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The numeraire portfolio in semimartingale financial models

Ioannis Karatzas, Constantinos Kardaras

arXiv:0803.1877v1q-fin.PRmath.OCmath.PR

TL;DR

The paper studies when a numeraire portfolio exists in general semimartingale markets with predictable convex constraints. It derives characteristic-based criteria and shows that NUPBR, rather than full NFLVR, suffices for utility optimization and is equivalent to equivalent supermartingale deflators.

  • Problem

    The paper asks how to identify a trading strategy whose wealth dominates others in relative terms through a supermartingale property in general frictionless semimartingale markets.

  • Method

    The authors derive necessary and sufficient numeraire-portfolio conditions from the predictable characteristics of stock-price returns and use stochastic-integral semimartingale machinery.

  • Results

    NUPBR is equivalent to the existence of equivalent supermartingale deflators, which are closely related to but strictly weaker than equivalent martingale measures.

  • Takeaways & Limitations

    NUPBR is the minimal a-priori assumption needed for utility maximization, and its predictable-characteristic formulation provides a direct way to check it.

  • Takeaways & Limitations

    Absolute continuity of the drift with respect to predictable quadratic variation is necessary but not sufficient for absence of immediate arbitrage in general jump models.

Abstract

from arXiv · show

We study the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numeraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numeraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the "No Free Lunch with Vanishing Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with Bounded Risk" (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.

0. Introduction

The paper studies numeraire portfolios in general semimartingale markets with predictable convex constraints, seeking conditions for their existence and links to no-free-lunch concepts. It argues that NUPBR, rather than NFLVR, is the minimal assumption relevant for utility optimization and is directly checkable through predictable characteristics.

  • 0.1. Background and Discussion of Results: The model treats asset prices as semimartingales and allows dynamic stochastic optimization under frictionless trading.Semimartingales decompose prices into finite-variation signal and local-martingale noise.
  • 0.1. Background and Discussion of Results: A numeraire portfolio is essentially unique and makes every other portfolio’s relative wealth process a supermartingale.Its existence is characterized using the predictable characteristics of stock-price returns.
  • 0.1. Background and Discussion of Results: The paper strengthens earlier results by allowing predictable closed convex constraints, dropping finite expected log-utility assumptions, and not imposing NFLVR.The numeraire portfolio may exist even when classical NA fails.
  • 0.1. Background and Discussion of Results: NUPBR is equivalent to boundedness in probability of attainable terminal wealths and is presented as the minimal assumption needed for utility maximization.When NUPBR fails, utility optimization cannot be performed for any utility function; when it holds, utility maximization can proceed.
  • 0.1. Background and Discussion of Results: The paper shows that NUPBR is equivalent to the existence of equivalent supermartingale deflators, which are weaker than equivalent martingale measures.This provides a mathematical connection between numeraire portfolios and free-lunch conditions.

1. The Market, Investments, and Constraints

The market consists of strictly positive discounted semimartingale stock prices, with predictable integrable portfolios subject to convex constraints and strict-positivity requirements for wealth. The returns process is described through its predictable characteristics, which provide the main mathematical language for the model.

  • 1. The Market, Investments, and Constraints: The model has d strictly positive semimartingale stocks and a bank account normalized to one, so stock prices are already discounted.The returns process X, rather than the stock-price process S directly, is used in the analysis.
  • 1. The Market, Investments, and Constraints: The framework permits a possibly infinite stopping-time horizon and allows jumps at predictable times by permitting G to have jumps.The model otherwise works with strictly positive asset prices, while noting that negative prices are excluded from the setup.
  • 1. The Market, Investments, and Constraints: The returns process is summarized by predictable characteristics (B, C, η), equivalently represented by the predictable triplet (b, c, ν) relative to an operational clock G.These components describe drift, continuous covariation, and jump-measure intensity.
  • 1. The Market, Investments, and Constraints: A portfolio is a predictable, X-integrable process whose components represent proportions of wealth invested in stocks and the money market.Wealth starts from normalized capital W0 = 1.
  • 1. The Market, Investments, and Constraints: Admissibility requires wealth and its left-continuous version to remain strictly positive, preventing doubling strategies through a zero credit limit.The condition π⊤∆X > −1 ensures that wealth remains strictly positive.
  • 1. The Market, Investments, and Constraints: Predictable convex constraints encode restrictions such as prohibitions on short selling and borrowing, and may depend on time and the path.C-constrained portfolios satisfy π(ω,t) ∈ C(ω,t) throughout the planning horizon.

2. The Num´eraire Portfolio: Definitions, General Discussion, and Predictable Characterization

The numeraire portfolio is defined by supermartingale relative wealth and, when it exists, is essentially unique. The paper gives predictable-characteristic conditions for existence under closed convex constraints and relates the portfolio to several optimality criteria.

  • Definitions and general discussion: A numeraire portfolio ρ makes W^π/W^ρ a supermartingale for every admissible portfolio π, and its generated wealth process is unique.If two such portfolios exist, Jensen’s inequality implies their wealth ratios are equal.
  • Definitions and general discussion: The existence problem is to find necessary and sufficient conditions expressed through the predictable characteristics of the returns or stock-price process.These characteristics comprise drift, volatility, and jump-intensity information.
  • Preliminary characterization: Relative wealth is analyzed through the rate rel(π | ρ), with W^π/W^ρ a supermartingale exactly when rel(π | ρ) ≤ 0 almost everywhere.This converts the portfolio comparison into a predictable drift condition.
  • Preliminary characterization: Under constraints C ⊆ C0, ρ is the numeraire portfolio if and only if the relative-rate inequality holds for every predictable C-valued process, while ρ is predictable and X-integrable.The characterization applies beyond the initially admissible class for the inequality condition.
  • Predictable characterization: Existence requires the predictable set I ∩ Ĉ to be null; when it is null, a unique candidate solves the concave pointwise problem ρ = arg max_{π∈C∩N⊥} g(π).If the candidate additionally satisfies (ψρ · G)t < +∞, it is X-integrable and is the numeraire portfolio; conversely, an existing numeraire portfolio satisfies these conditions.
  • Predictable characterization: The paper summarizes existence as finiteness of the deterministic increasing functional Ψ(B, C, η) on [[0, T]], with further integrability conditions governing the constructed portfolio.In log-integrable Lévy settings, the numeraire portfolio is characterized by pointwise maximization of g; otherwise auxiliary approximating problems are used.

3. Unbounded Profits with Bounded Risks, Supermartingale Deflators, and the Num´eraire Portfolio

This section relates no-free-lunch conditions to equivalent supermartingale measures under predictable cone constraints, emphasizing NUPBR as a weaker condition than NFLVR.

  • NAC excludes arbitrage, while NUPBRC excludes sequences of wealth processes whose terminal values become unbounded with bounded risk.
  • UPBR means producing substantial terminal wealth from portfolios requiring progressively less initial capital, with a fixed positive probability.
  • NFLVRC holds exactly when both NAC and NUPBRC hold under predictable closed convex cone constraints.
  • For cone constraints, NFLVRC is equivalent to existence of an equivalent C-supermartingale measure.

3.3. Beyond the Fundamental Theorem of Asset Pricing.

The section shows that classical FTAP conclusions can fail beyond cone constraints and that predictable characteristics can instead characterize NUPBRC and construct free lunches.

  • For non-conic constraints, NFLVRC may hold even when no equivalent supermartingale measure exists.A two-stock example has NFLVRC but no ESMMC because its conic hull removes the relevant restriction.
  • Predictable characteristics can determine whether NUPBRC holds and can be used to construct an UPBR when it fails.The functional Ψ distinguishes the events where NUPBRC fails or holds.
  • The three-dimensional Bessel-process market has a numeraire portfolio and arbitrage, showing that NFLVR is not necessary for numeraire or log-optimal portfolios.
  • No deterministic functional of predictable characteristics can characterize NA in general, because identical characteristics can correspond to markets with different arbitrage properties.
  • NUPBR is identified as the minimal no-free-lunch-type condition needed to ensure utility maximization can proceed.

3.4. Supermartingale deflators.

This section introduces equivalent supermartingale deflators and connects them to admissible wealth processes, utility optimization, and the numeraire portfolio.

  • An equivalent supermartingale deflator D is positive, starts at one, and makes D W^π a supermartingale for every constrained portfolio.
  • A deflator of the form D*=1/W^ρ is tradeable and exists exactly when the numeraire portfolio ρ exists, subject to finite terminal numeraire wealth.
  • The deflator class, rather than the equivalent supermartingale-measure class, is the condition needed for utility maximization.
  • Existence of an equivalent supermartingale deflator implies that every admissible wealth process is a semimartingale up to the horizon and has a limit at infinity.
  • A tradeable supermartingale deflator has a dual minimal reverse relative-entropy property corresponding to numeraire log-optimality.

3.5. The second main result.

The paper places the numeraire portfolio in the context of arbitrage and proves equivalences among its existence, equivalent supermartingale deflators, and NUPBR under predictable closed convex constraints.

  • Theorem 3.12: Theorem 3.12 states that numeraire-portfolio existence, a non-empty set of equivalent supermartingale deflators, and NUPBR are equivalent.The result applies to a stock-price process with predictable closed convex constraints.
  • Theorem 3.12: The implication from a deflator to NUPBR follows because deflated admissible wealth processes are positive supermartingales and their terminal values are bounded in probability.The bound is uniform over all admissible portfolios.
  • Theorem 3.12: The converse analyzes failure of the numeraire portfolio and constructs either an unbounded increasing profit or an UPBR.Thus, failure of numeraire-portfolio existence contradicts NUPBR.
  • Measure changes: Existence of the numeraire portfolio is invariant under equivalent changes of probability measure, although the portfolio itself changes.This invariance does not follow directly from the definition.
  • Consequences: Under NUPBRC, wealth-process limit assumptions used in the definitions of NA and NFLVR become superfluous.Theorem 3.12 and Proposition 3.11 establish this implication.

3.6. Consequences of non-existence of the num´eraire portfolio.

When the numeraire portfolio does not exist, predictable-characteristic conditions identify either unbounded increasing profits or UPBRs, including gains obtainable by approximating a non-integrable candidate.

  • Non-integrability: Proposition 3.16 shows that if the candidate is not X-integrable up to T, suitable bounded-support approximations can generate arbitrarily large gains with fixed positive probability.The approximations are ρ_n := θ_nρ, with θ_n converging to the indicator I.
  • Construction: The construction uses truncations such as θ_n := I_Σn, where Σ_n limits time, portfolio magnitude, and the candidate's support.One stated choice is Σ_n := {(ω,t) ∈ [[0,T∧n]] | |ρ(ω,t)| ≤ n}.
  • Predictable characteristics: When NUPBRC fails, an UPBR can be constructed using the triplet of predictable characteristics.This is equivalent to failure of numeraire-portfolio existence or to a positive probability of infinite terminal numeraire wealth.
  • Failure modes: Failure of numeraire-portfolio existence occurs through unbounded increasing profit or failure of the constructed predictable process to be X-integrable up to T.The second case produces an UPBR through Proposition 3.16.
  • Singularities: A time-zero singularity can prevent forward investment in an otherwise favorable candidate, making the proof of Proposition 3.16 non-trivial.For dX_t = t^-1/2dt + dβ_t, the relevant accumulated quantity is infinite for every t > 0 and τ = 0.
  • Related boundary: For continuous-path models without constraints, prior work constructs instant arbitrage, whereas its construction in the presence of jumps remains open in the cited discussion.The authors state that they could not construct this instant arbitrage with jumps.

3.7. Application to Utility Optimization.

The paper shows that NUPBR is the minimal condition for utility optimization: without it, optimization fails for every utility function, while under it standard arguments apply subject to finiteness and utility regularity.

  • Main conclusion: NUPBR is identified as the minimal condition that allows the utility maximization problem to be solved.The paper treats this as the central consequence of the utility-optimization analysis.
  • Failure of NUPBR: If NUPBRC fails, every utility maximization problem either has no solution or has infinitely many solutions.When U(∞) < ∞ there is no solution; when U(∞) = +∞, the value is infinite and attainment yields infinitely many solutions.
  • Assumptions: The positive-case argument assumes that utility is continuously differentiable and satisfies the Inada conditions U′(0) = +∞ and U′(+∞) = 0.These assumptions are introduced after the analysis of failed NUPBRC.
  • NUPBR holds: When NUPBRC holds, the numeraire portfolio makes all relative wealth processes supermartingales and supports a bipolar relationship for positive terminal claims.When the value is finite, this relationship implies existence of a utility-optimization solution.
  • Additive model: The results also extend to nonpositive stock-price semimartingales by shifting prices and imposing positivity on wealth and pre-wealth.The multiplicative representation remains available through π := (1/W_−)θ, although π loses its earlier interpretation.
  • Constraints: Constraints imposed on investment proportions lead to the transformed constraint set bC rather than directly to C.The alternative formulation uses (θ_iS_i−/W_−)_{1≤i≤d} ∈ C.

4. Proof of Proposition 2.10 on the NUIP Condition

The proof characterizes immediate arbitrage directions through predictable set-valued processes derived from predictable characteristics, enabling measurable selection and construction of unbounded increasing profits.

  • NUIP characterization: An unbounded increasing profit implies a predictable direction whose wealth and stochastic-integral processes are non-decreasing and nonconstant with positive probability.The associated direction satisfies the sign, continuous-variation, and drift requirements defining immediate arbitrage.
  • Predictable approximations: The set of immediate arbitrage directions is not closed, so the proof introduces closed predictable approximations I_a to apply measurable selection results.Closedness is essential for selecting predictable processes.
  • Properties of I_a: Each I_a is increasing in a and takes values in closed convex subsets, with I represented as the union of the I_a.Consequently, intersection with the constraint set can be tested using sufficiently large a.
  • Measurability: Predictability of the relevant set-valued process follows from predictable measurability, continuity in the portfolio variable, and closedness arguments.The resulting intersection {I ∩ Č ≠ ∅} is predictable.
  • Construction: If {I ∩ Č ≠ ∅} has positive measure, measurable selection produces a bounded admissible strategy whose wealth is non-decreasing and exceeds one with positive probability.This constructs the required unbounded increasing profit.

5. Proof of the Main Theorem 2.15

The proof reduces the numeraire condition to a deterministic convex-analytic problem, then establishes predictability and integrability of the resulting portfolio process.

  • Deterministic characterization: Lemma 5.1 equates the relevant no-free-lunch condition with existence of a unique constrained vector satisfying rel(π | ρ) ≤ 0.The vector lies in C ∩ N⊥ and satisfies ν[ρ⊤x ≤ −1] = 0.
  • Deterministic characterization: When ν integrates the logarithm, ρ maximizes g over C ∩ N⊥; otherwise, approximating Lévy measures yield convergent optimizers.The limiting construction handles cases where direct logarithmic integration is unavailable.
  • Proof strategy: The deterministic implication is established in prior work, while failure of the no-free-lunch condition rules out the numeraire condition.The proof invokes the cited Lévy-model analysis for the difficult direction.
  • Integrability: Theorem 5.2 characterizes X-integrability through G-integrability of three predictable processes associated with continuous variation, jumps, and drift.The first controls continuous quadratic variation, the second small-jump variation and large-jump intensity, and the third the bounded-jump drift.
  • Integrability: Under ν[ρ⊤x ≤ −1] = 0 and rel(0 | ρ) ≤ 0, Lemma 5.3 reduces X-integrability to finiteness of an increasing predictable process.For integrability up to T, the corresponding terminal value must also be finite.
  • Predictable construction: The predictable candidate is constructed pointwise on the logarithm-integrable set and elsewhere as the pointwise limit of predictable optimizers for approximating measures.Measurable-selection results provide predictability, after which the integrability criterion completes the proof.

6. On Rates of Convergence to Zero for Positive Supermartingales

The section characterizes when a positive supermartingale converges to a positive finite limit or to zero using predictable characteristics, and gives an exact decay rate under an additional jump bound.

  • Convergence criterion: Every positive supermartingale converges, and Proposition 6.1 uses the increasing process H to distinguish finite positive limits from convergence to zero.H combines predictable drift, continuous quadratic variation, and a compensator term for jumps.
  • Financial interpretation: For Wπ/Wρ, the abstract components of H correspond to relative drift, squared continuous-volatility distance, and a jump compensator.Thus the abstract proposition applies directly to relative wealth processes.
  • Decay rate: If ΔZ ≥ −1 + δ for some δ > 0, then H_t^-1 log Y_t converges to −1 when H diverges.This identifies the exact rate at which log Y tends to −∞.
  • Proof strategy: The proof decomposes log Y into continuous and purely discontinuous local-martingale components and analyzes finite- versus infinite-variation events separately.The jump analysis splits contributions into left and right regions and applies predictable quadratic-variation estimates.
  • Jump analysis: When the jump compensator is finite, the relevant jump expression has a finite limit; when it is infinite, normalized jump terms have nonpositive asymptotic bounds.The two cases are controlled through the auxiliary processes E and F.
  • Proof strategy: The continuous and jump estimates together imply Proposition 6.1 through the definition of H.The conclusion follows after combining the two local-martingale analyses.

7. Proof of Proposition 3.16

The proof establishes the equivalence between unbounded semimartingale returns and unbounded terminal stochastic exponentials under a supermartingale assumption, then applies it to portfolio wealth processes.

  • Portfolio decomposition: The proof separates paths where the candidate portfolio is integrable up to T from paths where its integrability criterion diverges.On the integrable set, conditioning preserves predictability and integrability.
  • Integrable region: On the integrable set, dominated convergence and the stochastic exponential formula show that terminal wealth limits are independent of the approximating portfolio sequence.The convergence holds in probability under both the conditional and original measures.
  • Nonintegrable region: On the nonintegrable set, bounded predictable scalings produce integrable portfolios whose terminal gains are unbounded in probability.The stochastic-exponential argument then yields terminal wealth unbounded in probability on a set of positive probability.
  • Definitions: The paper defines unboundedness in probability through the supremum of absolute process values over the time interval.One-sided versions use corresponding one-sided suprema.
  • Stochastic exponential lemma: Lemma 7.1 proves that, for returns with jumps greater than −1 and inverse stochastic exponentials that are positive supermartingales, process unboundedness is equivalent to terminal wealth unboundedness.The proof uses four steps involving logarithmic bounds, stopping, and quadratic variation.
  • Stochastic exponential lemma: The stopping argument shows that a class bounded below is also bounded above under the stated supermartingale conditions.This supports the reduction from general unboundedness to unboundedness from above.
  • Stochastic exponential lemma: Unbounded returns force unbounded stochastic exponentials through either logarithmic decline or unbounded quadratic variation.The converse fails without the inverse-exponential supermartingale assumption, as the Brownian example demonstrates.
  • Limitation: Without the supermartingale assumption, a process can be bounded below and unbounded above while its stochastic exponential remains bounded above.The example uses R_t = at + β_t with a ∈ (0, 1/2).

Appendix A. Measurable Random Subsets

The appendix develops measurability tools for closed convex random subsets and measurable optimization, enabling predictable selection of portfolio candidates.

  • Random subsets: A random subset of R^d is measurable through a σ-algebra generated by distance-to-set mappings.The convention is dist(z, ∅) = +∞.
  • Random subsets: Measurability is equivalently characterized by whether intersections with compact, closed, or open sets are measurable.The compact-set formulation directly yields measurability of random-set intersection events.
  • Random subsets: For singleton-valued random subsets, random-set measurability is equivalent to measurability of the selected point.This connects the set-valued framework to ordinary measurable functions.
  • Closure properties: Unions and intersections of measurable random subsets remain within the measurable framework.The appendix introduces these closure properties before developing measurable optimization.
  • Measurable optimization: A Carathéodory function on a measurable closed random set produces a closed measurable sublevel or constraint set.The result assumes the target subset is closed and the random set is closed and convex.
  • Measurable optimization: The measurable maximum theorem makes the value function measurable and, under a unique optimizer, provides a measurable optimizer.A measurable selector also exists for any nonempty measurable closed convex random set.

Appendix B. Semimartingales and Stochastic Integration up to +∞

Appendix B defines semimartingales and stochastic integration up to infinity by transforming an infinite time horizon to [0,1], and distinguishes this property from merely having a limit at infinity. It also records decomposition and reciprocal-process results used later.

  • Definitions: A process with a limit at infinity is a semimartingale up to infinity when its time-transformed version on [0,1] is a semimartingale under the transformed filtration.The transformation maps t to t/(1−t), with the value at 1 defined as the process limit at infinity.
  • Definitions: An integrand is X-integrable up to infinity when its stochastic integral with respect to X is a semimartingale up to infinity.
  • Definitions: Having a limit at infinity does not ensure semimartingale behavior up to infinity: X_t = t^-1 sin t has limit 0 but infinite total variation.Because the example is deterministic, semimartingale status up to infinity would require finite variation.
  • Structural results: Every semimartingale up to infinity decomposes into a finite-variation-up-to-infinity process and a local martingale up to infinity.The local martingale component admits localization by stopping times whose explosion events increase to the whole sample space.
  • Structural results: A positive supermartingale is a special semimartingale up to infinity; if its terminal limit is positive, its inverse and the stochastic integral of its inverse are semimartingales up to infinity.The proof uses Doob–Meyer decomposition and Itô’s formula for the inverse function.
  • Scope: The appendix formulates results for infinite horizons, while finite stopping-time versions follow by applying them to stopped processes; differences arise when the stopping time can equal infinity.

Appendix C. σ-Localization

Appendix C introduces σ-localization as localization along increasing predictable sets and relates σ-localized semimartingale classes to predictable characteristics. It derives criteria connecting σ-supermartingales with local and ordinary supermartingales.

  • σ-localization: σ-localization defines a class by requiring membership in the original class after restriction to an increasing sequence of predictable sets covering the time-space domain.
  • σ-localization: Stopping-time localization is a special case of σ-localization when the predictable sets are intervals [[0,τ_n]].
  • Localized classes: The class U consists of class-(D) semimartingales, while Uloc corresponds to special semimartingales with locally integrable finite-variation structure.Class-(D) processes admit a Doob–Meyer decomposition with integrable total variation and a uniformly integrable martingale component.
  • Characteristics: Predictable characteristics provide an interpretation of σ-localized classes, including criteria for membership in Uloc and U.Proposition C.2 states these criteria using the drift, continuous covariance, jump compensator, and operational clock.
  • Characteristics: σ-localization permits direct discussion of drift rates even when those rates cannot be integrated into an ordinary drift process.The appendix characterizes the sign of the drift rate for σ-supermartingales and σ-submartingales.
  • σ-supermartingales: For σ-supermartingales, integrability of the relevant negative-jump term is equivalent to being a local supermartingale.The proof identifies a decreasing predictable finite-variation part after characteristic-based integrability is established.
  • σ-supermartingales: A σ-supermartingale bounded below by a constant is a local supermartingale, and with integrable initial positive part it is a supermartingale.The corresponding statement also applies to σ-martingales and local martingales.
  • Related results: The σ-martingale case of the proposition is known as the Ansel–Stricker theorem, while related σ-supermartingale results are attributed to Kallsen.
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