)-contingent claim \(X\) is represented by the time \(T\)-contingent claim \(G_{T}=X-F_{X}(t,T)\) that satisfies the following conditions.
- \(F_{X}(t,T)\) is a \({\cal F}_{t}\)-measurable random variable.
- The arbitrage price at time \(t\) of a contingent claim \(G_{T}\) equals zero, i.e., \(\Pi_{t}(G_{T})=0\). \(\sharp\)
The random variable \(F_{X}(t,T)\) is referred to as the forward price of a contingent claim \(X\) at time \(t\) for the settlement date \(T\). The contingent claim \(X\) may be defined in particular as a preassigned amount of the underlying financial asset to be delivered at the settlement date. For instance, if the underlying asset of a forward contract is one share of a stock \(S\), then clearly \(X=S_{T}\). Similarly, if the asset to be delivered at time \(T\) is a zero-coupon bond of maturity \(U\geq T\), we have \(X=p(T,U)\). Note that both \(S_{T}\) and \(p(T,U)\) are attainable contingent claim in the market model.
\begin{equation}{\label{musl1321}}\tag{67}\mbox{}\end{equation}
Proposition \ref{musl1321}. The forward price \(F_{X}(t,T)\) at time \(t\leq T\), for the settlement date \(T\), of an attainable contingent claim \(X\) equals
\begin{equation}{\label{museq1327}}\tag{68}
F_{X}(t,T)=\frac{ $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[X/B_{T}|{\cal F}_{t}]}
{ $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[1/B_{T}|{\cal F}_{t}]}=\frac{\Pi_{t}(X)}{p(t,T)}.
\end{equation}
Proof. It is suffices to observe that
\[0=\Pi_{t}(G_{T})=B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{G_{T}}{B_{T}}
\right |{\cal F}_{t}\right ]=B_{t}\left ( \(\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}}\right |{\cal F}_{t}\right ]-F_{X}(t,T)\cdot
\)latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{1}{B_{T}}\right |{\cal F}_{t}\right ]\right )\]
by using \(G_{T}=X-F_{X}(t,T)\). The proof is complete from (\ref{museq1325}) and (\ref{museq1326}). \(\blacksquare\)
Let us examine the two typical cases of forward contracts mentioned above. If the underlying asset for delivery at time \(T\) is a zero-coupon bond of maturity \(U\geq T\), then (\ref{museq1327}) becomes
\begin{equation}{\label{museq1328}}\tag{69}
F_{p(T,U)}(t,T)=\frac{p(t,U)}{p(t,T)}\mbox{ for all }t\in [0,T].
\end{equation}
On the other hand, the forward price of a stock \(S\) equals
\[F_{S_{T}}(t,T)=\frac{S_{t}}{p(t,T)}\mbox{ for all }t\in [0,T].\]
For the sake of brevity, we will write \(F_{p}(t,U,T)\) and \(F_{S}(t,T)\) instead of \(F_{p(T,U)}(t,T)\) and \(F_{S_{T}}(t,T)\), respectively. More generally, for any tradeable asset \(A\), we write \(F_{Z}(t,T)\) to denote the forward price of the asset, that is,
\begin{equation}{\label{museq70}}\tag{70}
F_{Z}(t,T)=Z_{t}/p(t,T)
\end{equation}
for all \(t\in [0,T]\).
Forward Risk-Neutral Martingale Measure.
We assume that we are given an arbitrage-free family \(p(t,T)\) of bond prices and the related savings account \(B\). Note that by assumption, \(0<p(0,T)= \)latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[1/B_{T}]<\infty$. We now investigate the relation between risk-neutral \(\mathbb{P}^{*}\) (using \(B_{t}\) as the numeriare) and forward risk-neutral \(P_{T}\) (using \(p(t,T)\) as the numeraire) for any \(T\leq T^{*}\) martingale measures. Suppose that we model the forward rates under \(\mathbb{P}^{*}\) by integrating the forward rate dynamics (\ref{binch5eq24}) as follows.
\begin{equation}{\label{binch5eq25}}\tag{71}
r_{t}=f(t,t)=f(0,T)+\int_{0}^{t}\alpha (s,T)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,T)d{\bf W}^{*}_{s}.
\end{equation}
By the change of numeraire technique in Theorem \ref{binp60} and Example \ref{bine3}, we find, from (\ref{museq23}),
\begin{equation}{\label{museq1331}}\tag{72}
\eta_{t}=\left .\frac{d\mathbb{P}_{T}}{d\mathbb{P}^{*}}\right |_{{\cal F}_{t}}=
\frac{p(t,T)}{p(0,T)B_{t}}=\exp\left (\int_{0}^{t}{\bf S}(s,T)
d{\bf W}^{*}_{s}-\frac{1}{2}\int_{0}^{t}\parallel {\bf S}(s,T)\parallel^{2}ds\right ).
\end{equation}
Observe that a change of measure using \(\eta\) implies
\[{\bf W}^{*}_{t}={\bf W}^{T}_{t}+\int_{0}^{t}{\bf S}(s,T)ds,\]
where \({\bf W}_{t}^{T}\) follows a standard Brownian motion under the forward martingale measure \(\mathbb{P}_{T}\). Using formula (\ref{binch5eq25}) and condition (\ref{binch5eq26}) we find that under \(\mathbb{P}_{T}\)
\[r_{t}=f(0,T)+\int_{0}^{T}\boldsymbol{\sigma}(s,T)d{\bf W}^{T}_{s}.\]
This shows that for every \(T\), the unbiased forward rate form of the expectation hypothesis under a forward martingale measure \(\mathbb{P}_{T}\) (we can use any \(T\leq T^{*}\)) is true, i.e., \(\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[r_{t}]=f(0,T)\).
\begin{equation}{\label{musp1322}}\tag{73}\mbox{}\end{equation}
Proposition \ref{musp1322}. The forward price at \(t\) for the date \(T\) of an attainable contingent claim \(X\) which settles at time \(T\) equals
\begin{equation}{\label{museq1333}}\tag{74}
F_{X}(t,T)=\mathbb{E}_{P_{T}}[X|{\cal F}_{t}]\mbox{ for all }t\in [0,T],
\end{equation}
provided that \(X\) is \(P_{T}\)-integrable. In particular, the forward price process \(F_{X}(t,T)\), for \(t\in [0,T]\), follows a martingale under the forward martingale measure \(\mathbb{P}_{T}\).
Proof. We see that, from (\ref{museq1331}),
\begin{equation}{\label{museq28}}\tag{75}
\eta_{t}= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[\eta_{T}|{\cal F}_{t}]\mbox{ and }\eta_{T}=\frac{1}{B_{T}\cdot p(0,T)}.
\end{equation}
The Bayes formula in Proposition \ref{musa04} and (\ref{museq28}) yields
\begin{align*}
\mathbb{E}_{P_{T}}[X|{\cal F}_{t}] & =\frac{ \(\mathbb{E}_{\scriptsize \mathbb{P}^{*}}[\eta_{T}\cdot X|{\cal F}_{t}]}{ \)latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[\eta_{T}|{\cal F}_{t}]}\\
& =\frac{B_{t}\cdot p(0,T)}{p(t,T)}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}\cdot p(0,T)}\right |{\cal F}_{t}\right ]\\
& =\frac{1}{p(t,T)}\cdot\left (B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}}\right |{\cal F}_{t}\right ]\right ).
\end{align*}
Combining (\ref{museq1325}) and (\ref{museq1327}), we obtain the desired result. \(\blacksquare\)
Under (\ref{museq1331}), the equation (\ref{museq1333}) can be given a more explicit from, namely
\[\mathbb{E}_{P_{T}}[X|{\cal F}_{t}]= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .X\cdot\exp\left (\int_{t}^{T}{\bf S}(s,T)
d{\bf W}^{*}_{s}-\frac{1}{2}\int_{t}^{T}\parallel {\bf S}(s,T)\parallel^{2}ds\right )\right |{\cal F}_{t}\right ].\]
The equalities \(F_{p}(t,T,U)=\mathbb{E}_{P_{T}}[p(T,U)|{\cal F}_{t}]\) for \(0<T\leq U\leq T^{*}\) and \(F_{S}(t,T)=\mathbb{E}_{P_{T}}[S_{T}|{\cal F}_{t}]\) for every \(t\in [0,T^{*}]\) are immediate consequences of the last proposition. More generally, the relative price of any traded security (which pays no coupons or dividends) follows a local martingale under the forward martingale measure \(P_{T}\) provided that the price of a bond which matures at time \(T\) is taken as a numeraire.
\begin{equation}{\label{musl1323}}\tag{76}\mbox{}\end{equation}
Proposition \ref{musl1323}. The arbitrage price of an attainable contingent claim \(X\) which settles at time \(T\) is given by the formula
\begin{equation}{\label{museq1335}}\tag{77}
\Pi_{t}(X)=p(t,T)\cdot \mathbb{E}_{P_{T}}[X|{\cal F}_{t}]
\end{equation}
for all \(t\in [0,T]\).
Proof. The desired result is an immediate consequence of (\ref{museq1327}) and (\ref{museq1333}). \(\blacksquare\)
The following proposition deals with a contingent claim which settles at time \(U\neq T\). The aim is to express the value of this claim in terms of the forward martingale measure for the date \(T\).
Proposition. Let \(X\) be an arbitrary attainable contingent claim which settles at time \(U\). We have the following properties
(i) If \(U\leq T\), then the price of \(X\) at time \(t\leq U\) equals
\begin{equation}{\label{museq1336}}\tag{78}
\Pi_{t}(X)=B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{U}}\right |
{\cal F}_{t}\right ]=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [\left .\frac{X}{p(U,T)}\right |{\cal F}_{t}\right ].
\end{equation}
(ii) If \(U\geq T\) and \(X\) is \({\cal F}_{T}\)-measurable, then, for any \(t\leq U\), we have
\begin{equation}{\label{museq1337}}\tag{79}
\Pi_{t}(X)=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [\left .X\cdot p(T,U)\right |{\cal F}_{t}\right ].
\end{equation}
Proof. In part (i), we invest at time \(U\) a \({\cal F}_{U}\)-measurable payoff \(X\) in zero-coupon bonds which mature at time \(T\). For case (ii), observe that in order to replicate a \({\cal F}_{T}\)-measurable claim \(X\) at time \(U\), it is enough to purchase, at time \(T\), \(X\) units of a zero-coupon bond maturing at time \(U\). Both strategies are manifestly self-financing, and thus the result follows. Now we give an alternative proof. For part (i), from (\ref{museq1325}), we have
\begin{equation}{\label{museq32}}\tag{80}
\Pi_{t}(X)=B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{U}}\right |{\cal F}_{t}\right ].
\end{equation}
Now, we assume that the claim \(Y=X/p(U,T)\) settles at time \(T\). Then, from (\ref{museq1325}) again, we have
\begin{equation}{\label{museq33}}\tag{81}
\Pi_{t}(Y)=B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{Y}{B_{T}}\right |{\cal F}_{t}\right ]=.
B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}\cdot p(U,T)}\right |{\cal F}_{t}\right ]
\end{equation}
Since \(X\) is \({\cal F}_{U}\)-measurable and
\begin{equation}{\label{museq31}}\tag{82}
p(U,T)=\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .e^{-\int_{U}^{T}r_{s}ds}\right |
{\cal F}_{U}\right ]=\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{U}}{B_{T}}\right |{\cal F}_{U}\right ]
\end{equation}
by (\ref{bineq101}), we have for every \(t\in [0,U]\)
\begin{align*}
B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{T}\cdot p(U,T)}\right |{\cal F}_{t}\right ] & =
B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left\{\left .\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X\cdot B_{U}}{B_{U}\cdot B_{T}\cdot p(U,T)}\right |{\cal F}_{U}\right ]\right |{\cal F}_{t}\right\}\\
& =B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left\{\left .\frac{X}{B_{U}\cdot p(U,T)}\cdot
\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{U}}{B_{T}}\right |{\cal F}_{U}\right ]\right |{\cal F}_{t}\right\}\\
& =B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{U}}\right |{\cal F}_{t}\right ]\mbox{(from (\ref{museq31}))}.
\end{align*}
From (\ref{museq32}) and (\ref{museq33}), we have \(\Pi_{t}(X)=\Pi_{t}(Y)\). This means that the claim \(X\) that settles at time \(U\) has, at any date \(t\in [0,U]\), an identical arbitrage price to the claim \(Y\) that settles at time \(T\). Formula (\ref{museq1336}) now follows from relation (\ref{museq1335}) applied to the claim \(Y\). Similarly, to prove formula (\ref{museq1337}), we observe that since \(X\) is \({\cal F}_{T}\)-measurable, we have for \(t\in [0,T]\)
\[B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{U}}\right |{\cal F}_{t}
\right ]=B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left\{\left .\frac{X}{B_{T}}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{B_{T}}{B_{U}}\right |{\cal F}_{T}\right ]\right |
{\cal F}_{t}\right\}=B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X\cdot p(T,U)}
{B_{T}}\right |{\cal F}_{t}\right ].\]
We conclude once again that a \({\cal F}_{T}\)-measurable claim \(X\) which settles at time \(U\geq T\) is essentially equivalent to a claim
$Y=X\cdot p(T,U)$ which settles at time \(T\). \(\blacksquare\)
Gaussian HJM Model.
In this subsection, we assume that the volatility \(\boldsymbol{\sigma}\) of the forward rate is deterministic; such a case will be referred to as the Gaussian HJM model. This terminology refers to the fact that the forward rate \(f(t,T)\) and the spot rate \(r_{t}\) have Gaussian distributions under the martingale measure \(\mathbb{P}^{*}\) (ref. (\ref{museq1315}) and (\ref{museq1316})). Proposition \ref{binch5p14} also describes the bond price dynamics. The aim is to show that the arbitrage price of any attainable interest rate-sensitive claim can be evaluated by each of the following procedures.
- We start with arbitrary dynamics of the forward rate such that condition (\ref{museq1312}) (or condition (\ref{binch5eq15})) is satisfied. We then find a martingale measure \(\mathbb{P}^{*}\), and apply the risk-neutral valuation formula.
- We assume instead that the underlying probability measure \(\mathbb{P}\) is actually the spot (resp. forward) martingale measure. In other words, we assume that condition (\ref{binch5eq15}) (resp. condition (\ref{museq1312})) is satisfied with the process \(\boldsymbol{\lambda}\) (resp. \(\boldsymbol{\gamma}\)) equal to zero.
Since both procedures give the same valuation results, we conclude that the specification of the risk premium is not relevant in the context of arbitrage valuation of interest rate-sensitive derivatives in the Gaussian HJM framework.
Since both procedures give the same valuation results (ref. the next Proposition \ref{musp1331})), we conclude that the specification of the risk premium is not relevant in the context of arbitrage valuation of interest rate-sensitive derivatives in the Gaussian HJM framework. We now consider Theorems \ref{binch5p14} and \ref{binch5p11}, where the dynamics under \(P\) of the instantaneous forward rate \(f_{0}(t,T)\) are
\[f_{0}(t,T)=f(0,T)+\int_{0}^{t}\alpha_{0}(s,T)ds+\int_{0}^{t}\boldsymbol{\sigma}(s,T)\cdot d{\bf W}_{s}\]
and the bond price
\[p_{0}(t,T)=\exp\left (-\int_{t}^{T}f_{0}(t,s)ds\right )\]
from (\ref{museq114}). For any collection of maturities \(0<T_{1}<T_{2}<\cdots <T_{k}=T^{*}\), we write \({\bf T}\) to denote the vector \((T_{1},\cdots ,T_{k})\); similarly, \({\bf p}(t,{\bf T})\) stands for the \(\mathbb{R}^{k}\)-valued process \((p(t,T_{1}),\cdots p(t,T_{k}))\). Let us put
\[{\bf Z}(t,{\bf T})=\frac{{\bf p}(t,{\bf T})}{B_{t}}\mbox{ and }{\bf Z}_{0}(t,{\bf T})=\frac{{\bf p}_{0}(t,{\bf T})}{B_{t}}.\]
Then we have the following proposition
\begin{equation}{\label{musp1331}}\tag{83}\mbox{}\end{equation}
Proposition \ref{musp1331}. Suppose that the coefficient \(\boldsymbol{\sigma}\) is deterministic. Then for any choice \({\bf T}\) of maturity dates and of a spot martingale measure \(\mathbb{P}^{*}\), the probability law of the process \({\bf Z}(t,{\bf T})\) for \(t\in [0,T^{*}]\) under the martingale measure \(\mathbb{P}^{*}\) coincides with the probability law of the process \({\bf Z}_{0}(t,{\bf T})\) for \(t\in [0,T^{*}]\) under \(\mathbb{P}\).
Proof. The assertion follows from Girsanov’s theorem. Indeed, for any fixed \(0<T\leq T^{*}\), the dynamics of \({\bf Z}(t,{\bf T})\) under a spot martingale measure \(\mathbb{P}^{*}\equiv \mathbb{P}^{\scriptsize (\boldsymbol{\lambda})}\) are (ref. (\ref{binch5eq16}) for \({\bf Z}(t,{\bf T})\) being a martingale)
\[dZ(t,T)=Z(t,T)\cdot {\bf S}(t,T)\cdot d{\bf W}_{t}^{(\boldsymbol{\lambda})},\]
where \({\bf W}_{t}^{(\boldsymbol{\lambda})}\) follows a standard Brownian motion under \(P^{\scriptsize (\boldsymbol{\lambda})}\). On the other hand, under \(\mathbb{P}\) we have
\[dZ_{0}(t,T)=Z_{0}(t,T)\cdot {\bf S}(t,T)\cdot d{\bf W}_{t}.\]
Therefore we have
\begin{equation}{\label{museq1339}}\tag{84}
Z(t,T)=Z(0,T)+\int_{t}^{T}Z(t,s){\bf S}(t,s)\cdot d{\bf W}_{s}^{(\boldsymbol{\lambda})}
\end{equation}
and
\begin{equation}{\label{museq1340}}\tag{85}
Z_{0}(t,T)=Z_{0}(0,T)+\int_{t}^{T}Z_{0}(t,s){\bf S}(t,s)\cdot d{\bf W}_{s}.
\end{equation}
Moreover, for every \(0<T\leq T^{*}\)
\[Z(0,T)=p(0,T)=\exp\left (-\int_{0}^{T}f(0,s)ds\right )=p_{0}(0,T)=Z_{0}(0,T).\]
Since \(\boldsymbol{\sigma}\) is deterministic, the assertion follows from (\ref{museq1339}) and (\ref{museq1340}). \(\blacksquare\)
It seems plausible that \(\boldsymbol{\sigma}\) depends explicitly on the level of the foreard rate. This corresponds to the following forward interest rate dynamics
\begin{equation}{\label{museq1319}}\tag{86}
df(t,T)=\alpha (t,T,f(t,T))dt+\boldsymbol{\sigma}(t,T,f(t,T))\cdot d{\bf W}_{t},
\end{equation}
where the deterministic function \(\boldsymbol{\sigma}\) is sufficiently regular so that (\ref{museq1319}) (with \(\alpha\) given by (\ref{museq1320})) admits a unique strong solution for any fixed maturity \(T\). Recall that the drift coefficients \(\alpha\) satisfies, under the martingale measure \(\mathbb{P}^{*}\)
\begin{equation}{\label{museq1320}}\tag{87}
\alpha (t,T,f(t,T))=\boldsymbol{\sigma}(t,T,f(t,T))\cdot\int_{t}^{T}\boldsymbol{\sigma}(t,s,f(t,s))ds.
\end{equation}
Consequently, under the martingale measure \(\mathbb{P}^{*}\), (\ref{museq1319}) may be written in the following way
\[df(t,T)=\boldsymbol{\sigma}(t,T,f(t,T))\cdot\left (\left (\int_{t}^{T}\boldsymbol{\sigma}(t,s,f(t,s))ds\right )dt+
d{\bf W}_{t}^{*}\right ).\]
To show the independence of the arbitrage pricing of the market prices for risk (i.e. \(\boldsymbol{\lambda}\)) in a slightly more general setting, it is convenient to make use of the savings account. Since
\[p(t,T)=\exp\left (-\int_{t}^{T}f(t,s)ds\right )\mbox{ for all }t\in [0,T],\]
and
\[B_{t}=\exp\left (\int_{0}^{t}f(s,s)ds\right )\mbox{ for all }t\in [0,T^{*}],\]
it is clear that the joint probability distribution of processes \(p(\cdot ,T)\) and \(B\) is uniquely determined under \(\mathbb{P}^{*}\), and thus the arbitrage price of any attainable European claim \(X\) depending on short-term rate and bond prices (or on the forward rates), which equals
\[\Pi_{t}(X)=B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{X}{B_{t}}\right |{\cal F}_{t}\right ],\]
is obviously independent of the market prices for risk. This does not mean that the market prices for risk are neglected altogether in the HJM approach. They are still present in the specification of the actual bond price fluctuations. However, in contrast to traditional models of the short-term rate, in which the dynamics of the short-term rate and bond processes are not jointly specified, the HJM methodology assumes a simultaneous influence of market prices for risk on the dynamics of the short-term rate and all bond prices. Consequently in this case, the market prices for risk drop out altogether from arbitrage values of interest rate-sensitive derivatives.
By the risk-neutral valuation principle, we know that we can evaluate attainable contingent claim \(X\), computing discounted expectation with respect to equivalent martingale measures. Now consider the special case \(X=r_{T}\). Using Corollary \ref{binc7}, we can relate the forward risk-neutral (using \(\mathbb{P}_{T}\) as measure with numeraire \(p(t,T)\)) pricing formula with the risk-neutral pricing formula (using \(\mathbb{P}^{*}\) as measure with numeraire \(B_{t}\)) and get
\[B_{t}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{r_{T}}{B_{T}}\right |{\cal F}_{t}
\right ]=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [\left .\frac{r_{T}}{p(T,T)}\right |{\cal F}_{t}\right ],\]
that is,
\begin{equation}{\label{binch5eq27}}\tag{88}
$latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .r_{T}\cdot e^{-\int_{t}^{T}r_{s}ds}\right |
{\cal F}_{t}\right ]=p(t,T)\cdot \mathbb{E}_{P_{T}}[r_{T}|{\cal F}_{t}].
\end{equation}
If \(r_{T}=1\), then, from (\ref{binch5eq27}), we have
\begin{equation}{\label{binch5eq29}}\tag{89}
\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .e^{-\int_{t}^{T}r_{s}ds}\right |{\cal F}_{t}\right ]=p(t,T).
\end{equation}
This implies
\begin{align*}
\mathbb{E}_{\scriptsize \mathbb{P}^{*}}[r_{T}|{\cal F}_{t}] & =-\frac{1}{p(t,T)}\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [
\left .\frac{\partial}{\partial T}e^{-\int_{t}^{T}r_{s}ds}\right |{\cal F}_{t}\right ]\\
& =-\frac{1}{p(t,T)}\cdot\frac{\partial}{\partial T}\mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [
\left .e^{-\int_{t}^{T}r_{s}ds}\right |{\cal F}_{t}\right ]\\
& =-\frac{p_{T}(t,T)}{p(t,T)}\mbox{ (by equation (\ref{binch5eq29}))}\\
& =f(t,T).
\end{align*}
Since \(r_{T}=f(T,T)\), we see that the forward rate is a martingale under \(\scriptsize \mathbb{P}^{*}\).
We now give a brief indication about the question of market completeness for the bond market. In contrast to previous models we now have (theoretically) a continum of traded assets (one for each time of maturity \(T\)). Rather than going into detail on how to resolve the problems arising from the infinite number of basic securities, we will try to generate the finite situation familiar to us. We would like to choose “basic maturities” \(0<T_{1}<T_{2}<\cdots <T_{d}\), and use the corresponding bond price processes as basic undelying processes. We have to investigate whether there exist maturities such that the market is complete with respect to trading strategies involving only these bonds and the savings account. This question is linked to uniqueness of the martingale measure.
Proposition. The following statements are equivalent.
(a) The risk-neutral martingale measure is unique.
(b) There exist maturities \(T_{1},\cdots ,T_{d}\) such that the matrix \(D(t;T_{1},\cdots ,T_{d})_{ij}=\{\sigma_{i}(t,T_{j})\}\) is nonsingular.
(c) There exist maturities \(T_{1},\cdots ,T_{d}\) such that the matrix \(H(t;T_{1},\cdots ,T_{d})_{ij}=\{S_{i}(t,T_{j})\}\) is nonsingular. \(\sharp\)
\begin{equation}{\label{binch5p24}}\tag{90}\mbox{}\end{equation}
Proposition \ref{binch5p24}. Assume that the functions \(\sigma_{1}(t,T),\cdots ,\sigma_{d}(t,T)\) are deterministic are real-analytic in the \(t\) and \(T\) variables, and furthermore, for each \(t\) are independent as functions of \(T\). Then it is possible to choose maturities \(T_{1},\cdots ,T_{d}\) such that the matrix \(\{S_{i}(t,T_{j})\}^{d}_{ij}\) is nonsingular. Apart from a finite set of “forbidden’ points these volatilities can be chosen freely as long as they are distinct. \(\sharp\)
Theorem. Under the conditions of Proposition \ref{binch5p24} and with respect to bond trading strategies involving only bonds of maturities \(T_{1},\cdots , T_{d}\) described by Proposition \ref{binch5p24} (and the savings account) the bond market is complete. \(\sharp\)
