The continuous behavior of the numéraire portfolio

The continuous behavior of the numéraire portfolio under small changes in information structure, probabilistic views and investment constraints

Constantinos Kardaras Constantinos Kardaras, Mathematics and Statistics Department, Boston University, 111 Cummington Street, Boston, MA 02215, USA.
July 15, 2019

The numéraire portfolio in a financial market is the unique positive wealth process that makes all other nonnegative wealth processes, when deflated by it, supermartingales. The numéraire portfolio depends on market characteristics, which include: (a) the information flow available to acting agents, given by a filtration; (b) the statistical evolution of the asset prices and, more generally, the states of nature, given by a probability measure; and (c) possible restrictions that acting agents might be facing on available investment strategies, modeled by a constraints set. In a financial market with continuous-path asset prices, we establish the stable behavior of the numéraire portfolio when each of the aforementioned market parameters is changed in an infinitesimal way.

Key words and phrases:
Information, investment constraints, log-utility maximization, mathematical finance, numéraire portfolio, semimartingales, stability, well-posed problems.
2000 Mathematics Subject Classification:
60H99, 60G44, 91B28, 91B70

0. Introduction

Within the class of expected utility maximization problems in the theory of Financial Economics, the one involving expected logarithmic utility plays a central role. Its importance can be understood by going as back as [14], where the optimal exponential growth for a gambler’s wealth was discovered from an information-theoretic point of view. In general semimartingale models, it is the only case of utility where an explicit solution can be given in terms of the triplet of predictable characteristics, as was carried out in [9].

The log-optimal portfolio, when it exists, is the numéraire portfolio (an appellation that was introduced in [17]) according to the definition in [4]: all other wealth processes, when discounted by the log-optimal one, become supermartingales under the historical (statistical, real world) probability. In fact, the numéraire portfolio can exist even in cases where the log-optimal problem does not have a unique solution, which happens when the value of the log-optimal problem is infinite.

The numéraire portfolio depends on the stochastic nature of the financial market. As the output of an optimization problem, it is of importance to ensure that it is stable under small changes in the market parameters. Here, focus is given on the following characteristics:

  • Available information that economic agents have access to.

  • Statistical (or even subjective) views on the possible future outcomes.

  • Investment constraints (usually, institutionally-enforced) that agents face.

Institutionally-enforced constraints can involve, for example, prevention of short sales. Another important restriction that agents face is that of a finite credit limit; their wealth has to remain positive in order to avoid bankruptcy.

The purpose of this work is to guarantee that small deviations from the above market characteristics do not lead to radical changes in the structure of the numéraire portfolio. Naturally, part of the problem is to rigorously define what is meant by “deviations” of the market characteristics. In turn, this means that in order to achieve the desired continuous behavior of the numéraire portfolio, certain economically-reasonable topological structures have to be placed on filtrations, probabilities and constraint sets.

Stability of the numéraire portfolio is a qualitative study; there are, however, good quantitative reasons to undertake such study. Lately, there has been significant interest in quantifying the value of insider information, as measured via the increase in the log-utility of an insider with respect to a non-informed trader. One can check, for example, [2] and the wealth of references therein. It then becomes plausible to examine marginal values of insider information, or of investment freedom. The last question is intimately related to differentiability of the numéraire portfolio (or at least of the value of the log-utility maximization problem) with respect to market parameters. Such differentiability would give a first-order approximation of the behavior of the numéraire portfolio. Before seeking conditions ensuring differentiability, which is a possible topic for further research, a zeroth-order study concerning continuity has to be carried out. In the present work, we only scratch the very surface of the problem of differentiability of the numéraire portfolio and the calculation of its derivative.

The structure of the paper as follows. Section 1 sets up the model with continuous asset-price processes, where markets are parameterized via a triplet of data, including information flows, statistical structure and investment constraints that agents face. A “proximity” concept for the market parameters is introduced by defining modes of convergence for the three data inputs. Theorem 1.3 is the result which establishes continuity of the numéraire portfolio in a rather strong sense under convergence of market’s data. Then, Section 2 is dedicated to proving Theorem 1.3.

The workable expression that is obtained for the numéraire portfolios allows for a bare-hands approach to proving Theorem 1.3. This should be contrasted with the treatment in [16] and [13], where passage to the dual problem, as described in [15], is necessary. There, unnatural (from an economical point of view) uniform integrability conditions have to be assumed involving the class of equivalent martingale measures of the market.

The assumption of continuity of the asset-price processes is made for simplifying the presentation. (It should be noted however that an elementary example in §1.4 shows that the result of Theorem 1.3 is not valid without additional control on the agent’s constraints.) Even by assuming continuity of the asset-price processes, one cannot completely avoid dealing with jumps in the proof of Theorem 1.3. Changing from the probability measure of one market to the one of another, as has to be done, results in the appearance of martingale density processes with possible jump components. These technical complications make the proof of Theorem 1.3 somewhat lengthy.

1. The Result on the Continuous Behavior of the Numéraire Portfolio

1.1. The set-up

Every stochastic process in the sequel is defined on a stochastic basis . Here, is a probability on , where is a -algebra that will make all involved random variables measurable. Further, is a “large” filtration that will dominate all other filtrations that will appear. Of course, for all and is assumed to satisfy the usual hypotheses of right-continuity and saturation by -null sets.

1.1.1. Assets and investing

The price-processes of traded financial assets, where , are denoted by . All processes , , are -adapted and are assumed to have been discounted by a “baseline” asset that will act as a deflator for the denomination of all wealth processes.

The minimal filtration that makes adapted and satisfies the usual hypotheses will be denoted by . Since is -adapted, . In the sequel, the information flow of economic agents acting in the market will be modeled via elements such that


We shall also model statistical, or subjective, views of economic agents via , where


The following innocuous assumption on the structure of the will be in force throughout:


For a pair satisfying (INFO) and (P-LOC-EQUIV), (CON-SEMI-MART) implies that is a -semimartingale. Therefore, one can define the class of all possible nonnegative wealth processes starting from (normalized) unit initial capital for a market in which the information-probability structure is given by :


The dependence on from in (1.1) above is suppressed, simply because there is no dependence in view of (P-LOC-EQUIV). The following structural assumption on the class of wealth processes will be in force throughout.


(Note that “” denotes a nonincreasing limit.) In other words, the set is bounded in -probability for all . For a pair satisfying (INFO) and (P-LOC-EQUIV), (NUPBR) implies that is bounded in -probability for all .

Remark 1.1.

According to [11], condition (NUPBR), an acronym for No Unbounded Profit with Bounded Risk, is equivalent to existence of the numéraire portfolio (see §1.1.4 below) for any pair that satisfies (INFO) and (P-LOC-EQUIV). Since this work is aimed at studying stability of the numéraire portfolio, (NUPBR) is a minimal structural assumption.

1.1.2. Constraints on investment

Fix some pair corresponding to the information-probability structure of the financial market. Agents in this market might be facing constraints on possible investment strategies, which we now formally describe. Consider a set-valued process , where denotes the class of Borel subsets of . A process in will be called -constrained if for all ; in short, . (Investment constraints of this kind, but where no dependence of the constraint sets on is involved, appear in an Itô-process modeling context in the literature in [6].) We denote by the class of all -constrained wealth processes in ; namely, . For , both and are -predictable. It makes sense, both from a mathematical and a financial point of view, to give the constraints set a predictable structure as well. A set-valued process will be called -predictable if is an -predictable set for all compact . For more information on this kind of measurability, see Appendix 1 of [11], or Chapter 17 of [1] for a more general treatment. Further, it is financially reasonable to put some closedness and convexity structure on . We call closed and convex if has these properties for all .

1.1.3. Financial market data

Before the formal definition of the financial market’s data is given, we tackle degeneracies that might appear in the asset-price process. Call , where “” denotes the trace operator on matrices and denotes the continuous, -matrix-valued quadratic covariation process of . It is straightforward that is -predictable and nondecreasing. There exists a -nonnegative-definite-matrix-valued, -predictable process such that (in obvious matrix notation). Define , where the dependence of on is suppressed; is -predictable and takes values in linear subspaces of . We denote by the orthogonal complement of ; this is also a -predictable, -subset-valued process. (The facts that and are -predictable follow by the results of Appendix 1 of [11].) Now, pick any -predictable, -integrable and -valued process . The gains process has null quadratic variation. Under (NUPBR), is identically equal to zero. Therefore, any agent should be free to invest in these -valued strategies, since they result in zero wealth. In other words, we should have, in compact notation, .

We are ready to give the modeling structure of the financial market environment.

Definition 1.2.

A triplet will be called financial market data, if satisfies (INFO), satisfies (P-LOC-EQUIV), and is an -predictable, convex and closed -set valued process such that .

1.1.4. Numéraire portfolios

Under (CON-SEMI-MART), and for a pair that satisfies (INFO) and (P-LOC-EQUIV), decompose , where is an -adapted, continuous process of locally finite variation, and is a -local -martingale. Assumption (NUPBR) implies that there exists an -predictable process such that


(This last fact was already present in [7], although not stated this way. The previous structural conditions (1.2) have also appeard in [3] and [19].)

In the financial market with data , the numéraire portfolio is the unique wealth process with the property that is -supermartingale for all . (For a complete list of the properties of the numéraire portfolio in connection to what is described here, one could check [11].) It can be shown that the numéraire portfolio is the one that maximizes the growth of the wealth process, where the -growth of a wealth process with is defined to be the finite variation part of in its -semimartingale decomposition.

We shall now give a more concrete description of the numéraire portfolio. Start with some such that , and consider the -predictable, -dimensional process defined implicitly via . Using Itô’s formula and (1.2), the -growth of is easily seen to be equal to . As discussed previously, if is to be the numéraire portfolio, it must have maximal growth. Therefore, let be the unique -predictable, -dimensional, -valued process that satisfies


for all . (If , .) The process is well-defined; this follows from the fact that the maximization problem (1.3) defining is strictly concave and coercive on the closed convex set . Its -predictability follows from the corresponding property of the inputs , , ; again, we send the interested reader to Appendix 1 of [11]. It then follows that the -numéraire portfolio satisfies and the dynamics for . In other words, in logarithmic terms,


Indeed, it is straightforward to check that as defined above is such that is a -supermartingale for all .

1.2. Convergence assumptions

In order to formulate the question of continuous behavior of the numéraire portfolio, several markets will be considered. For each , the market structure will be modeled via the data . The limiting behavior of the data triplets will be given in the paragraphs that follow. What is sought after is convergence, as , of the market’s numéraire portfolio to the numéraire portfolio of the market corresponding to .

First, convergence of filtrations is settled. Let us give some intuition. Assume, for simplicity, that all markets work under that same probabilistic structure, given by . For any , an agent with information can only project at each time the the conditional probability that will happen or not. A natural way to define convergence of then would be to require that converges in -probability to , at least pointwise for all . We ask something somewhat weaker.


Note that (F-CONV) certainly holds in the case where converges monotonically to , in the sense that or for all , in view of the martingale convergence theorems.

The assumption on convergence of to is:


Note that, as a consequence of Scheffe’s lemma, (P-CONV) is equivalent to saying that converges in total variation to on for all .

We turn to the constraints sets. For two subsets and define their Hausdorff distance


For , let . For a collection of subsets of , define

Note that this convergence is weaker than requiring and that it is equivalent to saying that is the closed limit of the sequence (see Definition 3.66, page 109 of [1]). We then ask that


1.3. Stability of the numéraire portfolio

Continuity of the log-wealth of the numéraire portfolios will be obtained with respect to a strong convergence notion, which is now defined. Consider a collection , each element being a continuous -semimartingale. For , write , where is -adapted, continuous and of finite variation and is a -local -martingale. We say that -converges to and write if and only if as well as hold for all . By the treatment in [18], it can be shown that -convergence is equivalent to (local, in time) convergence in the semimartingale topology on that was introduced in [8]. In particular, -convergence is stronger than the uniform convergence on compacts in probability: implies , the last equality meaning , for all .

Theorem 1.3.

Consider a collection of markets, each with data , indexed by . Assume that all (CON-SEMI-MART), (NUPBR), (F-CONV), (P-CONV) and (C-CONV) are valid. For , let , be the numéraire portfolio. Then,

The proof of Theorem 1.3 is given in Section 2. It is easy to argue why Theorem 1.3 is true, and this somewhat sets the plan for the proof. For notational simplicity, let and for all . Under (P-CONV) and (F-CONV) one would expect that converges in some sense to . Then, (C-CONV) and (1.3) should imply that converges (again, in some sense) to . After that, (1.4) makes it very plausible that should converge to . Of course, one has to give precise meaning to these “senses” of convergence of the predictable processes. The details of the proof are technical, but more or less follow the above intuitive steps.

Remark 1.4.

The result of Theorem 1.3, given all its notation and assumptions, implies


as well as


Both of the above limiting relationships are incarnations of the fact that small deviations from information, probability and investment constraints structures will lead to a small relative change in the numéraire portfolio. While (1.6) is from the point of view of the limiting market, (1.7) takes the viewpoint of the approximating markets.

1.4. The case of asset-prices with jumps

Theorem 1.3 need not hold in the case where jumps are present in the asset-price process. A simple discrete one-time-period counterexample is given below; after that, a discussion follows on what the issue is, along with a possible resolution.

Example 1.5.

Consider a one-time-period discrete stochastic basis , where . Suppose that is rich enough to accommodate a sequence of independent standard normal random variables, as well as some random variable , independent of the previous Gaussian sequence with , where . Define a collection of filtrations via , for all (the information at the terminal date is the same in all markets), as well as

Of course, converges monotonically upwards to .

The financial market has one risky asset: . With , set and . The classical No Arbitrage condition holds for the market with information , which is the equivalent of (NUPBR) for discrete-time models. The probabilistic structure is the same in all the markets, given by . Further, no institutionally-enforced constraints are present for agents acting in the market indexed by any .

For the limiting market with -information, the model is just a (conditional) binomial one, since is -measurable. We have . Since , it is easy to see (optimizing the expected log-utility) that the limiting market’s numéraire portfolio is such that . If , .

Consider now the market with information for some . Conditional on , is independent of and its law is Gaussian with mean and variance . Since the conditional law of is supported on the whole real line, we get . Therefore, for each , every approximating market’s numéraire portfolio satisfies . This obviously does not converge to , if .

In the previous example, all the nonnegative wealth process sets are trivial, but the limiting is non-trivial. Even though there are no institutionally-enforced constraints in the markets, agents still have to face the natural constraints , , that ensure the positivity of the wealth process. As it turns out, for all , while . Such behavior is of course absent in the case of continuous-path price processes.

A possible resolution to the previous problem could be the following. In a general discrete-time model, if denotes the natural positivity constraints of the market with information , then holds for all and in in view of (INFO). If one forces from the beginning the additional assumption for all , the problem encountered at Example 1.5 ceases to exist, and one should be able to proceed as before.

1.5. First-order analysis

Once continuity of the numéraire portfolio is established, the next natural step is to study the direction of change given specific changes of the inputs. We provide here a first insight on how the numéraire portfolio changes when we alter only the probabilistic structure of the problem, keeping the information fixed and working on the non-constrained case. In more general situations the problem is expected to be rather involved.

For the purposes of this subsection, we shall change the notation slightly. We simply use , instead of , to denote the common filtration of all agents. Let be the “limiting” probability (the one that we previously denoted by ). Furthermore, let be some probability that is equivalent to , and let . Write for the Doob-Meyer decomposition of under ; here, and for all .

Define ; since is a strictly positive -martingale, one can write

where is a local -martingale that is strongly orthogonal to . (This multiplicative decomposition of follows in a straightforward way from its corresponding additive decomposition — see Theorem III.4.11 of [10].) It follows that satisfies

where , and is a local -martingale that is strongly orthogonal to . With the above notation, and according to Girsanov’s theorem, we have .

Let denote the numéraire portfolio under market data . Since there are no constraints on investment, satisfies and for ; in other words, and using the Doob-Meyer decomposition of under , we have

From we get ; therefore,


Given the above equality, and using the fact that , it is straightforward that

In a similar manner, one can proceed to higher-order -derivatives of at . For example, (1.8) and simple algebra (remembering that ) gives

after which it is straightforward that

2. Proof of Theorem 1.3

Throughout the proof, all the assumptions of Theorem (1.3) are in force. Without loss of generality, and for notational convenience, it is assumed that . Then, with for all , holds for all .

2.1. Setting out the plan

The first step towards proving Theorem 1.3 will involve the fixed-probability case, where for all . Then, the general case where (P-CONV) is assumed will be dealt with.

In order to lighten notation, we set


For the fixed probability case, we also consider


Since , we have . For each , is the numéraire portfolio for an agent with data . In order to prove Theorem 1.3, first we shall show that


and then that


2.2. A deterministic concave maximization problem

For fixed and , all defined in (2.1) and defined in (2.2) appear as solutions to a deterministic concave maximization problem of the form


for some , where the pseudo-inner-product on is defined via (remember that is the usual Euclidean inner-product) for all vectors and of , where is a -nonnegative-definite matrix. Of course, denotes the pseudo-norm generated by the last pseudo-inner-product . In (2.5), and we suppose that , so there is a unique solution to (2.5), and is well-defined.

It makes sense then to study the deterministic problem (2.5). Only for this subsection, all elements involved, including and will be assumed deterministic. For the -nonnegative-definite matrix we shall be assuming that , which implies in particular that for all , where is the Euclidean norm. Observe that the -predictable process satisfies since, formally, .

The dependence of of (2.5) on and will be now examined. Remember that for , as well as the definition of “” from (1.5).

Proposition 2.1.

Let , be vectors in and , be closed and convex subsets of with and . With the notation of problem (2.5), we have

  1. .

  2. .

  3. .


(1) Let and . First-order conditions for optimality imply that and . Adding the previous two inequalities gives , or, equivalently, , which proves the result.

(2) Let . Since , first-order conditions give . In other words,