No arbitrage of the first kind and local martingale numéraires

Abstract

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind ($\mbox{NA}_{1}$) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under $\mbox{NA}_{1}$, a local martingale numéraire may fail to exist. In this work, we establish that under $\mbox{NA}_{1}$, a supermartingale numéraire under the original probability $P$ becomes a local martingale numéraire for equivalent probabilities arbitrarily close to $P$ in the total variation distance.

Appendix: No-arbitrage conditions, revisited

1.1 A.1 Condition $\mbox{NA}_{1}$: equivalent formulations

We discuss equivalent forms of the condition $\mbox{NA}_{1}$ in the context of a general abstract setting, where the model is given by specifying the set of wealth processes. The advantage of this generalization is that we can use only elementary properties without any reference to stochastic calculus and integration theory.

Let $\mathcal{X}^{1}$ be a convex set of càdlàg processes $X$ with $X\ge-1$ and $X_{0}=0$, containing the zero process. For $x\ge0$, we define the set $\mathcal{X}^{x}=x\mathcal{X}^{1}$ and note that $\mathcal {X}^{x} \subseteq\mathcal{X}^{1}$ when $x\in[0,1]$. Put $\mathcal{X}:=\mathrm{cone}\,\mathcal{X}^{1}={{\mathbb {R}}}_{+}\mathcal{X}^{1}$ and define the sets of terminal random variables $\mathcal{X}^{1}_{T}:=\{X_{T}\colon\ X\in\mathcal{X}^{1}\}$ and $\mathcal {X}_{T}:=\{X_{T}\colon\ X\in\mathcal{X}\}$. In this setting, the elements of $\mathcal{X}$ are interpreted as admissible wealth processes starting from zero initial capital; the elements of $\mathcal {X}^{x}$ are called $x$-admissible.

Remark A.1

(“Standard” model)

In the typical example, a $d$-dimensional semimartingale $S$ is given and $\mathcal{X}^{1}$ is the set of stochastic integrals $H\cdot S$, where $H$ is $S$-integrable and $H\cdot S\ge-1$. Though our main result deals with the standard model, it is natural to discuss the basic definitions and their relations with concepts of arbitrage theory in a more general framework.

Define the set of strictly 1-admissible processes $\mathcal{X}^{1}_{>} \subseteq\mathcal{X}^{1}$ as composed of those $X\in \mathcal{X}^{1}$ such that $X > -1$ and $X_{-}>-1$. The sets $x+\mathcal{X}^{x}$, $x+\mathcal{X}^{x}_{>}$, and so on, $x\in{{\mathbb {R}}}_{+}$, have obvious interpretations. We are particularly interested in the set $1+\mathcal{X}^{1}_{>}$. Its elements are strictly positive wealth processes starting with unit initial capital and may be thought of as tradable numéraires.

For $\xi\in L^{0}_{+}$, define the superhedging value $x (\xi) := \inf\{x:\ \xi\in x+\mathcal{X}^{x}_{T}-L^{0}_{+}\}$. We say that the wealth-process family $\mathcal{X}$ satisfies the condition $\mbox{NA}_{1}$ (no arbitrage of the first kind) if $x(\xi)>0$ for every $\xi \in L^{0}_{+} \setminus\{0\}$. Alternatively, the condition $\mbox{NA}_{1}$ can be defined via

$$\bigg(\bigcap_{x>0}\big\{ x+\mathcal{X}^{x}_{T}-L^{0}_{+}\big\} \bigg)\cap L^{0}_{+}=\{0\}. $$

The family $\mathcal{X}$ is said to satisfy the condition $\mbox{NAA}_{1}$ (no asymptotic arbitrage of the first kind) if for any sequence $(x^{n})_{n}$ of positive numbers with $x^{n}\downarrow0$ and any sequence of value processes $X^{n} \in\mathcal{X}$ such that $x^{n}+X^{n}\ge0$, we have

$$\limsup_{n\to\infty} P[x^{n}+X^{n}_{T}\ge1]=0. $$

Finally, we say that the family $\mathcal{X}$ satisfies the condition NUPBR (no unbounded profit with bounded risk) if the set $\{ X_{T}:\ X\in\mathcal{X}^{1}_{>}\}$ is bounded in $L^{0}$. Since we have $(1/2)\mathcal{X}^{1}_{T}=\mathcal{X}^{1/2}_{T}\subseteq\{X_{T}:\ X\in \mathcal{X}^{1}_{>}\}$, the set $\{X_{T}:\ X\in\mathcal{X}^{1}_{>}\}$ is bounded in $L^{0}$ if and only if the set $\mathcal{X}^{1}_{T}$ is bounded in $L^{0}$.

The next result shows that all three market viability notions introduced above coincide.

Lemma A.1

$\textit{NAA}_{1} \Longleftrightarrow \textit{NUPBR} \Longleftrightarrow \textit{NA}_{1}$.

Proof

$\mbox{NAA}_{1} \Rightarrow \mbox{NUPBR}$: If the family $\{X_{T}:\ X\in\mathcal {X}^{1}_{>}\}$ fails to be bounded in $L^{0}$, then $P[1+\tilde{X}^{n}_{T}\ge n]\ge{\varepsilon}>0$ for a sequence $(\tilde{X}^{n})\subseteq\mathcal{X}^{1}_{>}$, and we obtain a violation of $\mbox{NAA}_{1}$ with $n^{-1}+n^{-1}\tilde{X}^{n}_{T}$.

$\mbox{NUPBR} \Rightarrow \mbox{NA}_{1}$: If $\mbox{NA}_{1}$ fails, then there exist $\xi \in L^{0}_{+}\setminus\{0\}$ and a sequence $(X^{n})$ with $X^{n}\in \mathcal {X}^{1/n}$ such that $1/n+X^{n}\ge\xi$. Then the sequence $nX^{n}_{T}\in\mathcal {X}^{1}$ fails to be bounded in $L^{0}$, in violation of NUPBR.

$\mbox{NA}_{1} \Rightarrow \mbox{NAA}_{1}$: If the implication fails, then there are sequences $x^{n}\downarrow0$ and $X^{n}\ge-x^{n}$ such that $P[x^{n}+X^{n}_{T}\ge1]\ge 2{\varepsilon}>0$. By the von Weizsäcker theorem (see [18] or [11, Theorem A.2.3]), any sequence of random variables bounded from below contains a subsequence converging in the Cesarò sense a.s. as well as all its further subsequences. We may assume without loss of generality that for $\xi^{n}:=x^{n}+X^{n}_{T}$, the sequence $\bar{\xi}^{n}:=(1/n)\sum _{i=1}^{n}\xi_{i}$ converges to $\xi\in L^{0}_{+}$. Note that $\xi\neq0$. Indeed,

$$\begin{aligned} {\varepsilon}(1-P[\bar{\xi}^{n}\ge{\varepsilon}]) \ge& \frac{1}{n}\sum _{i=1}^{n}E\xi^{i}I_{\{\bar{\xi}^{n}< {\varepsilon}\}}\ge\frac{1}{n}\sum _{i=1}^{n}E\xi^{i}I_{\{\xi^{i}\ge1,\;\bar{\xi}^{n}< {\varepsilon}\}}\\ \ge& \frac{1}{n}\sum_{i=1}^{n}P[\xi^{i}\ge1,\;\bar{\xi}^{n}< {\varepsilon }]\ge\frac{1}{n}\sum_{i=1}^{n}(P[\xi^{i}\ge1]-P[\bar{\xi}^{n}\ge {\varepsilon}])\\ \ge&2{\varepsilon}-P[\bar{\xi}^{n}\ge{\varepsilon}]. \end{aligned}$$

It follows that $P[\bar{\xi}^{n}\ge{\varepsilon}] \ge{\varepsilon }/(1-{\varepsilon})$. Thus,

$$E[\xi\wedge1]=\lim_{n} E[\bar{\xi}^{n}\wedge1]\ge{\varepsilon }^{2}/(1-{\varepsilon})>0. $$

It follows that there exists $a>0$ such that $P[\xi\ge2a]>0$. In view of Egorov’s theorem, we can find a measurable set $\varGamma \subseteq\{\xi\ge a\}$ with $P[\varGamma]>0$ on which $x^{n}+X^{n}\ge a$ for all sufficiently large $n$. But this means that the random variable $aI_{\varGamma}\neq0$ can be superreplicated starting with arbitrarily small initial capital, in contradiction with the assumed condition $\mbox{NA}_{1}$. □

Remark A.2

(On terminology and bibliography)

The conditions $\mbox{NAA}_{1}$ and $\mbox{NA}_{1}$ have clear financial meanings, whereas the boundedness in $L^{0}$ of the set $\mathcal{X}^{1}_{T}$, at first glance, looks like a technical condition—see [5]. The concept of $\mbox{NAA}_{1}$ first appeared in [10] in a much more general context of large financial markets, along with another fundamental notion $\mbox{NAA}_{2}$ (no asymptotic arbitrage of the second kind). The boundedness in $L^{0}$ of $\mathcal{X}^{1}_{T}$ was discussed in [9] (as the BK-property), in the framework of a model given by value processes; however, it was overlooked that it coincides with $\mbox{NAA}_{1}$ for the “stationary” model, that is, when the stochastic basis and the price process do not depend on $n$. The same condition appeared under the acronym NUPBR in [12] and was shown to be equivalent to $\mbox{NA}_{1}$ in [13].

1.2 A.2 $\mbox{NA}_{1}$ and NFLVR

Remaining in the framework of the abstract model of the previous subsection, we provide here results on the relation of the condition $\mbox{NA}_{1}$ with other fundamental notions of arbitrage theory; compare with [9].

Define the convex sets $C:=(\mathcal{X}_{T}-L^{0}_{+})\cap L^{\infty}$ and denote by $\bar{C}$ and $\bar{C}^{*}$ the norm-closure and weak^∗ closure of $C$ in $L^{\infty}$, respectively. The conditions NA, NFLVR, and NFL are defined correspondingly via

$$C\cap L^{\infty}_{+}=\{0\},\qquad \bar{C}\cap L^{\infty}_{+}=\{0\}, \qquad\bar{C}^{*}\cap L^{\infty}_{+}=\{0\}. $$

Consecutive inclusions induce a hierarchy of these properties via

$$ \textstyle\begin{array}{ccccc} C &\subseteq& \bar{C}&\subseteq& \bar{C}^{*} ,\\ \textit{NA} &\ \Longleftarrow\ & \textit{NFLVR} &\ \Longleftarrow\ & \textit{NFL}. \end{array} $$

Lemma A.2

$\textit{NFLVR} \Longrightarrow \textit{NA} \& \textit{NA}_{1}$.

Proof

Assume that NFLVR holds. Condition NA follows trivially. If $\mbox{NA}_{1}$ fails, then there exists a $[0,1]$-valued $\xi\in L^{0}_{+} \setminus\{0 \}$ such that for each $n\ge1$, we can find $X^{n}\in\mathcal{X}^{1/n}$ with $1/n+X^{n}_{T}\ge\xi$. Then the random variables $X^{n}_{T}\wedge\xi$ belong to $C$ and converge uniformly to $\xi$, contradicting NFLVR. □

To obtain the converse implication in Lemma A.2, we need an extra property. We call a model natural if the elements of $\mathcal {X}$ are adapted processes and for any $X\in\mathcal{X}$, $s\in [0,T)$ and $\varGamma\in\mathcal{F}_{s}$, the process $\tilde{X}:=I_{\varGamma\cap\{X_{s}\le0\}}I_{[s,T]}(X-X_{s})$ is an element of $\mathcal{X}$. In words, a model is natural if an investor deciding to start trading at time $s$ when the event $\varGamma$ happened can use from this time, if $X_{s}\le0$, an investment strategy that leads to a value process with the same increments as $X$.

Lemma A.3

Suppose that the model is natural. If NA holds, then any $X\in\mathcal{X}$ admits the bound $X\ge- \lambda$, where $\lambda=\|X_{T}^{-}\|_{\infty}$.

Proof

If $P[X_{s}<-\lambda]>0$, then $\tilde{X}:=I_{\{X_{s}<-\lambda\}}I_{[s,T]}(X-X_{s})$ belongs to $\mathcal {X}$ and satisfies $\tilde{X}_{T}\ge0$ and $P[\tilde{X}_{T}>0]>0$, in violation of NA. □

Proposition A.4

Suppose that the model is natural and in addition that for every $n\ge 1$ and $X\in\mathcal{X}$ with $X \ge- 1/n$, the process $n X$ is in $\mathcal{X}^{1}$. Then

$$\textit{NFLVR}\ \Longleftrightarrow\ \textit{NA}\ \&\ \textit{NA}_{1}. $$

Proof

By Lemma A.2, we only have to show the implication “⇐”. If NFLVR fails, then there are $\xi_{n}\in C$ and $\xi\in L^{\infty}_{+}\setminus\{0\}$ such that $\|\xi_{n}-\xi\|_{\infty}\le n^{-1}$. By definition, we have $\xi_{n}\le \eta_{n}=X^{n}_{T}$, where $X^{n}\in\mathcal{X}$. Obviously, $\|\eta_{n}^{-}\|_{\infty}\le n^{-1}$, and since NA holds, $nX^{n}\in\mathcal{X}^{1}$ by virtue of Lemma A.3 and our hypothesis. By the von Weizsäcker theorem, we may assume that $\eta_{n}\to\eta$ a.s. Since $P[\eta>0]>0$, the sequence $nX^{n}_{T}\in \mathcal{X}^{1}_{T}$ tends to infinity with strictly positive probability, violating condition NUPBR or, equivalently, $\mbox{NA}_{1}$. □

Examples showing that the conditions NFLVR, NA, and $\mbox{NA}_{1}$ are all different, even for the standard model (satisfying, of course, the hypotheses of the above proposition) can be found in [7].

Assume now that $\mathcal{X}^{1}$ is a subset of the space $\mathcal {S}$ of semimartingales, equipped with the Emery topology given by the quasi-norm

$$\mathbf{D}(X):=\sup\{E[1\wedge|H\cdot X_{T}|]:\ H\ \hbox{is predictable, }\ |H|\le1\}. $$

Define the condition ESM as the existence of a probability $\tilde{P}\sim P$ such that $\tilde{E}X_{T}\le0$ for all processes $X\in\mathcal{X}$. A probability $\tilde{P}$ with this property is referred to as an equivalent separating measure. According to the Kreps–Yan separation theorem [11, Theorem 2.1.4], the conditions NFL and ESM are equivalent. The next result is proved in [9] on the basis of the paper [5], where this theorem was established for the “standard” model; see also [3].

Theorem A.3

Suppose that $\mathcal{X}^{1}$ is closed in $\mathcal{S}$ and that the following concatenation property holds: for any $X,X'\in\mathcal {X}^{1}$ and any bounded predictable processes $H,G\ge0$ such that $HG=0$, the process $\tilde{X}:=H\cdot X+G\cdot X'$ belongs to $\mathcal {X}^{1}$ if it satisfies the inequality $\tilde{X}\ge-1$. Then, under the condition NFLVR, it holds that $C=\bar{C}^{*}$, and as a corollary, we have

$$\textit{NFLVR} \ \Longleftrightarrow\ \textit{NFL} \ \Longleftrightarrow\ \textit{ESM}. $$

Remark A.4

It is shown in [15, Theorem 1.7] that the condition $\mbox{NA}_{1}$ is equivalent to the existence of the (unique) supermartingale numéraire in a setting where the wealth-process sets are abstractly defined via a requirement of predictable convexity (also called fork-convexity).

In the case of the “standard” model with a finite-dimensional semimartingale $S$ describing the prices of the basic risky securities, we have the following: If $S$ is bounded (resp. locally bounded), a separating measure is a martingale measure (resp. local martingale measure). Without any local boundedness assumption on $S$, we have the following result from [6], a short proof of which is given in [9] and which we use here.

Theorem A.5

In any neighbourhood in total variation of a separating measure, there exists an equivalent probability under which $S$ is a $\sigma$-martingale.

It follows that if NFLVR holds, then the process $S$ is a $\sigma$-martingale with respect to some probability measure $P'\sim P$ with density process $Z'$. Therefore, for any process $X = H\cdot S$ from $\mathcal{X}^{1}$, the process $1+X$ is a local martingale with respect to $P'$, or equivalently, $Z'(1+X)$ is a local martingale with respect to $P$; therefore, $Z'$ is a local martingale deflator.

Remark A.6

A counterexample in [4, Sect. 6] involving a simple one-step model shows that Theorem A.5 is not valid in markets with countably many assets. As a corollary, the condition NFLVR (equivalent in this one-step model to $\mbox{NA}_{1}$) is not sufficient to ensure the existence of a local martingale measure or a local martingale deflator.