Mathematical Finance In Continuous Time

Charles Haigh Wood (1856-1927) was an English painter.

The topics are

Standard Market Models
Multidimensional Black-Scholes Model
The Greeks
Approximations of Continuous Time Market Models
Futures Markets

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Standard Market Models.

Uncertainty in the financial market is modelled by means of a family of complete filtered probability space $(\Omega ,{\cal F},\mathbb{P})$, $\mathbb{P}\in {\cal P}$, where ${\cal P}$ is a collection of mutually equivalent probability measures on $(\Omega,{\cal F}_{T})$. Events in the financial market are revealed over time according to the filtration $\{{\cal F}_{t}\}_{0\leq t\leq T}$ which is assumed to satisfy the usual conditions of right-continuity and completeness; that is, ${\cal F}_{t}=\cap_{s>t}{\cal F}_{s}$ for every $t\leq T$ and ${\cal F}_{0}$ contains all null sets, i.e., if $B\subset A\in {\cal F}_{0}$ and $\mathbb{P}(A)=0$, then $B\in {\cal F}_{0}$. It is convenient to assume that the $\sigma$-field ${\cal F}_{0}$ is $\mathbb{P}$-trivial (for some $\mathbb{P}$, and thus for all $\mathbb{P}\in {\cal P}$); that is, for every $A\in {\cal F}_{0}$ either $\mathbb{P}(A)=0$ or $\mathbb{P}(A)=1$, and that ${\cal F}_{T}={\cal F}$.

There are $d+1$ primary traded assets, whose price processes are given by stochastic processes $S^{(0)},S^{(1)},\cdots ,S^{(d)}$. We assume that ${\bf S}=(S^{(0)},S^{(1)},\cdots ,S^{(d)})$ follows an adapted, right-continuous with left-limits (RCLL) and strictly positive semimartingale on the filtered probability space $(\Omega,\{{\cal F}_{t}\}_{0\leq t\leq T},\mathbb{P})$ for some $\mathbb{P}$ and thus for all $\mathbb{P}\in {\cal P}$. This means that each process $S^{(i)}$ admits a unique decomposition $S^{(i)}=S_{0}^{(i)}+M^{(i)}+A^{(i)}$, where $M^{(i)}$ is a local martingale, and $A^{(i)}$ is an adapted process of finite variation with $M_{0}^{(i)}=A_{0}^{(i)}=0$. We assume that the processes $S^{(0)}, S^{(1)},\cdots ,S^{(d)}$ represent the prices of some traded assets (stocks, bonds, or options).

Standard Spot Market.

There was an implicit numeraire behind the prices $S^{(0)},,S^{(1)},\cdots , S^{(d)}$; it is the numeraire relevant for domestic transactions at time $t$. A {\bf numeraire} is a price process $X_{t}$ almost surely strictly positive for each $t\in [0,T]$. We assume now that $S_{t}^{(0)}$ is a non-dividend paying asset, which is almost surely strictly positive and use $S^{(0)}$ as numeraire. Historically the money market account $B_{t}$, given by $B_{t}=e^{r_{t}}$ with a positive deterministic process $r_{t}$ and $r_{0}=0$, was used as a numeraire, and we may think of $S_{t}^{(0)}$ as being $B_{t}$.

The principal task will be the pricing and hedging of contingent claims, which we model as ${\cal F}_{T}$-measurable random variables. This implies that the contingent claims specify a stochastic cash-flow at time $T$ and that they may depend on the whole path of the underlying in $[0,T]$ because ${\cal F}_{T}$ contains all that information. We will often have to impose further integrability conditions on the contingent claims under consideration. The fundamental concept in pricing and hedging contingent claims is the interplay of self-financing replicating portfolios and risk-neutral probabilities.

We call an $\mathbb{R}^{d+1}$-valued predictable, locally bounded process
\[\boldsymbol{\phi}_{t}=(\phi^{(0)}_{t},\phi^{(1)}_{t},\cdots ,\phi^{(d)}_{t})\mbox{ for }t\in [0,T]\]
a trading strategy} (or dynamic portfolio process). The conditions ensure that the stochastic integral $\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d{\bf S}_{u}$ exists. Here $\phi_{t}^{(i)}$ denotes the number of shares of asset $i$ held in the portfolio at time $t$, which is to be determined on the basis of information available before time $t$, i.e., the investor selects his time $t$ portfolio after observing the prices ${\bf S}(t-)$. The components $\phi_{t}^{(i)}$ may assume negative as well as positive values, reflecting the fact that we allow short sales and assume that the assets are perfectly divisible.

Definition. The value of the portfolio $\boldsymbol{\phi}$ at time $t$ is given by the scalar product
\begin{equation}{\label{museq101}}\tag{1}
V_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}
=\sum_{i=0}^{d}\phi_{t}^{(i)}\cdot S_{t}^{(i)}\mbox{ for all }t\in [0,T].
\end{equation}
The process $V_{t}(\boldsymbol{\phi})$ is called the value process, or wealth process, of the trading strategy $\boldsymbol{\phi}$. The gains process $G_{t}(\boldsymbol{\phi})$ is defined by
\begin{equation}{\label{museq102}}\tag{2}
G_{t}(\boldsymbol{\phi})=\int_{0}^{t}\boldsymbol{\phi}_{u}
\cdot d{\bf S}_{u}=\sum_{i=0}^{d}\int_{0}^{t}\phi_{u}^{(i)}dS_{u}^{(i)}\mbox{ for all }t\in [0,T].
\end{equation}

A trading strategy $\boldsymbol{\phi}$ is called elf-financing when the wealth process $V_{t}(\boldsymbol{\phi})$ satisfies
\[V_{t}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})+G_{t}(\boldsymbol{\phi})\mbox{ for all }t\in [0,T].\sharp\]

We will write $\boldsymbol{\phi}$ to denote the class of all self-financing trading strategies and use ${\cal M}=({\bf S}, \boldsymbol{\phi})$ to denote the financial market. Observe that since $G_{t}(\boldsymbol{\phi})$ models the gains or losses realized up to and including time $t$, it is clear that we implicitly assume that the securities do not generate any revenue such as dividends. If the dividends of the risky security are paid continuously at the rate $\boldsymbol{\kappa}_{t}=(\kappa_{t}^{(0)},\cdots , \kappa_{t}^{(d)})$, the gains process includes also the accumulated dividend gains and thus
\[G_{t}(\boldsymbol{\phi})=\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d{\bf S}_{u}+\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot
{\bf S}_{u}^{\boldsymbol{\kappa}}du\mbox{ for all }t\in [0,T],\]
where ${\bf S}_{t}^{\boldsymbol{\kappa}}=(\kappa_{t}^{(0)}S_{t}^{(0)},\cdots ,\kappa_{t}^{(d)}S_{t}^{(d)})$. For convenience, we assume that $\boldsymbol{\kappa}={\bf 0}$.

The financial interpretation of equalities (\ref{museq101}) and (\ref{museq102}) is that all changes in the wealth of the portfolio are due to capital gains, as opposed to withdrawals of cash or infusions of new funds. They reflect also the fact that the market is implicitly assumed to be frictionless, meaning that there are no transaction costs and no restrictions on short-selling. Equality (\ref{museq102}) assumes that the process $\boldsymbol{\phi}$ is sufficiently regular so that the stochastic integral in (\ref{museq102}) exists. This property of $\boldsymbol{\phi}$ is also invariant with respect to an equivalent change of the underlying probability measure. More exactly, if for some predictable process $\boldsymbol{\phi}$ the stochastic integral in (\ref{museq102}) exists under some probability measure $\mathbb{P}$, then it exists also under any probability measure $\bar{\mathbb{P}}$ equivalent to $\mathbb{P}$; furthermore, the integrals evaluated under $\mathbb{P}$ and $\bar{\mathbb{P}}$ coincide; that is,
\[\mathbb{P}\mbox{-}\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d{\bf S}_{u}=
\bar{\mathbb{P}}\mbox{-}\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d{\bf S}_{u} \mbox{ for all }t\in [0,T].\]

To avoid technicalities, we assume throughout, unless otherwise specified, that the trading process $\boldsymbol{\phi}$ is locally bounded (this restriction becomes inconvenient when the concept of completeness of a market model is examined). Let us comment briefly on the implications of the assumption. If ${\bf S}$ is a $\mathbb{P}$-local martingale, then the integral
\[N_{t}=\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d{\bf S}_{u}\mbox{ for all }t\in [0,T]\]

is known to follow a $\mathbb{P}$-local martingale (in general, the stochastic integral of a predictable process with respect to a local martingale is not necessarily a local martingale).

\begin{equation}{\label{binp57}}\tag{3}\mbox{}\end{equation}

Theorem (Numeraire Invariance Theorem). Self-financing portfolios remain self-financing after a numeraire change.

Proof. From the product rule (\ref{bineq48}), we get
\begin{equation}{\label{bineq50}}\tag{4}
d\left (\frac{S_{t}^{(i)}}{X_{t}}\right )=S_{t}^{(i)}\cdot d\left (\frac{1}
{X_{t}}\right )+\frac{1}{X_{t}}\cdot dS_{t}^{(i)}+d\langle S_{i},1/X\rangle (t),
\end{equation}
where $\langle S_{i},1/X\rangle (t)$ denotes the quadratic covariation between the semimartingales $S_{i}$ and $1/X$. In the same manner
\[d\left (\frac{V_{t}(\boldsymbol{\phi})}{X_{t}}\right )=V_{t}(\boldsymbol{\phi})\cdot d\left (\frac{1}{X_{t}}\right )+
\frac{1}{X_{t}}\cdot dV_{t}(\boldsymbol{\phi})+d\langle V(\boldsymbol{\phi}),1/X\rangle\mathbb{E}_{t}.\]
Since
\[dV_{t}(\boldsymbol{\phi})=d(V_{0}(\boldsymbol{\phi})+G_{t}(\boldsymbol{\phi}))=dG_{t}(\boldsymbol{\phi})=
\sum_{i=0}^{d}\phi_{t}^{(i)}dS_{t}^{(i)},\]
we have
\begin{align*}
d\left (\frac{V_{t}(\boldsymbol{\phi})}{X_{t}}\right ) & =\left (\sum_{i=0}^{d}\phi_{t}^{(i)}S_{t}^{(i)}\right )\cdot d\left (\frac{1}
{X_{t}}\right )+\frac{1}{X_{t}}\cdot\sum_{i=0}^{d}\phi_{t}^{(i)}dS_{t}^{(i)}+
d\left\langle\sum_{i=0}^{d}\phi_{i}S_{i},\frac{1}{X}\right\rangle\mathbb{E}_{t}\\
& =\sum_{i=0}^{d}\phi_{t}^{(i)}\left [S_{t}^{(i)}\cdot d\left (\frac{1}
{X_{t}}\right )+\frac{1}{X_{t}}\cdot dS_{t}^{(i)}+d\langle S_{i},1/X\rangle\mathbb{E}_{t}\right ]\\
& =\sum_{i=0}^{d}\phi_{t}^{(i)}\cdot d\left (\frac{S_{t}^{(i)}}{X_{t}}\right )\mbox{ (by (\ref{bineq50}))}.
\end{align*}
That is,
\[\frac{V_{t}(\boldsymbol{\phi})}{X_{t}}=\frac{V_{0}(\boldsymbol{\phi})}{X_{0}}+\sum_{i=0}^{d}
\int_{0}^{t}\phi_{t}^{(i)}d\left (\frac{S_{t}^{(i)}}{X_{t}}\right )=\frac{V_{0}(\boldsymbol{\phi})}{X_{0}}+
\frac{G_{t}(\boldsymbol{\phi})}{X_{t}}.\]
Thus the portfolio expressed in the new numeraire remains self-financing. $\blacksquare$

Turning back to the special numeraire $S_{t}^{(0)}$ we consider the discounted price process
\[\bar{{\bf S}}_{t}=\frac{{\bf S}_{t}}{S_{t}^{(0)}}=(1,\bar{S}_{t}^{(1)},\cdots ,\bar{S}_{t}^{(d)})\]
with $\bar{S}_{t}^{(i)}=S_{t}^{(i)}/S_{t}^{(0)}$ for $i=1,2,\cdots ,d$. Furthermore, the discounted wealth process $\bar{V}_{t}(\boldsymbol{\phi})$ is given by
\[\bar{V}_{t}(\boldsymbol{\phi})=\frac{V_{t}(\boldsymbol{\phi})}{S_{t}^{(0)}}=\phi^{(0)}_{t}+
\sum_{i=1}^{d}\phi_{t}^{(i)}\bar{S}_{t}^{(i)}\]
and the discounted gains process $\bar{G}_{t}(\boldsymbol{\phi})$ is
\[\bar{G}_{t}(\boldsymbol{\phi})=\sum_{i=1}^{d}\int_{0}^{t}\phi_{t}^{(i)}d\bar{S}_{t}^{(i)}.\]
Observe that $\bar{G}_{t}(\boldsymbol{\phi})$ does not depend on the numeraire component $\phi^{(0)}$.

Proposition. Let $\boldsymbol{\phi}$ be a trading strategy. Then $\boldsymbol{\phi}$ is self-financing if and only if $\bar{V}_{t}(\boldsymbol{\phi})=\bar{V}_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi})$. Of course, $V_{t}(\boldsymbol{\phi})\geq 0$ if and only if $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$.

Proof. The proof follows by the numeraire invariance theorem using $S^{(0)}$ as numeraire. $\blacksquare$

\begin{equation}{\label{binr612}}\tag{5}\mbox{}\end{equation}

Remark \ref{binr612}. The above result shows that a self-financing strategy is completely determined by its initial value and the components $\phi^{(1)},\cdots ,\phi^{(d)}$. In other words, any set of predictable processes $\phi^{(1)},\cdots ,\phi^{(d)}$ such that the stochastic integrals $\int\phi^{(i)}d\bar{S}^{(i)}$ for $i=1,\cdots ,d$ exist can be uniquely extended to a self-financing strategy $\boldsymbol{\phi}$ with specified initial value $\bar{V}_{0}(\boldsymbol{\phi})=v$ by setting
\begin{equation}{\label{bineq3}}\tag{6}
\phi^{(0)}_{t}=v+\sum_{i=1}^{d}\int_{0}^{t}\phi_{u}^{(i)}d\bar{S}_{u}^{i}-
\sum_{i=1}^{d}\phi_{t}^{(i)}\bar{S}_{t}^{(i)}\mbox{ for }t\in [0,T]. \sharp
\end{equation}

Definition. A self-financing trading strategy $\boldsymbol{\phi}$ is called an arbitrage opportunity when the wealth process $V(\boldsymbol{\phi})$ satisfies the following conditions:
\[V_{0}(\boldsymbol{\phi})=0,\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1
\mbox{ and }\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})>0\right\}>0. \sharp\]

\begin{equation}{\label{r135}}\tag{7}\mbox{}\end{equation}

Remark \ref{r135}. We know that $V_{T}(\boldsymbol{\phi})= V_{0}(\boldsymbol{\phi})+G_{T}(\boldsymbol{\phi})$. If $\boldsymbol{\phi}$ is an arbitrage opportunity then $V_{T}(\boldsymbol{\phi})=G_{T}(\boldsymbol{\phi})$; that is, if $\boldsymbol{\phi}$ is an arbitrage opportunity then $G_{0}(\boldsymbol{\phi})=0$, $G_{T}(\boldsymbol{\phi})\geq 0$ a.s. and $\mathbb{P}\left\{G_{T}(\boldsymbol{\phi})>0\right\}>0$. $\sharp$

Arbitrage opportunities represent the limitless creation of wealth through risk-free profit and thus they should not exist in a well-functioning market (in practice, they should disappear rapidly). We say that a probability measure $\bar{\mathbb{P}}$ is equivalent to ${\cal P}$ if it is equivalent to some probability measure $\mathbb{P}$ from ${\cal P}$, and thus, of course, to any probability measure $\mathbb{P}$ from ${\cal P}$.

Definition. We say that a probability measure $\bar{\mathbb{P}}$ defined on $(\Omega , {\cal F}_{T})$, which is equivalent to ${\cal P}$, is called a

(resp. strong) equivalent martingale measure for $\bar{{\bf S}}$ when the discounted price process $\bar{{\bf S}}$ is a $\bar{\mathbb{P}}$-local martingale (resp. martingale).
(resp. strong) equivalent martingale measure for the market ${\cal M}$ when the discounted wealth process $\bar{V}(\boldsymbol{\phi})$ follows a $\bar{\mathbb{P}}$-local martingale measure (resp. martingale). $\sharp$

Let us denote by ${\cal P}(\bar{{\bf S}})$ and by ${\cal P}({\cal M})$ the class, possibly empty, of all equivalent martingale measures for $\bar{{\bf S}}$ and ${\cal M}$, respectively. Then we have the following simple result, which shows that ${\cal P}(\bar{{\bf S}})$ and ${\cal P}({\cal M})$ coincide.

\begin{equation}{\label{musl1011}}\tag{8}\mbox{}\end{equation}

Proposition \ref{musl1011}. A probability measure $\bar{\mathbb{P}}$ is a equivalent martingale measure for $\bar{{\bf S}}$ if and only if it is a equivalent martingale measure for ${\cal M}$.

Proof. (Note that the proof is similar to that of Theorem \ref{binp57}.) For the inclusion ${\cal P}(\bar{{\bf S}})\subseteq {\cal P}({\cal M})$, note that using Ito’s integration by parts formula, we obtain
\begin{align*}
d\bar{V}_{t}(\boldsymbol{\phi}) & =\frac{1}{S_{t}^{(0)}}dV_{t}(\boldsymbol{\phi})+V_{t}(\boldsymbol{\phi})
d\left (\frac{1}{S_{t}^{(0)}}\right )+d\left\langle V(\boldsymbol{\phi}),\frac{1}{S^{(0)}}\right\rangle\mathbb{E}_{t}\\
& =\boldsymbol{\phi}_{t}\cdot\left (\frac{1}{S_{t}^{(0)}}d{\bf S}_{t}+{\bf S}_{t}d\left (\frac{1}{S_{t}^{(0)}}\right )+
d\left\langle {\bf S},\frac{1}{S^{(0)}}\right\rangle\mathbb{E}_{t}\right )\\
& =\boldsymbol{\phi}_{t}\cdot d\bar{{\bf S}}_{t}\end{align*}

since $dV_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot d{\bf S}_{t}$. Put another way, we have
\begin{equation}{\label{museq103}}\tag{9}
\bar{V}_{t}(\boldsymbol{\phi})=\bar{V}_{0}(\boldsymbol{\phi})+\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}=
\bar{V}_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi}),
\end{equation}
where $\bar{G}_{t}(\boldsymbol{\phi})$ follows a local $\bar{\mathbb{P}}$-martingale (recall that $\boldsymbol{\phi}$ is a locally bounded process), so $\bar{V}_{t}(\boldsymbol{\phi})$ is a local $\bar{\mathbb{P}}$-martingale. For the converse inclusion, it is enough to observe that buy-and-hold strategies of the form $(0,\cdots ,1,\cdots ,0)$. $\blacksquare$

Proposition. Assume $S_{t}^{(0)}=B_{t}=e^{r_{t}}$, then $\bar{\mathbb{P}}$ equivalent to $\mathbb{P}$ is a martingale measure if and only if every asset price process $S^{(i)}$ has price dynamics under $\bar{\mathbb{P}}$ of the form
\[dS_{t}^{(i)}=r_{t}S_{t}^{(i)}dt+dM_{t}^{(i)},\]
where $M_{t}^{(i)}$ is a $\bar{\mathbb{P}}$-local martingale. $\sharp$

A self-financing trading strategy $\boldsymbol{\phi}$ is called tame relative to the numeraire $S^{(0)}$ if $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$ for all $t\in [0,T]$. We use the notation $\bar{\boldsymbol{\phi}}$ for the set of tame trading strategies. (Bingham and Kiesel \cite[p.175]{bin})Let us consider another definition. Also, a strategy is said to be tame relative to $S^{(0)}$ if there exists $m\in \mathbb{R}$ such that the discounted wealth $\bar{V}_{t}(\boldsymbol{\phi})\geq m$ for all $t\in [0,T]$. Note that the class of tame strategies is manifestly invariant with respect to an equivalent change of a probability measure, however it is not invariant with respect to the choice of the numeraire asset (Musiela and Rutkowski \cite[p.235]{mus}).

\begin{equation}{\label{binp52}}\tag{10}\mbox{}\end{equation}

Proposition \ref{binp52}. If $\boldsymbol{\phi}$ is a tame trading strategy, then $\bar{V}_{t}(\boldsymbol{\phi})$ is a nonnegative local martingale, and also a supermartingale, under each equivalent martingale measure.

Proof. Since $\bar{{\bf S}}_{t}$ is a local martingale and $\boldsymbol{\phi}\in\bar{\boldsymbol{\phi}}$ is predictable and locally bounded, it follows that the stochastic integral $\bar{G}_{t}(\boldsymbol{\phi})=\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}$ is a local martingale. Now $\bar{V}_{t}(\boldsymbol{\phi})=\bar{V}_{0}(\boldsymbol{\phi})+\bar{G}_{t}(\boldsymbol{\phi})$, so $\bar{V}_{t}(\boldsymbol{\phi})$ is a local martingale, and nonnegative since $\boldsymbol{\phi}$ is tame. The nonnegative local martingales are supermartingales. $\blacksquare$

\begin{equation}{\label{binp53}}\tag{11}\mbox{}\end{equation}

Theorem \ref{binp53}. Assume ${\cal P}(\bar{{\bf S}})\neq\emptyset$. Then the market model contains no arbitrage opportunities in $\bar{\boldsymbol{\phi}}$.

Proof. For any $\boldsymbol{\phi}\in\bar{\boldsymbol{\phi}}$ and under any equivalent martingale measure $\bar{\mathbb{P}}$, $\bar{V}_{t}(\boldsymbol{\phi})$ is a supermartingale by Proposition \ref{binp52}. That is,
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\bar{V}_{t}(\boldsymbol{\phi})\right |
{\cal F}_{u}\right ]\leq\bar{V}_{u}(\boldsymbol{\phi})\mbox{ for }u\leq t\leq T.\]
For $\boldsymbol{\phi}\in\bar{\boldsymbol{\phi}}$ to be an arbitrage opportunity, we must have $\bar{V}_{0}(\boldsymbol{\phi})=V_{0}(\boldsymbol{\phi})=0$. So
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\bar{V}_{t}(\boldsymbol{\phi})\right ]=
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left.\bar{V}_{t}(\boldsymbol{\phi})\right | {\cal F}_{0}\right
]\leq\bar{V}_{0}(\boldsymbol{\phi})=0 \mbox{ for all }0\leq t\leq T\]

since ${\cal F}_{0}=\{\emptyset ,\Omega\}$. Now $\boldsymbol{\phi}$ is tame, so $\bar{V}_{t}(\boldsymbol{\phi})\geq 0$ for $0\leq t\leq T$, which implies $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\bar{V}_{t}(\boldsymbol{\phi})\right ] =0$ for $0\leq t\leq T$. In particular, $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\bar{V}_{T}(\boldsymbol{\phi})\right ]=0$. Since $V_{T}(\boldsymbol{\phi})=S_{t}^{(0)}\cdot\bar{V}_{T}(\boldsymbol{\phi})$, and $S_{t}^{(0)}$ is a strictly positive random variable $\bar{\mathbb{P}}$-a.s., it is easily seen that $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [V_{T}(\boldsymbol{\phi})\right ]=0$. But an arbitrage opportunity $\boldsymbol{\phi}$ also satisfies $\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1$. Since $\bar{\mathbb{P}}$ and $\mathbb{P}$ are equivalent, this means $\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1$. Therefore $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [V_{T}(\boldsymbol{\phi}) \right ]=0$ and $\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1$ imply $0=\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})>0 \right\}=\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})>0\right\}$. The contradiction occurs and the proof is completed. $\blacksquare$

Let $\bar{\mathbb{P}}$ be an equivalent martingale measure. We now restrict our attention to contingent claims $X$ such that $X/S_{T}^{(0)}\in L^{1}({\cal F},\bar{\mathbb{P}})$. A self-financing trading strategy $\boldsymbol{\phi}$ is called $\bar{\mathbb{P}}$-admissible when the discounted gains process
\[\bar{G}_{t}(\boldsymbol{\phi})=\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}\]

is a $\bar{\mathbb{P}}$-martingale. The class of all $\bar{\mathbb{P}}$-admissible trading strategies is denoted by $\boldsymbol{\phi}(\bar{\mathbb{P}})$. We also denote by ${\cal M}(\bar{\mathbb{P}})=({\bf S}, \boldsymbol{\Phi}(\bar{\mathbb{P}}))$ the spot market model. We do not assume that admissible trading strategies are tame. However, since we only used the supermartingale property of $\bar{G}_{t}$ in the proof of Theorem \ref{binp53}, we can repeat the argument to obtain the following theorem.

A contingent claim $X$ is called attainable when there exists at least an admissible trading strategy satisfying $V_{T}(\boldsymbol{\phi})=X$. We call such a trading strategy $\boldsymbol{\phi}$ a replicating strategy for $X$. The financial market model ${\cal M}$ is said to be complete when any contingent claim is attainable. We say that a contingent claim $X$ is attainable in ${\cal M}(\bar{\mathbb{P}})$ if $X$ is $\boldsymbol{\phi}(\bar{\mathbb{P}})$-attainable.

\begin{equation}{\label{musp1011}}\tag{12}\mbox{}\end{equation}

Theorem \ref{musp1011}. The spot market model ${\cal M}(\bar{\mathbb{P}})$ contains no arbitrage opportunities in $\boldsymbol{\phi}(\bar{\mathbb{P}})$; that is, for any equivalent martingale measure $\bar{\mathbb{P}}\in {\cal P}({\cal M})$, the spot market model ${\cal M}(\bar{\mathbb{P}})$ is arbitrage-free. Any contingent claim $X$ attainable in ${\cal M}(\bar{\mathbb{P}})$ admits a unique replicating process in $\boldsymbol{\phi}(\bar{\mathbb{P}})$.

Proof. To prove the first statement, it is enough to verify that the class $\boldsymbol{\phi}(\bar{\mathbb{P}})$ of trading strategies does not contain arbitrage opportunities. For any $\boldsymbol{\phi}\in\boldsymbol{\phi}(\bar{\mathbb{P}})$, we must have $\bar{V}_{0}(\boldsymbol{\phi})= V_{0}(\boldsymbol{\phi})=0$. So, from Remark \ref{r135}, we have $\bar{V}_{t}(\boldsymbol{\phi})=\bar{G}_{t}(\boldsymbol{\phi})$, where $\bar{G}(\boldsymbol{\phi})$ follows a martingale under $\bar{\mathbb{P}}$ (by $\mathbb{P}$-admissibility) with $\bar{G}_{0}(\boldsymbol{\phi})=0$, i.e., $\bar{V}(\boldsymbol{\phi})$ follows a martingale under $\bar{\mathbb{P}}$. Therefore, we have
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\bar{V}_{t}(\boldsymbol{\phi})\right ]=
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\bar{V}_{t}(\boldsymbol{\phi})\right | {\cal F}_{0}\right
]=\bar{V}_{0}(\boldsymbol{\phi})=0 \mbox{ for all }0\leq t\leq T\]

since ${\cal F}_{0}=\{\emptyset ,\Omega\}$. In particular, $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[\bar{V}_{T}(\boldsymbol{\phi})]=0$. Since $V_{T}(\boldsymbol{\phi})=S_{t}^{(0)}\cdot\bar{V}_{T}(\boldsymbol{\phi})$, and $S_{t}^{(0)}$ is a strictly positive random variable $\bar{\mathbb{P}}$-a.s., it is easily seen that $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [V_{T}(\boldsymbol{\phi})\right ]=0$. But an arbitrage opportunity $\boldsymbol{\phi}$ also satisfies $\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1$. Since $\bar{\mathbb{P}}$ and $\mathbb{P}$ are equivalent, this means $\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})\geq 0\right\}=1$. Therefore $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [V_{T}(\boldsymbol{\phi}) \right ]=0$ and $\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})\geq 0 \right\}=1$ imply $0=\bar{\mathbb{P}}\left\{V_{T}(\boldsymbol{\phi})>0 \right\}=\mathbb{P}\left\{V_{T}(\boldsymbol{\phi})>0\right\}$. The contradiction occurs. For the second assertion, note that if the uniqueness of a replicating process were violated, it would be possible to construct an arbitrage opportunity along the same lines in the proof of Proposition \ref{musp311}, by investing in the $S^{(0)}$ asset (note that the discounted price of this asset is manifestly a $\bar{\mathbb{P}}$-martingale. $\blacksquare$

Under the assumption that no arbitrage opportunities exist, the question of pricing and hedging a contingent claim reduces to the existence of replicating self-financing trading strategies. Again we emphasize that this depends on the class of trading strategies. On the other hand, it does not depend on the numeraire. We can show that if a contingent claim is attainable in a given numeraire it is also attainable in any other numeraire and the replicating strategies are the same. If a contingent claim $X$ is attainable, $X$ can be replicated by a portfolio $\boldsymbol{\phi}\in\boldsymbol{\phi}(\bar{\mathbb{P}})$. This means that holding the portfolio and holding the contingent claim are equivalent from a financial point of view. In the absence of arbitrage, the arbitrage price process $\Pi_{t}(X|\bar{\mathbb{P}})$ of the contingent claim must therefore satisfy $\Pi_{t}(X|\bar{\mathbb{P}})= V_{t}(\boldsymbol{\phi})$.

\begin{equation}{\label{binp56}}\tag{13}\mbox{}\end{equation}

Theorem \ref{binp56} (Risk-Neutral Valuation Formula). The arbitrage price process of any attainable claim is given by the risk-neutral valuation formula
\[\Pi_{t}(X|\bar{\mathbb{P}})=S_{t}^{(0)}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ].\]

Proof. Since $X$ is attainable, there exists a replicating strategy $\boldsymbol{\phi}\in\boldsymbol{\phi}(\bar{\mathbb{P}})$ such that $V_{T}(\boldsymbol{\phi})=X$ and $\Pi_{t}(X|\bar{\mathbb{P}})= V_{t}(\boldsymbol{\phi})$. Since $\boldsymbol{\phi}\in \boldsymbol{\phi}(\bar{\mathbb{P}})$, the discounted value process $\bar{V}_{t}(\boldsymbol{\phi})$ is a martingale, and hence
\begin{align*}
\Pi_{t}(X|\bar{\mathbb{P}}) & =V_{t}(\boldsymbol{\phi})=S_{t}^{(0)}\cdot\bar{V}_{t}(\boldsymbol{\phi})\\
& =S_{t}^{(0)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\bar{V}_{T}(\boldsymbol{\phi})\right |{\cal F}_{t}\right ]=S_{t}^{(0)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{V_{T}(\boldsymbol{\phi})}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\\ & =S_{t}^{(0)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ].\end{align*}
This completes the proof. $\blacksquare$

Corollary. For any two replicating portfolios $\boldsymbol{\phi}, \boldsymbol{\psi}\in\boldsymbol{\phi}(\bar{\mathbb{P}})$ we have $V_{t}(\boldsymbol{\phi})=V_{t}(\boldsymbol{\psi})$. $\sharp$

The choice of $\bar{\mathbb{P}}$ was itself completely arbitrary, and we could have several equivalent martingale measures $\bar{\mathbb{P}}$. This makes the definition of admissible strategies with respect to different martingale measures somewhat unsatisfactory. Therefore, we need to show that, for any two equivalent martingale measures $\bar{\mathbb{P}}_{1}$ and $\bar{\mathbb{P}}_{2}$, their arbitrage prices agree.

Definition. A self-financing trading strategy $\boldsymbol{\phi}$ is said to be admissible relative to $S^{(0)}$ when it is a tame strategy and is $\bar{\mathbb{P}}$-admissible for some equivalent martingale measure $\bar{\mathbb{P}}\in {\cal P}({\cal M})$. We write $\boldsymbol{\phi}_{0}$ to denote the class of all strategies that are admissible relative to $S^{(0)}$. The pair ${\cal M}=({\bf S},\boldsymbol{\phi}_{0})$ is referred to as the standard spot market model. $\sharp$

Proposition. The standard spot market model ${\cal M}= ({\bf S},\boldsymbol{\phi}_{0})$ is arbitrage-free. The arbitrage price of any contingent claim attainable in ${\cal M}$ is well-defined. If a contingent claim $X$, which settles at time $T$, is $\boldsymbol{\phi}_{0}(\bar{\mathbb{P}}_{i})$-attainable for $i=1,2$, then for every $t\in [0,T]$
\[\Pi_{t}(X|\bar{\mathbb{P}}_{1})=\Pi_{t}(X|\bar{\mathbb{P}}_{2}),\]
or equivalently,
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{1}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}
\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ].\]

Proof. Using the similar proofs in Theorem \ref{binp53} or Theorem \ref{musp1011}, it shows that there are no arbitrage opportunities in $\boldsymbol{\phi}_{0}$. To prove the second statement, consider a $\boldsymbol{\phi}_{0}$-attainable contingent claim $X$ which settles at time $T$. Let $\boldsymbol{\phi},\boldsymbol{\psi} \in\boldsymbol{\phi}_{0}$ be two strategies such that $V_{T}(\boldsymbol{\phi})=V_{T}(\boldsymbol{\psi})=X$. Let $\boldsymbol{\phi}\in\boldsymbol{\phi}_{0}(\bar{\mathbb{P}}_{1})$ and $\boldsymbol{\psi}\in\boldsymbol{\phi}_{0}(\bar{\mathbb{P}}_{2})$. Then
\[V_{t}(\boldsymbol{\phi})=S_{t}^{(0)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{1}}
\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\mbox{ and }
V_{t}(\boldsymbol{\psi})=S_{t}^{(0)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}
\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ].\]
On the other hand, from (\ref{museq103}), we get
\[\bar{V}_{T}(\boldsymbol{\phi})=\bar{V}_{t}(\boldsymbol{\phi})+
\int_{t}^{T}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}=
\bar{V}_{t}(\boldsymbol{\phi})+\bar{G}_{T}(\boldsymbol{\phi})- \bar{G}_{t}(\boldsymbol{\phi}),\]

where $\bar{G}(\boldsymbol{\phi})$ is a $\bar{\mathbb{P}}_{1}$-martingale. Since $\bar{G}(\boldsymbol{\phi})$ follows a supermartingale under $\bar{\mathbb{P}}_{2}$, we have
\begin{align*}
\bar{V}_{t}(\boldsymbol{\psi}) & =\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}\left [\left .
\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\\ & =\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}
\left [\left .\frac{V_{T}(\boldsymbol{\phi})}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\\ & =\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}\left [\left .\bar{V}_{T}(\boldsymbol{\phi})\right |{\cal F}_{t}\right ]\\
& =\bar{V}_{t}(\boldsymbol{\phi})+\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{2}}\left [
\bar{G}_{T}-\bar{G}_{t}|{\cal F}_{t}\right ]\leq\bar{V}_{t}(\boldsymbol{\phi}),
\end{align*}
and thus $V_{t}(\boldsymbol{\psi})\leq V_{t}(\boldsymbol{\phi})$. Interchanging the roles, we obtain $V_{t}(\boldsymbol{\phi})\leq V_{t}(\boldsymbol{\psi})$ and thus the equality $V_{t}(\boldsymbol{\psi})=V_{t}(\boldsymbol{\phi})$ is satisfied for every $t<T$. $\blacksquare$

The common value of an arbitrage price $\Pi_{t}(X|\bar{\mathbb{P}})$ is denoted by $\Pi_{t}(X)$ and is referred to as the arbitrage price of $X$ in ${\cal M}=({\bf S},\boldsymbol{\phi}_{0})$.

Proposition. Let $X$ be a contingent claim which settles at time $T$ and is attainable in the market model ${\cal M}=({\bf S},\boldsymbol{\phi}_{0})$. The aribitrage price of $X$ satisfies
\begin{equation}{\label{museq108}}\tag{14}
\Pi_{0}(X)=\sup_{\bar{\mathbb{P}}\in {\cal P}({\cal M})}S_{0}^{(0)}\cdot
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\frac{X}{S_{T}^{(0)}}\right ]=
\inf_{\boldsymbol{\phi}\in\theta (X)}V_{0}(\boldsymbol{\phi}),
\end{equation}
where $\theta (X)$ is the class of all tame trading strategies replicating $X$.

Proof. Since a claim $X$ is $\boldsymbol{\phi}_{0}(\bar{\mathbb{P}}_{1})$-attainable for some martingale measure $\bar{\mathbb{P}}_{1}\in {\cal P}({\cal M})$, there exists a strategy satisfying
\[\frac{X}{S_{T}^{(0)}}=\frac{V_{T}(\boldsymbol{\phi})}{S_{T}^{(0)}}=
\bar{V}_{T}(\boldsymbol{\phi})=\bar{V}_{0}(\boldsymbol{\phi})+ \int_{0}^{T}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}= \frac{\Pi_{0}(X)}{S_{0}^{(0)}}+\bar{G}_{T}(\boldsymbol{\phi}).\]

Under the assumptions, the process $\bar{G}(\boldsymbol{\phi})$ follows a supermartingale under any martingale measure $\bar{\mathbb{P}}\in {\cal P}({\cal M})$, and thus
\begin{equation}{\label{museq109}}\tag{15}
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\frac{X}{S_{T}^{(0)}}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}
\left .\left [\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{0}\right ]\leq
\frac{\Pi_{0}(X)}{S_{0}^{(0)}}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{1}}\left [\frac{X}{S_{T}^{(0)}}\right ],
\end{equation}
since $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left [\bar{G}_{T}(\boldsymbol{\phi}) \right |{\cal F}_{0}\right ]\leq \bar{G}_{0}(\boldsymbol{\phi})=0$. This ends the proof of the first equality. For the second, assume that $\boldsymbol{\psi}$ is an arbitrary tame strategy belonging to $\theta (X)$ so that
\[\frac{X}{S_{T}^{(0)}}=\bar{V}_{0}(\boldsymbol{\psi})+\int_{0}^{T}
\boldsymbol{\psi}_{u}\cdot d\bar{{\bf S}}_{u}=\bar{V}_{0}(\boldsymbol{\psi})+\bar{G}_{T}(\boldsymbol{\psi}).\]
Once again, $\bar{G}(\boldsymbol{\psi})$ is a supermartingale under any martingale measure $\bar{\mathbb{P}}\in {\cal P}({\cal M})$ so that

\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\bar{G}_{T}(\boldsymbol{\psi})\right ]= \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left[\bar{G}_{T}(\boldsymbol{\psi})\right | {\cal F}_{0}\right]=\bar{G}_{0}(\boldsymbol{\psi})\leq 0.\] Using (\ref{museq109}), we obtain
\begin{equation}{\label{museq1010}}\tag{16}
\bar{V}_{0}(\boldsymbol{\psi})\geq\sup_{\bar{\mathbb{P}}\in
{\cal P}({\cal M})}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\frac{X}{S_{t}^{(0)}}\right ];
\end{equation}
that is,
\[\inf_{\boldsymbol{\phi}\in\theta (X)}\bar{V}_{0}(\boldsymbol{\phi})\geq\sup_{\bar{\mathbb{P}}\in
{\cal P}({\cal M})}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\frac{X}{S_{t}^{(0)}}\right ].\]

Since $X$ is attainable in ${\cal M}$, there exists a tame strategy $\boldsymbol{\psi}$ and a martingale measure $\bar{\mathbb{P}}$ such that the equality holds in (\ref{museq1010}). This proves the second equality in (\ref{museq108}). $\blacksquare$

Proposition. Assume that the discounted contingent claim $X/S_{T}^{(0)}$ is $\bar{\mathbb{P}}$-integrable. If the $\bar{\mathbb{P}}$-martingale $M$ defined by
\[M_{t}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{X}{S_{T}^{(0)}}\right |{\cal F}_{t}\right ]\]
admits an integral representation of the form
\[M_{t}=x+\sum_{i=1}^{d}\int_{0}^{t}\phi_{u}^{(i)}d\bar{S}_{u}^{i},\]
with $\phi^{(1)},\cdots ,\phi^{(d)}$ predictable and locally bounded, then $X$ is attainable.

Proof. We are looking for a trading strategy $\boldsymbol{\phi}\in\boldsymbol{\phi}_{0}$ such that $V_{T}(\boldsymbol{\phi})=X$. We use the components $\phi^{(1)},\cdots ,\phi^{(d)}$ from the definition of the integral representation of $M$ and then define $\phi^{(0)}$ as in Remark \ref{binr612} with $v=x$. Then $\boldsymbol{\phi}=(\phi^{(0)},\phi^{(1)},\cdots ,\phi^{(d)})$ is a self-financing trading strategy with
\[\bar{V}_{t}(\boldsymbol{\phi})=v+\bar{G}_{t}(\boldsymbol{\phi})
=x+\sum_{i=1}^{d}\int_{0}^{t}\phi_{i}(u)d\bar{S}_{i}(u)=M_{t}.\]
By definition $M$ is a nonnegative martingale, so $\boldsymbol{\phi}\in\boldsymbol{\phi}_{0}$ follows, and also by definition
\[V_{T}(\boldsymbol{\phi})=S_{T}^{(0)}\cdot
\bar{V}_{T}(\boldsymbol{\phi})=S_{T}^{(0)}\cdot M_{T}=S_{T}^{(0)}\cdot\frac{X}{S_{T}^{(0)}}=X.\]
This completes the proof. $\blacksquare$

\begin{equation}{\label{bint1}}\tag{17}\mbox{}\end{equation}

Theorem \ref{bint1}. If the strong martingale measure $\bar{\mathbb{P}}$ is the unique martingale measure for the financial market model ${\cal M}$, then ${\cal M}$ is complete, in the restricted sense that every contingent claim $X$ satisfying $X/S_{T}^{(0)}\in L^{1}({\cal F},\bar{\mathbb{P}})$ is attainable. $\sharp$

Futures Market.

Generally speaking, given an arbitrage-free model of the spot market, the futures price of any traded security can be, in principle, derived using no-arbitrage arguments. In some circumstances, one may find it preferable to impose conditions directly on the futures price dynamics of certain assets. For this reason, we shall now comment on a direct construction of an arbitrage-free model which involves the futures prices of $k-1$ assets and, in adiition, the spot price of one traded security. Let us denote $(f^{(1)},\cdots ,f^{(k-1)})={\bf f}$. In financial interpretation, each process $f^{(i)}$ is assumed to represent the futures price of a certain asset corresponding to the delivery date $T_{i}\geq T$. As before, the process $S^{(k)}$ stands for the spot price of some traded security. For convenience, we assume that $S^{(k)}$ and $1/S^{(k)}$ follow continuous, strictly positive semimartingales. Taking into account the specific features of futures contracts, we modify the definition of a self-financing trading strategy as follows.

Definition. An $\mathbb{R}^{k}$-valued predictable process $\boldsymbol{\phi}=(\phi^{(1)},\cdots ,\phi^{(k)})$ is a self-financing futures trading strategy when the wealth process $V^{({\bf f})}(\boldsymbol{\phi})$, which equals
\[V^{({\bf f})}(\boldsymbol{\phi})=\phi^{(k)}\cdot S^{(k)},\]
satisfies
\[V^{({\bf f})}(\boldsymbol{\phi})=V_{0}^{({\bf f})}(\boldsymbol{\phi})+G^{({\bf f})}(\boldsymbol{\phi}),\]
where gains process $G^{({\bf f})}(\boldsymbol{\phi})$ is given by the formula
\[G_{t}^{({\bf f})}(\boldsymbol{\phi})\equiv\int_{0}^{t}\boldsymbol{\phi}^{({\bf f})}_{u}\cdot d{\bf f}_{u}+\int_{0}^{t}
\phi_{u}^{(k)}dS_{u}^{(k)}\mbox{ for all }t\in [0,T],\]
where $\boldsymbol{\phi}^{({\bf f})}=(\phi^{(1)},\cdots ,\phi^{(k-1)})$. We write $\boldsymbol{\phi}^{({\bf f})}$ to denote the class of all self-financing futures trading strategies. $\sharp$

As for the spot market model, the futures market ${\cal M}^{({\bf f})}= ({\bf S},\boldsymbol{\phi}^{({\bf f})})$ is based on the specification of the class $\boldsymbol{\phi}^{({\bf f})}$ of adimissible trading strategies. The concepts of a contingent claim, replicating process, arbitrage opportunity and an arbitrage-free market remain valid. We shall formulate explicitly the definitions of martingale measures. First, the martingale measure for ${\bf f}$ is any probability measure $\bar{\mathbb{P}}$ equivalent to ${\cal P}$ such that ${\bf f}$ follows a local martingale under $\bar{\mathbb{P}}$. Second, a probability measure $\bar{\mathbb{P}}$, equivalent to ${\cal P}$, is called a martingale measure for the futures market model ${\cal M}^{({\bf f})}$ if, for arbitrary futures trading strategy $\boldsymbol{\phi}\in \boldsymbol{\phi}^{({\bf f})}$, the relative wealth process $\bar{V}^{({\bf f})}(\boldsymbol{\phi})=V^{({\bf f})}(\boldsymbol{\phi})/S^{(k)}$ follows a local martingale under $\bar{\mathbb{P}}$. We denote by ${\cal P}({\bf f})$ and ${\cal P}({\cal M}^{({\bf f})})$ the class of all martingale measures for the process ${\bf f}$ and for the market model ${\cal M}^{({\bf f})}$, respectively.

Proposition. Assume that for every $i=1,\cdots ,k-1$ we have
\begin{equation}{\label{museq1014}}\tag{18}
\left\langle f^{(i)},\frac{1}{S^{(k)}}\right\rangle\mathbb{E}_{t}=0\mbox{ for all }t\in [0,T].
\end{equation}
Then any probability measure $\bar{\mathbb{P}}$ from ${\cal P}({\bf f})$ is a martingale measure for the futures market model ${\cal M}^{({\bf f})}$, that is, ${\cal P}({\bf f})\subseteq {\cal P}({\cal M}^{({\bf f})})$.

Proof. From Ito’s formula, we have
\[d(1/S_{t}^{(k)})=-(1/S_{t}^{(k)})^{2}dS_{t}^{(k)}+(1/S_{t}^{(k)})^{3}d\langle S^{(k)},S^{(k)}\rangle\mathbb{E}_{t}.\]
This implies that $d\langle S^{(k)},1/S^{(k)}\rangle\mathbb{E}_{t}=-(S_{t}^{(k)})^{-2}d\langle S^{(k)},S^{(k)}\rangle\mathbb{E}_{t}$, since
\[0=d(S_{t}^{(k)}\cdot (1/S_{t}^{(k)}))=(1/S_{t}^{(k)})dS_{t}^{(k)}+
S_{t}^{(k)}d(1/S_{t}^{(k)})+d\langle S^{(k)},1/S^{(k)}\rangle\mathbb{E}_{t}.\]
An applictaion of Ito’s integration by parts formula yileds
\begin{align*}
d\bar{V}_{t}^{({\bf f})} & =(1/S_{t}^{(k)})dV_{t}^{({\bf f})}+V_{t}^{({\bf f})}
d(1/S_{t}^{(k)})+d\langle V^{({\bf f})},1/S^{(k)}\rangle\mathbb{E}_{t}\\
& =(S_{t}^{(k)})^{-1}\cdot (\boldsymbol{\phi}_{t}^{({\bf f})}\cdot
d{\bf f}_{t}+\phi_{t}^{k}dS_{t}^{(k)})+\phi_{t}^{k}\cdot S_{t}^{(k)}\cdot
d(1/S_{t}^{(k)})+\boldsymbol{\phi}_{t}^{({\bf f})}\cdot\langle {\bf f},
1/S^{(k)}\rangle\mathbb{E}_{t}+\phi_{t}^{k}d\langle S^{(k)},1/S^{(k)}\rangle\mathbb{E}_{t}\\
& =(1/S_{t}^{(k)})\cdot\boldsymbol{\phi}_{t}^{({\bf f})}\cdot
d{\bf f}_{t}+\boldsymbol{\phi}_{t}^{({\bf f})}\cdot d\langle {\bf f},1/S^{(k)}\rangle\mathbb{E}_{t},
\end{align*}
where
\[\boldsymbol{\phi}_{t}^{({\bf f})}\cdot d\langle {\bf f},1/S^{(k)}
\rangle\mathbb{E}_{t}=\sum_{i=1}^{k-1}\phi_{t}^{(i)}d\langle f^{(i)},1/S^{(k)}\rangle\mathbb{E}_{t}.\]
Since by assumption the process $\langle f^{(i)},1/S^{(k)}\rangle$ vanishes for every $i$, we have
\[\bar{V}_{t}^{({\bf f})}=\bar{V}_{0}+\int_{0}^{t}\boldsymbol{\phi}^{({\bf f})}_{u}\cdot d{\bf f}_{u}=
\bar{V}_{0}^{({\bf f})}+N_{t}^{({\bf f})}\mbox{ for all }t\in [0,T],\]

where the process $N_{t}^{({\bf f})}$ follows a $\bar{\mathbb{P}}$-local martingale (since $\bar{\mathbb{P}}\in {\cal P}({\bf f})$). This shows that $\bar{V}_{t}^{({\bf f})}$ follows a $\bar{\mathbb{P}}$-local martingale. $\blacksquare$

The Black model of the futures market assumes deterministic interest so that (\ref{museq1014}) is trivially satisfied, and indeed we know that ${\cal P}({\bf f})={\cal P}({\cal M}^{({\bf f})})$. Condition (\ref{museq1014}) need not holds, in general, when the stochastic character of the inetrest rates is acknowledged. Therefore, under uncertainty of interest rates, the property ${\cal P}({\bf f})\subseteq {\cal P}({\cal M}^{({\bf f})})$ is not necessarily valid.

Changes of Numeraire.

We assume that a contingent claim is represented by a nonnegative ${\cal F}_{T}$-measurable random variable. The price processes $S^{(0)},S^{(1)},\cdots ,S^{(d)}$ are assumed to be continuous, strictly positive semimartingales, so that processes $1/S^{(0)},1/S^{(1)},\cdots , 1/S^{(d)}$ also follow continuous martingales. It is clear that any security $S^{(i)}$ can be chosen as the numeraire asset.

By a positive trading strategy we mean a strategy $\boldsymbol{\phi}$ such that the wealth $V(\boldsymbol{\phi})$ follows a nonnegative process. We denote by $\boldsymbol{\phi}_{+}$ the class of all positive strategies. The class $\boldsymbol{\phi}_{+}$ is invariant not only with respect to an equivalent change of probability measure, but also with respect to the choice of a numeraire asset. In particular, if $\boldsymbol{\phi}$ belongs to $\boldsymbol{\phi}_{+}$, then for any fixed $i$, the discounted wealth $V(\boldsymbol{\phi})/S^{(i)}$ follows a nonnegative local martingale under any martingale measure for the process ${\bf S}/S^{(i)}$, since (referring the proof of Proposition \ref{musl1011})
\begin{equation}{\label{museq1015}}\tag{19}
\frac{V_{t}(\boldsymbol{\phi})}{S_{t}^{(i)}}=\frac{V_{0}(\boldsymbol{\phi})}{S_{0}^{(i)}}+\int_{0}^{t}
\boldsymbol{\phi}_{u}\cdot d({\bf S}_{u}/S_{u}^{(i)})
\end{equation}
for every $t\in [0,T]$. In financial interpretation, equality (\ref{museq1015}) means that a self-financing trading strategy remains self-financing after the change of the numeraire asset in which we express the prices of all other securities. Furthermore, if a strategy $\boldsymbol{\phi}$ replicates the claim $X$ that settles at time $T$, then it replicates the claim $X/S_{t}^{(i)}$.

In the context of stochastic interest rate models, it is frequently convenient to take the price of the $T$-maturity zero-coupon bond as the
numeraire asset. The market model obtained in this way is referred to as the $T$-forward market, and the corresponding martingale measure is called the forward measure for the date $T$. Let $F_{S^{(i)}}(t,T)=S_{t}^{(i)}/p(t,T)$ stand for the forward price of the $i$th asset for settlements at time $T$. For brevity, we sometimes write $F^{(i)}(t,T)=F_{S^{(i)}}(t,T)$ and ${\bf F}_{{\bf S}}(t,T)= (F^{(0)}(t,T),F^{(1)}(t,T),\cdots ,F^{(d)}(t,T))$. We write $\bar{V}(\boldsymbol{\psi})$ to denote the {\bf forward wealth} of $\boldsymbol{\psi}$; that is,
\[\bar{V}_{t}(\boldsymbol{\psi})=\boldsymbol{\psi}_{t}\cdot {\bf F}_{{\bf S}}(t,T)=
\frac{\boldsymbol{\psi}_{t}\cdot {\bf S}_{t}}{p(t,T)}=\frac{V_{t}(\boldsymbol{\phi})}{p(t,T)},\]
where $\boldsymbol{\psi}=\boldsymbol{\phi}$ is the corresponding spot trading strategy and, as usual, $V_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}$.

Proposition. A trading strategy $\boldsymbol{\psi}$ is a self-financing forward strategy; that is, the forward wealth $\bar{V} (\boldsymbol{\psi})$ satisfies
\begin{equation}{\label{museq1016}}\tag{20}
d\bar{V}_{t}(\boldsymbol{\psi})=\sum_{i=0}^{d}\psi_{t}^{0}
dF^{(i)}(t,T)=\boldsymbol{\psi}_{t}\cdot d{\bf F}_{{\bf S}}(t,T)
\end{equation}
if and only if the wealth process $V_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot {\bf S}_{t}$ of the spot trading strategy $\boldsymbol{\phi}=\boldsymbol{\psi}$ satisfies
\[dV_{t}(\boldsymbol{\phi})=\sum_{i=0}^{d}\phi_{t}^{(i)}dS_{t}^{(i)}=\boldsymbol{\phi}_{t}\cdot d{\bf S}_{t}.\]

Proof. We shall prove the “only if” case. We need to show that
\begin{equation}{\label{museq1017}}\tag{21}
dV_{t}(\boldsymbol{\psi})=\boldsymbol{\psi}_{t}\cdot d(p(t,T)\cdot {\bf F}_{{\bf S}}(t,T)).
\end{equation}
Since $V_{t}(\boldsymbol{\psi})=p(t,T)\cdot\bar{V}_{t}(\boldsymbol{\psi})$, the Ito’s formula yields
\[dV_{t}(\boldsymbol{\psi})=p(t,T)\cdot d\bar{V}_{t}(\boldsymbol{\psi})+
\bar{V}_{t}(\boldsymbol{\psi})\cdot dp(t,T)+d\langle p(\cdot ,T),
\bar{V}(\boldsymbol{\psi})\rangle\mathbb{E}_{t}\equiv I_{1}+I_{2}+I_{3}.\]
From (\ref{museq1016}), it follows that
\[I_{1}=p(t,T)\cdot\boldsymbol{\psi}_{t}\cdot d{\bf F}_{{\bf S}}(t,T).\]
Furthermore,
\[I_{2}=\bar{V}_{t}(\boldsymbol{\psi})dp(t,T)=\boldsymbol{\psi}_{t}\cdot {\bf F}_{{\bf S}}(t,T)dp(t,T).\]
Finally, once again by (\ref{museq1016}), we obtain
\[I_{3}=d\langle p(\cdot ,T),\bar{V}_{t}(\boldsymbol{\psi})
\rangle\mathbb{E}_{t}=\boldsymbol{\psi}_{t}\cdot d\langle p(\cdot ,T),
{\bf F}_{{\bf S}}(t,T)\rangle\mathbb{E}_{t},\]
where
\[\langle p(\cdot ,T),{\bf F}_{{\bf S}}(t,T)\rangle =(\langle p(\cdot ,T),F^{(0)}(t,T)\rangle ,
\langle p(\cdot ,T),F^{(1)}(t,T)\rangle ,\cdots ,\langle p(\cdot ,T),F^{(d)}(t,T)\rangle ).\]
Since for any $i=0,1,\cdots ,d$
\[d(p(t,T)\cdot F^{(i)}(t,T))=p(t,T)dF^{(i)}(t,T)+F^{(i)}(t,T)\cdot dp(t,T)+
d\langle p(\cdot ,T),F^{(i)}(\cdot ,T)\rangle\mathbb{E}_{t},\]
it is clear that equality (\ref{museq1017}) is indeed satisfied. $\blacksquare$

\begin{equation}{\label{binp55}}\tag{22}\mbox{}\end{equation}

Proposition \ref{binp55} (Bayes Formula). Assume $\bar{\mathbb{P}}$ is absolutely continuous with respect to $\mathbb{P}$ and $Z$ is its Radon-Nikodym derivative. If $Y$ is bounded $($or $\bar{\mathbb{P}}$-integrable$)$ and ${\cal F}_{t}$-measurable, then
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left [Y\right |{\cal F}_{u}\right ]=
\frac{1}{Z_{u}}\cdot \mathbb{E}_{\mathbb{P}}\left .\left [Y\cdot Z_{t}\right |{\cal F}_{u}\right ]\mbox{ a.s. $u\leq t$}.\]

\begin{equation}{\label{binp60}}\tag{23}\mbox{}\end{equation}

Theorem \ref{binp60}. Let $X$ be a non-dividend-paying numeraire such that $X/S^{(0)}$ is a $\bar{\mathbb{P}}$-martingale. Then there exists a probability measure $\mathbb{P}^{*}$, defined by its Radon-Nikodym derivative $\eta$ with respect to $\bar{\mathbb{P}}$,
\begin{equation}{\label{bineqa927}}\tag{24}
\eta_{t}=\left .\frac{d\mathbb{P}^{*}}{d\bar{\mathbb{P}}}\right |_{{\cal F}_{t}}=\frac{X_{t}}{X_{0}S_{t}^{(0)}},
\end{equation}
such that

(i) the basic security prices discounted with respect to $X$ are $\mathbb{P}^{*}$-martingales;

(ii) if a contingent claim $Y$ is attainable under $(S_{t}^{(0)}, \bar{\mathbb{P}})$, then it is attainable under $(X_{t},\mathbb{P}^{*})$, and the replicating portfolio is the same. So the arbitrage price processes, given by the risk-neutral valuation formula, coincide.

Proof. To prove part (i), denote by $\bar{S}_{t}^{(i)}=S_{t}^{(i)}/X_{t}$ the relative price of a security $S^{(i)}$ with respect to the numeraire $X_{t}$. Since $X_{t}/S_{t}^{(0)}$ is a $\bar{\mathbb{P}}$-martingale, $\eta_{t}=X_{t}/(X_{0}S_{t}^{(0)})$ is a $\bar{\mathbb{P}}$-martingale. We then can use the Bayes formula in Proposition \ref{binp55}. For $0\leq u\leq t$, we have
\begin{align*}
\mathbb{E}_{\mathbb{P}^{*}}\left .\left [\bar{S}_{t}^{(i)}\right |{\cal F}_{u}
\right ] & =\frac{1}{\eta_{u}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left [
\bar{S}_{t}^{(i)}\eta_{t}\right |{\cal F}_{u}\right ]\mbox{ (by Bayes formula)}\\
& =\frac{X_{0}S_{u}^{(0)}}{X_{u}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{S_{t}^{(i)}}
{X_{t}}\cdot\frac{X_{t}}{X_{0}S_{t}^{(0)}}\right |{\cal F}_{u}\right ]
=\frac{S_{u}^{(0)}}{X_{u}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left [\bar{S}_{t}^{(i)}\right |{\cal F}_{u}\right ]\\
& =\frac{S_{u}^{(0)}}{X_{u}}\cdot\tilde{S}_{u}\mbox{ (since $\tilde{S}$ is a $\bar{\mathbb{P}}$-martingale)}\\
& =\frac{S_{u}^{(0)}}{X_{u}}\cdot\frac{S_{u}^{(i)}}{S_{u}^{(0)}}=\frac{S_{u}^{(i)}}{X_{u}}=\bar{S}_{u}^{i}.
\end{align*}
So the basic securities are martingales, and hence so is any portfolio.

To prove part (ii), if $Y$ is attainable under $\bar{\mathbb{P}}$, then there exists a $\bar{\mathbb{P}}$-admissible portfolio $\boldsymbol{\phi}$ satisfying $V_{t}(\boldsymbol{\phi})=Y$, and by the risk-neutral valuation formula in Theorem \ref{binp56}
\[V_{t}(\boldsymbol{\phi})=S_{t}^{(0)}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{Y}{S_{t}^{(0)}}\right |
{\cal F}_{t}\right ].\]
Using again the Bayes formula, we obtain
\begin{align*}
\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\frac{Y}{X_{t}}\right |{\cal F}_{t}\right ] & =
\frac{1}{\eta_{t}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{Y}{X_{t}}\cdot\eta_{t}\right |{\cal F}_{t}\right ]
=\frac{X_{0}S_{t}^{(0)}}{X_{t}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .
\frac{Y}{X_{t}}\cdot\frac{X_{t}}{X_{0}S_{t}^{(0)}}\right |{\cal F}_{t}\right ]\\
& =\frac{1}{X_{t}}\cdot S_{t}^{(0)}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .
\frac{Y}{S_{t}^{(0)}}\right |{\cal F}_{t}\right ]=\frac{V_{t}(\boldsymbol{\phi})}{X_{t}}.
\end{align*}
That is,
\[V_{t}(\boldsymbol{\phi})=X_{t}\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\frac{Y}{X_{t}}\right |{\cal F}_{t}\right ].\]

Since self-financing portfolios remain self-financing under a change of numeraire (by the numeriare invariance theorem in Theprem \ref{binp57}), and since by part (i), $V_{t}(\boldsymbol{\phi})$ is a $\mathbb{P}^{*}$-martingale, it follows that $\boldsymbol{\phi}$ is $\mathbb{P}^{*}$-adimissible. $\blacksquare$

\begin{equation}{\label{binc7}}\tag{25}\mbox{}\end{equation}

Corollary \ref{binc7}. Let $X$ and $Y$ be numeraires satisfying the assumptions of the above proposition and $Z$ a contingent claim. Then we have the change of numeraire formula
\[X_{t}\cdot \mathbb{E}_{\mathbb{P}^{*}_{X}}\left [\left .\frac{Z}{X_{T}}\right |
{\cal F}_{t}\right ]=Y_{t}\cdot \mathbb{E}_{\mathbb{P}^{*}_{Y}}\left [\left .\frac{Z}{Y_{T}}\right |{\cal F}_{t}\right ].\]

Proof. We can use the second part of the above proposition, or use the Radon-Nikodym derivative
\[\left .\frac{d\mathbb{P}^{*}_{X}}{d\mathbb{P}^{*}_{Y}}\right |_{{\cal F}_{T}}=
\frac{\left .\frac{d\mathbb{P}^{*}_{X}}{d\bar{\mathbb{P}}}\right |_{{\cal F}_{T}}}
{\left .\frac{d\mathbb{P}^{*}_{Y}}{d\bar{\mathbb{P}}}\right |_{{\cal F}_{T}}}=
\frac{\frac{X_{T}}{X_{0}S_{T}^{(0)}}}{\frac{Y_{T}}{Y_{0}S_{T}^{(0)}}}=\frac{X_{T}/Y_{T}}{X_{0}/Y_{0}}\]
and repeat the argument. $\blacksquare$

Since any security $S^{(i)}$ can be chosen as the numeraire asset, in this way, we formally obtain not a single market model, but rather a finite family of standard spot market models ${\cal M}^{(i)}$ for $i=0,1,\cdots ,d$. Let us stress that we may take the price process of a portfolio of primary securities (that is, the wealth process of a self-financing trading strategy) as the numeraire asset, provided that it follows a strictly positive process. Given such a strategy $\boldsymbol{\psi}$, it is not difficult to verify that
\[\frac{V_{t}(\boldsymbol{\phi})}{V_{t}(\boldsymbol{\psi})}=\frac{V_{0}(\boldsymbol{\phi})}{V_{0}(\boldsymbol{\psi})}+
\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d({\bf S}_{u}/V_{u}(\boldsymbol{\psi})),\]
where $\boldsymbol{\phi}$ is an arbitrary self-financing strategy. Summarizing, we have a considerable degree of freedom in the choice of the numeraire asset .

Proposition. Let the class ${\cal P}({\cal M}^{(i)})$ be nonempty for some $i$. More precisely, we assume that the relative price processes $S^{(j)}/S^{(i)}$ are $\bar{\mathbb{P}}$-martingales for some $\bar{\mathbb{P}}\in {\cal P}({\cal M}^{(i)})$. Then, we have the following properties.

(i) For any $j\neq i$, if the process $S^{(j)}/S^{(i)}$ follows a $\bar{\mathbb{P}}$-martingale, the class ${\cal P}({\cal M}^{(j)})$ of martingale measures is nonempty;

(ii) For any trading strategy $\boldsymbol{\phi}$ such that the relative wealth process $V(\boldsymbol{\phi})/S^{(i)}$ follows a strictly positive martingale under $\bar{\mathbb{P}}$, there exists a martingale measure for the relative price process ${\bf S}/V(\boldsymbol{\phi})$.

Proof. Since buy-and-hold strategies are manifestly self-financing, the first statement follows from the second. To prove (ii), we may in fact regard $V(\boldsymbol{\phi})$ as an arbitrary strictly positive process $N$ such that $N/S^{(i)}$ is a $\bar{\mathbb{P}}$-martingale. We define a measure $\mathbb{P}^{*}$ on $(\Omega ,{\cal F}_{T})$ by setting
\begin{equation}{\label{museq1018}}\tag{26}
\frac{\mathbb{P}^{*}}{\bar{\mathbb{P}}}=\frac{S_{0}^{(i)}\cdot N_{T}}
{N_{0}\cdot S_{T}^{(i)}}\equiv\eta_{T}\mbox{, }\bar{\mathbb{P}}\mbox{-a.s.}
\end{equation}
The random variable $\eta_{T}$ is strictly positive and
\[\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\eta_{T}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left [
\eta_{T}\right |{\cal F}_{0}\right ]=\frac{S_{0}^{(i)}}{N_{0}}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left .\left
[\frac{N_{T}}{S_{T}^{(i)}}\right |{\cal F}_{0} \right ]=\frac{S_{0}^{(i)}}{N_{0}}\cdot\frac{N_{0}}{S_{0}^{(i)}}=1\]

using the martingale property of the process $N/S^{(i)}$ under $\bar{\mathbb{P}}$. Consequently,
\[\mathbb{P}^{*}(\Omega )=\int_{\Omega}d\mathbb{P}^{*}=\int_{\Omega}\eta_{T}
d\bar{\mathbb{P}}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\eta_{T}\right ]=1,\]

it says that $\mathbb{P}^{*}$ is indeed a probability measure. We wish to show that, under $\mathbb{P}^{*}$, all relative price processes $S^{(j)}/N$ follow (local) martingales. Let us assume that the random variable $S_{T}^{(j)}/N_{T}$ is $\mathbb{P}^{*}$-integrable. Using the Bayes rule, we obtain
\[\mathbb{E}_{\mathbb{P}^{*}}\left .\left [\frac{S_{T}^{(j)}}{N_{T}}\right |{\cal F}_{t}
\right ]=\frac{\mathbb{E}_{\mathbb{P}^{*}}\left .\left [\frac{\eta_{T}\cdot S_{T}^{(j)}}
{N_{T}}\right |{\cal F}_{t}\right ]}{\mathbb{E}_{\mathbb{P}^{*}}\left .\left [\eta_{T}
\right |{\cal F}_{T}\right ]}\equiv\frac{J_{1}}{J_{2}}.\]
Using (\ref{museq1018}), we have
\[J_{1}=\mathbb{E}_{\mathbb{P}^{*}}\left .\left [\frac{S_{0}^{(i)}\cdot S_{T}^{(j)}}
{N_{0}\cdot S_{T}^{(i)}}\right |{\cal F}_{t}\right ]=\frac{S_{0}^{(i)}\cdot S_{t}^{(j)}}{N_{0}\cdot Z_{t}^{(i)}},\]

where we have used the martingale property of $S^{(j)}/S^{(i)}$ under $\bar{\mathbb{P}}$. For $J_{2}$, we have $J_{2}=(N_{t}\cdot S_{0}^{(i)})/ (N_{0}\cdot S_{t}^{(i)})$, and thus $J_{1}/J_{2}=S_{t}^{(j)}/N_{t}$ as expected. $\blacksquare$

Proposition. Let $X$ be a contingent claim which can be priced by arbitrage in standard market models ${\cal M}^{(i)}$ and ${\cal M}^{(j)}$. More specifically, we assume that $X$ is $\bar{\mathbb{P}}_{i}$-attainable and $\bar{\mathbb{P}}_{j}$-attainable, where $\bar{\mathbb{P}}_{i}\in {\cal P}({\cal M}^{(i)})$ and $\bar{\mathbb{P}}_{j}\in {\cal P}({\cal M}^{(j)})$. Then for every $t\in [0,T]$, we have
\[\Pi_{t}^{(i)}(X)=S_{t}^{(i)}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{i}}\left .\left [
\frac{X}{S_{T}^{(i)}}\right |{\cal F}_{t}\right ]=S_{t}^{(j)}\cdot
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}_{j}}\left .\left [\frac{X}{S_{T}^{(j)}}\right |{\cal F}_{t}\right ]=\Pi_{t}^{(j)}(X).\sharp$\]

Example. (The money market account as numeraire). Assuming that a riskless asset (e.g. bank account) exists, it is natural to take it as a numeraire. More precisely, define $B_{t}$ as the value at date $t$ of a fund created by investing one money unit at time $t=0$ and continuously reinvested at the (instantaneously riskless) intataneous interest rate $r_{t}$. Under technical assumptions, we have
\[B_{t}=\exp\left (\int_{0}^{t}r_{u}du\right ).\]
The discounted price process of a security with respect to the numeraire $B_{t}$ is simply
\[\bar{S}_{t}=\exp\left (-\int_{0}^{t}r_{u}du\right )\cdot S_{t}.\]
Historically, $B_{t}$ was used as the numeraire $S_{t}^{(0)}$ and then $\mathbb{P}^{*}_{B}=\bar{\mathbb{P}}$. $\sharp$

\begin{equation}{\label{bine3}}\tag{27}\mbox{}\end{equation}

Example \ref{bine3} (Zero-coupon bonds as numeraire). A zero-coupon bond is the natural choice of numeraire if one looks at the time $t$ price of an asset giving the right to a single cash-flow at a well-defined future time $T$. The simplest such asset is a zero-coupon bond with cash-flow $1$ at time $T$. We denote its time $t$ price by $p(t,T)$. We assume that the money-market account $B_{t}$ (as defined above) was used to define $\bar{\mathbb{P}}$. Since $p(T,T)=1$, the risk-neutral valuation formula gives
\[p(t,T)=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\exp\left (-\int_{t}^{T}r_{u}du\right )\right |{\cal F}_{t}\right ].\]

So $p(t,T)$ is a $\bar{\mathbb{P}}$-martingale, and by Theorem \ref{binp60}
\begin{equation}{\label{binaeq294}}\tag{28}
\frac{d\mathbb{P}^{*}}{d\bar{\mathbb{P}}}=\frac{p(t,T)}{p(0,T)B_{t}}=
\frac{p(t,T)}{p(0,T)}\cdot\exp\left (-\int_{0}^{t}r_{u}du\right ).
\end{equation}
The relative price process of a basic asset $S_{t}$ with respect to $p(t,T)$, i.e., $\bar{S}(t)=S_{t}/p(t,T)$, is called the forward price $F_{S}(t)$ of the security $S_{t}$. By Theorem \ref{binp60} (i), we have
\[F_{S}(t)=\bar{S}(t)=\mathbb{E}_{\mathbb{P}^{*}}[\bar{S}(T)|{\cal F}_{t}]=
\mathbb{E}_{\mathbb{P}^{*}}[S_{T}/P(T,T)|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}^{*}}[S_{T}| {\cal F}_{t}].\]

To rephrase this: the forward price, relative to time $T$, of a security which pays no dividends up to time $T$, is equal to the expectation of the value at time $T$ of this security under the “forward neutral” martingale measure $\bar{\mathbb{P}}$. $\sharp$

From Example \ref{bine3}, we have the following result.

\begin{equation}{\label{binp614}}\tag{29}\mbox{}\end{equation}

Proposition \ref{binp614}. Denote by $T$ and $K$, respectively, the maturity and exercise price of a European call option on an asset $S$. The time $t=0$ price $C_{0}$ of the call option can be written as
\[C_{0}=p(0,T)\cdot \mathbb{E}_{\mathbb{P}^{*}}[(S_{T}-K)^{+}]\]
or
\[C_{0}=S_{0}\cdot \mathbb{P}^{*}_{S}(A)-K\cdot p(0,T)\cdot \mathbb{P}^{*}(A),\]
where $A=\{\omega :S_{T}(\omega )>K\}$, $\mathbb{P}^{*}$ is given in (\ref{binaeq294}) and $\mathbb{P}^{*}_{S}$ is defined by (\ref{bineqa927}) when $S$ is taken as a numeraire.

Proof. From Theorem \ref{binp60} (i), $\bar{C}_{t}=C_{t}/p(t,T)$ is a $\mathbb{P}^{*}$-martingale. Since ${\cal F}_{0}=\{\emptyset,\Omega\}$ and $C_{T}=(S_{T}-K)^{+}$, we have
\[\frac{C_{0}}{p(0,T)}=\bar{C}_{0}=\mathbb{E}_{\mathbb{P}^{*}}[\bar{C}_{T}|{\cal F}_{0}]=\mathbb{E}_{\mathbb{P}^{*}}\left [
\frac{C_{T}}{P(T,T)}\right ]=\mathbb{E}_{\mathbb{P}^{*}}[(S_{T}-K)^{+}].\]
This proves the first expression. Now, we write
\begin{align*}
\frac{C_{0}}{p(0,T)} & =\mathbb{E}_{\mathbb{P}^{*}}[(S_{T}-K)^{+}]=\mathbb{E}_{\mathbb{P}^{*}}[(S_{T}-K)\cdot 1_{A}]\\
& =\mathbb{E}_{\mathbb{P}^{*}}[S_{T}\cdot 1_{A}]-K\cdot \mathbb{E}_{\mathbb{P}^{*}}
[1_{A}]=\mathbb{E}_{\mathbb{P}^{*}}[S_{T}\cdot 1_{A}]-K\cdot \mathbb{P}^{*}(A).
\end{align*}
That is,
\[C_{0}=p(0,T)\cdot \mathbb{E}_{\mathbb{P}^{*}}[S_{T}\cdot 1_{A}]-K\cdot p(0,T)\cdot \mathbb{P}^{*}(A).\]
Now we use the change of numeriare formula in Corollary \ref{binc7} by considering $p(t,T)$ as a numeraire to replace the role $X_{t}$ and $S_{t}$ as a numeraire to replace the role $Y_{t}$, and the contingent claim in the formula is replaced by $S_{T}\cdot 1_{A}$. Then, we have
\begin{align*}
p(0,T)\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\frac{S_{T}\cdot 1_{A}}{p(T,T)}\right ] & =p(0,T)\cdot \mathbb{E}_{\mathbb{P}^{*}}\left [\left .\frac{S_{T}\cdot 1_{A}}{p(T,T)}\right |{\cal F}_{0}\right ]=S_{0}
\cdot \mathbb{E}_{\mathbb{P}^{*}_{S}}\left [\left .\frac{S_{T}\cdot 1_{A}}{S_{T}}\right |{\cal F}_{0}\right ]\\
& =S_{0}\cdot \mathbb{E}_{\mathbb{P}^{*}_{S}}[1_{A}]=S_{0}\cdot\mathbb{P}^{*}_{S}(A).\end{align*}
This proves the second expression. $\blacksquare$

The proof of Proposition \ref{binp614} gives the following generalization.

Proposition. Consider the option to exchange asset $Y$ against to asset $X$ at time $T$, which gives one the right to the cash flow $(X_{T}-K\cdot Y_{T})^{+}$ $($with $K=1)$. Using $Y$ as a numeraire, the European call option price $C_{0}(X)$ on an asset $X$ is such that
\[C_{0}(X)=Y_{0}\cdot \mathbb{E}_{\mathbb{P}^{*}_{Y}}\left [\left (
\frac{X_{T}}{Y_{T}}-K\right )^{+}\right ],\]
or
\[C_{0}(X)=X_{0}\cdot \mathbb{P}^{*}_{X}(A)-K\cdot Y_{0}\cdot \mathbb{P}^{*}_{Y}(A),\]
where $A=\{\omega :X_{T}(\omega )>K\cdot Y_{T}(\omega )\}$. $\sharp$

Existence of a Martingale Measure.

Let us examine the problem of the existence of a martingale measure for a given stochastic process $Z$; more precisely, we assume that $Z$ follows a continuous semimartingale under $\mathbb{P}$. We start by studying the form of the Radon-Nikodym derivative process. Since the underlying filtration is not necessarily Brownian, the density process may be discontinuous, in general. Let us consider a real-valued RCLL semimartingale $U$ defined on $(\Omega ,{\bf F},\mathbb{P})$ with $U_{0}=U_{0-}=0$. We write ${\cal E}(U)$ to denote the Doléans exponential of $U$; that is, the unique solution of the SDE
\begin{equation}{\label{museq1019}}\tag{30}
d{\cal E}_{t}(U)={\cal E}_{t-}(U)dU_{t}
\end{equation}
with ${\cal E}_{0}(U)=1$. Let us define the quadratic variation of $U$ by setting
\[[U]_{t}=U_{t}^{2}-2\int_{0}^{t}U_{u-}dU_{u}\mbox{ for all }t\in [0,T].\]
The solution to equation (\ref{museq1019}) is explicitly known, namely
\[{\cal E}_{t}(U)=\exp\left (U_{t}-\frac{1}{2}\cdot [U]_{t}^{c}\right )
\cdot\prod_{u\leq t}(1+\Delta_{u}U)\cdot\exp (-\Delta_{u}U),\]
where $\Delta_{u}U=U_{u}-U_{u-}$, and $[U]^{c}$ is the path-by-path continuous part of $[U]$. (It is well known that $[U]^{c}=\langle U^{c}\rangle$, where $U^{c}$ is the continuous local martingale part of $U$.) Suppose that
\begin{equation}{\label{museq1020}}\tag{31}
\Delta_{t}U>-1\mbox{ and }\mathbb{E}_{P}\left [{\cal E}_{T}(U)\right ]=1.
\end{equation}
Then ${\cal E}(U)$ follows a strictly positive, uniformly integrable martingale under $\mathbb{P}$. In such a case, we may introduce a probability measure $\bar{\mathbb{P}}_{U}$ on $(\Omega ,{\cal F}_{T})$, equivalent to $\mathbb{P}$, by postulating that the Radon-Nikodym derivative equals
\[\frac{d\bar{\mathbb{P}}_{U}}{d\mathbb{P}}={\cal E}_{T}(U)\mbox{ $\mathbb{P}$-a.s.}\]
Conversely, if $\bar{\mathbb{P}}$ is a probability measure on $(\Omega ,{\cal F}_{T})$ equivalent to $\mathbb{P}$, we define by $\eta$ the RCLL version of the conditional expectation
\begin{equation}{\label{museq1021}}\tag{32}
\eta_{t}=\mathbb{E}_{\mathbb{P}}\left [\left .\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}
\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
The process $\eta$ follows a strictly positive, uniformly integrable martingale. Also, it coincides with the Doléan exponential ${\cal E}(U)$ of a local martingale $U$, which in turn equals
\[U_{t}=\int_{0}^{t}\frac{1}{\eta_{s-}}d\eta_{s}\mbox{ for all }t\in [0,T].\]
To summarize, a probability measure $\bar{\mathbb{P}}$ on $(\Omega,{\cal F}_{T})$ is equivalent to the underlying probability measure $\mathbb{P}$ if and only if $\bar{\mathbb{P}}=\bar{\mathbb{P}}_{U}$ for some local martingale $U$ such that condition (\ref{museq1020}) is satisfied and $U_{0}=U_{0-}=0$. Let $\bar{\mathbb{P}}$ be a probability measure equivalent to $\mathbb{P}$ and let $\eta$ be given by (\ref{museq1021}). Consider a continuous, real-valued $\mathbb{P}$-semimartingale $Z$ with the canonical decomposition $Z=Z_{0}+M+A$. Since $M$ follows a continuous martingale, it is well known that the cross-variation $\langle\eta ,M\rangle$ exists. By virtue of Girsanov’s theorem, $Z$ is a continuous semimaringale under $\bar{\mathbb{P}}$, and its canonical decomposition under $\bar{\mathbb{P}}$ is $Z=Z_{0}+N+B$, where $N=M-\langle M,U\rangle$ follows a continuous local martingale under $\bar{\mathbb{P}}$, and $B=A+\langle M,U\rangle$ is a continuous process of finite variation. The following result provides teh basic criteria for the existence of a martingale measure. A semimartingale $Z$ is special when it admits a (unique, i.e., canonical) decomposition $Z=Z_{0}+M+A$, where $M$ is a local martingale and $A$ is a predictable process of finite variation. A continuous semimartingale is special.

Proposition. Let $Z$ be a real-valued special semimartingale under $\mathbb{P}$ with the canonical decomposition $Z=Z_{0}+M+A$. Then a local martingale $U$, satisfying $(\ref{museq1020})$, defines a martingale measure for $Z$ if and only if
\begin{equation}{\label{museq1022}}\tag{33}
A_{t}+\langle M,U\rangle\mathbb{E}_{t}=0\mbox{ for all }t\in [0,T].
\end{equation}
If $(\ref{museq1022})$ holds, there exists a real-valued predictable process $\zeta$ satisfying
\[\mathbb{P}\left\{\int_{0}^{T}|\zeta_{u}|d\langle M\rangle\mathbb{E}_{u}<+\infty\right\}=1,\]
and such that $A$ admits the representation
\[A_{t}=\int_{0}^{t}\zeta_{u}d\langle M\rangle\mathbb{E}_{u}.\]

Proof. The first assertion follows from Girsanov’s theorem. The second can be proved using Kunita-Watanabe inequality. $\blacksquare$

Fundamental Theorem of Asset Pricing.

Having seen that the absence of arbitrage opportunities is implied by the existence of an equivalent martigale measure, a natural question is to ask whether the converse is true, which would yield a fundamental theorem of asset pricing. By a fundamental theorem of asset pricing, we mean a result which establishes the equivalence of the absence of an arbitrage opportunity in the stochastic model of financial market, and the existence of a martingale measure. The definition of an arbitrage opportunity can be formulated in many different ways; in particular, it depends essentially on the choice of topology on the space of random variables. We shall give here only a result which deals with continuous, but possibly unbounded, processes. Let us say that a process $S$ admits a strict martingale measure if it admits a martingale measure $\bar{\mathbb{P}}$ such that $S$ is a martingale under $\bar{\mathbb{P}}$. Note that we do not assume a priori that (relative) security prices follow semimartingales. Therefore, the Ito’s integration theory is not at hand. To circumvent this difficulty, by a trading strategy we mean a simple predictable process (i.e. piecewise constant between predictable stopping times), so that the integral is trivially well-defined. Formally, a simple predictable trading strategy is a predictable process which can be represented as a finite linear combination of stochastic processes of the form $\psi\cdot 1_{(\tau_{1},\tau_{2}]}$, where $\tau_{1}$ and $\tau_{2}$ are stopping times and $\psi$ is an ${\cal F}_{\tau_{1}}$-measurable random variable. We say that a simple predictable trading strategy is $\delta$-admissible if the relative wealth process $V_{t}(\boldsymbol{\phi})\geq -\delta$ for every $t\in [0,T]$.

A price process $S$ satisfies NFLVR (no free lunch with vanishing risk) if for any sequence $\{\phi_{n}\}$ of simple trading strategies such that $\phi_{n}$ is $\delta_{n}$-admissible and the sequence $\delta_{n}$ tends to zero, we have $V_{T}(\boldsymbol{\phi}_{n})\rightarrow 0$ in probability as $n\rightarrow\infty$.

Theorem (Fundamental Theorem of Asset Pricing). There exists an equivalent martingale measure for the financial market model ${\cal M}$ if and only if the condition NFLVR holds true. $\sharp$

A price process $S$ satisfies NFLBR (no free lunch with bounded risk) if there does not exist a sequence of simple strategies $\{\phi_{n}\}_{n\geq 1}$ and a ${\cal F}_{T}$-measurable random variable $X$ with values in $\mathbb{R}_{+}\cup\{+\infty\}$ such that
$V_{T}(\phi_{n})\geq -1$, $\lim_{n\rightarrow\infty}V_{T}(\phi_{n})=X$ and $\mathbb{P}\{X>0\}>0$. Delbaen \cite{del92} showed that for a bounded, continuous process, condition NFLBR is equivalent to the existence of a strict martingale measure (in a discrete-time setting, this equivalence was established by Schachermayer \cite{sch93} without the boundedness assumption). If $S$ is an unbounded, continuous process, condition NFLBR implies the existence of a martingale measure (not necessarily strict). A slightly stronger no-arbitrage condition, referred to as NFLVR (no free lunch with vanishing risk), is obtained if the inequality $V_{T}(\phi_{n})\geq -1$ in the above definition is replaced by the following condition
\[V_{T}(\phi_{n})\geq -\delta_{n}\mbox{ for some sequence $\delta_{n}\downarrow 0$}.\]
Put another way, for any sequence $\{\phi_{n}\}_{n\geq 1}$ of simple predictable strategies such that $\phi_{n}$ is $\delta_{n}$-admissible and the sequence $\delta_{n}$ tends to zero, we have $V_{T}(\phi_{n})\rightarrow 0$ in probability as $n\rightarrow\infty$. Delbaen and Schachermayer \cite{del94b} showed that NFLVR is equivalent to the existence of a martingale measure, even when the continuity assumption is relaxed (we still need to assume that $S$ is a locally bounded semimartingale).

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

Multidimensional Black-Scholes Model.

The classic Black-Scholes model is now extended in several directions. First, the number of risky primary securities can be greater than $1$. Second, the underlying Brownain motion is assumed to be multidimensional, rather than one-dimensional. Finally, the volatility coefficient is no longer assumed to be constant (or deterministic). We consider a financial market ${\cal M}_{BS}$ in which $d+1$ assets are traded continuously. The first of these is an asset without systematic risk, called the bank account (or sometimes the bond), with price process defined by
\[dB_{t}=B_{t}\cdot r_{t}dt\mbox{ with }B_{0}=1.\]
The remaining $d$ assets (usually referred to as stocks) are subject to systematic risk. The price process $S_{t}^{(i)}$ for $0\leq t\leq T$ of the stock $1\leq i\leq d$ is modelled by the linear stochastic differential equation
\[dS_{t}^{(i)}=S_{t}^{(i)}\left (\mu_{t}^{(i)}dt+\boldsymbol{\sigma}_{t}^{(i)}\cdot d{\bf W}_{t}\right )\]
with $S_{0}^{(i)}>0$, or equivalently
\[dS_{t}^{(i)}=S_{t}^{(i)}\left (\mu_{t}^{(i)}dt+\sum_{j=1}^{n}\sigma_{t}^{(ij)}dW_{t}^{(j)}\right ), S_{0}^{(i)}=p_{i}\in
(0,\infty ),\]

where ${\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)})$ for $0\leq t\leq T$ is a standard $n$-dimensional Brownian motion, defined on a filtered probability space $(\Omega ,{\cal F},\mathbb{P},{\bf F})$. We assume that the underlying filtration ${\bf F}=\{{\cal F}_{t}:0\leq t\leq T\}$ is the Brownian filtration (basically ${\cal F}_{t}=\sigma ({\bf W}_{s},0\leq s\leq t)$ slightly enlarged to satisfy the usual conditions). We interpret the process $\{r_{t},0\leq t\leq T\}$ as the interest rate process (the short rate or spot rate), i.e., $r_{t}$ is the instantaneous interest rate. The vector process

\[\{\boldsymbol{\mu}_{t}=(\mu_{t}^{(1)},\cdots ,\mu_{t}^{(d)}),0\leq t\leq T\}\]

is the vector of appreciation rates for the stocks, measuring the instantaneous rate of change of ${\bf S}$ at time $t$. Finally the matrix process of volatilities

\[\{\boldsymbol{\sigma}_{t}=(\sigma_{t}^{(ij)}),1\leq i\leq d,1\leq j\leq n,0\leq t\leq T\}\]

models the instantaneous intensity with which the $j$ source of uncertainty influences the price of the $i$th stock at time $t$. The processes $r_{t}$, $\boldsymbol{\mu}_{t}$, and $\boldsymbol{\sigma}_{t}$ are referred to as the coefficients of the model ${\cal M}_{BS}$. We assume that these coefficients are progressively measurable with respect to ${\cal F}$ and satisfy the integrability conditions
\[\int_{0}^{T}(|r_{t}|+\parallel \boldsymbol{\mu}_{t}\parallel +\parallel\boldsymbol{\sigma}_{t}\parallel^{2})dt<\infty ,\mbox{a.s.}\]
The class of progressively measurable processes is a slight enlargement of the class of adapted processes with (right- or left-) continuous paths, which we will mainly encounter. $r_{t}$, $\boldsymbol{\mu}_{t}$ and $\boldsymbol{\sigma}_{t}$ adapted to ${\bf F}$ implies that anticipation of the future is precluded and allows for dependence of the stock prices on their past. The case $d=n=1$ and constant coefficients is the classical Black-Scholes model.

Example (Martingale Measure). We now consider the following model
\[\begin{array}{ll}
dB_{t}=rB_{t}dt, & B_{0}=1,\\
dS_{t}=S_{t}(\mu dt+\sigma dW_{t}), & S^{(0)}=p\in (0,\infty ),
\end{array}\]
with constant coefficients $\mu\in \mathbb{R}$, $r,\sigma\in \mathbb{R}_{+}$. We write as usual $\bar{S}_{t}=S_{t}/B_{t}$ for the discounted stock price process (with the bank account being the natural numeraire). Let $f(t,x)=x/B_{t}$. Then
\[f_{t}=-\frac{x}{(B_{t})^{2}}\cdot B’_{t}=-\frac{x}{(B_{t})^{2}}\cdot rB_{t}=-\frac{rx}{B_{t}}\mbox{, }f_{x}=\frac{1}{B_{t}}\mbox{ and }f_{xx}=0.\]
Using the Ito’s formula in Theorem \ref{binp65}, we have
\[d\bar{S}_{t}=\left (-\frac{rS_{t}}{B_{t}}+S_{t}\cdot\mu\cdot\frac{1}{B_{t}}
\right )dt+S_{t}\cdot\sigma\cdot\frac{1}{B_{t}}dW_{t}.\]
That is,
\begin{equation}{\label{bineq71}}\tag{34}
d\bar{S}_{t}=\bar{S}_{t}[(\mu -r)dt+\sigma dW_{t}].
\end{equation}
Because we use the Brownian filtration any pair of equivalent probability measures $\mathbb{P}$ and $\bar{\mathbb{P}}$ on ${\cal F}_{T}$ is a Girsanov pair, i.e.,
\[\left .\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}\right |_{{\cal F}_{t}}=L_{t}=
\exp\left\{-\int_{0}^{t}\gamma_{s}dW_{s}-\frac{1}{2}\int_{0}^{t}\gamma_{s}^{2}ds\right\},\]
where $\{\gamma_{t}:0\leq t\leq T\}$ is a measurable, adapted $d$-dimensional process with $\int_{0}^{T}\gamma_{t}^{2}dt<\infty$. By Girsanov’s theorem \ref{binp66}, we have
\begin{equation}{\label{bineq72}}\tag{35}
dW_{t}=d\bar{W}_{t}-\gamma_{t}dt,
\end{equation}
where $\bar{W}$ is a $\bar{\mathbb{P}}$-Brownian motion. Thus, from (\ref{bineq71}) and (\ref{bineq72}), the $\bar{\mathbb{P}}$-dynamics for $\bar{S}$ are
\[d\bar{S}_{t}=\bar{S}_{t}\left [(\mu -r-\sigma\gamma_{t})dt+\sigma d\bar{W}_{t}\right ].\]

Since $\bar{S}$ has to be a local martingale under $\bar{\mathbb{P}}$ (zero drift) we must have
\[\mu -r-\sigma\gamma_{t}=0\mbox{ for }t\in [0,T],\]
and so we must choose
\[\gamma_{t}\equiv\gamma =\frac{\mu -r}{\sigma}.\]
In this case, we have
\begin{equation}{\label{bineq77}}\tag{36}
d\bar{S}_{t}=\bar{S}_{t}\cdot\sigma\cdot d\bar{W}_{t}
\end{equation}
Since $S_{t}=\bar{S}_{t}B_{t}$, using the product rule (\ref{bineq48}) (note that $\langle M,N\rangle =0$ if $N$ is of finite variation), we have
\begin{align}
dS_{t} & =\bar{S}_{t}dB_{t}+B_{t}d\bar{S}_{t}\nonumber\\
& =\bar{S}_{t}rB_{t}dt+B_{t}\bar{S}_{t}\sigma d\bar{W}_{t}\mbox{ (by (\ref{bineq77}))}\nonumber\\
& =S_{t}(rdt+\sigma d\bar{W}_{t}).\label{bineq75}\tag{37}
\end{align}
We see that the appreciation rate $\mu$ is replaced by the interest rate $r$, hence the terminology risk-neutral martingale measure. $\sharp$

To ensure the absence of arbitrage opportunities, we postulate the existence a $n$-dimensional progressively measurable process $\boldsymbol{\gamma}$ such that the quality
\begin{equation}{\label{museq1023}}\tag{38}
r_{t}-\mu_{t}^{(i)}=\sum_{j=1}^{n}\sigma_{t}^{(ij)}\gamma_{t}^{(j)}=
\boldsymbol{\sigma}_{t}^{(i)}\cdot\boldsymbol{\gamma}_{t}
\end{equation}
is satisfied simultaneously for every $i=1,\cdots ,d$ (for Lebesgue a.e. $t\in [0,T]$, with probability one). Note that the market price of risk is not uniquely determined in general. Indeed, the uniqueness of a solution to (\ref{museq1023}) holds only if $n\leq d$, and the volatility matrix $\boldsymbol{\sigma}$ has full rank for all $t\in [0,T]$. For instance, if $n=d$ and the volatility $\boldsymbol{\gamma}$ is non-singular (for Lebesgue a.e. $t\in [0,T]$, with probability one), then
\[\boldsymbol{\gamma}_{t}=\boldsymbol{\gamma}_{t}^{-1}\cdot (r_{t}{\bf 1}_{n}-\boldsymbol{\mu}_{t})\]

for all $t\in [0,T]$, where ${\bf 1}_{n}$ denotes the $n$-dimensional vector with all entries $1$. Given any process $\boldsymbol{\gamma}$ satisfying (\ref{museq1023}), we introduce a measure $\bar{\mathbb{P}}$ on $(\Omega , {\cal F}_{T})$ by setting
\begin{equation}{\label{museq1024}}\tag{39}
\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=\exp\left (\int_{0}^{T}\boldsymbol{\gamma}_{u}\cdot dW_{u}-
\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right ),\mbox{ $\mathbb{P}$-a.s.},
\end{equation}
provided that the right-hand in (\ref{museq1024}) is well-defined. The Doléans exponential
\[\eta_{t}={\cal E}_{t}\left (\int_{0}^{\cdot}\boldsymbol{\gamma}_{u}
\cdot dW_{u}\right )=\exp\left (\int_{0}^{T}\boldsymbol{\gamma}_{u}\cdot dW_{u}-
\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\]
is known to follow a strictly positive supermartingale (but not necessarily a martingale) under $\mathbb{P}$, so that the case $\mathbb{E}_{\mathbb{P}}\left [ \eta_{T}\right ]<1$ is not excluded a priori. We conclude that formula (\ref{museq1024}) defines a probability measure $\bar{\mathbb{P}}$ equivalent to $\mathbb{P}$ if and only if $\eta$ follows a $\mathbb{P}$-martingale. It is thus essential to check whether a given process $\boldsymbol{\gamma}$ gives rise to an asociated equivalent martingale measure (for the last property to hold, it is enough (but not necessary) that $\boldsymbol{\gamma}$ follows a bounded process). We assume that the class of martingale measures is nonempty. By virtue of Girsanov’s theorem, the process
\[\bar{\bf W}_{t}={\bf W}_{t}-\int_{0}^{t}\boldsymbol{\gamma}_{u}du\]
for all $t\in [0,T]$ is a $n$-dimensional standard Brownian motion under $\bar{\mathbb{P}}$. Using Ito’s formula, it is easy to verify that the discounted stock price $S_{t}^{(i)}/B_{t}$ satisfies, under $\bar{\mathbb{P}}$,
\[d\left (\frac{S_{t}^{(i)}}{B_{t}}\right )=\left (\frac{S_{t}^{(i)}}
{B_{t}}\right )\cdot\boldsymbol{\sigma}_{t}^{(i)}\cdot d\bar{\bf W}_{t}\]

for any $i=1,\cdots ,d$. Put another way, the discounted prices of all stocks follow local martingales under $\bar{\mathbb{P}}$, so that any probability measure defined by means of (\ref{museq1023}) and (\ref{museq1024}) is a martingale measure for this model (corresponding to the choice of the savings account as the numeraire). For any fixed $\bar{\mathbb{P}}$, we define the class $\boldsymbol{\phi}(\bar{\mathbb{P}})$ of admissible trading strategies, and the class of tame strategies relative to $B$. Let us emphasize that we no longer assume that a process $\boldsymbol{\phi}$ representing a trading strategy is necessarily locally bounded (in the present setup, if the wealth process follows a (local) martingale under some martingale measure $\bar{\mathbb{P}}$, then it is necessarily a local martingale under any equivalent martingale measure). The standard market model obtained in this way is referred to as the multidimensional Black-Scholes market. The classic multidimensional Black-Scholes model assumes that $n=d$, the constant volatility $\boldsymbol{\sigma}$ is nonsibgular and the appreciation rate $\mu_{i}$ and the continuously compounded interest rate $r$ are constant. It is easily seen that under these assumptions, the martingale measure exists and is unique. We also have the following result.

\begin{equation}{\label{bint2}}\tag{40}\mbox{}\end{equation}

Theorem \ref{bint2}. If ${\cal M}_{BS}$ is arbitrage-free with respect to tame strategies, then there exists a progressively measurable process $\boldsymbol{\gamma}:[0,T]\times\Omega\rightarrow \mathbb{R}^{n}$, called the market price of risk, such that
\begin{equation}{\label{bineq4}}\tag{41}
\boldsymbol{\mu}_{t}-r_{t}{\bf 1}_{n}=\boldsymbol{\sigma}_{t}
\cdot\boldsymbol{\gamma}_{t}\mbox{ for }0\leq t\leq T\mbox{ a.s.},
\end{equation}
where ${\bf 1}_{n}$ denotes the $n$-dimensional vector with all entries $1$. Conversely, if such a process $\boldsymbol{\gamma}_{t}$ exists and satisfies, in addition to the above requirements,
\begin{equation}{\label{bineq73}}\tag{42}
\int_{0}^{T}\parallel\boldsymbol{\gamma}_{t}\parallel^{2}dt<\infty\mbox{ a.s.}
\end{equation}
and
\begin{equation}{\label{bineq74}}\tag{43}
\mathbb{E}_{\mathbb{P}}\left [\exp\left (-\int_{0}^{T}\boldsymbol{\gamma}_{u}
\cdot dW_{u}-\frac{1}{2}\int_{0}^{T}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\right ]=1,
\end{equation}
then ${\cal M}_{BS}$ is arbitrage-free.

Proof. To prove the converse, we recall that by Theorem \ref{binp53} the existence of an equivalent martingale measure rules out arbitrage. Therefore, we are going to define such an equivalent martingale measure $\bar{\mathbb{P}}$ (this will be seen in (\ref{binaeq758})). The above conditions (\ref{bineq73}) and (\ref{bineq74}) ensure that the exponential process
\[L_{t}=\exp\left (-\int_{0}^{t}\boldsymbol{\gamma}_{u}\cdot dW_{u}-
\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\gamma}_{u}\parallel^{2}du\right )\mbox{ for }0\leq t\leq T\]
is a martingale under $\mathbb{P}$, and thus
\begin{equation}{\label{binaeq758}}\tag{44}
\bar{\mathbb{P}}(A)\equiv \mathbb{E}_{\mathbb{P}}[L_{T}\cdot 1_{A}]=\int_{A}L_{T}d\mathbb{P}
\end{equation}
for $A\in {\cal F}_{T}$ defines an equivalent probability measure $\bar{\mathbb{P}}$ with Radon-Nikodym derivative
\[\left .\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}\right |_{{\cal F}_{t}}=L_{t}\mbox{ for }0\leq t\leq T.\]

$\bar{\mathbb{P}}$ is the risk-neutral equivalent martingale measure. Under $\bar{\mathbb{P}}$,
\[\bar{W}_{t}\equiv W_{t}+\int_{0}^{t}\boldsymbol{\gamma}_{u}du\mbox{ for }0\leq t\leq T\]

is a Brownian motion. The stock price dynamics under $\bar{\mathbb{P}}$ are
\[dS_{t}^{(i)}=S_{t}^{(i)}\cdot\left (r_{t}dt+\sum_{j=1}^{n}\sigma_{t}^{(ij)}
d\bar{W}_{t}^{(j)}\right )\mbox{ for }i=1,\cdots ,d ,\]
or equivalently for the discounted price processes (using the product rule (\ref{bineq48}) as in (\ref{bineq75}))
\[d\bar{S}_{t}^{(i)}=\bar{S}_{t}^{(i)}\cdot\left (\sum_{j=1}^{n}
\sigma_{t}^{(ij)}d\bar{W}_{t}^{(j)}\right )\mbox{ for }i=1,\cdots ,d.\]

So $\bar{S}$ is a local $\bar{\mathbb{P}}$-martingale and therefore $\bar{\mathbb{P}}$ is an equivalent martingale measure. $\blacksquare$

As an illustration of the effect of different choices of $\boldsymbol{\gamma}$ we consider a simple model with two securities $S^{(1)}$ and $S^{(2)}$. Let the price process dynamics be given as
\[dS_{t}^{(1)}=S_{t}^{(1)}\cdot (\mu_{t}^{(1)}dt+\sigma_{t}^{(1)}dW_{t})
\mbox{ and }dS_{t}^{(2)}=S_{t}^{(2)}\cdot (\mu_{t}^{(2)}dt+\sigma_{t}^{(2)}dW_{t}).\]
Assume there is a process $\gamma_{t}$ (which we might use to define a change of measure) satisfying
\[\frac{\mu_{t}^{(1)}-r_{t}}{\sigma_{t}^{(1)}}=\frac{\mu_{t}^{(2)}-r_{t}}{\sigma_{t}^{(2)}}=\gamma_{t}.\]
Observe that in the numerators we have the excess rate of return of the risky assets over the riskfree rate and in the denominators the volatility of the assets. So these quotients can be interpreted as the risk premium per unit of volatility. This ratio is often called the market price of risk. We can rewrite the above as
\[dS_{t}^{(1)}=S_{t}^{(1)}\cdot [(r_{t}+\gamma_{t}\sigma_{t}^{(1)})dt+
\sigma_{t}^{(1)}dW_{t}]\mbox{ and }dS_{t}^{(2)}=S_{t}^{(2)}\cdot [(r_{t}+
\gamma_{t}\sigma_{t}^{(2)})dt+\sigma_{t}^{(2)}dW_{t}].\]
If, for example, $\gamma\equiv 0$, then
\[dS_{t}^{(1)}=S_{t}^{(1)}\cdot (r_{t}dt+\sigma_{t}^{(1)}dW_{t})\mbox{ and }
dS_{t}^{(2)}=S_{t}^{(2)}\cdot (r_{t}dt+\sigma_{t}^{(2)}dW_{t})\]

and we see that $\bar{S}_{i}=S_{i}/B$ for $i=1,2$ are (local) martingales under $\mathbb{P}$ (using the product rule (\ref{bineq48}) as in (\ref{bineq75}), the drift will be zero), so we are already in our usual risk-neutral setting. If we set $\gamma=\sigma^{(2)}$, we get (doing some stochastic calculus)
\[d\left [\frac{S_{t}^{(1)}}{S_{t}^{(2)}}\right ]=(\sigma_{t}^{(1)}-
\sigma_{t}^{(2)})\cdot\frac{S_{t}^{(1)}}{S_{t}^{(2)}}\cdot dW_{t}.\]
So $S^{(1)}/S^{(2)}$ is a (local) martingale under $\mathbb{P}$ in this setting (zero drift). Since the attitude towards risk is described by $\sigma^{(2)}(=\gamma )$ (the risk in holding the asset $S^{(2)}$), $\mathbb{P}$ is called a risk-neutral measure with respect to $S^{(2)}$ ($S^{(2)}$ can be regarded as a numeraire).

Market Completeness.

We now turn to the question of market completeness. Basically, it is defined in much that same way as for a finite market model, except that certain technical restrictions are imposed on contingent claims.

Definition. The multidimensional Black-Scholes model is complete when any $\bar{\mathbb{P}}$-integrable contingent claim $X$ which is bounded from below is attainable; that is, if for any such claim $X$ there exists an admissible trading strategy $\boldsymbol{\phi}$ satisfying $X=V_{T}(\boldsymbol{\phi})$. In the opposite case, the market model is said to be incomplete. $\sharp$

Again we start by looking at the classical Black-Scholes model. We already know that we have a unique martingale measure $\bar{\mathbb{P}}$ (recall $\gamma =(\mu -r)/\sigma$ in Girsanov’s transformation). Given a contingent claim $X\in L^{1}(\Omega , {\cal F},\mathbb{P})$, then $X\in L^{1}(\Omega ,{\cal F},\bar{\mathbb{P}})$ also, and we can define the $\bar{\mathbb{P}}$-martingale
\[M_{t}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .e^{-rT}X\right |{\cal F}_{t}\right ].\]
Using the martingale representation theorem \ref{binp68} for one-dimensional Brownian motion, we know that under $\bar{\mathbb{P}}$
\begin{equation}{\label{bineq82}}\tag{45}
M_{t}=M_{0}+\int_{0}^{t}H_{u}d\bar{W}_{u}.
\end{equation}
Now the $\bar{\mathbb{P}}$-dynamics of $\bar{S}$ are
\[d\bar{S}_{t}=\bar{S}_{t}\cdot\sigma d\bar{W}_{t}\]
by (\ref{bineq77}), so in fact
\[M_{t}=M_{0}+\int_{0}^{t}\phi_{u}^{(1)}d\bar{S}_{u},\]
with
\begin{equation}{\label{bineq5}}\tag{46}
\phi^{(1)}_{t}=H_{t}/\sigma\bar{S}_{t}.
\end{equation}
Using (\ref{bineq3}) and defining
\begin{equation}{\label{bineq6}}\tag{47}
\phi^{(0)}_{t}=M_{t}-\phi^{(1)}_{t}\bar{S}_{t}=M_{t}-\frac{H_{t}}{\sigma},
\end{equation}
we obtain a self-financing replicating trading strategy (in terms of the discounted values $(1,\bar{S})$). We have thus shown that $X$ is attainable, and since $X$ was arbitrary the model is complete.

Turning to the general case, we restrict consideration to contingent claims $X$ with $X/B_{t}\in L^{1}(\Omega ,{\cal F}_{T},\bar{\mathbb{P}})$. From Theorem \ref{bint1} we know that uniqueness of the martingale measure implies that a financial model is complete. In ${\cal M}_{BS}$ all equivalent martingale measures are given by means of Girsanov’s transformation, and hence are characterized by functions $\boldsymbol{\gamma}$ satisfying (use Theorem \ref{bint2})
\[\boldsymbol{\mu}_{t}-r_{t}{\bf 1}_{n}=\boldsymbol{\sigma}_{t}
\cdot\boldsymbol{\gamma}_{t}\mbox{ for }0\leq t\leq T.\]
Keeping this in mind, a characterization of completeness in ${\cal M}_{BS}$ will involve conditions on the coefficients of the model.

\begin{equation}{\label{bint3}}\tag{48}\mbox{}\end{equation}

Theorem \ref{bint3}. The following statements are equivalent.

(a) There exists a unique equivalent martingale measure for the discounted stock price process $\bar{{\bf S}}$.

(b) The multidimensional Black-Scholes model ${\cal M}_{BS}$ is complete $($in the restricted sense that every contingent claim $X$ with $X/B_{t}\in L^{1}(\Omega ,{\cal F},\bar{\mathbb{P}})$ is attainable$)$.

(c) Inequality $n\leq d$ holds and the volatility $\boldsymbol{\sigma}$ has full rank for Lebesgue a.e. $t\in [0,T]$, with probability $1$.

(d) Equality $n=d$ holds and the volatility matrix $\boldsymbol{\sigma}_{t}(\omega )$ is $(\lambda\times \mathbb{P})$-a.e. nonsingular. $\sharp$

If $n\geq d$ and the matrix $\boldsymbol{\sigma}$ has full rank we can reduce the number of stocks by duplicating some of them as $(t,\omega )$-dependent linear combinations of others. We assume now that $d=n$, that the coefficients of the model satisfy the integrability conditions, that $\boldsymbol{\sigma}$ is non-singular and $\boldsymbol{\gamma}$ in (\ref{bineq4}) exists. Then, the financial market model admits a unique equivalent martingale measure $\bar{\mathbb{P}}$ with Radon-Nikodym derivative given by the Girsanov transformation
\[L_{t}=\exp\left\{-\int_{0}^{t}\boldsymbol{\gamma}_{u}\cdot dW_{u}-\frac{1}{2}\int_{0}^{t}\parallel\boldsymbol{\gamma}_{u}\parallel^{2} du\right\}\mbox{ for }0\leq t\leq T\]

and $\boldsymbol{\gamma}_{t}=(\boldsymbol{\sigma}_{t})^{-1}\cdot (\boldsymbol{\mu}_{t}-r_{t}{\bf 1}_{d})$. By Theorems \ref{bint2} and \ref{bint3}, the model is free of arbitrage and complete. We call such a model {\bf standard}. Recall that a contingent claim $X$ is a ${\cal F}_{T}$-measurable random variable such that $X/B_{t}\in L^{1}(\Omega ,{\cal F}_{T},\bar{\mathbb{P}})$.

Theorem (Risk-Neutral Valuation Formula). Let ${\cal M}_{BS}$ be a standard multidimensional Black-Scholes model and $X$ a contingent claim. The arbitrage price process of $X$ is given by the risk-neutral valuation formula
\[\Pi_{t}(X)=B_{t}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .\frac{X}{B_{t}}\right |{\cal F}_{t}\right ]=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}\left [\left .Xe^{-\int_{t}^{T}r_{u}du}\right |{\cal F}_{t}\right ].\]

Proof. Since the model is complete, any such contingent claim is attainable, and the result follows from Theorem \ref{binp56}. $\blacksquare$

We now look at the special case of the classical Black-Sholes model. By the risk-neutral valuation principle the price of a contingent claim $X$ is given by
\begin{equation}{\label{bineq80}}\tag{49}
\Pi_{t}(X)=e^{-r(T-t)}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[X|{\cal F}_{t}],
\end{equation}
with $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}$ given via the Girsanov density
\[L_{t}=\exp\left [-\left (\frac{b-r}{\sigma}\right )W_{t}-\frac{1}{2}\left (\frac{b-r}{\sigma}\right )^{2}t\right ].\]
Furthermore if $X$ is of form $X=\zeta (S_{T})$ with a sufficiently integrable function $\zeta$, then the price process is also given by $\Pi_{t}(X)=F(t,S_{t})$, where $F$ solves the Black-Sholes partial differential equation
\begin{equation}{\label{bineq8}}\tag{50}
\begin{array}{rcl}
{\displaystyle
F_{t}(t,s)+rsF_{s}(t,s)+\frac{1}{2}\sigma^{2}s^{2}F_{ss}(t,s)-rF(t,s)}
& = & 0\\
F(T,s) & = & \zeta (s).
\end{array}
\end{equation}

To obtain the partial differential equation (\ref{bineq8}), we use the Feynman-Kac formula in Theorem \ref{binp79}. We see that
\[G(t,S_{t})=e^{-rT}\cdot \mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[\zeta (S_{T})|{\cal F}_{t}]=
\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[e^{-rT}\cdot\zeta (S_{T})|{\cal F}_{t}]\]
is a $\bar{\mathbb{P}}$-martingale. Recall that $dS=S\cdot r\cdot dt+S\cdot\sigma\cdot d\tilde{W}$. Using Theorem \ref{binp79} with $h(x)=e^{-rT}\cdot\zeta (x)$, $\mu (t,S_{t})=r\cdot S_{t}$ and $\sigma (t,S_{t})=S_{t}\cdot\sigma$, we have the partial
differential equation
\begin{equation}{\label{bineq81}}\tag{51}
G_{t}(t,s)+rs\cdot G_{s}(t,s)+\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot G_{ss}(t,s)=0
\end{equation}
with final condition $G(T,s)=h(s)=e^{-rT}\cdot\zeta (s)$. From (\ref{bineq80}), we have $F(t,S_{t})=\Pi_{t}(X)=e^{rt}\cdot G(t,S_{t})$, i.e., $G(t,S_{t})=r^{-rt}\cdot F(t,S_{t})$. Then $G_{t}(t,s)=-re^{-rt}\cdot F(t,s)+e^{-rt}\cdot F_{t}(t,s)$, $G_{s}(t,s)=e^{-rt}\cdot F_{s}(t,s)$ and $G_{ss}(t,s)=e^{-rt}\cdot F_{ss}(t,s)$. From (\ref{bineq81}), we can get the partial differential equation (\ref{bineq8}). Considering a European call with strike $K$ and maturity $T$ on the stock $S$, i.e., $\zeta (S_{T})=(S_{T}-K)^{+}$, we can evaluate the above expected value (which is easier than solving the Black-Sholes partial differential equation) and obtain

Proposition (Black-Sholes Formula). The Black-Scholes price process of a European call is given by
\[C_{t}=S_{t}\cdot N(d_{1}(S_{t},T-t))-Ke^{-r(T-t)}\cdot N(d_{2}(S_{t},T-t)).\]
The functions $d_{1}(s,t)$ and $d_{2}(s,t)$ are given by
\[d_{1}(s,t)=\frac{\log (s/K)+(r+\frac{\sigma^{2}}{2})t}{\sigma\sqrt{t}}\mbox{ and }d_{2}(s,t)=d_{1}(s,t)-\sigma\sqrt{t}=
\frac{\log (s/K)+(r-\frac{\sigma^{2}}{2})t}{\sigma\sqrt{t}}.\sharp\]

To obtain a replicating portfolio, we use Ito’s lemma to find the dynamics of the $\bar{\mathbb{P}}$-martingale $M_{t}=G(t,S_{t})$
\[dM_{t}=\sigma S_{t}G_{s}(t,S_{t})d\tilde{W}_{t}.\]
Using this representation we get in terms of the notation in (\ref{bineq82})
\[H_{t}=\sigma S_{t}G_{s}(t,S_{t}),\]
which gives for the stock component of the replicating portfolio using the discounted assets $\bar{S}_{t}=S_{t}/B_{t}$ and (\ref{bineq5})
\[\phi^{(1)}_{t}=H_{t}/\sigma\bar{S}_{t}=G_{s}(t,S_{t})B_{t},\]
and using (\ref{bineq6}) the cash component is
\[\phi^{(0)}_{t}=M_{t}-\phi^{(1)}_{t}\bar{S}_{t}=G(t,S_{t})-
G_{s}(t,S_{t})B_{t}\bar{S}_{t}=G(t,S_{t})-G_{s}(t,S_{t})S_{t}.\]
To transfer this portfolio to undiscounted value we multiply it with the discount factor, i.e., $F(t,S_{t})=B_{t}G(t,S_{t})$ and get the following result.

Proposition. The replicating strategy in the classical Black-Sholes model is given by
\[\phi_{0}=\frac{F(t,S_{t})-F_{s}(t,S_{t})S_{t}}{B_{t}}\mbox{ and }\phi_{1}=F_{s}(t,S_{t}).\sharp\]

In the original paper by Black and Scholes used an arbitrage pricing approach (rather than the risk-neutral valuation approach) to deduce the price of a European call as the solution of a partial differential equation (We call this the PDE approach). The idea is as follows: start by assuming that the option price $C_{t}$ is given by $C_{t}=f(t,S_{t})$ for some sufficiently smooth function $f:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}$. By Ito’s formula we find for the dynamics of the options price process (observe that we work under $\mathbb{P}$ so $dS=S(\mu dt+\sigma dW$))
\begin{equation}{\label{bineq7}}\tag{52}
dC=\left [f_{t}(t,S)+f_{s}(t,S)S\mu +\frac{1}{2}f_{ss}(t,S)S^{2}\sigma^{2}\right ]dt+f_{s}S\sigma dW.
\end{equation}
Consider a portfolio $\psi$ consisting of a short position in $\psi_{1}(t)=f_{s}(t,S_{t})$ stocks and a long position in $\psi_{2}(t)=1$ call and assume the portfolio is self-financing. Then its value process is
\[V_{\psi}(t)=-\psi_{1}(t)S_{t}+C_{t},\]
and by the self-financing condition we have
\begin{align*}
dV_{\psi} & =-\psi_{1}dS+dC\\
& =-f_{s}(S\mu dt+S\sigma dW)+\left (f_{t}+f_{s}S\mu +\frac{1}{2}f_{ss}S^{2}\sigma^{2}\right )dt+f_{s}S\sigma dW\\
& =\left (f_{t}+\frac{1}{2}f_{ss}S^{2}\sigma^{2}\right )dt.
\end{align*}
So the dynamics of the value process of the portfolio do not have any exposure to the driving Brownian motion, and its appreciation rate in an arbitrage-free world must therefore equal the risk-free rate, i.e.,
\[dV_{\psi}(t)=rV_{\psi}(t)dt=(-rf_{s}S+rC)dt.\]
Comparing the coefficients and using $C_{t}=f(t,S_{t})$, we must have
\[-rSf_{s}+rf=f_{t}+\frac{1}{2}\sigma^{2}S^{2}f_{ss}.\]
This leads to the Black-Scholes partial differential equation for $f$, i.e.,
\[f_{t}+rSf_{s}+\frac{1}{2}\sigma^{2}S^{2}f_{ss}-rf=0.\]
Since $C_{T}=(S_{T}-K)^{+}$ we need to impose the terminal condition $f(s,T)=(s-K)^{+}$ for all $s\in \mathbb{R}_{+}$.

One point in the justification of the above argument is missing: we have to show that the trading strategy short $\psi_{1}$ stocks and long one call is self-financing. In fact, this is not true, since $\psi_{1}=\psi_{1}(t,S_{t})$ is dependent on the stock price process. Formally, for the self-financing condition to be true we must have
\[dV_{\psi}(t)=d(\psi_{1}(t)S_{t})+dC_{t}=\psi_{1}(t)dS_{t}+dC_{t}.\]
Now $\psi_{1}(t)=\psi_{1}(t,S_{t})$ depends on the stock price and so we have
\[d(\psi_{1}(t,S_{t})S_{t})=\psi_{1}(t)dS_{t}+S_{t}d\psi_{1}(t,S_{t})+d\langle\psi_{1},S\rangle (t).\]
We see that the portfolio $\psi_{1}$ is self-financing if
\[S_{t}d\psi_{1}(t,S_{t})+d\langle\psi_{1},S\rangle (t)=0.\]
It is an exercise in Ito calculus to show that this is not the case.

Variance-Minimizing Hedging.

The mean-variance is the approach to the hedging of non-attainable claims in an incomplete market. In a continuous-time framework, typical optimization problems are the following.

(MV1). For a fixed $c\in \mathbb{R}$, minimize
\[J_{1}(\boldsymbol{\phi})=\mathbb{E}_{P}\left [\left (\bar{X}-c-\int_{0}^{T}
\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}\right )^{2}\right ]\]
over all self-financing trading strategies $\boldsymbol{\phi}$.
(MV2). Minimize
\[J_{2}(\boldsymbol{\phi})=\mathbb{E}_{P}\left [\left (\bar{X}-c-\int_{0}^{T}
\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}\right )^{2}\right ]\]
over all $c\in \mathbb{R}$ and all self-financing trading strategies $\boldsymbol{\phi}$.
(MV3). Minimize
\[J_{3}(\boldsymbol{\phi})=Var_{P}\left (\bar{X}-\int_{0}^{T}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}\right )\]
over all self-financing trading strategies $\boldsymbol{\phi}$.

Note that problem (MV2.) is related to the closeness, in the space $L^{2}(P)$ of ${\cal F}_{T}$-measurable random variables, of the following set
\[{\cal F}_{T}=\left\{\int_{0}^{T}\boldsymbol{\phi}_{u}\cdot
d\bar{{\bf S}}_{u}:\int_{0}^{\cdot}\boldsymbol{\phi}_{u}\cdotd\bar{{\bf S}}_{u}\in {\cal H}_{P}^{2}\right\}.\]
In the last formula, we write ${\cal H}_{P}^{2}$ to denote the class of all real-valued special semimartingales with finite ${\cal H}_{P}^{2}$-norm, where
\[\parallel S\parallel_{{\cal H}_{P}^{2}}\equiv\left |\!\left ||S^{(0)}|+
\langle M,M\rangle\mathbb{E}_{T}^{1/2}\right |\!\right |_{L^{2}(P)}+
\left |\!\left |\int_{0}^{T}|dA_{u}|\right |\!\right |_{L^{2}(P)}\]
and $S=S^{(0)}+M+A$ is the canonical decomposition of $S$. If the discounted price $\bar{S}$ follows a square-integrable martingale under $P$, the closeness of the space ${\cal G}_{T}$ of stochastic integrals is straightforward since Ito’s integral defines an isometry.

Risk-Minimizing Hedging.

A different approach to the hedging of non-attainable claims, referred to as risk-minimizing hedging, starts by enlarging the class of trading strategies in order to allow for additional transfers of funds, referred to as costs. We fix a probability measure $\mathbb{P}$ and we assume that the canonical decomposition of $\bar{S}$ under $\mathbb{P}$ is $\bar{S}=\bar{S}_{0}+M+A$. For a given claim $X$, we consider the class of all replicating strategies, not necessarily self-financing, such that the discounted wealth process follows a real -valued semimartingale of class ${\cal H}_{\mathbb{P}}^{2}$. By definition, the (discounted) cost process $C(\boldsymbol{\phi})$ of a strategy $\boldsymbol{\phi}$ equals (Note that the cost process does not involve the initial cost, but only the additional transfers of funds, i.e., withdrawals and infusions.)
\[C_{t}(\boldsymbol{\phi})=\bar{V}_{t}(\boldsymbol{\phi})-
\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}_{u}\mbox{ for all }t\in [0,T],\]
where $\bar{V}_{t}(\boldsymbol{\phi})=\boldsymbol{\phi}_{t}\cdot\bar{{\bf S}}_{t}$. As in a discrete-time setting, a local risk minimization is based on the suitably defined minimality of the cost process. Schweizer \cite{sch91} shows that, under mild technical assumptions, a replicating strategy $\bar{\boldsymbol{\phi}}$ is locally risk-minimizing if it is mean self-financing and the cost process $C(\bar{\boldsymbol{\phi}})$ follows a square-integrable martingale strongly orthogonal to the martingale part of $\bar{{\bf S}}$ (Two martingales are said to be strongly orthogonal if their product follows a martingale). A strategy $\boldsymbol{\phi}$ is said to be mean self-financing under $\mathbb{P}$ if the cost process $C(\boldsymbol{\phi})$ follows a martingale under $\mathbb{P}$. The following generalization of the Kunita-Watanabe decomposition appears to be useful.

Definition. We say that a square-integrable random variable $\bar{X}$ admits the Föllmer-Schweizer decomposition with respect to $\bar{S}$ under $\mathbb{P}$ if
\begin{equation}{\label{museq1029}}\tag{53}
\bar{X}=c_{0}+\int_{0}^{T}\boldsymbol{\xi}_{u}\cdot d\bar{{\bf S}}+L_{T},
\end{equation}
where $c_{0}$ is a real number, $\boldsymbol{\xi}$ is a predictable process such that the stochastic integral $\int_{0}^{t}\boldsymbol{\xi}_{u}\cdot d\bar{{\bf S}}$ belongs to the class ${\cal H}_{\mathbb{P}}^{2}$ and $L$ is a square-integrable martingale strongly orthogonal to $M$ under $\mathbb{P}$ with $L_{0}=0$. $\sharp$

The following result emphasizes the relevance of the Föllmer-Schweizer decomposition in the present context. Basically, it says that a locally risk-minimizing replicating strategy is determined by the process $\boldsymbol{\xi}$ in representation (\ref{museq1029}).

\begin{equation}{\label{musp1022}}\tag{54}\mbox{}\end{equation}

Proposition \ref{musp1022}. Let $X$ be a European claim which settles at time $T$ such that $\bar{X}$ is square-integrable under $P$. Then, the following statements are equivalent.

(a) $X$ admits a locally risk-minimizing replicating strategy.

(b) $\bar{X}$ admits a Föllmer-Schweizer decomposition with respect to the stock price $\bar{{\bf S}}$ under $\mathbb{P}$.

Moreover, if (b) holds, then there exists a locally risk-minimizing replicating strategy $\bar{\boldsymbol{\phi}}$ satisfying $\bar{\boldsymbol{\phi}}^{i}=\xi^{i}$ for $i=1,\cdots ,d$, where $\boldsymbol{\xi}$ is given by $(\ref{museq1029})$. Finally, for any locally risk-minimizing strategy $\boldsymbol{\phi}$, we have
\[\int_{0}^{t}\boldsymbol{\phi}_{u}\cdot d\bar{{\bf S}}=
\int_{0}^{t}\boldsymbol{\xi}_{u}\cdot d\bar{{\bf S}}\mbox{ for all }t\in [0,T]. \sharp\]

In view of Proposition \ref{musp1022}, it is clear that the Föllmer-Schweizer decomposition provides a neat method of searching for the locally risk-minimizing strategy. It appears that the Föllmer-Schweizer decomposition under $\mathbb{P}$ corresponds to the Kunita-Watanabe decomposition under the so-called minimal martinagle measure $\hat{\mathbb{P}}$ associated with $\mathbb{P}$. Furthermore, if $X$ admits the locally risk-minimizing replicating strategy $\bar{\boldsymbol{\phi}}$, then the initial cost equals
\[V_{0}(\bar{\boldsymbol{\phi}})=C_{0}(\bar{\boldsymbol{\phi}})=\mathbb{E}_{\hat{P}}[\bar{X}]\]
(recall that $B_{0}=1$). More generally, the following version of the risk-neutral valuation formula holds
\begin{equation}{\label{museq1030}}\tag{55}
V_{t}(\bar{\boldsymbol{\phi}})=B_{t}\cdot \mathbb{E}_{\hat{\mathbb{P}}}\left [\left .
\bar{X}\right |{\cal F}_{t}\right ]\mbox{ for all }t\in [0,T].
\end{equation}
Despite its resemblance to the standard risk-neutral valuation formula, equality (\ref{museq1030}) is esentially weaker since the right-hand side manifestly depends on the choice of the minimal martingale measure (through the choice of the actual probability $P$). On the other hand, when applied to an attainable contingent claim, it gives the right result, that is, the arbitrage price of $X$ in the Black-Scholes market. We introduce the notion of a minimal martingale measure for a continuous, $\mathbb{R}^{d}$-valued semimartingale $\bar{{\bf S}}$, with the canonical decomposition $\bar{{\bf S}}=\bar{{\bf S}}_{0}+{\bf M}+{\bf A}$.

Definition. A martingale measure $\hat{\mathbb{P}}$ for $\bar{{\bf S}}$ is called a minimal martingale measure associated with $\mathbb{P}$ if any local $\mathbb{P}$-martingale strongly orthogonal under $\mathbb{P}$ to each local martingale $M^{i}$ for $i=1,\cdots ,k$ remains a local martingale under $\hat{\mathbb{P}}$. $\sharp$

Hofmann et al. \cite{hof} gave the applications to option pricing in a stochastic volatility model.

\begin{equation}{\label{musp1023}}\tag{56}\mbox{}\end{equation}

Proposition \ref{musp1023}. A minimal martingale measure $\hat{\mathbb{P}}$ associated with $\mathbb{P}$ exists if and only if there exists a progressively measurable $\mathbb{R}^{d}$-valued process $\hat{\boldsymbol{\gamma}}$ such that

(i) for every $i=1,\cdots ,k$
\[r_{t}-\mu_{t}^{(i)}=\boldsymbol{\sigma}_{t}^{i}\cdot\hat{\boldsymbol{\gamma}}\mbox{ $l\otimes \mathbb{P}$-a.e. on $[0,T]\times\Omega$};\]

(ii) the Doléans exponential ${\cal E}(U^{\hat{\boldsymbol{\gamma}}})$ of the process
\[U^{\hat{\boldsymbol{\gamma}}}=\int_{0}^{t}\hat{\boldsymbol{\gamma}}_{u}\cdot d{\bf W}_{u}\]
is a martingale under $\mathbb{P}$;

(iii) with probability one, for almost every $t$ we have

\[\hat{\boldsymbol{\gamma}}_{t}\in\mbox{Im}(\boldsymbol{\sigma}_{t})^{T}=\mbox{Ker}(\boldsymbol{\sigma}_{t})^{\perp},\]

where $\boldsymbol{\sigma}^{T}$ is the transpose of $\boldsymbol{\sigma}$. $\sharp$

The proof of Proposition \ref{musp1023} is omitted (see El Karoui and Quenez \cite{kar}). Condition (iii), which corresponds to the concept of minimality of $\hat{\mathbb{P}}$, says, essentially, that there exists a process $\boldsymbol{\zeta}$ satisfying $\hat{\boldsymbol{\gamma}}_{t}=\boldsymbol{\sigma}_{t}\cdot\boldsymbol{\zeta}_{t}$. In a typical example, when one starts with the complete Black-Scholes model and assumes that only some stocks are accessible for trading, under the minimal martingale measure the returns on traded stocks equal the risk-free rate of return, and the returns on non-traded stocks remain unchanged, that is, they are the same under the original probability measure $\mathbb{P}$ and under the associated minimal martingale measure $\hat{\mathbb{P}}$ (ref. Lamberton and Lapeyre \cite{lam}). We shall now present such an example in some detail.

We place ourselves in the multidimensional Black-Scholes setting with constant coefficients so that the price $B$ of a riskless bond, and prices $S^{i}$, $i=1,\cdots ,d$, of risky stock satisfy
\[\left\{\begin{array}{l}
dB_{t}=r\cdot B_{t}dt, B_{0}=1\\
dS_{t}^{(i)}=S_{t}^{(i)}\cdot (\mu^{i}dt+\boldsymbol{\sigma}^{i}\cdot
d{\bf W}_{t}), S_{0}^{(i)}>0,
\end{array}\right .\]
where ${\bf W}$ follows a $d$-dimensional standard Brownian motion on a probability space $(\Omega ,{\bf F},P)$, the appreciation rates $\mu^{i}$ are constants, and the volatility coefficients $\boldsymbol{\sigma}^{i}$, $i=1,\cdots ,d$, are linearly independent vectors in $\mathbb{R}^{d}$. We define the stock index process $I$ by setting
\[I_{t}=\sum_{i=1}^{d}w_{i}\cdot S_{t}^{(i)}\mbox{ for all }t\in [0,T],\]
where $w_{i}>0$ are constants such that $\sum_{i=1}^{d}w_{i}=1$. The stock index is thus the weighted arithmetic average of prices of all traded stocks. We consider a European call option written on the stock index, which corresponds to the contingent claim $C_{T}=(I_{T}-K)^{+}$. Assume first that $r=0$ (this assumption will be subsequently relaxed). It is clear that under the present hypotheses, there exists the unique martinagle measure $\bar{\mathbb{P}}$ for the multidimensional Black-Scholes model. $\bar{\mathbb{P}}$ is determined by the unique solution $\bar{\boldsymbol{\gamma}}\in \mathbb{R}^{d}$ to the equation
\[\mu^{i}+\boldsymbol{\sigma}^{i}\cdot\bar{\boldsymbol{\gamma}}=0\mbox{ for }i=1,\cdots ,k,\]
through the Doléans exponential
\[\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=\exp\left (\bar{\boldsymbol{\sigma}}\cdot
{\bf W}_{T}-\frac{1}{2}\parallel\bar{\boldsymbol{\sigma}}\parallel^{2}\cdot T\right )\mbox{ $\mathbb{P}$-a.s.}\]
Assume for the moment that all stocks can be used for hedging; we thus deal with a complete model of a security market. It is clear that the arbitrage price of the stock index call option in the complete Black-Scholes market equals
\[\Pi_{t}(C_{T})=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[C_{T}|{\cal F}_{t}]=h(S_{t}^{(1)},\cdots ,
S_{t}^{(d)},T-t)\mbox{ for all }t\in [0,T],\]
where $h:\mathbb{R}^{d}\times [0,T]\rightarrow \mathbb{R}$ is a certain smooth function. Moreover, the strategy which replicates the option satisfies
\[\phi_{t}^{(i)}=h_{S^{i}}(S_{t}^{(i)},\cdots ,S_{t}^{(d)},T-t),\]
for $i=1,\cdots ,d$, and
\[\phi_{t}^{0}=h(S_{t}^{(1)},\cdots ,S_{t}^{(d)},T-t)-\sum_{i=1}^{d}\phi_{t}^{(i)}\cdot S_{t}^{(i)}.\]

The option’s price and the replicating strategy $\boldsymbol{\phi}$ are independent of the drift coefficients $\mu^{i}$. This follows immediately from the fact the dynamics of stock prices under the martingale measure $\bar{\mathbb{P}}$ are
\[dS_{t}^{(i)}=S_{t}^{(i)}\cdot\boldsymbol{\sigma}^{i}\cdot d\bar{{\bf W}}_{t},\]

where the process $\bar{{\bf W}}_{t}={\bf W}_{t}- \bar{\boldsymbol{\gamma}}\cdot t$ follows a $d$-dimensional standard Brownian motion under $\bar{\mathbb{P}}$.

From now on, we assume that only some stocks, say $S^{(1)},\cdots ,S^{m}$ with $1\leq m<k$ are accessible for a particular trader, whose assessment of the future market behavior is reflected by the subjective probability measure $\mathbb{P}$. Such a specification is now essential since a solution to the locally risk-minimization problem will depend on appreciation rates $\mu^{i}$, which in turn depend on the choice of $\mathbb{P}$. Let us observe that the martingale measure for the $m$-dimensional process $(S^{(1)},\cdots ,S^{m})$ is not unique. Indeed, any solution $\boldsymbol{\gamma}\in \mathbb{R}^{d}$ to the equation
\[\mu^{i}+\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\gamma}=0,i=1,\cdots ,m\]
defines a martingale measure $\mathbb{P}_{\boldsymbol{\gamma}}$ for the process $(S^{(1)},\cdots ,S^{m})$, namely,
\[\frac{d\mathbb{P}_{\boldsymbol{\gamma}}}{d\mathbb{P}}=\exp\left (\boldsymbol{\gamma}\cdot {\bf W}_{T}-\frac{1}{2}\parallel
\boldsymbol{\gamma}\parallel^{2}\cdot T\right )\mbox{ $\mathbb{P}$-a.s.}\]
The minimal martingale measure $\hat{\mathbb{P}}$ for the process $\hat{{\bf S}}=(S^{(1)},\cdots ,S^{m})$, associated with $\mathbb{P}$, according to that vector $\hat{\boldsymbol{\gamma}}\in \mathbb{R}^{d}$ which satisfies
\[\left\{\begin{array}{l}
\mu^{i}+\boldsymbol{\sigma}^{i}\cdot\hat{\boldsymbol{\gamma}},i=1,\cdots ,m\\
\hat{\boldsymbol{\gamma}}\in\mbox{Im $\tilde{\boldsymbol{\sigma}}^{T}$}\end{array}\right .\]
where $\tilde{\boldsymbol{\sigma}}$ stands for the matrix with rows $\boldsymbol{\sigma}^{1},\cdots ,\boldsymbol{\sigma}^{d}$. Let us now examine the Föllmer-Schweizer decomposition of the stock index process $I$ with respect to $\hat{{\bf S}}$ under the probability measure $\mathbb{P}$. First, observe that under $\bar{\mathbb{P}}$ we have
\[dI_{t}=\sum_{i=1}^{d}w_{i}dS_{t}^{(i)}=\boldsymbol{\gamma}_{t}^{I}\cdot d\bar{{\bf W}}_{t},\]
where
\[\boldsymbol{\gamma}^{I}=\sum_{i=1}^{d}w_{i}\cdot S_{t}^{(i)}\cdot\boldsymbol{\gamma}^{i}.\]
Let ${\bf v}^{m+1},\cdots ,{\bf v}^{d}\in \mathbb{R}^{d}$ be any orthonormal basis in $\mbox{Ker}( \tilde{\boldsymbol{\sigma}})$. Then, we have (note that $\tilde{\boldsymbol{\sigma}}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{m}$, and thus $\tilde{\boldsymbol{\sigma}}^{T}:\mathbb{R}^{m}\rightarrow \mathbb{R}^{d}$)
\[\boldsymbol{\sigma}_{t}^{I}=\boldsymbol{\nu}_{t}^{\perp}+\boldsymbol{\nu}_{t}=\boldsymbol{\nu}_{t}^{\perp}+
\hat{\boldsymbol{\sigma}}_{t}\cdot\boldsymbol{\psi}_{t},\]
where
\[\boldsymbol{\nu}_{t}^{\perp}=\sum_{j=m+1}^{d}(\boldsymbol{\sigma}_{t}^{I}\cdot {\bf v}^{j}){\bf v}^{j}=
\sum_{i,j=m+1}^{d}w_{i}\cdot S_{t}^{(i)}\cdot (\boldsymbol{\sigma}^{i} \cdot {\bf v}^{j}){\bf v}^{j},\]

and $\boldsymbol{\psi}$ is some $\mathbb{R}^{m}$-valued adapted process. Consequently, under $\bar{\mathbb{P}}$ we have
\[dI_{t}=\boldsymbol{\nu}_{t}^{\perp}\cdot d\bar{{\bf W}}_{t}+\sum_{i=1}^{m}\psi_{t}^{i}dS_{t}^{(i)}.\]
On the other hand, under the original probability measure $P$ the dynamics of $I$ are
\[dI_{t}=\boldsymbol{\nu}_{t}^{\perp}\cdot d{\bf W}_{t}-\boldsymbol{\nu}_{t}^{\perp}\cdot
(\bar{\boldsymbol{\gamma}})^{\perp}dt+\sum_{i=1}^{m}\psi_{t}^{i}dS_{t}^{(i)},\]
where $(\bar{\boldsymbol{\sigma}})^{\perp}\in \mathbb{R}^{d}$ stands for the orthogonal projection of $\bar{\boldsymbol{\gamma}}$ on $\mbox{Ker}(\boldsymbol{\sigma})$, that is,
\[(\bar{\boldsymbol{\gamma}})^{\perp}=\sum_{j=m+1}^{d}(\bar{\boldsymbol{\sigma}}\cdot {\bf v}^{j}){\bf v}^{j}.\]
Concluding, under the minimal martingale measure $\hat{\mathbb{P}}$ associated with $\mathbb{P}$ we have
\[dI_{t}=\boldsymbol{\nu}_{t}^{\perp}\cdot d\hat{{\bf W}}_{t}-\boldsymbol{\nu}_{t}^{\perp}\cdot
(\bar{\boldsymbol{\gamma}})^{\perp}dt+\sum_{i=1}^{m}\psi_{t}^{i}dS_{t}^{(i)},\]
since $\hat{\boldsymbol{\gamma}}\perp\boldsymbol{\nu}_{t}^{\perp}$. Similar decomposition can be established for the process $\Pi_{t}(C_{T})$ which represents the option’s price. Note, however, that the knowledge of these representations is not sufficient to determine the locally risk-minimizing hedging of an option. Note that the locally-minimizing value of an option can be found from the formula
\[\hat{\Pi}_{t|P}(X)=\mathbb{E}_{\hat{\mathbb{P}}}[C_{T}|{\cal F}_{t}]\]
since the dynamics of $(S^{(1)},\cdots ,S^{(d)})$ under the minimla martingale measure $\hat{\mathbb{P}}$ are easily seen to be
\[\left\{\begin{array}{l}
dS_{t}^{(i)}=S_{t}^{(i)}\cdot\boldsymbol{\sigma}^{i}\cdot d\hat{{\bf W}}_{t}, i=1,\cdots ,m\\
dS_{t}^{(i)}=S_{t}^{(i)}\cdot\left ((\mu^{i}+\boldsymbol{\sigma}^{i}\cdot
\hat{\boldsymbol{\gamma}}dt)+\boldsymbol{\sigma}^{i}\cdot
d\hat{{\bf W}}_{t}\right ), i=m+1,\cdots ,d.\end{array}\right .\]
To find a locally risk-minimizing replicating strategy under $\mathbb{P}$, we proceed as follows. First, we introduce auxiliary processes $G^{j}$, $j=m+1,\cdots ,d$, by setting
\[dG_{t}^{j}=G_{t}^{j}\cdot {\bf v}^{j}\cdot d\hat{{\bf W}}_{t},G_{0}^{j}>0.\]
It is essential to observe that the unique martingale measure for the $d$-dimensional semimartingale $(S^{(1)},\cdots ,S^{m},G^{m+1},\cdots , G^{d})$ is easily seen to coincide with the minimal martingale measure $\hat{\mathbb{P}}$. The self-financing replicating strategy for $C_{T}$, with $S^{i}$, $i=1,\cdots ,m$, and $G^{j}$, $j=m+1,\cdots ,d$, playing the role of hedging assets, can be found by proceeding along the same lines as in the case of a complete market. Indeed, the arbitrage price of the option in this fictitious market equals
\[\hat{\Pi}_{t|\mathbb{P}}(X)=\mathbb{E}_{\hat{\mathbb{P}}}[C_{T}|{\cal F}_{t}]=\hat{h}(S_{t}^{(1)},
\cdots ,S_{t}^{m},G_{t}^{m+1},\cdots ,G_{t}^{d},T-t)\]
for some function $\hat{h}(x_{1},\cdots ,x_{d},T-t)$. Furthermore, the self-financing replicating strategy $\boldsymbol{\psi} =(\psi^{0},\psi^{1},\cdots ,\psi^{d})$ equals
\[\psi_{t}^{i}=\frac{\partial\hat{h}}{\partial x_{i}}(S_{t}^{(1)},\cdots ,S_{t}^{m},G_{t}^{m+1},\cdots ,G_{t}^{d},T-t)\]
for $i=1,\cdots ,d$, and
\[\psi_{t}^{0}=\hat{\Pi}_{t|\mathbb{P}}(X)-\sum_{i=1}^{m}\psi_{t}^{i}\cdot S_{t}^{(i)}-\sum_{j=m+1}^{d}\psi_{t}^{j}\cdot G_{t}^{j}.\]
In this way, we arrive at the following formula
\[C_{T}=\hat{\Pi}_{0|\mathbb{P}}(X)+\sum_{i=1}^{m}\int_{0}^{t}\psi_{u}^{i}dS_{u}^{(i)}
+\sum_{j=m+1}^{d}\int_{0}^{t}\psi_{u}^{j}dG_{u}^{j},\]
which is the Kunita-Watanabe decomposition of $C_{T}$ under $\hat{\mathbb{P}}$ (notice that for every $i$ and $j$, processes $S^{i}$ and $G^{j}$ are square-integrable martingales mutually orthogonal under $\hat{\mathbb{P}}$, since $\boldsymbol{\sigma}^{i}\perp {\bf v}^{j}$ if $i\leq m$ and $j>m$). We conclude that the process $(\psi^{1},\cdots ,\psi^{m})$ represents the locally risk-minimizing replicating strategy of the option, associated with the original probability measure $\mathbb{P}$. Furthermore, the corresponding cost process $C(\boldsymbol{\psi})$ satisfies
\[C_{t}(\boldsymbol{\psi})=\hat{\Pi}_{0|\mathbb{P}}(X)+\sum_{j=m+1}^{d}
\int_{0}^{t}\psi_{u}^{j}dG_{u}^{j}\mbox{ for all }t\in [0,T].\]
From now on, we relax the assumption that $r=0$. In a general case, we find easily that
\[dS_{t}^{(i)}=S_{t}^{(i)}\cdot (\hat{\mu}^{i}dt+\boldsymbol{\sigma}^{i}\cdot d\hat{{\bf W}}_{t}),\]
under the minimal martingale measure $\hat{\mathbb{P}}$, where $\hat{\mu}^{i}=r$ for $i=1,\cdots ,m$, and
\[\hat{\mu}^{i}=\mu^{i}+\boldsymbol{\sigma}^{i}\cdot\hat{\boldsymbol{\gamma}}\]
for $i=m+1,\cdots ,d$. Furthermore, we now set
\[dG_{t}^{j}=G_{t}^{j}\left (rdt+{\bf v}^{j}\cdot d\hat{{\bf W}}_{t}\right )\]
for $i=m+1,\cdots ,d$. The goal is to find a quasi-explicit expression for the valuation function $\hat{h}$. The first step is to evaluate the following conditional expectation
\[\mathbb{E}_{\hat{\mathbb{P}}}\left [\left .e^{-r(T-t)}\cdot\left (\sum_{i=1}^{d}w_{i}
S_{t}^{(i)}-K\right )^{+}\right |{\cal F}_{t}\right ]=g(S_{t}^{(1)},\cdots ,S_{t}^{(d)},T-t).\]
As soon as the function $g$ is known, to find the function $\hat{h}$, it is sufficient to express processes $S^{m+1},\cdots ,S^{(d)}$ in terms of $G^{m+1},\cdots ,G^{d}$. This can be done using explicit formulas which are available for all these processes.

First Step. Note that processes $S^{i}$ are given by the following explicit formula
\[S_{t}^{(i)}=S_{0}^{(i)}\cdot\exp\left ((\mu^{i}-\frac{1}{2}\cdot\parallel\boldsymbol{\sigma}^{i}\parallel^{2})\cdot t+
\boldsymbol{\sigma}^{i}\cdot\hat{{\bf W}}_{t}\right ).\]
Let us denote by $\rho$ the following function
\[\rho (w_{1},\cdots ,w_{d},\boldsymbol{\sigma}^{1},\cdots ,\boldsymbol{\sigma}^{d},K)=Q\left\{\sum_{i=1}^{d}w_{i}\cdot\exp \left (\boldsymbol{\sigma}^{i}\cdot\boldsymbol{\xi}\right )\geq K\right\},\]

where the random vector $\boldsymbol{\xi}=(\xi_{1},\cdots ,\xi_{k})$ has the standard $d$-dimensional Gaussian probability distribution under $\bar{\mathbb{P}}$. Using Proposition 2.2 in Lamberton and Lapeyre \cite{lam} (or by direct computations), we obtain
\[g(s_{1},\cdots ,s_{d},T-t)=e^{-r(T-t)}\cdot f(\tilde{s}_{1},\cdots ,\tilde{s}_{d},\sqrt{T-t}\cdot\boldsymbol{\sigma}^{1},\cdots ,\sqrt{T-t}\cdot\boldsymbol{\sigma}^{d},K),\]
where
\[\tilde{s}_{i}=s_{i}\cdot\exp\left ((\mu^{i}-\frac{1}{2}\cdot\parallel
\boldsymbol{\sigma}^{i}\parallel^{2})\cdot (T-t)\right )\]
and
\begin{align*}
f(x_{1},\cdots ,x_{d},{\bf b}^{1},\cdots ,{\bf b}^{d},K) & =\sum_{i=1}^{d}w_{i}\cdot x_{i}\cdot e^{\parallel {\bf b}^{i}\parallel^{2}/2}\cdot\rho (w_{1}\cdot x_{1}\cdot {\bf b}^{1}\cdot {\bf b}^{i},\cdots ,w_{d}\cdot x_{d}\cdot {\bf b}^{d}\cdot {\bf b}^{i},{\bf b}^{1},\cdots ,{\bf b}^{d},K)\\
& -K\cdot\rho (w_{1}\cdot x_{1},\cdots ,w_{d}\cdot x_{d},{\bf b}^{1},\cdots ,{\bf b}^{d},K)\end{align*}
for every $x_{1},\cdots ,x_{d}\in \mathbb{R}$ and ${\bf b}^{1},\cdots ,{\bf b}^{d}\in \mathbb{R}^{d}$.

Second Step. As already mentioned, to find the function $\hat{h}$ it is sufficient to express processes $S^{m+1},\cdots ,S^{(d)}$ in terms of auxiliary processes $G^{m+1},\cdots ,G^{d}$.

Market Imperfections

Under market imperfections, such as the presence of transaction costs, different lending/borrowing rates or short sales constraints, the problem of arbitrage pricing becomes much more involved. We shall comment briefly on the two most relevant techniques used in the context of the Black-Scholes model with imperfections, namely backward stochastic differential equations and stochastic optimal control.

Backward SDE’s

Assume first that the market is perfect, but possibly incomplete. Let $\boldsymbol{\phi}$ be a self-financing trading strategy. Then the wealth process $V=V(\boldsymbol{\phi})$ satisfies
\begin{equation}{\label{museq1031}}\tag{57}
dV_{t}=r_{t}\cdot V_{t}dt+\boldsymbol{\zeta}_{t}\cdot\left (
(\mu_{t}-r_{t}{\bf 1})dt+\boldsymbol{\sigma}_{t}\cdot d{\bf W}_{t}\right ),
\end{equation}
where the $i$th component of $\boldsymbol{\zeta}_{t}$ denotes the amount of cash invested in the $i$th stock at time $t$. Given a process $\boldsymbol{\zeta}$, we may consider (\ref{museq1031}) as a linear SDE with one unknown process $V$. It is well known that such an equation can be explicitly solved for any initial condition $V_{0}$. However, in replication of contingent claims, we are given instead the terminal condition $V_{T}=X$. Also, the stock portfolio $\boldsymbol{\zeta}$ is not known a priori. Therefore it is more appropriate to treat (\ref{museq1031}) as a backward SDE with two unknown processes $V$ and $\boldsymbol{\zeta}$. Observe that, at the intuitive level, the concept of a backward SDE combines the predictable representation property with the linear (or, more generally, nonlinear) SDE. One may thus argue that no essential gain can be achieved by introducing this notion within the framework of a perfect (complete or incomplete) market. On the other hand, there is no doubt that the notion of a backward SDE appears to be a useful tool when dealing with market imperfections. To this end, one needs first to develop a theoretical background, including the existence and uniqueness results as well as the so-called comparison theorem. A comparison theorem, which basically states that solutions of backward SDEs are ordered if the drift coefficients are, allows one to deal with a situation where a perfect hedging strategy is not available as a solution of a particular backward SDE, but can be described as a limit of a monotone sequence of solutions of simpler backward SDEs. Let us write down an example of a backward SDE, which arises in the study of imperfect markets. If the lending and borrowing rates are different, say $R_{t}\geq r_{t}$ for every $t$, then (\ref{museq1031}) becomes
\[dv_{t}=r_{t}\cdot V_{t}dt+(R_{t}-r_{t})\cdot\left (\sum_{i=1}^{d}
\zeta_{t}^{i}-V_{t}\right )^{+}dt+\boldsymbol{\zeta}_{t}\cdot\left (
(\boldsymbol{\mu}_{t}-r_{t}{\bf 1})dt+\boldsymbol{\sigma}_{t}\cdot d{\bf W}_{t}\right ).\]
Note that the nonlinearity in the last equation appears in the drift term only, but depends on both the wealth process $V$ and the stock portfolio $\boldsymbol{\zeta}$. For further information, we may consult El Karoui et al. \cite{kar97}.

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

The Greeks.

We will now analyse the impact of the underlying parameters in the standard Black-Sholes model on the prices of call and put options. The Black-Scholes option values depend on the current tock price, the volatility, the time to maturity, the interest rate and the strike price. The sensitivities of the option price with respect to the first four parameters are called the Greeks and are widely used for hedging purposes. We can determine the impact of these parameters by taking partial derivatives. Recall the Black-Sholes formula for a European call
\[C_{0}=C(S,T,K,t,\sigma )=SN(d_{1}(S,T))-Ke^{-rT}N(d_{2}(S,T)),\]
with the functions $d_{1}(s,t)$ and $d_{2}(s,t)$ given by
\[d_{1}(s,t)=\frac{\log (s/K)+(r+\frac{\sigma^{2}}{2})t}{\sigma\sqrt{t}}\mbox{ and }d_{2}(s,t)=d_{1}(s,t)-\sigma\sqrt{t}=
\frac{\log (s/K)+(r-\frac{\sigma^{2}}{2})t}{\sigma\sqrt{t}}.\]
We therefore get
\begin{align*}
\Delta & =\frac{\partial C}{\partial S}=N(d_{1})>0\\
{\cal V} & =\frac{\partial C}{\partial\sigma}=S\sqrt{T}\cdot n(d_{1})>0\\
\Theta & =\frac{\partial C}{\partial T}=\frac{S\sigma}{2\sqrt{T}}\cdot n(d_{1})+Kre^{-rT}\cdot N(d_{2})>0\\
\rho & =\frac{\partial C}{\partial r}=TKe^{-rT}\cdot N(d_{2})<0\\
\Gamma & =\frac{\partial^{2}C}{\partial S^{2}}=\frac{n(d_{1})}{S\sigma\sqrt{T}}>0,
\end{align*}
where $N(\cdot )$ is the cumulative normal distribution function and $n(\cdot )$ is its density. From the definition it is clear that $\Delta$ (delta) measures the change in the value of the option compared with the change in the value of the underlying asset, ${\cal V}$ (vega) measures the change of the option compared with the change in the volatility of the underlying, and similar statements hold for $\Theta$ (theta) and $\rho$ (rho). Furthermore $\Gamma$ (gamma) measures the sensitivity of the portfilo to the change in the stock price.

The Black-Sholes partial differential equation (\ref{bineq8}) can be used to obtain the relation between the Greeks, i.e., (observe that $\Theta$ is the derivative of $C$ with respect to the time to maturity $T-t$, while in the Black-Scholes PDE the partial derivative with respect to the current time $t$ appears)
\[rC=\frac{1}{2}S^{2}\sigma^{2}\Gamma +rS\Delta -\Theta .\]
Let us now compute the dynamics of the call option’s price $C_{t}$ under the risk-neutral martingale $\bar{\mathbb{P}}$. Using formula
(\ref{bineq7}) we find
\[dC_{t}=rC_{t}dt+\sigma N(d_{1}(S_{t},T-t))S_{t}d\tilde{W}_{t}.\]
defining the {\bf elasticity coefficients} of the option’s price as
\[\eta_{t}^{c}=\frac{N(d_{1}(S_{t},T-t))}{C_{t}}\]
we can rewrite the dynamics as
\[dC_{t}=rC_{t}dt+\sigma\eta_{t}^{c}C_{t}d\tilde{W}_{t}.\]
So, as expected in the risk-neutral world, the appreciation rate of the call option equals the risk-free rate $r$. The volatility coefficient is $\sigma\eta^{c}$, and hence stochastic.

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

Approximations of Continuous Time Market Models.

Discrete time financial market models, where the underlying probability space is finite and trading occurs at only finitely many points in time, are already understood. The intuitive assumption of no arbitrage can be characterized by the probabilistically desirable martingale property of the security price process with respect to an equivalent martingale measure. Using the risk-neutral valuation principle we are able to price attainable contingent claims in a unique way. For such claims it is also possible to obtain risk-free hedging portfolios satisfying the needs of risk-management restrictions. In addition, we can explicitly characterize the property of completeness of the market via the uniqueness of the equivalent martingale measure.

As we have seen so far, the knowledge concerning continuous time securities markets is less satisfactory. The concept of no-arbitrage has to be replaced by the more involved concept of no free launch with vanishing risk, and true market completeness is hardly ever encountered in a continuous time setting (we are only able to replicate contingent claims in some reasonable large subset of the set of contingent claims).

Despite these shortcomings, continuous time models have become central to the financial community. The economic acceptability of continuous time models is usually motivated by some sort of limiting procedure (steadily increasing the number of trading dates leads to a continuous model). This naive approximation does not allow continuous time models to be viewed as proper limits of discrete time financial markets, because the fundamental properties of no arbitrage and completeness are in general not preserved by such limiting procedures. The following question is the key to our discussion: If an idealised economy (i.e., a continuous time security market model) has a certain property (say $\eta$), is it possible to approximate the idealized economy by a sequence of “real-life” economies (i.e., the finite securities market models) such that each finite economy has property $\eta$? If so, we say the approximation procedure has the structure-preserving property.

A property of considerable practical and theoretical interest is the property of “dynamic completeness” of a market model. For example, the Black-Scholes risk-neutral model is often criticized for its assumptions of continuous dynamic hedging. Practitionerslook for some tangible ideas to improve risk-management faced with violation of the Black-Scholes world by discrete-dynamic hedging. A proper convergence concept quantifying the general validity and robustness of the continuous security market will allow for such improvement. Furthermore, this convergence would allow for another way to tackle these problems numerically.

There are basically two approximation procedures which have the structure-preserving property. The first approximation method relies on pathwise approximation results for stochastic processes. Such pathwise approximations rely on the fine structure of the filtrations (flow of information) used and provide a natural approach to dynamic hedging and completeness. The drawback of this method is the relative complexity of the convergence results and the limited computational tractability (in general the number of nodes in the approximating tree grows exponentially with the number of trading days). The second approximation method is based on the notion of weak convergence for stochastic processes. In this framework the limiting stochastic process of the risky assets is approximated by finite Markov chain models. The concept of weak convergence of these objects is well-established and can readily be used. The computational efficiency of the approximation is, due to the Markovian nature of the approximating modles, very high. The obvious drawbacks is that we now use a weak convergence theory, which only controls the distributions rather than the particular paths. Furthermore, the theory applies only under the assumptions of a diffusion setting. In the sequel, we will use the second approximation method, which is capable of producing all relevant results for our purposes.

Finite Market Approximations.

We now want to develop a general theory of approximations of continuous time markets, restricted to the diffusion setting, by discrete time market models. The previous analysis showed that the defining properties of market models are the $(d+1)$-dimensional stochastic process ${\bf S}$, describing the price processes (discounted with respect to an appropriate numeraire) of the modeled securities and the equivalent martingale measure used for pricing (and hedging) purposes. We can characterize such a martingale measure by its Radon-Nikod\'{y}m derivative $L$ with respect to the original probability measure. To emphasise this we denote the financial market models by ${\cal M}({\bf S},L)$. We assume that the numeraire processes are bounded and to avoid notational distractions assume they are equal to one, i.e., we are working with the already discounted securities prices in this section.

Definition. A finite market approximation of the continuous time securities market model ${\cal M}({\bf S},L)$ is a sequence of discrete time market models $\left\{{\cal M}^{(n)}({\bf S}^{(n)},L^{(n)})\right\}_{n=1}^{\infty}$, where $\left\{{\bf S}_{t}^{(n)}:t\in [0,T]\right\}$ are $(d+1)$-dimensional and the $\left\{L_{t}^{(n)}:t\in [0,T]\right\}$ are real-valued stochastic processes defined on probability spaces $(\Omega^{(n)},{\cal F}^{(n)}, \mathbb{P}^{(n)})$ with the following properties.

For each $n\geq 0$, the sample paths of the process ${\bf S}^{(n)}$ are piecewise constant and RCLL $($right-continuous with left limits$)$ and jump only at times $t$ given by a grid ${\cal T}_{n}=\left\{0=t_{0}^{(n)}<t_{1}^{(n)}<\cdots <t_{k_{n}}^{(n)}=T\right\}$, where for $n\rightarrow\infty$ the grid mesh $\sup_{k}\left |t_{k}^{(n)}-t_{k-1}^{(n)}\right |\rightarrow 0$.
For each $n\geq 0$, $\left\{{\bf S}_{t}^{(n)}:t\in {\cal T}_{n}\right\}$ is a finite Markov chain.
For each $n\geq 0$, $\left\{L_{t}^{(n)}:t\in {\cal T}_{n}\right\}$ is the Radon-Nikod\'{y}m derivative $($used for the change of measure$)$ in the discrete time market model ${\cal M}^{(n)}$.
$\left ({\bf S}^{(n)},L^{(n)}\right )$ converges weakly to $({\bf S},L)$, that is for all bounded continuous functions $latex h:
D^{d+2}[0,T]\rightarrow \mathbb{R}$, we have
\[\mathbb{E}_{n}\left [h({\bf S}^{(n)},L^{(n)})\right ]\rightarrow\mathbb{E}[h({\bf S},L)],\] where $D^{d+2}[0,T]$ is the space of all
RCLL functions from $[0,T]$ to $\mathbb{R}^{d+2}$, and $\mathbb{E}_{n}$ and $\mathbb{E}$ denote the expectations under the probability measures $\mathbb{P}^{(n)}$ and $\mathbb{P}$, respectively. $\sharp$

We denote by $\bar{\mathbb{P}}^{(n)}$ and $\bar{\mathbb{P}}$ the martingale measures on $(\Omega^{(n)},{\cal F}^{(n)})$ and $(\Omega,{\cal F})$, respectively, which are induced by the Radon-Nikod\'{y}m derivative processes $L^{(n)}$ and $L$, respectively, i.e.,
\[\frac{d\bar{\mathbb{P}}^{(n)}}{d\mathbb{P}^{n}}=L^{(n)}\mbox{ and }\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}=L.\]
The above definition does not require the approximating markets to share the properties of the limiting market. Due to the importance of this, we give the following definition .

Definition. A finite market approximation is structure preserving with respect to the property $\eta$ of the continuous time, if each discrete time market has the property $\eta$. $\sharp$

The property we will focus on is dynamic completeness. By the theory of weak convergence, we have the following consequence for contingent claim pricing using finite market approximations to a continuous time market model.

\begin{equation}{\label{bint641}}\tag{58}\mbox{}\end{equation}

Theorem \ref{bint641}. Let ${\cal M}^{(n)}({\bf S}^{(n)},L^{(n)})$ be a finite market approximation of a continuous time market model ${\cal M}({\bf S},L)$ and assume that $\{L_{T}^{(n)}\}$ is uniformly integrable. For contingent claims $Y=f({\bf S})$ and $Y^{(n)}=f({\bf S}^{(n)})$ with $f:D^{d+1}[0,T]\rightarrow \mathbb{R}_{+}$ a bounded function, and time $t=0$ prices $\Pi_{Y}$ and $\Pi_{Y^{n}}$, respectovely, of the contingent claims in their respective markets, we have the following limiting relation
\[\Pi_{Y}=\lim_{n\rightarrow\infty}\Pi_{Y^{(n)}}.\]

Proof. We will use the risk-neutral valuation formula in the discrete and continuous time markets. Since $\{L_{T}^{(n)}\}$ is uniformaly integrable and $f$ is bounded, the sequence of random variables $\{L_{T}^{(n)}\cdot f({\bf S}^{(n)})\}$ is uniformly integrable. This and convergence in distribution give convergence of expectations
\[\Pi_{Y}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[f({\bf S})]=\mathbb{E}[L_{T}\cdot f({\bf S})]
=\lim_{n\rightarrow\infty}\mathbb{E}_{n}[L_{T}^{(n)}\cdot f({\bf S}^{(n)})]=
\lim_{n\rightarrow\infty}\mathbb{E}_{\scriptsize \bar{\mathbb{P}}^{(n)}}[f({\bf S}^{(n)})]=
\lim_{n\rightarrow\infty}\Pi_{Y^{(n)}}.\]
This completes the proof. $\blacksquare$

Observe that Theorem \ref{bint641} applies to all finite market approximations (not just to structure-preserving approximations). A simple sufficient condition for uniform integrability is uniform boundedness in $L^{p}$ for some $p>1$. As an example one can consider pricing a European put option in a continuous time model. The theorem tells us that we can price the put by pricing the corresponding put in the appropriate discrete time setting and then computing the limit of the discrete time model prices (a European call is then priced by put-call parity). The next result explores the consequences for trading strategies for contingent claims.

\begin{equation}{\label{bint642}}\tag{59}\mbox{}\end{equation}

Theorem \ref{bint642}. Let ${\cal M}^{(n)}({\bf S}^{(n)},L^{(n)})$ be a finite market approximation of a continuous time market model ${\cal M}({\bf S},L)$ and assume that $\{{\bf S}^{(n)}\}$ is a sequence of semimartingales. Assume that the sequence of discrete time trading strategies $\boldsymbol{\phi}^{n}$ converges weakly to a continuous time trading strategy $\boldsymbol{\phi}$. Then the discrete time gains $G_{\boldsymbol{\phi}_{n}}$ and wealth $V_{\boldsymbol{\phi}_{n}}$ converge weakly to their continuous time
counter parts $G_{\boldsymbol{\phi}}$ and $V_{\boldsymbol{\phi}}$. $\sharp$

The real work in applying Theorem \ref{bint642} is to show the goodness of the underlying approximating process $\{{\bf S}^{(n)}\}$ and the weak convergence of the trading strategies. Having established these facts, one can apply the theorem to justify “discrete time hedging of continuous time claims”. Suppose we construct a continuous time hedging strategy $\boldsymbol{\phi}$ to hedge a contingent claim $X$, then the theorem tells us that under appropriate conditions on the approximating discrete time models, we can hedge (in the limit) the contingent claim with the induced discrete time strategies. Given that in practice trading is discrete, this is very reassuring for risk managers trying to cover their positions with trading strategies deduced from continuous time models.

\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}

Futures Markets.

We have the existence of two parallel markets in some assets, the spot market, for assets traded in the present, and the futures market, for assets to be realized in the future. We may also consider the combined spot-futures market Futures prices, like spot prices, are determined on the floor of the exchange by supply and demand, and are quoted in the financial press. Futures contracts, however, contracts on assets traded in the futures markets, have various special characteristics. Parties to futures contracts are subject to a daily settlement procedure known as marking to market. The initial deposit, p-aid when the contract is entered into, is adjusted daily by margin payments reflecting the daily movement in futures prices. The underlying asset and prices are specified in the contract, as is the delivery date. Futures contracts are highly liquid, and indeed, are intended more for trading than for delivery. A transaction may instead, however, involve an agreement between two parties for the sale/purchase of a specified asset at some specified future time for some specified price. Such an agreement is a forward contract; unlike futures, forward contracts are not traded. They are intended for delivery, and are not liquid.

We shall write $t=0$ for the time when a contract, or a option, is written, $t$ for the present time, $T$ for the expiry time of the option, and $T^{*}$ for the delivery time specified in the futures (or forward) contract. We will have $T^{*}\geq T$, and in general $T^{*}>T$; beyond this, $T^{*}$ will not affect the pricing of options with expiry $T$. Despite their fundamental differences, futures prices $f_{S}(t,T)$ on a stock $S$ at time $t$ with expiry $T$, and the corresponding forward prices $F_{S}(t,T)$, are closely linked. We denote by $p(t,T)$ the bond price processes.

\begin{equation}{\label{binp21}}\tag{60}\mbox{}\end{equation}

Proposition \ref{binp21}. If the bond price process $p(t,T)$ is predictable, the combined spot-futures market is arbitrage-free if and only if the futures and forward prices agree: for every underlying $S$ and every $t\leq T$, $f_{S}(t,T)=F_{S}(t,T)$. $\sharp$

We turn now to the problem of extending the option pricing theory from spot markets to futures markets. We assume that the stock price dynamics $S$ are given by geometric Brownian motion
\[dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t},\]
and that interest rates are deterministic. We know that there exists a unique equivalent martingale measure $\bar{\mathbb{P}}$ for the discounted stock price processes with expectation $\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}$. Write $f_{t}=f_{S}(t,T^{*})$ for the futures price $f_{t}$ corresponding to the spot price $S_{t}$. Then risk-neutral valuation gives
\[f_{t}=\mathbb{E}_{\scriptsize \bar{\mathbb{P}}}[S_{T^{*}}|{\cal F}_{t}]\mbox{ for }t\in [0,T^{*}],\]
while forward prices are given in terms of bond prices by
\[F_{t}=\frac{S_{t}}{p(t,T^{*})}\mbox{ for }t\in [0,T^{*}].\]
So by Proposition \ref{binp21}, we have
\[f_{t}=F_{t}=S_{t}e^{r}(T^{*}-t)\mbox{ for }t\in [0,T^{*}].\]
So we can use the product rule to determine the dynamics of the futures price
\[df_{t}=(\mu -r)f_{t}dt+\sigma f_{t}dW_{t}, f_{0}=S_{0}e^{rT^{*}}.\]
We know that the unique equivalent martingale measure in this setting is given by menas of a Girsanov density
\[L_{t}=\exp\left [-\frac{\mu -r}{\sigma}W_{t}-\frac{1}{2}\left (\frac{\mu -r} {\sigma}\right )^{2}t\right ],\]

so the $\bar{\mathbb{P}}$-dynamics of the futures prices are
\[df_{t}=\sigma f_{t}d\tilde{W}_{t}\]
with $\tilde{W}_{t}$ a $\bar{\mathbb{P}}$-Brownian motion, so $f$ is a $\bar{\mathbb{P}}$-martingale. We use the same notation, the strike $K$, expiry $T$ as in the spot case, and write $N$ for the standard normal distribution function.

Theorem. The arbitrage price $C$ of a European futures call option is
\[C_{t}=c(f_{t},T-t),\]
where $c(f,t)$ is given by Black’s futures options formula
\[c(f,t)=e^{-rt}\cdot [f\cdot N(d_{1}(f,t))-K\cdot N(d_{2}(f,t))],\]
where
\[d_{1,2}(f,t)=\frac{\log (f/K)\pm\frac{1}{2}\sigma^{2}t}{\sigma\sqrt{t}}. \sharp\]

Standard Market Models.

Standard Spot Market.

Futures Market.

Changes of Numeraire.

Existence of a Martingale Measure.

Fundamental Theorem of Asset Pricing.

Multidimensional Black-Scholes Model.

Market Completeness.

Variance-Minimizing Hedging.

Risk-Minimizing Hedging.

Backward SDE’s

The Greeks.

Approximations of Continuous Time Market Models.

Finite Market Approximations.

Futures Markets.

Hsien-Chung Wu

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Standard Market Models.

Standard Spot Market.

Futures Market.

Changes of Numeraire.

Existence of a Martingale Measure.

Fundamental Theorem of Asset Pricing.

Multidimensional Black-Scholes Model.

Market Completeness.

Variance-Minimizing Hedging.

Risk-Minimizing Hedging.

Backward SDE’s

The Greeks.

Approximations of Continuous Time Market Models.

Finite Market Approximations.

Futures Markets.

Hsien-Chung Wu

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Bibliography

Fixed-Income Market

Exotic Options

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