Johan Mari Henri Ten Kate (1831-1910) was a Dutch painter.
Option Valuation in Gaussian Models.
In this section, the forward measure methodology is employed in arbitrage pricing of interest rate derivative securities in a Gaussian framework. By a Gaussian framework we mean any model of the term structure, either based on the short-term rates or on forward rates, in which all bond price volatilities (as well as the volatility of any other underlying asset) follow deterministic functions.
European Spot Options.
The explicit valuation of European options is to observe that Proposition \ref{musl1323} provides a simple formula which expresses the price of a European call option written on a tradable asset, \(Z\) say, in terms of the forward price process \(F_{Z}(t,T)\) (ref. Propositions \ref{musl1321}) and \ref{musp1322}) and the forward probability measure \(P_{T}\). Indeed, we have for every \(t\in [0,T]\)
\begin{equation}{\label{museq151}}\tag{133}
\Pi_{t}((Z_{T}-K)^{+})=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [(F_{Z}(T,T)-K)^{+}|{\cal F}_{t}\right ]
\end{equation}
since manifestly \(Z_{T}=F_{Z}(T,T)\).
\begin{equation}{\label{musl1511}}\tag{134}\mbox{}\end{equation}
Proposition \ref{musl1511}. For any fixed \(T>0\), the process \({\bf W}^{T}\) given by the formula
\[{\bf W}^{T}_{t}={\bf W}_{t}^{*}-\int_{0}^{t}{\bf b}(s,T)ds\mbox{ for all }t\in [0,T]\]
follows a standard \(d\)-dimensional Brownian motion under the forward measure \(\mathbb{P}_{T}\). The forward price process for the settlement date \(T\) of a zero-coupon bond which matures at time \(U\) satisfies
\begin{equation}{\label{museq152}}\tag{135}
dF_{p}(t,U,T)=F_{p}(t,U,T)({\bf b}(t,U)-{\bf b}(t,T))\cdot d{\bf W}^{T}_{t}
\end{equation}
subject to the terminal condition \(F_{p}(T,U,T)=p(T,U)\). The forward price of a stock \(S\) satisfies \(F_{S}(T,T)=S_{T}\) and
\[dF_{S}(t,T)=F_{S}(t,T)(\boldsymbol{\sigma}_{t}-{\bf b}(t,T))\cdot d{\bf W}^{T}_{t}. \sharp\]
The next result, which uses the HJM framework, shows that the yield-to-maturity expectation hypothesis is satisfied for any fixed maturity \(T\) under the corresponding forward probability measure \(\mathbb{P}_{T}\). For every maturity \(T\), the instantaneous forward rate \(f(0,T)\) is an unbiased estimate, under the actual probability \(P\), of the future short-term rate \(r_{T}\).
Proposition. For any fixed \(T\in [0,T^{*}]\), the forward rate \(f(0,T)\) is equal to the expected value of the spot rate \(r_{T}\) under the forward probability measure \(\mathbb{P}_{T}\). That is, \(\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[r_{T}]=f(0,T)\). \(\sharp\)
Bond Options.
At expiry date \(T\), the payoff of a European call option written on a zero-coupon bond which matures at time \(U\geq T\) equals
\[C_{T}=(p(T,U)-K)^{+}.\]
Since \(p(T,U)=F_{p}(T,U,T)\), the payoff \(C_{T}\) can alternatively be expressed in the following way
\[C_{T}=(F_{p}(T,U,T)-K)^{+}=F_{p}(T,U,T)\cdot I_{D}-K\cdot I_{D},\]
where \(D=\{p(T,U)>K\}=\{F_{p}(T,U,T)>K\}\) is the exercise set. We assume that the volatilities are bounded.
\begin{equation}{\label{musp1511}}\tag{136}\mbox{}\end{equation}
Proposition \ref{musp1511}. Assume that the bond price volatilities \({\bf b}(\cdot ,T)\) and \({\bf b}(\cdot ,U)\) are bounded deterministic functions. The arbitrage price at time \(t\in [0,T]\) of a European call option with expiry date \(T\) and strike price \(K\), written on a zero-coupon bond which matures at time \(U\geq T\), equals
\begin{equation}{\label{museq154}}\tag{137}
C_{t}=p(t,U)\cdot N\left (h_{1}(p(t,U),t,T)\right )-K\cdot p(t,T)\cdot N\left (h_{2}(p(t,U),t,T)\right ),
\end{equation}
where
\[h_{1,2}(q,t,T)=\frac{\ln q-\ln p(t,T)-\ln K\pm\frac{1}{2}v_{U}^{2}(t,T)}{v_{U}(t,T)}\]
for \((q,t)\in \mathbb{R}_{+}\times [0,T]\), and
\begin{equation}{\label{museq156}}\tag{138}
v_{U}^{2}(t,T)=\int_{t}^{T}\parallel {\bf b}(s,U)-{\bf b}(s,T)\parallel^{2}ds\mbox{ for all }t\in [0,T].
\end{equation}
The arbitrage price of the corresponding European put option written on a zero-coupon bond equals
\[P_{t}=K\cdot p(t,U)\cdot N\left (-h_{2}(p(t,U),t,T)\right )-p(t,T)\cdot N\left (-h_{1}(p(t,U),t,T)\right ).\]
Proof. In view of the general valuation formula (\ref{museq151}), it is clear that we have to evaluate the conditional expectations
\[C_{t}=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [F_{p}(T,U,T)\cdot I_{D}|{\cal F}_{t}
\right ]-K\cdot p(t,T)\cdot P_{T}\{D|{\cal F}_{t}\}=I_{1}-I_{2}.\]
We know from Proposition \ref{musl1511} that the dynamics of \(F_{p}(t,U,T)\) under \(\mathbb{P}_{T}\) are given by formula (\ref{museq152}) so that
\begin{equation}{\label{museq73}}\tag{139}
F_{p}(T,U,T)=F_{p}(t,U,T)\cdot\exp\left (\int_{t}^{T}\boldsymbol{\gamma}(s,U,T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}(s,U,T)\parallel^{2}ds\right ),
\end{equation}
where \(\boldsymbol{\gamma}(s,U,T)={\bf b}(s,U)-{\bf b}(s,T)\). This can be rewritten as follows
\[F_{p}(T,U,T)=F_{p}(t,U,T)\cdot\exp\left (\zeta (t,T)-\frac{1}{2}v_{U}^{2}(t,T)\right ),\]
where \(F_{p}(t,U,T)\) is \({\cal F}_{t}\)-measurable, and \(\zeta (t,T)=\int_{t}^{T}\boldsymbol{\gamma}(s,U,T)\cdot d{\bf W}_{s}^{T}\) is, under \(\mathbb{P}_{T}\), a real-valued Gaussian random variable, independent of the \(\sigma\)-field \({\cal F}_{t}\), with zero
expected value and the variance \(\mbox{Var}_{\scriptsize\mathbb{P}_{T}}(\zeta (t,T))=v_{U}^{2}(t,T)\). Let \(\eta (t,T)=-\zeta (t,T)\).
Then \(\eta (t,T)\) is also a Gaussian random variable with zero mean and variance \(v_{U}^{2}(t,T)\). Now we have
\begin{align*}
\mathbb{P}_{T}\{D|{\cal F}_{t}\} & =\mathbb{P}_{T}\left\{\left .F_{p}(t,U,T)\cdot\exp\left (\zeta (t,T)-
\frac{1}{2}v_{U}^{2}(t,T)\right )>K\right |{\cal F}_{t}\right\}\\
& =\mathbb{P}_{T}\left\{F_{p}(t,U,T)\cdot\exp\left (\zeta (t,T)-\frac{1}{2}v_{U}^{2}(t,T)\right )>K\right\}\\
& =\mathbb{P}_{T}\left\{-\zeta (t,T)<\ln F_{p}(t,U,T)-\ln K-\frac{1}{2}v_{U}^{2}(t,T)\right\}\\
& =\mathbb{P}_{T}\left\{\eta (t,T)<\ln F_{p}(t,U,T)-\ln K-\frac{1}{2}v_{U}^{2}(t,T)\right\}
\end{align*}
so that
\begin{equation}{\label{museq76}}\tag{140}
I_{2}=K\cdot p(t,T)\cdot N\left (\frac{\ln F_{p}(t,U,T)-\ln K-\frac{1}{2}v_{U}^{2}(t,T)}{v_{U}(t,T)}\right ).
\end{equation}
To evaluate \(I_{1}\), we introduce an auxiliary probability measure \(\tilde{\mathbb{P}}_{T}\sim\mathbb{P}_{T}\) on \((\Omega ,{\cal F}_{T})\) by setting
\begin{equation}{\label{museq75}}\tag{141}
\frac{\tilde{\mathbb{P}}_{T}}{\mathbb{P}_{T}}=\exp\left (\int_{0}^{T}\boldsymbol{\gamma}(s,U,T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}(s,U,T)\parallel^{2}ds\right )\equiv\tilde{\xi}_{t}.
\end{equation}
By Girsanov’s theorem, it is clear that the process \(\tilde{{\bf W}}^{T}\), which equals
\begin{equation}{\label{museq74}}\tag{142}
\tilde{{\bf W}}_{t}^{T}={\bf W}^{T}-\int_{0}^{t}\boldsymbol{\gamma}(s,U,T)ds\mbox{ for all }t\in [0,T],
\end{equation}
follows a standard Brownian motion under \(\tilde{\mathbb{P}}_{T}\). Note also that the forward price \(F_{p}(T,U,T)\) admits the following representation under \(\tilde{\mathbb{P}}_{T}\) from (\ref{museq73}) and (\ref{museq74})
\[F_{p}(T,U,T)=F_{p}(t,U,T)\cdot\exp\left (\int_{t}^{T}\boldsymbol{\gamma}(s,U,T)\cdot d\tilde{{\bf W}}_{s}^{T}+\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}(s,U,T)\parallel^{2}ds\right )\]
so that
\begin{equation}{\label{museq157}}\tag{143}
F_{p}(T,U,T)=F_{p}(t,U,T)\cdot\exp\left (\tilde{\zeta}(t,T)+\frac{1}{2}v_{U}^{2}(t,T)\right ),
\end{equation}
where we denote \(\tilde{\zeta}(t,T)=\int_{t}^{T}\boldsymbol{\gamma}(s,U,T)\cdot d\tilde{{\bf W}}_{s}^{T}\). The random variable \(\tilde{\zeta}(t,T)\) has, under \(\tilde{\mathbb{P}}_{T}\), a Gaussian law with zero mean and variance \(v_{U}^{2}(t,T)\), and it is also independent of the \(\sigma\)-field \({\cal F}_{t}\). From (\ref{museq73}) and using \(F_{p}(t,U,T)=p(t,U)/p(t,T)\), we have
\[I_{1}=p(t,U)\cdot \mathbb{E}_{P_{T}}\left [\left .I_{D}\cdot\exp\left (\int_{t}^{T}
\boldsymbol{\gamma}(s,U,T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}(s,U,T)\parallel^{2}ds\right )\right |{\cal F}_{t}\right ],\]
that is, from (\ref{museq75}),
\[I_{1}=p(t,U)\cdot \mathbb{E}_{P_{T}}\left [\left .\frac{\tilde{\xi}_{T}}{\tilde{\xi}_{t}}\cdot I_{D}\right |{\cal F}_{t}\right ].\]
From the proof of Proposition \ref{musp1322}, we see that \(\tilde{\xi}_{t}=\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[\tilde{\xi}_{T}|{\cal F}_{t}]\). Using the Bayes rule in Proposition \ref{musa04}, we have
\[\mathbb{E}_{\scriptsize \tilde{\mathbb{P}}_{T}}[I_{D}|{\cal F}_{t}]=\frac{\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[\tilde{\xi}_{T}\cdot I_{D}|{\cal F}_{t}]}{\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[\tilde{\xi}_{T}|{\cal F}_{t}]}=
\frac{\mathbb{E}_{\scriptsize \mathbb{P}_{T}}[\tilde{\xi}_{T}\cdot I_{D}|{\cal F}_{t}]}{\tilde{\xi}_{t}}=
\mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [\left .\frac{\tilde{\xi}_{T}}{\tilde{\xi}_{t}}\cdot I_{D}\right |{\cal F}_{t}\right ].\]
That is, \(I_{1}=p(t,U)\cdot\tilde{\mathbb{P}}_{T}\{D|{\cal F}_{t}\}\). Taking into account (\ref{museq157}), we conclude that
\[\tilde{\mathbb{P}}_{T}\{D|{\cal F}_{t}\}=\tilde{\mathbb{P}}_{T}\left\{-\tilde{\zeta}(t,T)<\ln F_{p}(t,U,T)-\ln K+\frac{1}{2}
v_{U}^{2}(t,T)\right\},\]
and thus
\begin{equation}{\label{museq79}}\tag{144}
I_{1}=p(t,T)\cdot N\left (\frac{\ln F_{p}(t,U,T)-\ln K+\frac{1}{2}v_{U}^{2}(t,T)}{v_{U}(t,T)}\right ).
\end{equation}
This completes the proof of the valuation formula (\ref{museq154}). The formula that gives the price of the put option can be established along the same line. Alternatively, to find the price of a European put option written on a zero-coupon bond, one may combine equality (\ref{museq154}) with the put-call parity relationship (\ref{museq1517}). \(\blacksquare\)
From equations (\ref{museq76}) and (\ref{museq79}), formula (\ref{museq154}) can be rewritten as follows
\begin{equation}{\label{museq158}}\tag{145}
C_{t}=p(t,T)\cdot\left (F_{p}(t,U,T)\cdot N(\tilde{d}_{1}(F_{p}(t,U,T),t,T))-K\cdot N(\tilde{d}_{2}(F_{p}(t,U,T),t,T))\right ),
\end{equation}
where
\begin{equation}{\label{museq159}}\tag{146}
\tilde{d}_{1,2}(q,t,T)=\frac{\ln q-\ln K\pm\frac{1}{2}v_{U}^{2}(t,T)}{v_{U}(t,T)}
\end{equation}
for \((q,t)\in \mathbb{R}_{+}\times [0,T]\), where \(v_{U}(t,T)\) is given by (\ref{museq156}). Note also that we have
\[\frac{\tilde{\mathbb{P}}_{T}}{d\mathbb{P}^{*}}=\frac{d\tilde{\mathbb{P}}_{T}}{d\mathbb{P}_{T}}\cdot
\frac{d\mathbb{P}_{T}}{d\mathbb{P}^{*}}=\exp\left (\int_{0}^{T}{\bf b}(s,U)\cdot
d{\bf W}_{s}^{*}-\frac{1}{2}\int_{0}^{T}\parallel {\bf b}(s,U)\parallel^{2}ds\right ).\]
It is thus apparent that the auxiliary probability measure \(\tilde{\mathbb{P}}_{T}\) is in fact the restriction of the forward measure \(\mathbb{P}_{U}\) to the \(\sigma\)-field \({\cal F}_{T}\). Since the exercise set \(D\) belongs to the \(\sigma\)-field \({\cal F}_{T}\), we have \(\tilde{\mathbb{P}}_{T}\{D|{\cal F}_{t}\}=\mathbb{P}_{U}\{D|{\cal F}_{t}\}\). Therefore, formula (\ref{museq154}) admits the following alternative representation
\[C_{t}=p(t,T)\cdot\mathbb{P}_{U}\{D|{\cal F}_{t}\}-K\cdot p(t,T)\cdot\mathbb{P}_{T}\{D|{\cal F}_{t}\}.\]
Stock Options.
The payoff at expiry of a European call option written on a stock \(S\) equals \(C_{T}=(S_{T}-K)^{+}\), where \(T\) is the expiry date and \(K\) denotes the strike price. We assume that the dynamics of \(S\) under the martingale measure \(\mathbb{P}^{*}\) \(dS_{t}=S_{t}(r_{t}dt+\sigma_{t}dW_{t}^{*})\), where \(\sigma :[0,T^{*}]\rightarrow \mathbb{R}\) is a deterministic function.
\begin{equation}{\label{musp1512}}\tag{147}\mbox{}\end{equation}
Proposition \ref{musp1512}. Assume that the bond price volatility \(b(\cdot ,T)\) and the stock price volatility \(\sigma\) are bounded deterministic functions. Then the arbitrage price of a European call option with expiry date \(T\) and exercise price \(K\), written on a stock \(S\), equals
\begin{equation}{\label{museq1510}}\tag{148}
C_{t}=S_{t}\cdot N(h_{1}(S_{t},t,T))-K\cdot p(t,T)\cdot N(h_{2}(S_{t},t,T)),
\end{equation}
where
\begin{equation}{\label{museq1511}}\tag{149}
h_{1,2}(q,t,T)=\frac{\ln q-\ln K-\ln p(t,T)\pm\frac{1}{2}v^{2}(t,T)}{v(t,T)}
\end{equation}
for \((q,t)\in \mathbb{R}_{+}\times [0,T]\), and
\[v^{2}(t,T)=\int_{t}^{T}|\sigma_{s}-b(s,T)|^{2}ds\mbox{ for all }t\in [0,T].\]
Proof. The proof goes along the same lines as the proof of Proposition \ref{musp1511}, therefore we merely sketch its main steps. It is clear that
\[C_{T}=(F_{S}(T,T)-K)^{+}=F_{S}(T,T)\cdot I_{D}-K\cdot I_{D},\]
where \(D=\{S_{T}>K\}=\{F_{S}(T,T)>K\}\). Therefore, it is enough to evaluate the conditional expectations
\[C_{t}=p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [F_{S}(T,T)\cdot I_{D}|{\cal F}_{t}
\right ]-K\cdot p(t,T)\cdot\mathbb{P}_{T}\{D|{\cal F}_{t}\}=I_{1}-I_{2},\]
where \(F_{S}(T,T)\) is given by the formula
\[F_{S}(T,T)=F_{S}(t,T)\cdot\exp \left (\int_{t}^{T}\gamma_{S}(s,T)
dW_{s}^{T}-\frac{1}{2}\int_{t}^{T}|\gamma_{S}(s,T)|^{2}ds\right )\]
and \(\gamma_{S}(s,T)=\sigma_{s}-b(s,T)\). Proceeding as for the proof of Proposition \ref{musp1511}, one finds that
\[I_{2}=K\cdot p(t,T)\cdot N\left (\frac{\ln S_{t}-\ln K-\ln p(t,T)-\frac{1}{2}v^{2}(t,T)}{v(t,T)}\right ).\]
We now define an auxiliary probability measure \(\widehat{\mathbb{P}}_{T}\) by setting
\[\frac{\widehat{\mathbb{P}}_{T}}{\mathbb{P}_{T}}=\exp\left (\int_{0}^{T}\gamma_{S}(s,T)
dW_{s}^{T}-\frac{1}{2}\int_{0}^{T}|\gamma_{S}(s,T)|^{2}ds\right ).\]
Then the process \(\tilde{W}^{T}\) given by the formula
\[\tilde{W}_{t}^{T}=W_{t}^{T}-\int_{0}^{t}\gamma_{S}(s,T)ds\mbox{ for all }t\in [0,T]\]
follows the standard Brownian motion under \(\widehat{\mathbb{P}}_{T}\). Furthermore, the forward price \(F_{S}(T,T)\) satisfies
\[F_{S}(T,T)=\frac{S_{t}}{p(t,T)}\cdot\exp\left (\int_{t}^{T}\gamma_{S}(s,T)
d\tilde{W}_{s}^{T}+\frac{1}{2}\int_{t}^{T}|\gamma_{S}(s,T)|^{2}ds\right )\mbox{ for all }t\in [0,T].\]
Since we have
\[I_{1}=S_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [\left .I_{D}\cdot\exp\left (\int_{t}^{T}
\gamma_{S}(s,T)dW_{s}^{T}-\frac{1}{2}\int_{t}^{T}|\gamma_{S}(s,T)|^{2}ds\right )\right |{\cal F}_{t}\right ],\]
from the Bayes rule, we get \(I_{1}=S_{t}\cdot\widehat{\mathbb{P}}_{T}\{D|{\cal F}_{t}\}\). Consequently, we obtain
\[I_{1}=S_{t}\cdot N\left (\frac{\ln S_{t}-\ln K-\ln p(t,T)+\frac{1}{2}v^{2}(t,T)}{v(t,T)}\right ).\]
This completes the proof. \(\blacksquare\)
Example. Let us examine a very special case of the pricing formula established in Proposition \ref{musp1512}. Let \({\bf W}^{*}=(W^{1*},W^{2*})\) be a two-dimensional standard Brownian motion given on a probability space \((\Omega ,{\cal F},\mathbb{P}^{*})\). We assume that the bond price \(p(t,T)\) satisfies, under \(\mathbb{P}^{*}\),
\[dp(t,T)=p(t,T)(r_{t}dt+\hat{b}(t,T)(\rho ,\sqrt{1-\rho^{2}})\cdot d{\bf W}_{t}^{*}),\]
where \(\hat{b}(t,T):[0,T]\rightarrow \mathbb{R}\) is areal-valued, bounded deterministic function, and the dynamics of the stock price \(S\) are
\[dS_{t}=S_{t}(r_{t}dt+(\hat{\sigma}(t),0)\cdot d{\bf W}_{t}^{*})\]
for some function \(\hat{\sigma}:[0,T^{*}]\rightarrow \mathbb{R}\). Let us introduce the real-valued stochastic processes \(\widehat{W}^{1}\) and \(\widehat{W}^{2}\) by setting \(\widehat{W}^{1}_{t}=W_{t}^{1*}\) and \(\widehat{W}^{2}=\rho W_{t}^{1*}+\sqrt{1-\rho^{2}}W_{t}^{2*}\). It is not hard to check that \(\widehat{W}^{1}\) and \(\widehat{W}^{2}\) follow standard one-dimensional Brownian motions under the martingale measure \(\mathbb{P}^{*}\), and their cross-variation equals \(\langle\widehat{W}^{1},\widehat{W}^{2}\rangle=\rho t\) for \(t\in [0,T^{*}]\). It is evident that
\begin{equation}{\label{museq1513}}\tag{150}
dp(t,T)=p(t,T)(r_{t}dt+\hat{b}(t,T)d\widehat{W}^{2}_{t})\mbox{ and }
dS_{t}=S_{t}(r_{t}dt+\hat{\sigma}(t)d\widehat{W}^{1}_{t}).
\end{equation}
An application of Proposition \ref{musp1512} yields the following result. Assume that the dynamics of a bond and a stock price are given by
(\ref{museq1513}). If the volatility coefficients \(\hat{b}\) and \(\hat{\sigma}\) are deterministic functions, then the arbitrage price of a European call option written on a stock is given by (\ref{museq1510}) and (\ref{museq1511}) with
\[v^{2}(t,T)=\int_{t}^{T}(\hat{\sigma}^{2}(s)-2\rho\cdot\hat{\sigma}(s)\cdot\hat{b}(s,T)+\hat{b}^{2}(s,T))ds. \sharp\]}
The dynamics of the spot price \(Z\) of a tradable asset are assumed to be given by the expression
\begin{equation}{\label{museq83}}\tag{151}
dZ_{t}=Z_{t}(r_{t}dt+\boldsymbol{\xi}_{t}\cdot d{\bf W}_{t}^{*}).
\end{equation}
It is essential to assume that the volatility \(\boldsymbol{\xi}_{t}-{\bf b}(t,T)\) of the forward price of \(Z\) for the settlement date \(T\) is deterministic.
Proposition. The arbitrage price of a European call option with expiry date \(T\) and exercise price \(K\), written on an asset \(Z\), is given by the expression
\[C_{t}=p(t,T)\cdot\left (F_{Z}(t,T)\cdot N(\tilde{d}_{1}(F_{Z}(t,T),t,T))-K\cdot N(\tilde{d}_{1}(F_{Z}(t,T),t,T))\right ),\]
where
\begin{equation}{\label{museq1515}}\tag{152}
\tilde{d}_{1,2}(q,t,T)=\frac{\ln q-\ln K\pm\frac{1}{2}v^{2}(t,T)}{v(t,T)}
\end{equation}
for \((q,t)\in \mathbb{R}_{+}\times [0,T]\), and
\begin{equation}{\label{museq1516}}\tag{153}
v^{2}(t,T)=\int_{t}^{T}\parallel\boldsymbol{\xi}_{s}-{\bf b}(s,T)\parallel^{2}ds
\end{equation}
for all \(t\in [0,T]\). \(\sharp\)
Let \(P_{t}\) stand for the price at time \(t\leq T\) of a European put option written on a asset \(Z\) with expiry date \(T\) and strike price \(K\).
\begin{equation}{\label{musc1513}}\tag{154}\mbox{}\end{equation}
Proposition \ref{musc1513}. The following put-call parity relationship is valid
\begin{equation}{\label{museq1517}}\tag{155}
C_{t}-P_{t}=Z_{t}-p(t,T)\cdot K
\end{equation}
for all \(t\in [0,T]\).
Proof. We see that \(C_{T}-P_{T}=Z_{T}-K=F_{Z}(T,T)-K\). From Proposition \ref{musl1323} or (\ref{museq151}), we have
\[C_{t}-P_{t}=p(t,T)\cdot \mathbb{E}_{P_{T}}\left [F_{Z}(T,T)-K|{\cal F}_{t}\right ],\]
Since \(F_{Z}(t,T)=Z_{t}/p(t,T)\) in (\ref{museq70}), we obtain
\[C_{t}-P_{t}=p(t,T)\cdot F_{Z}(t,T)-p(t,T)\cdot K=Z_{t}-p(t,T)\cdot K\]
for every \(t\in [0,T]\). \(\blacksquare\)
Option on a Coupon-Bearing Bond.
The aim is to value a European option written on a coupon-bearing bond. For a given selection of dates \(T_{1}<\cdots <T_{m}\leq T^{*}\), we consider a coupon-bearing bond whose value \(Z_{t}\) at time \(t\leq T_{1}\) is
\[Z_{t}=\sum_{j=1}^{m}c_{j}\cdot p(t,T_{j})\mbox{ for all }t\in [0,T_{1}],\]
where \(c_{j}\) are real numbers. We consider a European call option with expiry date \(T\leq T_{1}\), whose payoff at expiry has the following form
\[C_{T}=(Z_{T}-K)^{+}=\left (\sum_{j=1}^{m}c_{j}\cdot p(T,T_{j})-K\right )^{+}.\]
\begin{equation}{\label{musp1514}}\tag{156}\mbox{}\end{equation}
Proposition \ref{musp1514}. The arbitrage price of a European call option on a coupon-bearing bond is given by the formula
\[C_{t}=\sum_{j=1}^{m}c_{j}\cdot p(t,T_{j})\cdot J_{1}^{j}-K\cdot p(t,T)\cdot J_{2},\]
where
\[J_{1}^{j}=P_{T}\left\{\sum_{i=1}^{m}c_{i}p(t,T_{i})\cdot\exp (
\zeta_{i}+v_{ij}-v_{ii}/2)>K\cdot p(t,T)\right\}\mbox{ for }j=1,\cdots ,m,\]
\[J_{2}=P_{T}\left\{\sum_{i=1}^{m}c_{i}p(t,T_{i})\cdot\exp (\zeta_{i}-v_{ii}/2)>K\cdot p(t,T)\right\},\]
and \((\zeta_{1},\cdots ,\zeta_{m})\) is a vector of random variables whose law under \(\mathbb{P}_{T}\) is Gaussian with zero mean, and which has the following variance-covariance matrix
\[\mbox{Cov}_{\scriptsize \mathbb{P}_{T}}(\zeta_{i},\zeta_{j})\equiv v_{ij}=\int_{t}^{T}\boldsymbol{\gamma}(s,T_{i},T)\cdot
\boldsymbol{\gamma}(s,T_{j},T)ds\mbox{ for }i,j=1,\cdots ,m,\]
where \(\boldsymbol{\gamma}(s,T_{i},T)={\bf b}(t,T_{i})-{\bf b}(t,T)\).
Proof. We need to evaluate the conditional expectation
\[C_{t}=p(t,T)\cdot\sum_{j=1}^{m}c_{j}\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}[F_{p}(T,T_{j},T)
\cdot I_{D}|{\cal F}_{t}]-K\cdot p(t,T)\cdot\mathbb{P}_{T}\{D|{\cal F}_{t}\}=I_{1}-I_{2},\]
where \(D\) stand for the exercise set
\[D=\left\{\sum_{j=1}^{m}c_{j}\cdot p(T,T_{j})>K\right\}=
\left\{\sum_{j=1}^{m}c_{j}\cdot F_{p}(T,T_{j},T)>K\right\}.\]
Let us first examine the conditional probability \(\mathbb{P}_{T}\{D|{\cal F}_{t}\}\). By the virtue of Proposition \ref{musl1511}, the process \(F_{p}(t,T_{i},T)\) satisfies
\[F_{p}(T,T_{i},T)=F_{p}(t,T_{i},T)\cdot\exp\left (\int_{t}^{T}
\boldsymbol{\gamma}(s,T_{i},T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}(s,T_{i},T)\parallel^{2}ds\right ),\]
where \(\boldsymbol{\gamma}(s,T_{i},T)={\bf b}(s,T_{i})-{\bf b}(s,T)\). In other words,
\[F_{p}(T,T_{i},T)=F_{p}(t,T_{i},T)\cdot\exp (\xi_{i}^{T}-v_{ii}/2),\]
where \(\xi_{i}^{T}\) is a random variable independent of the \(\sigma\)-field \({\cal F}_{t}\), and such that the probability law of \(\xi_{i}^{T}\) under \(\mathbb{P}_{T}\) is the Gaussian law \(N(0,v_{ii})\). Therefore,
\[\mathbb{P}_{T}\{D|{\cal F}_{t}\}=\mathbb{P}_{T}\left\{\sum_{i=1}^{m}c_{i}\cdot p(t,T_{i})
\cdot\exp (\xi_{i}^{T}-v_{ii}/2)>K\cdot p(t,T)\right\}\]
since \(F_{p}(t,T_{i},T)=p(t,T_{i})/p(t,T)\). This proves that \(I_{2}=K\cdot p(t,T)\cdot J_{2}\). Let us show that \(I_{1}=\sum_{j=1}^{m}c_{j}\cdot p(t,T_{j})\cdot J_{1}^{j}\). To this end, it is sufficient to check that for any fixed \(j\) we have
\begin{equation}{\label{museq1518}}\tag{157}
p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}[F_{p}(T,T_{j},T)\cdot I_{D}|{\cal F}_{t}]=p(t,T_{j})\cdot J_{1}^{j}.
\end{equation}
This can be done by proceeding in much the same way as in the proof of Proposition \ref{musp1511}. Let us fix \(j\) and introduce an auxiliary probability measure \(\tilde{\mathbb{P}}_{T_{j}}\) on \((\Omega ,{\cal F}_{T})\) by setting
\[\frac{d\tilde{\mathbb{P}}_{T_{j}}}{d\mathbb{P}_{T}}=\exp\left (\int_{0}^{T}
\boldsymbol{\gamma}(s,T_{j},T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{0}^{T}\parallel\boldsymbol{\gamma}(s,T_{j},T)\parallel^{2}ds\right ).\]
Then, the process
\[\tilde{{\bf W}}_{t}^{j}={\bf W}_{t}^{T}-\int_{0}^{t}\boldsymbol{\gamma}(s,T_{j},T)ds\]
follows a standard Brownian motion under \(\tilde{\mathbb{P}}_{T_{j}}\). Recall that \(\tilde{\mathbb{P}}_{T_{j}}=\mathbb{P}_{T_{j}}\) on \({\cal F}_{T}\), hence we write simply \(\mathbb{P}_{T_{j}}\) in place of \(\tilde{\mathbb{P}}_{T_{j}}\) in what follows. For any \(i\), the forward price \(F_{p}(t,T_{i},T)\) has the following representation under \(\mathbb{P}_{T_{j}}\)
\begin{equation}{\label{museq1519}}\tag{158}
dF_{p}(t,T_{i},T)=F_{p}(t,T_{i},T)\cdot\boldsymbol{\gamma}(t,T_{j},T)
\cdot (d\tilde{{\bf W}}_{t}^{j}+\boldsymbol{\gamma}(t,T_{j},T)dt).
\end{equation}
For a fixed \(j\), we define the vector of random variables \((\xi_{1},\cdots ,\xi_{m})\) by the formula
\[\xi_{i}=\int_{t}^{T}\boldsymbol{\gamma}(s,T_{i},T)\cdot\tilde{{\bf W}}_{s}^{j}.\]
It is clear that the random vector \((\xi_{1},\cdots ,\xi_{m})\) is independent of \({\cal F}_{t}\) with Gaussian law under \(\mathbb{P}_{T_{j}}\). More precisely, the expected value of each random variable \(\xi_{i}\) is zero, and for every \(k,l=1,\cdots ,m\), we have
\[v_{kl}=Cov_{P_{T_{j}}}(\xi_{k},\xi_{l})=\int_{t}^{T}\boldsymbol{\gamma}(s,T_{k},T)\cdot\boldsymbol{\gamma}(s,T_{l},T)ds.\]
On the other hand, using (\ref{museq1519}), we find
\[F_{p}(T,T_{i},T)=F_{p}(t,T_{i},T)\cdot\exp\left (\xi_{i}-\frac{1}{2}v_{ii}+v_{ij}\right )\mbox{ for }i=1,\cdots ,m.\]
The Bayes formula in Proposition \ref{binch4p1} yields
\[\mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [F_{p}(T,T_{j},T)\cdot I_{D}|{\cal F}_{t}\right ]=
F_{p}(t,T_{j},T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T_{j}}}[I_{D}|{\cal F}_{t}],\]
that is,
\[p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [F_{p}(T,T_{j},T)\cdot I_{D}|{\cal F}_{t}
\right ]=p(t,T_{j})\cdot \mathbb{P}_{T_{j}}\{D|{\cal F}_{t}\}\]
since \(F_{p}(t,T_{j},T)=p(t,T_{j})/p(t,T)\). Furthermore, we have
\[\mathbb{P}_{T_{j}}\{D|{\cal F}_{t}\}=\mathbb{P}_{T_{j}}\left\{\sum_{i=1}^{m}c_{i}\cdot
p(t,T_{i})\exp\left (\xi_{i}-\frac{1}{2}v_{ii}+v_{ij}\right )>K\cdot p(t,T)\right\}.\]
By combining the last two equalities, we arrive at (\ref{museq1518}). \(\blacksquare\)
The following simple result suggests an alternative way to prove Proposition \ref{musp1514}.
Proposition. Let us denote \(D=\{Z_{T}>K\}\). Then, the arbitrage price of a European call option written on a coupon bonc satisfies
\[C_{t}=\sum_{j=1}^{m}c_{j}\cdot p(t,T_{j})\cdot \mathbb{P}_{T_{j}}\{D|{\cal F}_{t}\}-K\cdot p(t,T)\cdot \mathbb{P}_{T}\{D|{\cal F}_{t}\}.\]
Proof. We have
\[Z_{T}=\sum_{j=1}^{m}c_{j}\cdot B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{1}
{B_{T_{j}}}\right |{\cal F}_{t}\right ],\]
and thus
\begin{align*}
C_{t} & =B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{I_{D}}{B_{T}}\cdot
\left (\sum_{j=1}^{m}c_{j}\cdot B_{T}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{1}
{B_{T_{j}}}\right |{\cal F}_{t}\right ]-K\right )\right |{\cal F}_{t}\right ]\\
& =\sum_{j=1}^{m}c_{j}\cdot B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{I_{D}}
{B_{T_{j}}}\right |{\cal F}_{t}\right ]-K\cdot B_{t}\cdot
$latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{I_{D}}{B_{T}}\right |{\cal F}_{t}\right ]
\end{align*}
since \(D\in {\cal F}_{T}\). Using the Bayes formula in Proposition \ref{musa04}, we get for every \(j\)
\begin{align*}
B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{I_{D}}{B_{T_{j}}}\right |{\cal F}_{t}
\right ] & =B_{t}\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T_{j}}}\left [I_{D}|{\cal F}_{t}\right ]
\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}\left [\left .\frac{1}{B_{T_{j}}}\right |{\cal F}_{t}\right ]\\
& =p(t,T_{j})\cdot \mathbb{P}_{T_{j}}\{D|{\cal F}_{t}\}.
\end{align*}
Since a similar relation holds for the last term, this completes the proof. \(\blacksquare\)
Pricing of General Contingent Claims.
Let us consider a European contingent claim \(X\), which settles at time \(T\), of the form \(X=g(Z_{T}^{1},\cdots ,Z_{T}^{n})\), where \(g:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is a bounded Borel measurable function. Assume that the price process \(Z^{i}\) of the \(i\)th asset satisfies, under \(\mathbb{P}^{*}\),
\[dZ_{t}^{i}=Z_{t}^{i}(r_{t}dt+\boldsymbol{\xi}_{t}^{i}\cdot d{\bf W}_{t}^{*}.\]
Then
\[F_{Z^{i}}(T,T)=F_{Z^{i}}(t,T)\cdot\exp\left (\int_{t}^{T}\boldsymbol{\gamma}_{i}(s,T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}_{i}(s,T)\parallel^{2}ds\right ),\]
where \(\boldsymbol{\gamma}_{i}(s,T)=\boldsymbol{\xi}^{i}_{s}-{\bf b}(s,T)\), or in short
\[F_{Z^{i}}(T,T)=F_{Z^{i}}(t,T)\cdot\exp\left (\zeta_{i}(t,T)-\frac{1}{2}\gamma_{ii}\right ),\]
where
\[\zeta_{i}(t,T)=\int_{t}^{T}\boldsymbol{\gamma}_{i}(s,T)\cdot
d{\bf W}_{s}^{T}\mbox{ and }\gamma_{ii}=\int_{t}^{T}
\parallel\boldsymbol{\gamma}_{i}(s,T)\parallel^{2}ds.\]
The forward price \(F_{Z^{i}}(t,T)\) is a random variable measurable with respect to the \(\sigma\)-field \({\cal F}_{t}\), while the random variable \(\zeta_{i}(t,T)\) is independent of this \(\sigma\)-field. Moreover, it is clear that the probability law under the forward measure \(\mathbb{P}_{T}\) of the random vector
\[(\zeta_{1}(t,T),\cdots ,\zeta_{n}(t,T))=\left (\int_{t}^{T}\boldsymbol{\gamma}_{1}(s,T)\cdot d{\bf W}_{s}^{T},\cdots ,
\int_{t}^{T}\boldsymbol{\gamma}_{n}(s,T)\cdot d{\bf W}_{s}^{T}\right )\]
is Gaussian \(N({\bf 0},\Gamma )\), where the entries of the \(n\times n\) matrix \(\Gamma\) are
\[\gamma_{ij}=\int_{t}^{T}\boldsymbol{\gamma}_{i}(s,T)\cdot\boldsymbol{\gamma}_{j}(s,T)ds.\]
Introducing a \(k\times n\) matrix \(\Theta =[\boldsymbol{\theta}_{1},\cdots ,\boldsymbol{\theta}_{1}]\) such that \(\Gamma =\Theta^{t}\Theta\) leads to the following result.
Proposition. Assume that \(\boldsymbol{\gamma}_{i}\) is a deterministic function for \(i=1,\cdots ,n\). Then the arbitrage price at time \(t\in [0,T]\) of a European contingent claim \(X=g(Z_{T}^{1},\cdots ,Z_{T}^{n})\) which settles at time \(T\) equals
\[\Pi_{t}(X)=p(t,T)\cdot\int_{\mathbb{R}^{k}}g\left (\frac{Z_{t}^{1}\cdot n_{k}({\bf x}+\boldsymbol{\theta}_{1})}
{p(t,T)\cdot n_{k}({\bf x})},\cdots ,\frac{Z_{t}^{n}\cdot n_{k}({\bf x}+\boldsymbol{\theta}_{n})}
{p(t,T)\cdot n_{k}({\bf x})}\right )\cdot n_{k}({\bf x})d{\bf x},\]
where \(n_{k}({\bf x})\) is the standard \(k\)-dimensional Gaussian density
\[n_{k}({\bf x})=(2\pi )^{-k/2}\cdot\exp\left (-\frac{\parallel {\bf x}
\parallel^{2}}{2}\right )\mbox{ for all }{\bf x}\in \mathbb{R}^{k},\]
and the vectors \(\boldsymbol{\theta}_{1},\cdots ,\boldsymbol{\theta}_{n}\in \mathbb{R}^{k}\) are such that for every \(i,j=1,\cdots ,n\), we have
\[\boldsymbol{\theta}_{i}\cdot\boldsymbol{\theta}_{j}=
\int_{t}^{T}\boldsymbol{\gamma}_{i}(s,T)\cdot\boldsymbol{\gamma}_{j}(s,T)ds.\]
Proof. Since \(F_{Z^{i}}(T,T)=Z_{T}^{i}\), from Proposition \ref{musl1323}, we have
\[\Pi_{t}(X)=p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}\left [\left .g(F_{Z^{1}}(T,T),\cdots ,
F_{Z^{n}}(T,T)\right |{\cal F}_{t}\right ]\equiv p(t,T)\cdot J.\]
In view of the definition of the matrix \(\Theta\), it is clear that
\begin{align*}
J & =\int_{\mathbb{R}^{k}}g\left (F_{Z^{1}}(t,T)\cdot\exp\left (\boldsymbol{\theta}_{1}\cdot {\bf x}-
\frac{\parallel\boldsymbol{\theta}_{1}\parallel^{2}}{2}\right ),
\cdots ,F_{Z^{n}}(t,T)\cdot\exp\left (\boldsymbol{\theta}_{n}\cdot
{\bf x}-\frac{\parallel\boldsymbol{\theta}_{n}\parallel^{2}}{2}\right )\right )\cdot n_{k}({\bf x})d{\bf x}\\
& =\int_{\mathbb{R}^{k}}g\left (\frac{Z_{t}^{1}\cdot n_{k}({\bf x}+\boldsymbol{\theta}_{1})}
{p(t,T)\cdot n_{k}({\bf x})},\cdots ,\frac{Z_{t}^{n}\cdot n_{k}({\bf x}+\boldsymbol{\theta}_{n})}
{p(t,T)\cdot n_{k}({\bf x})}\right )\cdot n_{k}({\bf x})d{\bf x}.
\end{align*}
This completes the proof. \(\blacksquare\)
Representation \(\Gamma =\Theta^{t}\Theta\) can be easily obtained from the eigenvalues and the eigenvectors of the matrix \(\Gamma\). Let \(\delta_{1},\cdots ,\delta_{n}\) be the eigenvalues of \(\Gamma\), and \({\bf w}_{1},\cdots ,{\bf w}_{n}\) the corresponding orthonormal eigenvectors. Then with \(D=diag (\delta_{1},\cdots ,\delta_{n})\) and \(V=[{\bf w}_{1},\cdots ,{\bf w}_{n}]\), we have \(\Gamma =VDV^{t}=VD^{1/2}(VD^{1/2})^{t}\). Let \(k\leq n\) be such that \(\delta_{1}\geq\delta_{2}\geq\cdots\geq\delta_{k}\) are strictly positive numbers, and \(\delta_{k+1}=\cdots =\delta_{n}=0\). Then
\[VD^{1/2}=[\sqrt{\delta_{1}}{\bf w}_{1},\cdots ,\sqrt{\delta_{k}}{\bf w}_{k},0,\cdots ,0]\]
and with \(\Theta^{t}=[\sqrt{\delta_{1}}{\bf w}_{1},\cdots ,\sqrt{\delta_{1}}{\bf w}_{k}\)], we have \(\Gamma =\Theta^{t}\Theta\), as desired.
Replication of Options.
Consider a contingent claim \(X\) which settles at time \(T\), and is represented by a \(\mathbb{P}_{T}\)-integrable, strictly positive random variable \(X\). The forward price of \(X\) for the settlement date \(T\) satisfies, from Proposition \ref{musp1322},
\begin{equation}{\label{museq1521}}\tag{159}
F_{X}(t,T)=\mathbb{E}_{P_{T}}[X|{\cal F}_{t}]=F_{X}(0,T)+\int_{0}^{t}
F_{X}(s,T)\boldsymbol{\gamma}_{s}\cdot d{\bf W}_{s}^{T}
\end{equation}
for some predictable process \(\boldsymbol{\gamma}\). Assume, in addition, that \(\boldsymbol{\gamma}\) is a deterministic function. The aim is to show, by means of a replicating strategy, that the arbitrage price of a European call option written on a claim \(X\) with expiry date \(T\) and strike price \(K\) equals
\begin{equation}{\label{museq1522}}\tag{160}
C_{t}=p(t,T)\cdot\left (F_{X}(t,T)\cdot N(\tilde{d}_{1}(F_{X}(t,T),t,T))-K\cdot N(\tilde{d}_{2}(F_{X}(t,T),t,T))\right ),
\end{equation}
where \(\tilde{d}_{1}\) and \(\tilde{d}_{2}\) are given by (\ref{museq1515}) with
\[v^{2}(t,T)=v_{X}^{2}(t,T)=\int_{t}^{T}\parallel\boldsymbol{\gamma}_{s}\parallel^{2}ds.\]
Since \(F_{Z}(t,T)=Z_{t}/p(t,T)\) in (\ref{museq70}), equality (\ref{museq1522}) yields the following expression for the forward price of the option
\begin{equation}{\label{museq1523}}\tag{161}
\frac{C_{t}}{p(t,T)}=F_{C}(t,T)=F_{X}(t,T)\cdot N(\tilde{d}_{1}(F_{X}(t,T),t,T))-K\cdot N(\tilde{d}_{2}(F_{X}(t,T),t,T)).
\end{equation}
Note that by applying Ito’s formula to (\ref{museq1523}), we obtain
\[dF_{C}(t,T)=N(\tilde{d}_{1}(F_{X}(t,T),t,T))\cdot dF_{t}.\]
Forward asset/bond market. Let us consider a \(T\)-forward market, i.e., a financial market in which the forward contracts for settlement at time \(T\) play the role of primary securities. Consider a forward strategy $\boldsymbol{\psi}=(\psi^{1},\psi^{2})$, where \(\psi^{1}\) and \(\psi^{2}\) stand for the number of forward contracts on the underlying claim \(X\) and on the zero-coupon bond with maturity \(T\), respectively. The forward wealth process \(\tilde{V}\) of a \(T\)-forward market portfolio \(\boldsymbol{\psi}\) equals
\[\tilde{V}_{t}(\boldsymbol{\psi})=\psi_{t}^{1}\cdot F_{X}(t,T)+\psi_{t}^{2}\cdot F_{p}(t,T,T)\]
Since \(F_{p}(t,U,T)=p(t,U)/p(t,T)\), we see that \(F_{p}(t,T,T)=1\) for any \(t\in [0,T]\). Therefore we have
\begin{equation}{\label{museq77}}\tag{162}
\tilde{V}_{t}(\boldsymbol{\psi})=\psi_{t}^{1}\cdot F_{X}(t,T)+\psi_{t}^{2}.
\end{equation}
A portfolio \(\boldsymbol{\psi}\) is self-financing in the \(T\)-forward market it its forward wealth satisifies \(d\tilde{V}_{t}(\boldsymbol{\psi})=\psi^{1}_{t}\cdot dF_{X}(t,T)\). From (\ref{museq1521}), we have
\[d\tilde{V}_{t}(\boldsymbol{\psi})=\psi_{t}^{1}F_{X}(t,T)\boldsymbol{\gamma}_{t}\cdot d{\bf W}_{t}^{T}.\]
The aim is to find the forward portfolio \(\boldsymbol{\psi}\) that replicates the forward contract written on the option, and to subsequently re-derive pricing formulas (\ref{museq1522}) and (\ref{museq1523}). To replicate the forward contract written on the option, it is enough to take positions in forward contracts on a claim \(X\) and in forward contracts on \(T\)-maturity bonds. Suppose that the option’s forward price \(F_{C}(t,T)=u(F_{X}(t,T),t)\) for some function \(u\). One may derive the following PDE
\[u_{t}(x,t)+\frac{1}{2}\gamma_{t}^{2}x^{2}u_{xx}(x,t)=0,\]
with \(u(x,T)=(x-K)^{+}\) for \(x\in \mathbb{R}_{+}\). The solution \(u\) to this problem is given by the formula
\begin{equation}{\label{museq78}}\tag{163}
u(x,t)=x\cdot N(\tilde{d}_{1}(x,t,T)-K\cdot N(\tilde{d}_{2}(x,t,T)).
\end{equation}
The corresponding strategy \(\boldsymbol{\psi}=(\psi^{1},\psi^{2})\) in the \(T\)-forward market is
\begin{equation}{\label{museq1524}}\tag{164}
\psi^{1}_{t}=u_{x}(F_{X}(t,T),t)=N(\tilde{d}_{1}(F_{X}(t,T),t,T)).
\end{equation}
Since \(\tilde{V}_{t}(\boldsymbol{\psi})=F_{C}(t,T)=u(F_{X}(t,T,t)\), from (\ref{museq77}), we have \(\psi_{t}^{2}=u(F_{X}(t,T,t)-\psi_{t}^{1}\cdot F_{X}(t,T)\). It can be checked, using Ito’s formula, that the strategy \(\boldsymbol{\psi}\) is self-financing in the \(T\)-forward market; moreover, \(\tilde{V}_{T}(\boldsymbol{\psi})=V_{T}(\boldsymbol{\psi})=(X-K)^{+}\). The forward price of the option is thus given (\ref{museq78}) since \(F_{C}(t,T)=u(F_{X}(t,T),t)\), which is the same as (\ref{museq1523}). Since \(\tilde{V}_{t}(\boldsymbol{\psi})=F_{C}(t,T)=C_{t}/p(t,T)\), the spot price at time \(t\) equals
\begin{equation}{\label{museq1525}}\tag{165}
C_{t}=p(t,T)\cdot\tilde{V}_{t}(\boldsymbol{\psi})=p(t,T)\cdot u(F_{X}(t,T),t).
\end{equation}
The last formula agrees with (\ref{museq1522}).
Forward/spot asset/bond market. It may be convenient to replicate the terminal payoff of an option by means of a combined spot/forward trading strategy. Let the date \(t\) be fixed, but arbitrary. Consider an investor who purchases at time \(t\) the number \(F_{C}(t,T)\) of \(T\)-maturity bonds and holds them to maturity. In addition, at any date \(s\geq t\), we take \(\psi_{s}^{1}\) positions in \(T\)-maturity forward contracts on the underlying claim, where \(\psi_{s}^{1}\) is given by (\ref{museq1524}). The terminal wealth of such a strategy at the date \(T\) equals
\[F_{C}(t,T)+\int_{t}^{T}\psi_{s}^{1}dF_{X}(s,T)=F_{C}(t,T)+\tilde{V}_{T}(\boldsymbol{\psi})-\tilde{V}_{t}
(\boldsymbol{\psi})=(X-K)^{+}\]
since \(\tilde{V}_{t}(\boldsymbol{\psi})=F_{C}(t,T)\) and \(\tilde{V}_{T}(\boldsymbol{\psi})=(X-K)^{+}\).
Spot asset/bond market. To replicate an option in a spot market, we need to assume that it is written on an asset which is tradable in the spot market. As the second asset, we use a \(T\)-maturity bond with the spot price \(p(t,T)\). Assume that a claim \(X\) corresponds to the value \(Z_{T}\) of a tradeable asset whose spot price at time \(t\) equals \(Z_{t}\). To replicate an option in the spot market, we consider the spot trading strategy \(\boldsymbol{\phi}=\boldsymbol{\psi}\), where \(Z\) and a \(T\)-maturity bond are primary securities. We deduce easily from (\ref{museq1525}) that the wealth \(V(\boldsymbol{\phi})\) equals
\begin{equation}{\label{museq85}}\tag{166}
V_{t}(\boldsymbol{\phi})=\phi_{t}^{1}\cdot Z_{t}+\phi_{t}^{2}\cdot
p(t,T)=p(t,T)\cdot\tilde{V}_{t}(\boldsymbol{\psi})=C_{t}
\end{equation}
so that the strategy \(\boldsymbol{\phi}\) replicates the option value at any date \(t\leq T\). It remains to check that \(\boldsymbol{\phi}\) is self-financing. The following property is a general feature of self-financing strategy in the \(T\)-forawrd market: a \(T\)-forward trading strategy \(\boldsymbol{\psi}\) is self-financing if and only if the spot market strategy \(\boldsymbol{\phi}=\boldsymbol{\psi}\) is self-financing. Replication of a European call option with terminal payoff \((Z_{T}-K)^{+}\) can thus be done using the spot trading strategy \(\boldsymbol{\phi}=(\phi^{1},\phi^{2})\), where
\begin{equation}{\label{museq84}}\tag{167}
\phi^{1}_{t}=N(\tilde{d}_{1}(F_{Z}(t,T),t,T))\mbox{ and }\phi^{2}_{t}=\frac{C_{t}-\phi_{t}^{1}\cdot Z_{t}}{p(t,T)}=-K\cdot
N(\tilde{d}_{2}(F_{Z}(t,T),t,T)).
\end{equation}
Here \(\phi_{t}^{1}\) and \(\phi_{t}^{2}\) represent the number of units of the underlying aset and of \(T\)-maturity bonds held at time \(t\), respectively.
Spot asset/cash market. Let us show that since a savings account follows a process of finite variation, replication of an option written on \(Z\) in the spot asset/cash market is not always possible. Suppose that \(\widehat{\boldsymbol{\phi}}=(\widehat{\phi}^{1},\widehat{\phi}^{2})\) is an asset/cash self-financing trading strategy which replicates an option. In particular, we have
\begin{equation}{\label{museq1526}}\tag{168}
C_{t}=\widehat{\phi}^{1}_{t}\cdot Z_{t}+\widehat{\phi}^{2}_{t}\cdot B_{t}
\mbox{ and }dC_{t}=\widehat{\phi}^{1}_{t}\cdot dZ_{t}+\widehat{\phi}^{2}_{t}\cdot dB_{t}.
\end{equation}
From (\ref{museq85}), we have \(dC_{t}=\phi^{1}_{t}\cdot dZ_{t}+\phi^{2}_{t}\cdot dp(t,T)\). From (\ref{museq83}) and (\ref{museq1323}), we have
\[dZ_{t}=Z_{t}(r_{t}dt+\boldsymbol{\xi}_{t}\cdot d{\bf W}_{t}^{*})
\mbox{ and }dp(t,T)=p(t,T)(r_{t}dt+{\bf b}(t,T)\cdot d{\bf W}_{t}^{*}).\]
Thenerfore
\begin{equation}{\label{museq1527}}\tag{169}
dC_{t}=(\phi_{t}^{1}\cdot Z_{t}+\phi_{t}^{2}\cdot p(t,T))r_{t}dt+
(\phi_{t}^{1}\cdot Z_{t}\cdot\boldsymbol{\xi}_{t}+\phi_{t}^{2}\cdot p(t,T)\cdot {\bf b}(t,T))\cdot d{\bf W}_{t}^{*}
\end{equation}
A comparison of martingale parts in (\ref{museq1526}) and (\ref{museq1527}) yields
\[(\phi_{t}^{1}\cdot Z_{t}\cdot\boldsymbol{\xi}_{t}+\phi_{t}^{2}\cdot
p(t,T)\cdot {\bf b}(t,T))\cdot d{\bf W}_{t}^{*}=\widehat{\phi}^{1}_{t}
\cdot Z_{t}\cdot\boldsymbol{\xi}_{t}\cdot d{\bf W}_{t}^{*}\]
since \(B_{t}\) is of finite variation. When the underlying Brownian motion is multidimensional, we cannot solve the above equality for \(\phi_{t}^{1}\) in general. If, however, \({\bf W}^{*}\) is one-dimensional and process \(Z\) and \(\boldsymbol{\xi}\) are strictly positive, then we have
\[\widehat{\phi}^{1}_{t}=\phi_{t}^{1}+\frac{\phi_{t}^{2}\cdot {\bf b}(t,T)
\cdot p(t,T)}{\boldsymbol{\xi}_{t}\cdot Z_{t}}.\]
We put, in addition, from (\ref{museq1526}),
\[\widehat{\phi}^{2}_{t}=\frac{C_{t}-\widehat{\phi}^{1}_{t}\cdot Z_{t}}{B_{t}}.\]
It is clear that such a strategy replicate the option. Moreover, it is self-financing since simple calculations show that
\[\widehat{\phi}^{1}_{t}\cdot dZ_{t}+\widehat{\phi}_{t}^{2}\cdot dB_{t}
=(\phi_{t}^{1}\cdot Z_{t}+\phi_{t}^{2}\cdot p(t,T))r_{t}dt+
(\phi_{t}^{1}\cdot Z_{t}\cdot\boldsymbol{\xi}_{t}+\phi_{t}^{2}\cdot
p(t,T)\cdot {\bf b}(t,T))\cdot d{\bf W}_{t}^{*}=dC_{t}=dV_{t}(\widehat{\boldsymbol{\phi}}).\]
For instance, the stock/cash trading strategy that involves at time \(t\), from (\ref{museq84}),
\[\widehat{\phi}^{1}_{t}=N(\tilde{d}_{1}(F_{Z}(t,T),t,T))-
\frac{{\bf b}(t,T)\cdot p(t,T)}{\boldsymbol{\xi}_{t}\cdot Z_{t}}\cdot K\cdot N(\tilde{d}_{2}(F_{Z}(t,T),t,T))\]
shares of stock, and the amount \(C_{t}-\widehat{\phi}_{t}^{1}\cdot Z_{t}\) held in a savings account, is a self-financing strategy replicating a European call option written on \(Z\).
Futures Prices.
The next goal is to establish the relationship between forward and futures prices. We consider an arbirary tradable asset whose spot price \(Z\) has the dynamics given by the expression (\ref{museq83}). The forward price of \(Z\) for settlement at the date \(T\) is already known to satisfy
\[F_{Z}(T,T)=F_{Z}(t,T)\cdot\exp\left (\int_{t}^{T}\boldsymbol{\gamma}_{Z}(s,T)\cdot d{\bf W}_{s}^{T}-\frac{1}{2}
\int_{t}^{T}\parallel\boldsymbol{\gamma}_{Z}(s,T)\parallel^{2}ds\right ),\]
where \(\boldsymbol{\gamma}_{Z}(s,T)=\boldsymbol{\xi}_{s}-{\bf b}(s,T)\), and \({\bf W}_{t}^{T}={\bf W}_{t}^{*}-\int_{0}^{t}{\bf b}(s,T)ds\) is a Brownian motion under the forward measure \(\mathbb{P}_{T}\). Since the martingale measure \(\mathbb{P}^{*}\) for the spot market is assumed to be unique, it is natural to introduce the futures price by means of the following definition.
Definition. The futures price \(f_{Z}(t,T)\) of an asset \(Z\), in the futures contract that expires at time \(T\), is given by the formula
\begin{equation}{\label{museq1528}}\tag{170}
f_{Z}(t,T)= $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[Z_{T}|{\cal F}_{t}]
\end{equation}
for all \(t\in [0,T]\). \(\sharp\)
Equality (\ref{museq1528}) defines the futures price of a contingent claim \(Z_{T}\) which settles at time \(T\); hence it applies to any contingent claim which settles at time \(T\). We are in a position to establish the relationship between the forward and futures prices of an arbitrary asset.
Proposition. Assume that the volatility \(\boldsymbol{\gamma}_{Z}(t,T)=\boldsymbol{\xi}_{t}-{\bf b}(t,T)\) of the forward price process \(F_{Z}(t,T)\) follows a deterministic function. Then the futures price \(f_{Z}(t,T)\) equals
\begin{equation}{\label{museq1529}}\tag{171}
f_{Z}(t,T)=F_{Z}(t,T)\cdot\exp\left (\int_{t}^{T}({\bf b}(s,T)-\boldsymbol{\xi}_{s})\cdot {\bf b}(s,T)ds\right ).
\end{equation}
Proof. It is clear that
\[F_{Z}(T,T)=F_{Z}(t,T)\cdot\zeta_{t}\cdot\exp\left (\int_{t}^{T}
({\bf b}(s,T)-\boldsymbol{\xi}_{s})\cdot {\bf b}(s,T)ds\right ),\]
where \(\zeta_{t}\) stands for the following random variabel
\[\zeta_{t}=\exp\left (\int_{t}^{T}(\boldsymbol{\xi}_{s}-{\bf b}(s,T))
\cdot d{\bf W}_{s}^{*}-\frac{1}{2}\parallel\boldsymbol{\xi}_{s}-{\bf b}(s,T)\parallel^{2}ds\right ).\]
The random variable \(\zeta_{t}\) is independent of the \(\sigma\)-filed \({\cal F}_{t}\), and \($latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[\zeta_{t}]=1\). Since by definition
\begin{align*}
f_{Z}(t,T) & =\mathbb{E}_{\scriptsize \mathbb{P}^{*}}[Z_{T}|{\cal F}_{t}]=\mathbb{E}_{\scriptsize \mathbb{P}^{*}}[F_{Z}(T,T)|{\cal F}_{t}]\\
& =F_{Z}(t,T)\cdot\exp\left (\int_{t}^{T}({\bf b}(s,T)-
\boldsymbol{\xi}_{s})\cdot {\bf b}(s,T)ds\right )\cdot \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[\zeta_{t}|{\cal F}_{t}]\\
& \mbox{(since \(\boldsymbol{\xi}_{t}-{\bf b}(t,T)\) is a deterministic function)}\\
& =F_{Z}(t,T)\cdot\exp\left (\int_{t}^{T}({\bf b}(s,T)-\boldsymbol{\xi}_{s})\cdot {\bf b}(s,T)ds\right )\cdot $latex \mathbb{E}_{\scriptsize \mathbb{P}^{*}}[
\zeta_{t}]\mbox{ (by the independence)}
\end{align*}
This completes the proof. \(\blacksquare\)
Observe that the dynamics of the futures price process \(f_{Z}(t,T)\) for \(t\in [0,T]\), under the martingale measure \(\mathbb{P}^{*}\), are
\begin{equation}{\label{museq1530}}\tag{172}
df_{Z}(t,T)=f_{Z}(t,T)(\boldsymbol{\xi}_{t}-{\bf b}(t,T))\cdot d{\bf W}_{t}^{*}.
\end{equation}
It is interesting to note that the dynamics of the forward price \(F_{Z}(t,T)\) under the forward measure \(\mathbb{P}_{T}\) are given by the analogous expression
\[dF_{Z}(t,T)=F_{Z}(t,T)(\boldsymbol{\xi}_{t}-{\bf b}(t,T))\cdot d{\bf W}_{t}^{T}.\]
Futures Options.
The goal is to establish an explicit formula for the arbitrage price of a European call option written on a futures contract on a zero-coupon bond. Let us denote by \(f_{p}(t,U,T)\) the futures prices for settlement at the date \(T\) of a \(U\)-maturity zero-coupon bond. From (\ref{museq1530}), we have
\[df_{p}(t,U,T)=f_{p}(t,U,T)({\bf b}(t,U)-{\bf b}(t,T))\cdot d{\bf W}_{t}^{*}\]
subject to the terminal condition
\begin{equation}{\label{museq89}}\tag{173}
f_{p}(T,U,T)=p(T,U)=F_{p}(T,U,T)
\end{equation}
by (\ref{museq142}). Since it costs nothing to take a long or short position in a futures contract, The wealth process \(V^{f}(\boldsymbol{\psi})\) of any futures trading strategy \(\boldsymbol{\psi}=(\psi^{1},\psi^{2})\) equals
\begin{equation}{\label{museq86}}\tag{174}
V_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{2}\cdot p(t,T)
\end{equation}
We fix \(U\) and \(T\), and we write briefly \(f_{t}\) instead of \(f_{p}(t,U,T)\) in what follows. A futures trading strategy \(\boldsymbol{\psi}=(\psi^{1},\psi^{2})\) is said to be self-financing if its wealth process \(V^{f}(\boldsymbol{\psi})\) satisfies the standard relationship
\[V_{t}^{f}(\boldsymbol{\psi})=V_{0}^{f}(\boldsymbol{\psi})+
\int_{0}^{t}\psi_{s}^{1}\cdot df_{s}+\int_{0}^{t}\psi_{s}^{2}\cdot dp(t,T)\]
or
\begin{equation}{\label{museq87}}\tag{175}
dV_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{1}\cdot df_{t}+\psi_{t}^{2}\cdot dp(t,T).
\end{equation}
Let us consider the relative wealth process \(\tilde{V}_{t}^{f}(\boldsymbol{\psi})=V_{t}^{f}(\boldsymbol{\psi})\cdot p^{-1}(t,T)\). As one might expect, the relative wealth of a self-financing futures trading strategy follows a local martingale under the forward measure \(P_{T}\). Indeed, using Ito’s formula we get
\[d\tilde{V}_{t}^{f}=p^{-1}(t,T)dV_{t}^{f}+V_{t}^{f}dp^{-1}(t,T)+d\langle V^{f},p^{-1}(t,T)\rangl\mathbb{E}_{t}\]
so that
\begin{align*}
d\tilde{V}_{t}^{f} & =p^{-1}(t,T)\psi_{t}^{1}df_{t}+p^{-1}(t,T)\psi_{t}^{2}dp(t,T)+\psi_{t}^{2}p(t,T)dp^{-1}(t,T)\\
& +\psi_{t}^{1}d\langle f,p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t}+\psi_{t}^{2}d\langle
p(\cdot ,T),p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t}\mbox{ (by (\ref{museq86}) and (\ref{museq87}))}\\
& =p^{-1}(t,T)\psi_{t}^{1}df_{t}+\psi_{t}^{1}d\langle f,p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t}\\
& +\psi_{t}^{2}\cdot (p^{-1}(t,T)dp(t,T)+p(t,T)dp^{-1}(t,T)+d\langle (\cdot ,T),p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t})\\
& =p^{-1}(t,T)\psi_{t}^{1}df_{t}+\psi_{t}^{1}d\langle f,p^{-1}(\cdot ,T)
\rangle\mathbb{E}_{t}+\psi_{t}^{2}\cdot d(p^{-1}(t,T)\cdot p(t,T))\\
& =p^{-1}(t,T)\psi_{t}^{1}df_{t}+\psi_{t}^{1}d\langle f,p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t}.
\end{align*}
On the other hand, we have
\[d\langle f,p^{-1}(\cdot ,T)\rangle\mathbb{E}_{t}=-f_{t}\cdot p^{-1}(t,T)\cdot
({\bf b}(t,U)-{\bf b}(t,T))\cdot {\bf b}(t,T)dt.\]
Combining these formulas, we arrive at the expression
\[d\tilde{V}_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{1}f_{t}p^{-1}(t,T)
({\bf b}(t,U)-{\bf b}(t,T))\cdot (d{\bf W}_{t}^{*}-{\bf b}(t,T)dt),\]
which is valid under \(\mathbb{P}^{*}\), or equivalently, at the formula
\[d\tilde{V}_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{1}f_{t}p^{-1}(t,T)
({\bf b}(t,U)-{\bf b}(t,T))\cdot d{\bf W}_{t}^{T},\]
which in turn is satisfied under the forward measure \(\mathbb{P}_{T}\). We conclude that the relative wealth of any self-financing futures trading strategy follows a local martingale under the forward measure for the date \(T\). Let \(X=g(f_{t},T)\) be any \(P_{T}\)-integrable contingent claim and \(\Pi_{t}^{f}(X)\) be the arbitrage price of \(Z\) at time \(t\in [0,T]\). If \(\boldsymbol{\psi}\) is a futures trading strategy replicating \(X\), then the process \(\tilde{V}^{f}(\boldsymbol{\psi})\) is a \(\mathbb{P}_{T}\)-martingale, and thus
\[\mathbb{E}_{P_{T}}[X|{\cal F}_{t}]=\mathbb{E}_{P_{T}}[\tilde{V}_{T}^{f}(\boldsymbol{\psi})|{\cal F}_{t}]=\tilde{V}_{t}^{f}
(\boldsymbol{\psi})=p^{-1}(t,T)\cdot V_{t}^{f}(\boldsymbol{\psi})=p^{-1}(t,T)\Pi_{t}^{f}(X).\]
Therefore, we have
\begin{equation}{\label{museq1532}}\tag{176}
\Pi_{t}^{f}(X)=p(t,T)\cdot \mathbb{E}_{P_{T}}[X|{\cal F}_{t}]\mbox{ for all }t\in [0,T].
\end{equation}
One may argue that equality (\ref{museq1532}) is trivial since the equality \(\Pi_{t}(X)=p(t,T)\cdot \mathbb{E}_{P_{T}}[X|{\cal F}_{t}]\) was already established in Proposition \ref{musl1323}, and the arbitrage price of any attainable contingent claim is independent of the choice of financial instruments used in replication. For this simple argument to be formally valid, however, it is necessary to construct first a consistent financial model which encompasses both spot and futures contracts. For the sake of expositional simplicity, we assume here that the expiry date of an option coincides with the settlement date of the underlying futures contract.
\begin{equation}{\label{musp1522}}\tag{177}\mbox{}\end{equation}
Proposition \ref{musp1522}. Assume that \(U\geq T\). The arbitrage price at time \(t\in [0,T]\) of a European call option with expiry date \(T\) and exercise price \(K\) written on the futures contract for a \(U\)-maturity zero-coupon bond with delivery date \(T\) equals
\[C_{t}^{f}=p(t,T)\cdot\left (f_{t}\cdot h_{U}(t,T)\cdot N(g_{1}(f_{t},t,T)-K\cdot N(g_{2}(f_{t},t,T))\right ),\]
where
\begin{equation}{\label {museq1534}}
g_{1}(q,t,T)=\frac{\ln q-\ln K+\frac{1}{2}\int_{t}^{T}(\parallel {\bf b}(s,U)\parallel^{2}-\parallel {\bf b}(s,T)\parallel^{2})ds}
{v_{U}(t,T)}\mbox{ and }g_{2}(q,t,T)=g_{1}(q,t,T)-v_{U}(t,T),
\end{equation}
the function \(v_{U}(t,T)\) is given by \((\ref{museq156})\), and
\begin{equation}{\label{museq1535}}\tag{178}
h_{U}(t,T)=\exp\left (\int_{t}^{T}({\bf b}(s,U)-{\bf b}(s,T))\cdot {\bf b}(s,T)ds\right ).
\end{equation}
Proof. From (\ref{museq1532}) and (\ref{museq89}), we need to evaluate
\[C_{t}^{f}=p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}[(f_{p}(T,U,T)-K)^{+}|{\cal F}_{t}]=
p(t,T)\cdot \mathbb{E}_{\scriptsize \mathbb{P}_{T}}[(F_{p}(T,U,T)-K)^{+}|{\cal F}_{t}].\]
Proceeding as in the proof of Proposition \ref{musp1511}, we find (ref. (\ref{museq158}) and (\ref{museq159}))
\begin{equation}{\label{museq1536}}\tag{179}
C_{t}^{f}=p(t,T)\cdot\left (F_{p}(t,U,T)\cdot N(\tilde{d}_{1}
(F_{p}(t,U,T),t,T))-K\cdot N(\tilde{d}_{2}(F_{p}(t,U,T),t,T))\right ),
\end{equation}
where
\[\tilde{d}_{1}(q,t,T)=\frac{\ln q-\ln K+\frac{1}{2}v_{U}^{2}(t,T)}
{v_{U}(t,T)}\mbox{ and }\tilde{d}_{2}(q,t,T)=\tilde{d}_{1}(q,t,T)-v_{U}(t,T),\]
and \(v_{U}(t,T)\) is given by (\ref{museq156}). On the other hand, (\ref{museq1529}) yields
\[F_{p}(t,U,T)=f_{p}(t,U,T)\cdot\exp\left (-\int_{t}^{T}({\bf b}(s,T)-
{\bf b}(s,U))\cdot {\bf b}(s,T)ds\right )=f_{p}(t,U,T)\cdot h_{U}(t,T).\]
Substituting the above formula into (\ref{museq1536}), we find the desired formula. \(\blacksquare\)
We denote by \(f_{Z}(t,T)\) the futures price of a tradable security \(Z\). We assume that the volatilities \({\bf b}(t,T)\) and
$\boldsymbol{\gamma}_{Z}(t,T)$ are deterministic functions. The proof of the next result is similar to that of Proposition \ref{musp1522}.
\begin{equation}{\label{musp1523}}\tag{180}\mbox{}\end{equation}
Proposition \ref{musp1523}. The arbitrage price of a European call option with expiry date \(T\) and strike price \(K\) written on a futures contract which settles at time \(T\) for delivery of one unit of security \(Z\) is given by the formula
\[C_{t}^{f}=p(t,T)\cdot\left (f_{Z}(t,T)\cdot h(t,T)\cdot N(g_{1}(f_{Z}(t,T),t,T))-K\cdot N(g_{2}(f_{Z}(t,T),t,T))\right ),\]
where
\[g_{1,2}(q,t,T)=\frac{\ln q-\ln K+\ln h(t,T)\pm\frac{1}{2}v^{2}(t,T)}{v(t,T)}\]
for \((q,t)\in \mathbb{R}_{+}\times [0,T]\), the function \(v(t,T)\) is given by (\ref{museq1516}), and
\[h(t,T)=\exp\left (\int_{t}^{T}\boldsymbol{\gamma}_{Z}(s,T)\cdot {\bf b}(s,T)ds\right ). \sharp\]
Suppose that the volatility coefficient \({\bf b}(s,T)\) vanishes identically for \(s\in [0,T]\) and the volatility of \(Z\) is constant; that is, \(\xi_{s}=\sigma\) for some real number \(\sigma\). In such case, the valuation formula provided by Proposition \ref{musp1523} agrees with the standard Black futures formula. Let us examine the put-call parity for futures options. We write \(P_{t}^{f}\) to denote the price of a European put futures option with expiry date \(T\) and strike price \(K\).
\begin{equation}{\label{musp1524}}\tag{181}\mbox{}\end{equation}
Proposition \ref{musp1524}. Under the assumption of Proposition \ref{musp1523}, the following put-call parity relationship is valid
\[C_{t}^{f}-P_{t}^{f}=p(t,T)\cdot\left (f_{Z}(t,T)\exp\left (\int_{t}^{T}
\boldsymbol{\gamma}_{Z}(s,T)\cdot {\bf b}(s,T)ds\right )-K\right ). \sharp\]
The put-call parity relationship (\ref{museq1517}) for European spot options can be easily established by a direct construction of two portfolios. Let us consider the following portfolios: long call abd short putoption; and one unit of the underlying asset and \(K\) units of \(T\)-maturity zero-coupon bonds. The portfolios have manifestly the same value at time \(T\), namely \(X_{T}-K\). Consequently, if no arbitrage opportunities exist, they have the same value at any time \(t\leq T\); that is, (\ref{museq1517}) holds. Due to the specific daily marking to market feature of the futures market, this simple reasoning is no longer valid for futures options under stochastic interest rates. It appears that the pu-call parity for futures options has the form given in Proposition \ref{musp1524}. Note that this relationship depends on the assumed dynamics of the underlying price processes, that is, on the choice of a market model.
PDE Approach to Interest Rate Derivatives.
We now present the PDE approach to the hedging and valuation of contingent claims in the Gaussian HJM setting. The PDE exaimed here are directly related to the price dynamics of bonds and underlying assets. The arbitrage price of a derivative security is expressed in terms of the time parameter \(t\), the current price of an underlying asset and the price of a certain zero-coupon bond.
PDEs for Spot Derivatives.
We start by examining the case of a European call option with expiry date \(T\) written on a tradable asset \(Z\). We assume throughout that the dynamics of the (spot) price process of \(Z\) are governed under a probability measure \(\mathbb{P}\) by the expression
\begin{equation}{\label{museq1541}}\tag{182}
dZ_{t}=Z_{t}(\mu_{t}dt+\boldsymbol{\xi}_{t}\cdot d{\bf W}_{t}),\
\end{equation}
where \(\mu\) is a stochastic process. We also assume implicitly that \(Z\) follows a strictly positive process. It should be stressed that \(P\) is not necessarily a martingale measure. For a fixed date \(D\geq T\), the price of a bond which matures at time \(D\) is assumed to follow, under \(\mathbb{P}\),
\begin{equation}{\label{museq1542}}\tag{183}
dp(t,D)=p(t,D)(\lambda_{t}dt+{\bf b}(t,D)\cdot d{\bf W}_{t}),
\end{equation}
where \(\lambda\) is a stochastic process. Volatilities \(\boldsymbol{\xi}_{t}\) and \({\bf b}(t,D)\) can also follow stochastic processes; it is essential to assume, however, that the volatility \(\boldsymbol{\xi}_{t}-{\bf b}(t,D)\) of the forward price of \(Z\) is deterministic.
We consider a European option with expiry date \(T\) written on the forward price of \(Z\) for the date \(D\), where \(D\geq T\). More precisely, by definition the option’s payoff at expirt equals
\[C_{T}=p(T,D)\cdot (F_{Z}(T,D)-K)^{+}=(Z_{T}-K\cdot p(T,D))^{+}\]
since \(F_{Z}(t,T)=Z_{t}/p(t,T)\) in (\ref{museq70}). When \(D=T\), we deal with a standard option written on \(Z\) since \(p(T,T)=1\). For \(D>T\), the option can be interpreted either as an option written on the forward price of \(Z\) with deferred payoff at time \(D\), or simply as an option to exchange one unit of an asset \(Z\) for \(K\) units of \(D\)-maturity bonds. We wish to express the option price in terms of the spot prices of the underlying asset and the \(D\)-maturity bond. Suppose that the price \(C_{t}\) admits the representation \(C_{t}=v(Z_{t},p(t,D),t)\), where \(v:\mathbb{R}_{+}\times [0,1]\times [0,T]\rightarrow \mathbb{R}\) is an unknown function which satisfies the terminal condition (it is implicitly assumed that \(p(T,D)\leq 1\); this restrictions is not essential, however)
\[v(x,y,T)=y\cdot\left (\frac{x}{y}-K\right )^{+}\mbox{ for all }(x,y)\in \mathbb{R}_{+}\times (0,1].\]
Let \(\boldsymbol{\phi}=(\phi^{1},\phi^{2})\) be a self-financing trading strategy which assumes continuous trading in the option’s underlying asset and in \(D\)-maturity bonds. In financial interpretation, \(\phi_{t}^{1}\) and \(\phi_{t}^{2}\) stand for the number of shares of the underlying asset and the number of units of \(D\)-maturity bonds, respectively, which are held at time \(t\leq T\). Assume that the terminal wealth of the portfolio \(\boldsymbol{\phi}\) replicates the payoff of the option. Then the following chain of equalities is valid for any \(t\in [0,T]\)
\begin{equation}{\label{museq90}}\tag{184}
V_{t}(\boldsymbol{\phi})=\phi_{t}^{1}\cdot Z_{t}+\phi_{t}^{2}\cdot p(t,D)=v(Z_{t},p(t,D),t)=C_{t}
\end{equation}
Since \(\boldsymbol{\phi}\) is self-financing. its wealth process also satisfies
\[dV_{t}(\boldsymbol{\phi})=\phi_{t}^{1}\cdot dZ_{t}+\phi_{t}^{2}\cdot dp(t,D).\]
Substituting (\ref{museq1541}) and (\ref{museq1542}) into the above equality, we obtain
\begin{equation}{\label{museq93}}\tag{185}
dV_{t}(\boldsymbol{\phi})=\left (\phi_{t}^{1}\cdot\mu_{t}\cdot Z_{t}
+\phi_{t}^{2}\cdot\lambda_{t}\cdot p(t,D)\right )dt+\left (\phi_{t}^{1}\cdot
Z_{t}\boldsymbol{\xi}_{t}+\phi_{t}^{2}\cdot p(t,D)\cdot{\bf b}(t,D)\right )\cdot d{\bf W}_{t}.
\end{equation}
From (\ref{museq90}), \(\phi^{2}\) can be found from the following equality
\begin{equation}{\label{museq1543}}\tag{186}
\phi_{t}^{2}=p^{-1}(t,D)\cdot\left (v(Z_{t},p(t,D),t)-\phi_{t}^{1}\cdot Z_{t}\right ).
\end{equation}
We assume that the function \(v(x,y,t)\) is sufficiently smooth on the open domain \((0,\infty )\times (0,1)\times (0,T)\). From (\ref{museq90}), using the Ito’s formula, we obtain
\[dC_{t}=v_{t}dt+v_{x}dZ_{t}+v_{y}dp(t,D)+\frac{1}{2}\left (v_{xx}\cdot
d\langle Z\rangl\mathbb{E}_{t}+v_{yy}\cdot d\langle p(\cdot ,D)\rangl\mathbb{E}_{t}\right )+
v_{xy}\cdot d\langle Z,p(\cdot ,D)\rangl\mathbb{E}_{t}.\]
Consequently, substitution of the dynamics of \(Z\) and \(B\) in (\ref{museq1541}) and (\ref{museq1542}), respectively, yields
\begin{align}
dC_{t} & =\left (v_{t}+Z_{t}\cdot\mu_{t}\cdot v_{x}+p(t,D)\cdot\lambda_{t}
\cdot v_{y}+Z_{t}\cdot p(t,D)\cdot\boldsymbol{\xi}_{t}\cdot{\bf b}(t,D)\cdot v_{xy}\right )dt\nonumber\\
& +\frac{1}{2}\left (Z_{t}^{2}\parallel\boldsymbol{\xi}_{t}
\parallel^{2}\cdot v_{xx}+p^{2}(t,D)\cdot\parallel {\bf b}(t,D)\parallel^{2}\cdot v_{xy}\right )dt\nonumber\\
& +\left (Z_{t}\cdot\boldsymbol{\xi}_{t}\cdot v_{x}+p(t,D)\cdot{\bf b}(t,D)\cdot v_{y}\right )\cdot d{\bf W}_{t}.\label{museq96}\tag{187}
\end{align}
By equating the \(d{\bf W}_{t}\) part of equations (\ref{museq93}) and (\ref{museq96}), we obtain
\[\int_{0}^{t}\left (\phi_{s}^{1}\cdot Z_{s}\cdot\boldsymbol{\xi}_{s}
+\phi_{s}^{2}\cdot p(s,D)\cdot {\bf b}(s,D)-Z_{s}\cdot
\boldsymbol{\xi}_{s}\cdot v_{x}-p(s,D)\cdot {\bf b}(s,D)\cdot v_{y}\right )\cdot d{\bf W}_{s}=0\]
for every \(t\in [0,T]\). Eliminating \(\phi^{2}\) and rearranging using (\ref{museq1543}), we get
\begin{equation}{\label{museq1544}}\tag{188}
\int_{0}^{t}\left (Z_{s}\cdot\boldsymbol{\xi}_{s}\cdot (\phi_{s}^{1}
-v_{x})+{\bf b}(s,D)\cdot (v-\phi_{s}^{1}\cdot Z_{s}-p(s,D)\cdot v_{y})\right )\cdot d{\bf W}_{s}=0
\end{equation}
for every \(t\in [0,T]\). Suppose now that
\begin{equation}{\label{museq1545}}\tag{189}
\phi_{t}^{1}=v_{x}=v_{x}(Z_{t}.p(t,D),t)\mbox{ for all }t\in [0,T].
\end{equation}
For (\ref{museq1544}) to be satisfied, it is sufficient that the equality
\[v=Z_{t}\cdot v_{x}+p(t,D)\cdot v_{y},\]
or written as
\begin{equation}{\label{museq97}}\tag{190}
v(Z_{t}.p(t,D),t)=Z_{t}\cdot v_{x}(Z_{t}.p(t,D),t)+p(t,D)\cdot v_{y}(Z_{t}.p(t,D),t),
\end{equation}
is satisfied for every \(t\in [0,T]\). In terms of the function of \(v\), we thus have, from (\ref{museq97}),
\begin{equation}{\label{museq1546}}\tag{191}
v(x,y,t)=x\cdot v_{x}(x,y,t)+y\cdot v_{y}(x,y,t),
\end{equation}
which should hold for every \((x,y,t)\in (0,\infty )\times (0,1)\times (0,T)\). Combining (\ref{museq1545}) and (\ref{museq1546}) with (\ref{museq1543}), we find immediately that
\begin{equation}{\label{museq1547}}\tag{192}
\phi_{t}^{2}=v_{y}=v_{y}(Z_{t},p(t,D),t)\mbox{ for all }t\in [0,T]
\end{equation}
Furthermore, by taking partial derivatives of relatioship (\ref{museq1546}) with respect to \(x\) and \(y\), we obtain
\begin{equation}{\label{museq1548}}\tag{193}
\left\{\begin{array}{l}
x\cdot v_{xx}(x,y,t)+y\cdot v_{xy}(x,y,t)=0;\\
x\cdot v_{xy}(x,y,t)+y\cdot v_{yy}(x,y,t)=0.
\end{array}\right .
\end{equation}
Using (\ref{museq1545}), (\ref{museq1546}), (\ref{museq93}) and (\ref{museq96}) (the \(d{\bf W}_{t}\) part will be cancelled under the above settting), we find the following expression
\[C_{t}-V_{t}(\boldsymbol{\phi})=\int_{0}^{t}\left (v_{t}+\frac{1}{2}
Z_{s}^{2}\cdot\parallel\boldsymbol{\xi}_{s}\parallel^{2}\cdot v_{xx}+
\frac{1}{2}p^{2}(s,D)\cdot\parallel {\bf b}(s,D)\parallel^{2}\cdot v_{yy}
+Z_{s}\cdot p(s,D)\cdot\boldsymbol{\xi}_{s}\cdot {\bf b}(s,D)\cdot v_{xy}\right )ds.\]
Since we have assumed that \(C_{t}=V_{t}(\boldsymbol{\phi})\) in (\ref{museq90}), we arrive at the following PDE which is satisfied by the function \(v=v(x,y,t)\) by regarding \(Z_{s}\) as \(x\) and \(p(s,D)\) as \(y\)
\[v_{t}+\frac{1}{2}\parallel\boldsymbol{\xi}_{t}\parallel^{2}\cdot
x^{2}\cdot v_{xx}+\frac{1}{2}\parallel {\bf b}(t,D)\parallel^{2}\cdot
y^{2}\cdot v_{yy}+\boldsymbol{\xi}_{t}\cdot {\bf b}(t,D)\cdot xy \cdot v_{xy}=0.\]
We need to consider the above PDE together with the PDE (\ref{museq1546}). Making use of (\ref{museq1548}), we find that the function \(v\) solves
\begin{equation}{\label{museq1549}}\tag{194}
v_{t}+\frac{1}{2}\parallel\boldsymbol{\xi}_{t}-{\bf b}(t,D)\parallel^{2}\cdot x^{2}\cdot v_{xx}=0
\end{equation}
subject to the terminal condition \(v(x,y,T)=y\cdot (x/y-K)^{+}\) for every \((x,y)\in \mathbb{R}_{+}\times (0,1)\). Note that (\ref{museq1546}) implies that the function \(v\) admits the representation \(v(x,y,t)=y\cdot H(x/y,t)\), where \(H:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) (it is enough to put \(x=zy\), and to check that the function \(h(z,y,t)=v(zy,y,t)/y\) does not depend on the second argument since \(h_{y}(z,y,t)=0\), i.e., \(h\) is a constant with respect to the second argument). Substituting \(v(x,y,t)=y\cdot H(x/y,t)\) in (\ref{museq1549}), we find the PDE satisfied by an auxiliary function \(H\)
\[H_{t}(z,t)+\frac{1}{2}\parallel\boldsymbol{\xi}_{t}-{\bf b}(t,D)\parallel^{2}\cdot z^{2}\cdot H_{zz}(z,t)=0\]
with the terminal condition \(H(z,T)=(z-K)^{+}\) for every \(z\in \mathbb{R}_{+}\). The solution to the above problem is well known to be given by the formula
\[H(z,t)=z\cdot N(d_{1}(z,t,T)-K\cdot N(d_{2}(z,t,T)),\]
where
\[d_{1,2}(z,t,T)=\frac{\ln z-\ln K\pm\frac{1}{2}v_{z}^{2}(t,T)}{v_{z}(t,T)}
\mbox{ and }v_{z}^{2}(t,T)=\int_{t}^{T}\parallel\boldsymbol{\xi}_{s}-{\bf b}(s,D)\parallel^{2}ds.\]
It is now straightforward to derive the following formula for the function \(v(x,y,t)\)
\[v(x,y,t)=x\cdot N(k_{1}(x,y,t,T))-K\cdot y\cdot N(k_{2}(x,y,t,T)),\]
where
\[k_{1,2}(x,y,t,T)=\frac{\ln x-\ln K-\ln y\pm\frac{1}{2}v_{z}^{2}(t,T)}{v_{z}(t,T)}.\]
The replicating strategy \(\boldsymbol{\phi}\) of a call option equals \(\phi_{t}^{1}=v_{x}(Z_{t},p(t,D),t)\) and \(\phi_{t}^{2}=v_{y}(Z_{t},p(t,D),t)\). Put more explicitly, at any time \(t\leq T\), the replicating portfolio involves \(\phi_{t}^{1}=N(k_{1}(Z_{t}.p(t,D),t,T))\) units of the underlying asset combined with \(\phi_{t}^{2}=-K\cdot N(k_{2}(Z_{t},p(t,D),t,T))\) units of \(D\)-maturity zero-coupon bonds. The PDE approach can be extended to cover the case of more general European contingent claims which may depend on several assets, for instances, to European options on coupon-bearing bonds.
\begin{equation}{\label{musp1531}}\tag{195}\mbox{}\end{equation}
Proposition \ref{musp1531}. Assume that the price process \(Z\) and \(p(t,D)\) follow \((\ref{museq1541})\) and \((\ref{museq1542})\), respectively, and the volatilities \(\boldsymbol{\xi}_{t}-{\bf b}(t,D)\) of the forward price is deterministic. Consider a European contingent claim \(X\). of the form \(X=p(T,D)\cdot g(Z_{T}/p(T,D))\), which settles at time \(T\). The arbitrage price of \(X\) equals
\[\Pi_{t}(X)=v(Z_{t},p(t,D),t)=p(t,D)\cdot H(Z_{t}/p(t,D),t)\]
for every \(t\in [0,T]\), where the function \(H:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) solves the following PDE
\[H_{z}(z,t)+\frac{1}{2}\parallel\boldsymbol{\xi}_{t}-{\bf b}(t,D)\parallel^{2}\cdot z^{2}\cdot H_{zz}(z,t)=0\]
with the terminal condition \(H(z,T)=g(z)\) for every \(z\in \mathbb{R}_{+}\). \(\sharp\)
A savings account can be used in the replication of European claims which settles at time \(T\) and have the form \(X=B_{T}\cdot g(Z_{T}/B_{T})\) for some function \(g\). Let us consider the European option with expiry date \(T\) and terminal payoff \((Z_{T}-K\cdot B_{T})^{+}\). If the volatility of the underlying asset is deterministic, Proposition \ref{musp1531} is in force, and thus replication of such an option involves \(N(k_{1}(Z_{t},B_{t},t,T))\) units of the underlying asset combined with the amount \(-K\cdot B_{t}\cdot N(k_{2}(Z_{t},B_{t},t,T))\) held in a saving account. In general, a standard European option cannot be replicated using a savings account.
PDEs for Futures Derivatives.
Let us fix three dates \(T,D\) and \(R\) such that \(T\leq\min\{D,R\}\). The futures price of an asset \(Z\) in a contract which settles at time \(R\) satisfies
\begin{equation}{\label{museq1550}}\tag{196}
df_{Z}(t,R)=f_{Z}(t,R)(\boldsymbol{\xi}_{t}-{\bf b}(t,R)))\cdot
d{\bf W}_{t}=f_{Z}(t,R)\boldsymbol{\zeta}_{t}\cdot d{\bf W}_{t},
\end{equation}
where \(\boldsymbol{\zeta}_{t}=\boldsymbol{\xi}_{t}-{\bf b}(t,R)\). We have assumed that the drift coefficient in the dynamics of \(f_{Z}(t,R)\) vanishes. This is not essential, however. Indeed, suppose that a nonzero drift in the dynamics of the futures price is present. Then we may either modify all foregoing considertaions in a suitable way, or, more conveniently, we may first make, using Girsanov’s theorem, an equivalent change of an underlying probability measure in such a way that the drift of the futures price will disappear. The drift of bond price will thus change, however, it is arbitrary (ref. (\ref{museq1552})), and it does not enter the final result anyway. It is convenient to assume that the volatility \(\boldsymbol{\zeta}\) of the futures price and the bond price volatility \({\bf b}(t,D)\) are deterministic functions.
For convenience, we write \(f_{t}\) instead of \(f_{Z}(t,R)\). Consider a European option with the terminal payoff \(C_{t}^{f}=p(t,D)\cdot (f_{T}-K)^{+}\) at time \(T\), or equivalently \((f_{T}-K)^{+}\) at time \(D\) (hence it is a standard futures option with deferred payoff). Assume that the price at time \(t\) of such an option admits the representation \(C_{t}^{f}=v(f_{t},p(t,D),t)\) for some function \(v:\mathbb{R}_{+}\times [0,1]\times [0,T]\rightarrow \mathbb{R}\), which satisfies the terminal condition \(v(x,y,T)=y\cdot (x-K)^{+}\) for \((x,y)\in \mathbb{R}_{+}\times (0,1]\). We now consider a self-financing trading strategy \(\boldsymbol{\psi}=(\psi^{1},\psi^{2})\), where \(\psi^{1}\) and \(\psi^{2}\) represent positions in futures contracts and \(D\)=maturity zero-coupon bonds, respectively. It is apparent that the wealth process \(V^{f}(\boldsymbol{\psi})\) satisfies
\begin{equation}{\label{museq1551}}\tag{197}
V_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{2}\cdot p(t,D)=v(f_{t}.p(t,D),t),
\end{equation}
where the first equality is from (\ref{museq86}) and the second equality is a consequence of the assumption that the trading strategy \(\boldsymbol{\psi}\) replicates the value of the option. Furthermore, since \(\boldsymbol{\psi}\) is self-financing,its wealth process \(V^{f}(\boldsymbol{\psi})\) also satisfies
\[dV_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{1}\cdot df_{t}+\psi_{t}^{2}\cdot dp(t,D)\]
so that, from (\ref{museq1542}) and (\ref{museq1550}),
\begin{equation}{\label{museq1552}}\tag{198}
dV_{t}^{f}(\boldsymbol{\psi})=\psi_{t}^{2}\cdot\lambda_{t}\cdot
p(t,D)dt+\left (\psi_{t}^{1}\cdot f_{t}\cdot\boldsymbol{\zeta}_{t}+
\psi_{t}^{2}\cdot p(t,D)\cdot {\bf b}(t,D)\right )\cdot d{\bf W}_{t}.
\end{equation}
On the other hand, assuming that the function \(v(x,y,t)\) is sufficiently smooth, and applying Ito’s formula, we get
\[dC_{t}^{f}=v_{t}dt+v_{x}df_{t}+v_{y}\cdot dp(t,D)+\frac{1}{2}\left (
v_{xx}\cdot d\langle f\rangl\mathbb{E}_{t}+y_{yy}\cdot d\langle p(\cdot ,D)
\rangl\mathbb{E}_{t}\right )+v_{xy}\cdot d\langle f,p(\cdot ,D)\rangl\mathbb{E}_{t}.\]
Substituting the dynamics of \(f\) and \(p(t,D)\) using (\ref{museq1542}) and (\ref{museq1550}), we get
\begin{align}
dC_{t}^{f} & =\left (v_{t}+p(t,D)\cdot\lambda_{t}\cdot v_{y}+f_{t}\cdot
p(t,D)\cdot\boldsymbol{\zeta}_{t}\cdot {\bf b}(t,D)\cdot v_{xy}\right )dt\nonumber\\
& +\frac{1}{2}\left (f_{t}^{2}\cdot\parallel\boldsymbol{\zeta}_{t}
\parallel^{2}\cdot v_{xx}+p^{2}(t,D)\parallel {\bf b}(t,D)\parallel^{2}\cdot v_{yy}\right )dt\nonumber\\
& +\left (f_{t}\cdot\boldsymbol{\zeta}_{t}\cdot v_{x}+p(t,D)\cdot{\bf b}(t,D)\cdot v_{y}\right )\cdot d{\bf W}_{t}.\label{museq100}\tag{199}
\end{align}
Comparing the \(d{\bf W}_{t}\) part of equations (\ref{museq1552}) and (\ref{museq100}), we obtain the following equality
\[\int_{0}^{t}\left (\psi_{s}^{1}\cdot f_{s}\cdot\boldsymbol{\zeta}_{s}+\psi_{s}^{2}\cdot p(t,D)\cdot {\bf b}(s,D)-f_{s}\cdot
\boldsymbol{\zeta}_{s}\cdot v_{x}-p(t,D)\cdot {\bf b}(s,D)\cdot v_{y}\right )\cdot d{\bf W}_{s}=0,\]
which is valid for every \(t\in [0,T]\). Using (\ref{museq1551}), we find that
\begin{equation}{\label{museq1553}}\tag{200}
\int_{0}^{t}\left (\psi_{s}^{1}\cdot f_{s}\cdot\boldsymbol{\zeta}_{s}
+v\cdot {\bf b}(s,D)-f_{s}\cdot\boldsymbol{\zeta}_{s}\cdot v_{x}-
p(t,D)\cdot {\bf b}(s,D)\cdot v_{y}\right )\cdot d{\bf W}_{s}=0,
\end{equation}
Suppose now that
\begin{equation}{\label{museq1554}}\tag{201}
\psi_{t}^{1}=v_{x}(f_{t},p(t,D),t)\mbox{ for all }t\in [0,T].
\end{equation}
Then for (\ref{museq1553}) to hold, it is enough to assume that
\begin{equation}{\label{museq1555}}\tag{202}
v(f_{t}.p(t,D),t)-p(t,D)\cdot v_{y}(f_{t},p(t,D),t)=0,
\end{equation}
or equivalently, that the equality
\begin{equation}{\label{museq1556}}\tag{203}
v(x,y,t)=y\cdot v_{y}(x,y,t)
\end{equation}
is satisfied. Combining (\ref{museq1551}) and (\ref{museq1555}), we find
\begin{equation}{\label{museq2000}}\tag{204}
\psi_{t}^{2}=v_{y}(f_{t},p(t,D),t)\mbox{ for all }t\in [0,T].
\end{equation}
By taking partial derivative of (\ref{museq1556}) with respect to \(x\) and \(y\), we get
\begin{equation}{\label{museq1557}}\tag{205}
\left\{\begin{array}{l}
y\cdot v_{xy}(x,y,t)=v_{x}(x,y,t);\\
y\cdot v_{yy}(x,y,t)=0.
\end{array}\right .
\end{equation}
Using (\ref{museq1554}), (\ref{museq2000}), (\ref{museq1552}) and (\ref{museq100}), we find t
\[C_{t}^{f}-V_{t}^{f}(\boldsymbol{\zeta})=\int_{0}^{t}\left (v_{t}+
\frac{1}{2}f_{s}^{2}\cdot\parallel\boldsymbol{\zeta}_{s}
\parallel^{2}\cdot v_{xx}+\frac{1}{2}p^{2}(s,D)\cdot\parallel
{\bf b}(s,D)\parallel^{2}\cdot v_{yy}+f_{s}\cdot p(s,D)\cdot
\boldsymbol{\zeta}_{s}\cdot {\bf b}(s,D)\cdot v_{xy}\right )ds\]
for every \(t\in [0,T]\). Since \(C_{t}^{f}\) and \(V_{t}^{f}(\boldsymbol{\psi})\), this leads to the following PDE
\[v_{t}+\frac{1}{2}\parallel\boldsymbol{\zeta}_{t}\parallel^{2}\cdot
x^{2}\cdot v_{xx}+\frac{1}{2}\parallel {\bf b}(t,D)\parallel^{2}\cdot y^{2}
\cdot v_{yy}+\boldsymbol{\zeta}_{t}\cdot {\bf b}(t,D)\cdot xy\cdot v_{xy}=0.\]
Finally, taking into account (\ref{museq1557}), we find that \(v\) satisfies
\begin{equation}{\label{museq1558}}\tag{206}
v_{t}+\frac{1}{2}\parallel\boldsymbol{\zeta}_{t}\parallel^{2}\cdot
x^{2}\cdot v_{xx}+\boldsymbol{\zeta}_{t}\cdot {\bf b}(t,D)\cdot x\cdot v_{x}=0
\end{equation}
subject to the terminal condition \(v(x,y,T)=y\cdot (x-K)^{+}\) for \((x,y)\in\mathbb{R}_{+}\times (0,1]\). In view of (\ref{museq1556}), we restrict the attention to those solutions of (\ref{museq1558}) which admit the representation \(v(x,y,t)=y\cdot H(x,t)\) for a ceratin function \(H:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) (indeed, if a function \(v\) satisfies(\ref{museq1556}), then the function \(h(x,y,t)=v(x,y,t)/y\) does not depend on the second argument since the partial derivative \(h_{y}(x,y,t)\) vanishes). Using (\ref{museq1558}), we deduce that \(F\) satisfies
\[H_{t}(x,t)+\frac{1}{2}\parallel\boldsymbol{\zeta}_{t}\parallel^{2}
\cdot x^{2}\cdot H_{xx}(x,t)+\boldsymbol{\zeta}_{t}\cdot {\bf b}(t,D)\cdot x\cdot H_{x}(x,t)=0\]
with the terminal condition \(H(x,T)=(x-K)^{+}\). In order to simplify the above equation, we define the function \(L:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) by setting \(L(z,t)=H(z\cdot\exp (-\eta (t,T)),t)\), where
\[\eta (t,T)=\int_{t}^{T}\boldsymbol{\zeta}_{s}\cdot {\bf b}(s,D)ds\mbox{ for all }t\in [0,T].\]
It is straigntforward to verify that \(L\) solves the PDE
\[L_{t}(z,t)+\frac{1}{2}\parallel\boldsymbol{\zeta}_{t}\parallel^{2}\cdot z^{2}\cdot L_{zz}(z,t)=0\]
with th terminal condition \(L(z,t)=(z-K)^{+}\). Arguing along the same lines as in the previous discussions, we find that
\[L(z,t)=z\cdot N(d_{1}(z,t,T))-K\cdot N(d_{2}(z,t,T)),\]
where
\[d_{1,2}(z,t,T)=\frac{\ln z-\ln K\pm\frac{1}{2}v_{f}^{2}(t,T)}{v_{f}(t,T)}\]
with
\[v_{f}^{2}(t,T)=\int_{t}^{T}\parallel\boldsymbol{\zeta}_{s}
\parallel^{2}ds=\int_{t}^{T}\parallel\boldsymbol{\xi}_{s}-{\bf b}(s,R)\parallel^{2}ds.\]
Recalling that
\[v(x,y,t)=y\cdot H(x,t)=y\cdot L(x\cdot\exp (\eta (t,T)),t),\]
we obtain
\[v(x,y,t)=y\cdot\left (x\cdot\exp (\eta (t,T))\cdot N(l_{1}(x,t,T))-K\cdot N(l_{2}(x,t,T))\right )\]
for \((x,y,t)\in \mathbb{R}_{+}\times [0,1]\times [0,T]\), where
\[l_{1,2}(x,t,T)=\frac{\ln x-\ln K+\eta (t,T)\pm\frac{1}{2}v_{f}^{2}(t,T)}{v_{f}(t,T)}.\]
An any date \(t\), the replicating strategy \(\boldsymbol{\psi}\) involves \(\psi_{t}^{1}\) positions in the underlying futures contract, where \(\psi_{t}^{1}=v_{x}(f_{t},p(t,D),t)\), combined with holding \(\psi_{t}^{2}\) \(D\)-maturity zero-coupon bonds, where
\[\psi_{t}^{2}=v(f_{t},p(t,D),t)/p(t,D)=v_{y}(f_{t},p(t,D),t).\]
More explicitly, we have
\[\psi_{t}^{1}=p(t,D)\cdot\exp (\eta (t,T))\cdot N(l_{1}(f_{t},t,T))
\mbox{ and }\psi_{t}^{2}=f_{t}\cdot\exp (\eta (t,T))\cdot N(l_{1}(f_{t},t,T))-K\cdot N(d_{2}(f_{t},t,T)).\]
Example. Assume that a futures contract has a zero-coupon bond which matures at time \(U\geq R\) as the underlying asset. Then \(\boldsymbol{\xi}_{s}={\bf b}(s,U)\) and thus
\[v_{f}^{2}(t,T)=\int_{t}^{T}\parallel {\bf b}(s,U)-{\bf b}(s,R)\parallel^{2}ds.\]
Moreover, in the case we have
\[l_{1}(x,t,T)=\frac{\ln x-\ln K+\int_{t}^{T}\boldsymbol{\gamma}(s,U,R)\cdot {\bf b}(s,D)ds+\frac{1}{2}\int_{t}^{T}
\parallel\boldsymbol{\gamma}(s,U,R)\parallel^{2}ds}{v_{f}(t,T)},\]
where
\[\boldsymbol{\gamma}(s,U,R)={\bf b}(s,U)-{\bf b}(s,R)\mbox{ for all }s\in [0,T].\]
In a particular case, when \(D=R=T\), we obtain \(l_{1}(x,t,T)=g_{1}(x,t,T)\) and \(\eta (t,T)=h(t,T)\), where the functions \(g_{1}\) and \(h\) are given in (\ref{museq1534}) and (\ref{museq1535}), respectively. \(\sharp\)
We prove the following counterpart of Proposition \ref{musp1531}
Proposition. Suppose that the futures price \(f_{t}=f_{Z}(t,R)\) of an asset \(Z\) satisfies (\ref{museq1550}), where the volatility \(\boldsymbol{\zeta}_{t}=\boldsymbol{\xi}_{t}-{\bf b}(t,R)\) is such that \(\boldsymbol{\zeta}_{t}\cdot {\bf b}(t,D)\) is a deterministic function. Let \(X\) be a European contingent claim which settles at time \(T\) and has the form \(X=p(t,D)\cdot g(f_{T})\) for some function \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}\). The arbitrage price of \(X\) equals
\[\Pi_{t}^{f}(X)=v(f_{t},p(t,D),t)=p(t,D)\cdot L(f_{t}\cdot\exp (\eta (t,T)),t)\]
for every \(t\in [0,T]\), where
\[\eta (t,T)=\int_{t}^{T}(\boldsymbol{\xi}_{s}-{\bf b}(s,R))\cdot {\bf b}(s,D)ds\]
and the function \(L(z,t)\) solves the PDE
\[L_{t}(z,t)+\frac{1}{2}\parallel\boldsymbol{\zeta}_{t}\parallel^{2}\cdot z^{2}\cdot L_{zz}(z,t)=0\]
with the terminal condition \(L(z,T)=g(z)\) for every \(z\in \mathbb{R}_{+}\). \(\sharp\)
