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arxiv: 2508.13966 · v2 · submitted 2025-08-19 · 💱 q-fin.MF

Market Viability and Completeness for Multinomial Models

Nahuel I. Arca This is my paper

Pith reviewed 2026-05-18 22:44 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords equivalent martingale measuresmarket viabilitycompletenesstwo-period modelsconvex combinationsmultinomial marketsdiscrete-time modelsKorn-Kreer-Lenssen model
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The pith

Two-period finite markets have equivalent martingale measures that are convex combinations of a finite number of extreme measures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the set of equivalent martingale measures in two-period markets with finitely many outcomes as the convex combinations of a finite collection of such measures. This description supplies a concrete way to study whether the market is viable and complete under the no-arbitrage condition. An explicit algorithm is supplied for locating the finite collection, and the same technique is shown to apply to other problems of finding intersections of convex sets. The method is then used on a discrete-time version of the Korn-Kreer-Lenssen model to exhibit concrete ways in which discrete-time analysis diverges from its continuous-time counterpart.

Core claim

In a two-period market with finitely many possible terminal values, the equivalent martingale measures are precisely the convex combinations of a finite number of extreme martingale measures. An algorithm recovers these extreme measures by solving the linear conditions that enforce the martingale property at each node.

What carries the argument

Representation of the convex set of equivalent martingale measures as the convex hull of its finitely many extreme points.

If this is right

  • Viability of the market reduces to checking that at least one of the finite extreme measures exists.
  • Completeness can be read off from the linear independence of the payoff vectors under the finite set of extreme measures.
  • The algorithm supplies a practical method for enumerating all arbitrage-free prices in the two-period setting.
  • The same convex-geometry technique extends directly to computing intersections of other convex sets that arise in pricing problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite characterization makes numerical pricing feasible in multinomial trees by reducing the problem to a finite linear program.
  • Recursive application of the two-period result might yield a workable description for multi-period models, though the number of extreme measures would grow.
  • The observed gap between discrete and continuous versions suggests that path-dependent claims require separate continuous-time arguments even when the marginal distributions match.

Load-bearing premise

The market runs for exactly two periods and admits only finitely many possible outcomes.

What would settle it

Construct a two-period market with infinitely many terminal outcomes and exhibit an equivalent martingale measure that lies outside the convex hull of any finite collection of other equivalent martingale measures.

Figures

Figures reproduced from arXiv: 2508.13966 by Nahuel I. Arca.

Figure 1
Figure 1. Figure 1: Binomial model with 3 trading dates. Another variant of this model is the trinomial tree [2, 5, 7, 8, 14, 15]. In this model, time is still discrete but, at each instant of time, the future branches in three possibilities: those two of the binomial model, and a third one consisting in the price remaining constant. A version of this model with ud = 1 is shown in figure 2. In each one of these models, each p… view at source ↗
Figure 2
Figure 2. Figure 2: Trinomial tree model with 3 trading dates and [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Information tree of the binomial model with 3 trading dates. [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Information tree of the trinomial tree model with 3 trading dates. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The market is arbitrage-free if and only if (1 + [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The market is arbitrage-free if and only if (1 + [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: ∆ ∩ A is the convex hull of (∂∆) ∩ A. vλ are affine combinations of x and y, and because x, y ∈ A, then vλ ∈ A for all λ. Furthermore, because ∆ is bounded, there exists λ0 > 0 sufficiently large that satisfies vλ0 ∈/ ∆. Because v0 = x ∈ ∆, there exists λ1 ∈ [0, λ0) such that vλ1 ∈ ∂∆. Then vλ1 ∈ (∂∆) ∩ A and vλ1 = x + λ1(x − y) ⇒ 1 1 + λ1 vλ1 + λ1 1 + λ1 y = x . That is to say, x is a convex combination b… view at source ↗
Figure 8
Figure 8. Figure 8: The generators are p 1 and p 2 . This process can stop even before, with the help of the following results. Proposition 4. If ∆, A ⊂ R b , B is the affine space spanned by ∆, ∆ ∩ A ̸= ∅(⇒ B ∩ A ̸= ∅) and dim(A ∩ B) ≥ 1, then (∂∆) ∩ A ̸= ∅. Proof. Let x ∈ ∆ ∩ A. Let y ∈ A ∩ B such that y ̸= x. For each λ ∈ R, let vλ := x + λ(y − x) . Then vλ ∈ A ∩ B for all λ ∈ R. Furthermore, because ∆ is bounded, there ex… view at source ↗
read the original abstract

In this paper we aim to study viability and completeness in finite markets. In order to do that, we characterize the set of equivalent martingale measures of two-period markets as convex combinations of a finite number of martingale measures. We provide an algorithm for finding such measures, that can be applied in other problems of convex geometry, and represents the starting point for a study of such characterizations of convex sets' intersections. We apply these results to the study of a discrete-time version of the Korn-Kreer-Lenssen model, and give an example of the limitations of using discrete-time models to understand continuous-time ones.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript characterizes the set of equivalent martingale measures for two-period finite-outcome markets as the convex hull of a finite collection of extreme martingale measures. It supplies an algorithm for computing these measures, applies the framework to a discrete-time version of the Korn-Kreer-Lenssen model, and uses the example to illustrate limitations of discrete-time models relative to their continuous-time counterparts. The work is framed as a contribution to the study of viability and completeness in finite markets.

Significance. The central characterization is a direct consequence of the fact that the set of equivalent martingale measures is a polytope in a finite-dimensional probability simplex; the paper therefore adds little new mathematical content beyond standard convex geometry. The algorithm may nevertheless be of practical interest for explicit computation in small markets and for related convex-geometry problems. The discrete-time application usefully flags approximation issues, though the quantitative demonstration of those limitations is not yet detailed.

major comments (1)
  1. [Characterization of equivalent martingale measures] The abstract and introduction assert that the set of EMMs is the convex hull of finitely many measures, yet supply no derivation steps or explicit verification that the linear martingale constraints intersect the simplex in a polytope whose vertices are themselves EMMs. Because this is the load-bearing claim for all subsequent results, the manuscript must either provide a self-contained argument or cite the precise convex-analysis theorem invoked.
minor comments (3)
  1. [Algorithm section] The algorithm is described at a high level; a pseudocode listing or a fully worked numerical example on a small state space would make the procedure reproducible and would also serve the stated goal of applicability to other convex-geometry problems.
  2. [Application to discrete-time Korn-Kreer-Lenssen model] In the Korn-Kreer-Lenssen application, the specific transition probabilities, the number of outcomes per period, and the quantitative measure of the discrete-versus-continuous discrepancy should be stated explicitly so that readers can assess the claimed limitations.
  3. [Notation and setup] Notation for the finite outcome space and the two-period filtration should be introduced once and used consistently; occasional shifts between “multinomial” and “finite-outcome” terminology are mildly distracting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to incorporate a more explicit justification of the central characterization.

read point-by-point responses

  1. Referee: [Characterization of equivalent martingale measures] The abstract and introduction assert that the set of EMMs is the convex hull of finitely many measures, yet supply no derivation steps or explicit verification that the linear martingale constraints intersect the simplex in a polytope whose vertices are themselves EMMs. Because this is the load-bearing claim for all subsequent results, the manuscript must either provide a self-contained argument or cite the precise convex-analysis theorem invoked.

    Authors: We agree that the current exposition would be strengthened by an explicit derivation of the characterization. In the revised manuscript we will insert a short self-contained proposition (with proof) immediately after the model setup in Section 2. The argument runs as follows: the set of all probability measures on the finite outcome space is the standard simplex, a polytope. The martingale conditions amount to a finite collection of linear equality constraints on the probability vector. Their intersection with the simplex is therefore itself a polytope. By the Krein-Milman theorem every point of this polytope—including every equivalent martingale measure—can be written as a convex combination of its extreme points, which are themselves martingale measures. We will also add a reference to the relevant result in convex geometry (e.g., Rockafellar, Convex Analysis, Theorem 18.1, or Grünbaum, Convex Polytopes, §2.4). This addition directly addresses the referee’s request without altering the subsequent results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim follows from standard finite-dimensional convex geometry

full rationale

The paper's core result—that the set of equivalent martingale measures on a two-period finite-outcome market is the convex hull of finitely many extreme measures—follows directly from the maintained assumptions of finite time and finite state space. Under these conditions the set is a polytope (the probability simplex intersected with linear martingale constraints), a standard fact of convex geometry that requires no additional derivation inside the paper. The provided algorithm is a computational procedure for enumerating vertices and does not redefine or fit the target set. The application to the discrete Korn-Kreer-Lenssen model is an illustrative example rather than a load-bearing justification. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The work is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard no-arbitrage theory of finite discrete markets and on the geometric fact that the set of equivalent martingale measures is a convex polytope in two-period settings.

axioms (2)
  • domain assumption Existence of equivalent martingale measures is equivalent to viability in finite discrete markets
    Invoked when the paper links viability to the non-emptiness of the set of EMMs.
  • domain assumption The market has exactly two periods and finitely many outcomes
    Required for the convex-combination representation to involve only finitely many extreme measures.

pith-pipeline@v0.9.0 · 5620 in / 1348 out tokens · 37402 ms · 2026-05-18T22:44:47.155095+00:00 · methodology

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Reference graph

Works this paper leans on

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