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\begin{document}
\title{Fundamental Theorem of Asset Pricing in a Nutshell: With a View toward Num\'{e}raire Change}
\author{Yan Zeng}
\date{Version 1.0, last revised on 2009-09-27.}
\maketitle
\begin{abstract}
A survey of the Fundamental Theorem of Asset Pricing in mathematical finance.
\end{abstract}
\tableofcontents
\newpage
\section{Introduction}
This note serves as a summary of the num\'{e}raire change
techniques as presented in Brigo and Mercurio \cite{BM07} \S2.2 and \S2.3. The idea is to present the main results in a
logically ``natural" order so that we can easily remember them.
Roughly speaking, we are concerned with the following question: {\it
Is the Fundamental Theorem of Asset Pricing (FTAP) invariant under
num\'{e}raire change?}
The answer is {\it negative}. The key idea of this presentation is therefore the following: the version of FTAP as formulated in Delbaen and Schachermayer \cite{DS94} starts from relaxing the no-arbitrage condition, so that it states its most general results in terms of local martingales or even $\sigma$-martingales. This is insufficient for the practical usage of risk-neutral pricing, as we really need martingale property, not {\it local} martingale property. We should instead start from the other way around: insist on martingale property and derive the right formulation of no-arbitrage.
In Section \ref{sect_review_classical} and \ref{sect_review_2nd_classical}, we review the classical formulation of the first and second Fundamental Theorem of Asset Pricing (FTAP). Such a formulation is not invariant under num\'{e}raire change. In Section \ref{sect_review_right} and \ref{sect_review_2nd_right}, we review the ``right" formulation of Fundamental Theorem of Asset Pricing, insisting on martingale measures instead of local martingales measures. Such a formulation turns out to be invariant under num\'{e}raire change. In Section \ref{sect_Ito_process}, we give concrete formulas for the case of It\^{o} processes.
This note is based on a series of papers by Delbaen and
Schachermayer (\cite{DS94}, \cite{DS95},
\cite{DS97}, \cite{DS98a}, \cite{DS98b}), Schachermayer \cite{Schachermayer93}, Geman et al. \cite{GER95},
Yan et al. (\cite{Yan98}, \cite{LY99}, \cite{Yan02}, \cite{XY03}),
and Shiryaev \cite{Shiryaev99}, as well as the references therein.
\section{Fundamental Theorem of Asset Pricing: classical formulation}\label{sect_review_classical}
We first summarize the state of the art before Delbaen and
Schachermayer \cite{DS94}. The case when the time set is finite is completely settled in Dalang et al. \cite{DMW89} and the use of simple or even elementary integrands as trading strategies is no restrction at all. For the case of discrete but infinite time sets, the problem is solved in Schachermayer \cite{Schachermayer93}; the case of continuous and bounded processes in continuous time is solved in Delbaen \cite{Delbaen92}. In these two cases the theorems are stated in terms of simple integrands and limits of sequences and by using the concept of {\it no free lunch with bounded risk}.
To state the results of Delbaen and
Schachermayer \cite{DS94}, we consider a probability space $(\Omega, {\cal F}, P)$ and a
right-continuous filtration $\mathbb F = \{{\cal F}_t: 0\le t \le
T\}$ $(T\le \infty)$. In the given economy, $(K+1)$ non-dividend
paying securities are traded continuously from time $0$ until time
$T$. Their prices are modeled by a $(K+1)$-dimensional adapted, positive
semimartingale $S=\{(S^0_t, S^1_t, \cdots, S^K_t): 0\le t \le T\}$.
We assume $S^0\equiv 1$.\footnote{This assumption hides the fact that we need a num\'{e}raire (i.e. a positive discounter), since $1$ is used as the num\'{e}raire.}
\begin{definition}(Yan \cite{Yan98} p.661, p.662)
A {\bf trading strategy} is an $\mathbb R^{K+1}$-valued predictable
process $\phi=\{\phi_t: 0\le t \le T\}$ which is integrable w.r.t
semimartingale $S$.\footnote{We need here the notion of integration
w.r.t. a vector-valued semimartingale, which is is defined globally,
not componentwisely (see Jacod \cite{Jacod80}). This is because the
notion of componentwise stochastic integral is insufficient for
stating FTAP in the most general setting (see Shiryaeve
\cite{Shiryaev99} p.635, Shiryaev and Cherny \cite{SC02}). However,
when $\phi$ is locally bounded, componentwise integration is
sufficient for stating FTAP. See Cherny \cite{Cherny} for more
details.} The {\bf value process} associated with a strategy $\phi$
is defined by
\[
V_t(\phi)=\phi_tS_t = \sum_{k=0}^{K}\phi_t^kS_t^k, 0 \le t \le T,
\]
and the {\bf gains process} associated with a strategy $\phi$ is
defined by
\[
G_t(\phi) = (\phi \cdot S)_t := \int_0^t\phi_udS_u =
\sum_{k=0}^K\int_0^t \phi_u^kdS_u^k, 0 \le t \le T.
\]
A trading strategy $\phi$ is ${\bf self-financing}$ if
\[
V_t(\phi)= V_0(\phi) + G_t(\phi), 0 \le t < T.
\]
\end{definition}
\begin{definition}(Delbaen and
Schachermayer \cite{DS94} Definition 2.7)
A trading strategy $\phi$ is {\bf admissible} if $G_t(\phi)$ is bounded from below,
i.e. there is a constant $M$ such that $G_t(\phi) \ge -M$ a.s. for
all $t\ge 0$.
\end{definition}
\begin{definition}(Delbaen and Schachermayer \cite{DS97} or Shiryaev \cite{Shiryaev99} p.650, V\!I\!I \S2a.2 Definition 1)
We say that the vector of price processes $S$ satisfies the condition of {\bf no arbitrage} (NA)
at time $T$ if for each self-financing strategy $\phi$, we have
\[
P(G_T(\phi) \ge 0 ) = 1\Rightarrow P(G_T(\phi)=0)=1.
\]
\end{definition}
The above concept of no arbitrage is already sufficient for stating
FTAP in the discrete time case, but for continuous time case, we
need the following concept:
\begin{definition}(Delbaen and
Schachermayer \cite{DS94} Definition 2.8 or Shiryaev \cite{Shiryaev99} p.650, V\!I\!I \S2a.2 Definition 3)
Let
\[
K = \{ G_{T}(\phi) | \mbox{$\phi$ admissible and
$G_{\infty}(\phi)=\lim_{t\to\infty}G_t(\phi)$ exists a.s. if
$T=\infty$} \}
\]
and
\[
C = \{g\in L^{\infty}(\Omega, {\cal F}_T, P)| \; \mbox{$g\le f$ for
some $f\in K$}\}.
\]
We say that $S$ satisfies the condition of {\bf no free lunch with
vanishing risk} (NFLVR) for admissible strategies, if
\[
\bar C \cap L_+^{\infty}(\Omega, {\cal F}_T, P) = \{0\},
\]
where $\bar C$ denotes the closure of $C$ with respect to the norm
topology of $L^{\infty}(\Omega, {\cal F}_T, P)$.
\end{definition}
To understand intuitively the NFLVR condition, we note $S$ allows
for a free lunch with vanishing risk, if there is $f \in
L_+^{\infty}(\Omega, {\cal F}_T, P)\setminus \{0\}$, a sequence
$(G_{T}(\phi_n))_{n=0}^{\infty} \subset K$, where
$(\phi^n)_{n=0}^{\infty}$ is a sequence of admissible integrands, and
$(g_n)_{n=0}^{\infty} \subset L^{\infty}(\Omega, {\cal F}_T, P)$ satisfying $g_n\le G_{T}(\phi_n)$, such that
\[
\lim_{n\to\infty}|\!|f-g_n|\!|_{L^{\infty}(\Omega, {\cal F}_T,
P)}=0.
\]
In particular the negative parts $(G_{T}^-(\phi_n))_{n=0}^{\infty}$
and $(g^-_n)_{n=0}^{\infty}$ tend to zero uniformly, which explains
the term ``vanishing risk".
%We comment that the assumption $S^0\equiv 1$ hides the fact that
%NFLVR depends on num\'{e}raire, which makes the classical formulation of FTAP not invariant under
%num\'{e}raire change (Example \ref{example_bessel}).
The last piece of our vocabulary for stating FTAP is the following one.
\begin{definition}\label{definition_EMM}(Shiryaev \cite{Shiryaev99} p.652, V\!I\!I \S2b.1 and p.656, V\!I\!I \S2c.2)
An {\bf equivalent martingale measure} (EMM) is a probability
measure equivalent to $P$ and under which $S$ is a martingale. An
{\bf equivalent local martingale measure} (ELMM) is a probability
measure equivalent to $P$ and under which $S$ is a local martingale.
An {\bf equivalent $\sigma$-martingale measure} (E$\sigma$MM) is a
probability measure equivalent to $P$ and under which $S$ is a
$\sigma$-martingale, i.e. $S=S_0+H\cdot M$ with $M$ a martingale and
$H$ a positive predictable process integrable w.r.t $M$.
\end{definition}
%We comment again that the assumption $S^0\equiv 1$ hides the fact
%that EMM or ELMM is always w.r.t a num\'{e}raire: discounted by this
%num\'{e}raire, $S$ is a martingale or local martingale. There exists
%a counter-example where the existence of ELMM is not invariant under
%change of num\'{e}raire (Example \ref{example_bessel}).
\medskip
Now we are ready to state a list of results on the classical formulation of Fundamental
Theorem of Asset Pricing:
\begin{theorem}\label{theorem_FTAP_continuous}(Shiryaev \cite{Shiryaev99} p.655, V\!I\!I \S2c, Theorem 1, 2, and Corollary)
Let $S$ be defined as above.
a) If $S$ is bounded, then
\[
NFLVR \Leftrightarrow EMM.
\]
b) If $S$ is locally bounded, then
\[
NFLVR \Leftrightarrow ELMM.
\]
c) If $S$ is a general semimartingale, then
\[
NFLVR \Leftrightarrow E\sigma MM.
\]
For a clearer insight into the connection between the above results
and the corresponding results in the discrete-time case (see Theorem
\ref{theorem_FTAP_discrete}), we reformulate the theorem as follows.
In general semimartingale models $S_t=(1,S^1_t,\cdots, S^K_t)_{0\le
t \le T}$, $T<\infty$, we have
\[
EMM \Rightarrow ELMM \Rightarrow E\sigma MM \Leftrightarrow NFLVR.
\]
When $S$ is moreover locally bounded, we have
\[
EMM \Rightarrow ELMM \Leftrightarrow E\sigma MM \Leftrightarrow
NFLVR.
\]
When $S$ is further assumed to be bounded, we have
\[
EMM \Leftrightarrow ELMM \Leftrightarrow E\sigma MM \Leftrightarrow
NFLVR.
\]
\end{theorem}
As a comparison, we recall FTAP in the discrete-time case.
\begin{theorem} \label{theorem_FTAP_discrete}(Dalang et al. \cite{DMW89}. Also see Delbaen and Schachermayer \cite{DS97} Theorem 15)
In the discrete- and finite-time case (i.e. $t=0,1,\cdots,T<\infty$)
, we have
\[
EMM \Leftrightarrow ELMM \Leftrightarrow E\sigma MM \Leftrightarrow
NA.
\]
\end{theorem}
\begin{remark}
For Theorem \ref{theorem_FTAP_discrete} to hold, we
need no additional assumptions on trading strategies beside
predictability. We also comment that $T<\infty$ is essential for Theorem \ref{theorem_FTAP_discrete}; otherwise a counter example exists
(see Shiryaev \cite{Shiryaev99} p.415, V \S2b.3). For the case of $T=\infty$,
the NA condition needs to be modified to ``no free lunch with
bounded risk" (see Schachermayer \cite{Schachermayer93}).
\end{remark}
%We comment again that the assumption $S^0\equiv 1$ hides the fact
%that EMM or ELMM is always w.r.t a num\'{e}raire: discounted by this
%num\'{e}raire, $S$ is a martingale or local martingale. There exists
%a counter-example where the existence of ELMM is not invariant under
%change of num\'{e}raire (Example \ref{example_bessel}).
In the above statement of FTAP, we have set $S^0\equiv 1$. In practice, we usually do not have an asset whose price is
identically $1$. So FTAP as stated in Theorem \ref{theorem_FTAP_continuous} and \ref{theorem_FTAP_discrete} is really a mathematical
simplification: instead of a general semimartingale $S=(S^0, S^1, \cdots, S^K)$, we considered the discounted process:
\[
\frac{1}{S_0}\left(S^0,S^1, \cdots, S^K\right) = \left(1, \frac{S^1}{S^0},\cdots, \frac{S^N}{S^0}\right).
\]
Therefore we implicitly used $S^0$ as a num\'{e}raire, and
EMM or ELMM should be understood as w.r.t a num\'{e}raire: discounted by this
num\'{e}raire, $S$ is a martingale or local martingale. So measure and num\'{e}raire appear in a dual pair. Similarly, the notion of {\it admissibility} should be understood as ``in given num\'{e}raire". That is, we require $G_t(\phi)$ denominated in the num\'{e}raire is bounded from below. Formally, we have
\begin{definition}
A {\bf num\'{e}raire} is any strictly positive semimartingale.
\end{definition}
\begin{definition}\label{definition_EMM_numeraire}
An {\bf equivalent martingale measure $Q^N$ associated with the num\'{e}raire $N$} is a probability
measure equivalent to $P$ such that $S/N$ is a martingale under $Q^N$.
\end{definition}
\begin{definition}
A trading strategy $\phi$ is {\bf admissible under the num\'{e}raire $N$} if there is a constant $M$ such that $G_t(\phi)/N_t \ge -M$ a.s. for all $t\ge 0$.
\end{definition}
To reconcile any potential conceptual conflicts, we need the following
\begin{lemma}\label{lemma_self-financing}
$\phi$ is a self-financing strategy if and only if for any
num\'{e}raire $N$, we have
\[
d\left(\frac{V_t(\phi)}{N_t}\right) = \sum_{k=0}^K
\phi_t^kd\left(\frac{S_t^k}{N_t}\right).
\]
\end{lemma}
\begin{proof}
Sufficiency is obvious as we can take $N_t\equiv 1$. For necessity,
we note by integration-by-part formula
\begin{eqnarray*}
& & d\left(\frac{V_t(\phi)}{N_t}\right) \\
&=& \frac{dV_t(\phi)}{N_{t-}} +
V_{t-}(\phi)d\left(\frac{1}{N_t}\right) + d[V(\phi), 1/N]_t
= \sum_{k=0}^{K}\frac{\phi_t^kdS_t^k}{N_{t-}} + \sum_{k=0}^K
\phi_t^kS^k_{t-} d\left(\frac{1}{N_t}\right) +
\sum_{k=0}^{K}\phi_t^kd[S^k,1/N]_t \\
&=& \sum_{k=0}^{K}\phi_t^k \left\{\frac{dS_t^k}{N_{t-}} +
S^k_{t-}d\left(\frac{1}{N_t}\right) + d[S^k, 1/N]_t \right\} = \sum_{k=0}^{K}\phi_t^k d\left(\frac{S^k_t}{N_t}\right),
\end{eqnarray*}
where the second ``=" has used the observation that
\[
V_{t-}(\phi) = V_t(\phi)-\Delta V_t(\phi) =
\sum_{k=0}^K\phi_t^kS_t^k - \sum_{k=0}^K\phi_t^k\Delta S_t^k =
\sum_{k=0}^{K}\phi_t^kS^k_{t-}.
\]
\end{proof}
As we change num\'{e}raire, a question naturally arises: {\it Is Fundamental Theorem of Asset Pricing invariant under a num\'{e}raire change?} This question can be more precisely stated as follows:\footnote{Here, EMM and ELMM are always understood as being associated with a given num\'{e}raire. }
\medskip
\noindent 1) Under num\'{e}raire change, is NFLVR preserved?
\noindent 2) Under num\'{e}raire change, is the existence of EMM or ELMM preserved?
\noindent 3) Under num\'{e}raire change, if the existence of EMM or ELMM is preserved, is the uniqueness of such a measure also preserved?
\noindent 4) How are the equivalent martingale measures related to each other? For example, can we represent the Radon-Nikodym derivatives in terms of the num\'{e}raires?
\noindent 5) If we change the num\'{e}raire $N$ to another num\'{e}raire $U$, does the risk-neutral pricing formula still holds? That is, for an ${\cal F_T}$-measurable random variable $\xi$ satisfying suitable integrability conditions, do we have ($t