Remarks on multi-period martingale optimal transport

Brendan Pass; Joshua Hiew

REVIEW 4 major objections 4 minor 23 references

Remarks on multi-period martingale optimal transport

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves first uniqueness and structural results for three-period martingale optimal transport with fully pairwise-interacting costs, plus an essentially explicit conditional solution in the vanishing-interaction limit.

desk verdict Useful new tools and a clean linearization framework, but the headline three-period uniqueness theorems rest on unproved structural lemmas, especially Lemma 4.4. read the letter →

arxiv 2506.05505 v1 pith:S4DNSMLE submitted 2025-06-05 math.OC math.PR

Brendan Pass , Joshua Hiew This is my paper

classification math.OCmath.PR MSC 49Q2260G4291G20

keywords martingaleoptimaltransportmulti-periodthree-periodmodel-freepricingleft-monotonecouplinggluinglemmauniquenesspricebounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Martingale optimal transport (MOT) gives model-free upper and lower bounds on derivative prices, but until now the structure of optimizers was understood only when the payoff couples one fixed time to every other time; payoffs in which all pairs of times interact were essentially open. This paper attacks the three-period case, with cost $c(x,y,z)=c_1(x,y)+c_2(y,z)+c_3(x,z)$. It develops two martingale gluing lemmas that specify when two-period martingale couplings can be assembled into a three-period martingale and which assemblies are optimal, then proves uniqueness and structural results for three-period problems under several regimes of marginals: two-point, three-point, or discrete first-two marginals. It also shows that as the $x$--$z$ interaction vanishes, limits of optimizers are described by a new conditional transport problem whose solution is essentially explicit. If correct, these results turn some three-period model-free pricing problems into problems solvable by two-period computations, and the paper demonstrates the approximation on real option data.

What carries the argument

The carrying mechanism combines two tools. The martingale gluing lemmas (Lemmas 2.1 and 2.4) characterize when two two-period martingale couplings can be combined into a three-period one: Lemma 2.1 glues adjacent couplings $\pi_{XY}$ and $\pi_{YZ}$ by choosing, at each $y$, a conditional coupling between $\kappa^X_Y(y,\cdot)$ and $\kappa^Z_Y(y,\cdot)$ whose barycenter is $y$; Lemma 2.4 glues couplings sharing the first marginal when their conditional laws are in convex order. When the cost is a sum of pair costs, the lemmas imply that the two-period projections of any optimizer are themselves optimal, so the remaining pair cost is minimized fiberwise. The second tool is left-monotonicity: for costs whose $x$-derivative is strictly concave in $z$ (the martingale Spence--Mirrlees condition), $(\mathrm c,W)$-monotonicity forces the support of any optimizer to be left-monotone, meaning no pair with $x<x'$ has its $z$-value strictly between the two $z$-values coupled to $x$. Left-monotonicity, together with a two-point-support lemma, pins the optimizer onto two graphs $T_-:X\to Z_-$ and $T_+:X\to Z_+$ with weights determined by the barycenter condition; this two-graph structure is what the uniqueness proofs in the three-period theorems ultimately inherit.

What would settle it

Find a pair of marginals $\sigma_X,\sigma_Z$ of the two-interval type for which some $(\mathrm c,W)$-monotone optimal coupling in problem (3.2) has, on a set of positive $\sigma_X$-measure, conditional support on three points, or exhibit an $x$ of positive mass whose three-point couplings do not force $\sigma_Z((z_0,z_1))=0$. Either example would refute Lemma 3.7 and with it the two-graph uniqueness structure used in Theorems 3.6 and 4.2.

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Extended reading notes

Core claim

The paper claims the first uniqueness and structural results for the three-period martingale optimal transport problem when the cost is $c(x,y,z)=c_1(x,y)+c_2(y,z)+c_3(x,z)$, so that all three pairs of variables interact. Theorem 4.2 establishes uniqueness when the middle marginal $\mu_Y$ has two-point support, the $x$--$z$ cost satisfies the martingale Spence--Mirrlees condition, and $Z$ splits into two intervals bracketing the support of $\mu_Y$; the optimizer concentrates on two graphs $T_\pm$ over $(x,y)$. Theorem 4.5 adds uniqueness when $\mu_Y$ has three-point support and the cost is $f(x,y)z^2$ with $\partial_x f,\partial_y f<0$, by showing the induced coupling between $w=f(x,y)$ and $z$ is a graph and that $\nu=f_\#\pi_{XY}$ is uniquely determined. Theorem 4.8 gives uniqueness for discrete $\mu_X$ and $\mu_Y$ under a cost whose pairwise differences cross any given line only countably often. In the vanishing-interaction limit, Theorem 3.6 yields an essentially explicit solution: conditionally on the middle variable, the optimal coupling is supported on two monotone graphs, one in each of two intervals, with weights fixed by the barycenter equation $y=\lambda_-T_-(x)+\lambda_+T_+(x)$.

Load-bearing premise

The load-bearing premise is that every optimizer of the conditional transport problem is supported on exactly two points for almost every $x$, one in each interval; the proof of that premise (Lemma 3.7) relies on a countability step about exceptional intervals that the text does not fully justify.

Editorial extensions

If this is right

For costs of the form $c_1(x,y)+c_2(y,z)+c_3(x,z)$ with $\mu_Y$ supported on two points, the three-period optimizer is unique and concentrated on two graphs, so the model-free price bound can be computed by solving two-period problems.
For $\mu_Y$ supported on three points and cost $f(x,y)z^2$ with $\partial_x f,\partial_y f<0$, uniqueness holds and the coupling is concentrated on a graph in the $(f,z)$ coordinates, reducing the optimizer search to the one-dimensional law $\nu=f_\#\pi_{XY}$.
For discrete $\mu_X$ and $\mu_Y$ with a cost whose pairwise differences meet any given line only countably often, $\mu_Z$-almost every $z$ is assigned to a single pair $(x_i,y_j)$, giving a unique graph-supported optimizer.
As the $x$--$z$ interaction vanishes, the limiting optimizer is characterized fiberwise by the conditional problem (3.2), which yields a first-order linear approximation of the price bound around the decoupled problem.
In the two numerical examples (third moment of the sum and a basket of straddles), prices from tree-like martingale models lie inside the approximate bounds, and the exact linear-programming bounds are very close to the approximations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the two-graph structure is stable, the uniqueness theorems should imply quantitative stability of three-period price bounds under small changes in the marginals within the covered regimes; the paper itself notes that multi-marginal MOT is unstable in higher dimension.
Editorial inference: the fiberwise conditional problem behind Theorem 3.1 could be iterated to express the second derivative of the price bound at $\varepsilon=0$, giving a quadratic approximation for weakly interacting payoffs rather than only the linear one.
Editorial inference: Theorem 4.8 suggests a numerical recipe for discrete problems---solve the dual and read the optimal graph off complementary slackness---which may remain tractable when exact three-period solves are not.
Editorial inference: the gluing lemmas are stated for general $n$, so the zero-interaction limit characterization and its first-order approximation likely extend to four or more periods, although the uniqueness theorems themselves are period-three results.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does real work: the gluing lemmas (2.1, 2.4) are new and correct, the limit/linearization results (2.7, 2.10, 3.1) give a usable first-order approximation of the model-free price bound, and the localized problem in Section 3 is a nice contribution. The numerical section is honest: it compares the approximation against exact LP bounds and the fit is good. Theorem 4.8, for discrete marginals, has a clean duality-based proof and seems solid. There is a lot here worth engaging with.

The soft spots are where the paper makes its biggest claims. Lemma 3.7 is stated for any π in ΠBar but the proof invokes left-monotonicity, which only holds for optimal couplings. That is an overstatement, and the step "there are at most countably many intervals" is not justified; you need a disjointness argument that is not supplied. This gap threatens Theorems 3.6 and 4.2, though it looks patchable.

More serious is Lemma 4.4, which underpins Theorem 4.5. The proof says "very similar arguments to Theorem 3.5 and Theorem 3.6" yield the graph structure for γ = (f,z)#π. That does not follow. The induced cost is c(w,z) = wz^2, whose derivative ∂_w c = z^2 is strictly convex in z, so the left-monotonicity direction of Theorem 3.5 is not available. And the conditional mean of z given w is not a fixed function, unlike in the localized problem. The authors need to supply a real argument here, not a hand-wave. As written, Theorem 4.5's uniqueness claim is not established.

The converse direction of Proposition 2.5 is terse but actually works via the same projection decomposition; that is a minor style complaint, not a real flaw.

Who is this for? People working on multi-period MOT will want the gluing lemmas and the linearization technique, and the paper deserves a serious referee. But the main uniqueness theorems are not ready to be relied upon. I would send it to review with a strong request to fix Lemma 3.7 and either prove Lemma 4.4 properly or soften the claims.

Referee Report

4 major / 4 minor

Summary. The paper develops tools for multi-period martingale optimal transport (MOT), including gluing lemmas for two-period martingale couplings, results on the behavior of optimizers under vanishing cost perturbations, and a first-order approximation of the optimal cost. These tools are applied to three-period MOT problems with costs of the form c1(x,y)+c2(y,z)+c3(x,z). The main theoretical claims are structural and uniqueness results: Theorem 3.1 characterizes limits as the (x,z)-interaction vanishes; Theorems 3.5 and 3.6 give left-monotonicity and uniqueness for a localized conditional MOT problem; Theorems 4.2, 4.5, and 4.8 assert uniqueness of the three-period optimal coupling under different assumptions on the marginals and cost. The paper closes with a numerical application to Amazon option data, comparing first-order approximate bounds with exact linear-programming bounds and tree-based model prices.

Significance. If the main theorems are correct, the paper would provide some of the first structural and uniqueness results for multi-period MOT with costs in which all pairs of variables interact, a genuinely open and important direction. The gluing lemmas (Lemmas 2.1 and 2.4) and the perturbation results (Propositions 2.7 and 2.10) are clean, useful, and likely to be reusable independently. The numerical section is a strength: for small-scale problems the authors solve the full three-period MOT by linear programming and compare exact bounds with their first-order approximations, which gives a concrete check of the approximation scheme. However, the central uniqueness claims rest on two lemmas whose proofs have substantial gaps, so the theoretical contribution is currently only partially validated.

major comments (4)

[Section 3.2, Lemma 3.7] Lemma 3.7 is stated for an arbitrary π ∈ ΠBar(F, σX, σZ), but its proof invokes left-monotonicity, which is an optimality property established only for optimizers in Theorem 3.5. The statement should either assume π is left-monotone or optimal, or the proof should establish left-monotonicity from the stated assumptions. In addition, the step 'there are at most countably many intervals within Z satisfying this' requires a separate argument showing that the intervals (z0, z1) associated with different x can be chosen disjointly; without such an argument the conclusion that only countably many x have three-point conditional support does not follow. Since Theorem 3.6 and Theorem 4.2 rely on Lemma 3.7, this gap is load-bearing.
[Section 4.2, Lemma 4.4] The proof of Lemma 4.4 consists of a single sentence referring to 'very similar arguments' to Theorems 3.5 and 3.6, but this is not a routine adaptation. The induced problem has a nonconstant barycenter function m(w)=E[y | f(x,y)=w], whereas Theorem 3.5 concerns the constant barycenter F(x)=ȳ; the induced cost c(w,z)=w z^2 has ∂_w c = z^2, which is convex in z rather than strictly concave as required by the martingale Spence-Mirrlees condition; and the strict monotonicity and invertibility of T± on W are not proved. Because Theorem 4.5 uses Lemma 4.4 both to obtain the two-graph structure and to identify the conditional weights, the uniqueness claim in Theorem 4.5 is not established by the text.
[Section 4.2, proof of Theorem 4.5, Part 1] In the argument that ν determines πXY, the text asserts that supp(ν) ⊂ [x+y0, x+y2] and then analyzes level sets f(xi,yi)=w, stating that for w > f(x,y1) there is only one such point, (x0,y0). This is only valid for f(x,y)=x+y up to sign, but Assumption A4 only assumes ∂x f, ∂y f < 0. Either the proof silently specializes f, or the displayed interval and level-set analysis contain sign errors. As written, the bootstrap argument for uniqueness of q0, q1, q2 is not justified for the class of f allowed by A4.
[Section 2.1, Proposition 2.5] The converse implication in Proposition 2.5 is asserted without proof; the sentence 'The proof is similar...' does not address it. Unlike Proposition 2.3, one cannot replace a single projection Proj_{1,i} by an optimal two-period coupling while keeping the other projections fixed, because Lemma 2.4 requires the conditional convex-order condition κ^i_1 ⪯ κ^j_1 for the whole collection. A proof or a counterexample is needed before the proposition can be used as a basic tool.

minor comments (4)

[Section 1.1 and Acknowledgments] There are small typos: 'margingale' should be 'martingale' in Section 1.1, and 'and and' appears in the acknowledgments.
[Theorem 3.6] The notation Z = [z−, z−] ∪ [z+, z+] uses the same symbols z− and z+ for endpoints and for the support points in Definition 3.4; this is confusing and should be clarified, for example with underline/overline notation for the interval endpoints.
[Theorem 4.5] The proof refers to 'the three points {y1, y2, y3}', while Assumption A2 defines the support as {y0, y1, y2}; the indexing should be made consistent.
[Section 4.2, Lemma 4.4] The phrase 'very similar arguments to Theorem 3.5 and Theorem 3.6' is imprecise: Theorem 3.6 is a uniqueness statement, not a graph-structure statement; the intended reference appears to be Lemma 3.7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation chain is self-contained and imports its key external tools from outside the literature.

full rationale

The paper's central claims (Theorems 3.6, 4.2, 4.5, 4.8) are derived from explicit assumptions on marginals and costs, using external tools such as Strassen's theorem, Zaev's (c,W)-monotonicity characterization, and classical two-period MOT results from Beiglböck--Juillet and Henry-Labordère--Touzi. The internal propositions (2.3, 2.5, 2.7, 2.10, 3.1, 3.5, 3.6) are proved by direct arguments from the definitions. No parameter is fitted to data and then reported as a prediction; the numerical section compares first-order approximations against exact linear-programming values as independent benchmarks. Some arguments are terse, notably Lemma 4.4's appeal to 'very similar arguments' to Theorems 3.5 and 3.6, and Lemma 3.7 contains a countability claim that may need a separate justification. These are potential proof gaps or correctness risks, not circularity: the conclusions are not assumed in the hypotheses, and the paper does not reduce any central result to a self-citation chain or to a definitional identity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities and fits no free parameters in its derivations. It relies on standard theorems and explicit structural hypotheses on the measures and costs. The main proof gaps are in Lemma 3.7 and Lemma 4.4, which are not additional axioms but unsupported steps.

assumptions (5)

standard math Strassen's theorem: convex order μ_i ⪯c μ_{i+1} implies existence of martingale couplings.
Used to ensure ΠM is nonempty and in the proof of Lemma 2.4.
standard math Zaev's (c,W)-monotonicity characterization (Theorem 3.6 of [23]).
Used in Theorem 3.5 to conclude optimal couplings are (c,W)-monotone.
standard math Duality and existence of an optimal dual solution for multi-period MOT (Theorem 5.2 of [18]).
Used in Lemma 4.6 for complementary slackness.
domain assumption Domain assumptions: compact X,Y,Z ⊂ R, continuous costs, marginals in convex order.
Sets the problem framework for all results.
domain assumption Structural assumptions on marginals and cost: μY with two or three atoms, μX absolutely continuous or discrete, Z = Z− ∪ Z+ two intervals, c3 of martingale Spence-Mirrlees type, or c = f(x,y)z^2 with ∂x f, ∂y f < 0.
These are the explicit hypotheses of Theorems 3.6, 4.2, 4.5, 4.8; they limit the scope of the uniqueness claims.

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Cite this review

Pith. "Pith review of Remarks on multi-period martingale optimal transport." pith.science (2026). https://pith.science/paper/S4DNSMLE

@misc{pith2026250605505,
  author       = {Pith},
  title        = {Pith review of: Remarks on multi-period martingale optimal transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4DNSMLE}},
  note         = {Machine review of arXiv:2506.05505}
}

read the original abstract

We study the structural properties of multi-period martingale optimal transport (MOT). We develop new tools to address these problems, and use them to prove several uniqueness and structural results on three-period martingale optimal transport. More precisely, we establish lemmas on how and when two-period martingale couplings may be glued together to obtain multi-period martingales and which among these glueings are optimal for particular MOT problems. We use these optimality results to study limits of solutions under convergence of the cost function and obtain a corresponding linearization of the optimal cost. We go on to establish a complete characterization of limiting solutions in a three-period problem as the interaction between two of the variables vanishes. Under additional assumptions, we show uniqueness of the solution and a structural result which yields the solution essentially explicitly. For the full three-period problem, we also obtain several structural and uniqueness results under a variety of different assumptions on the marginals and cost function. We illustrate our results with a real world application, providing approximate model independent upper and lower bounds for options depending on Amazon stock prices at three different times. We compare these bounds to prices computed using certain models.

Figures

Figures reproduced from arXiv: 2506.05505 by the authors.

**Figure 1.** Zoomed view around ε = 0 for first-order approximation vs. tree-like method for the third moment of a sum (sum of cross term only). 5.2 Example: Basket of straddle options We consider a basket of forward start straddle options, combining payoffs over all pairwise periods (x, y), (y, z), and (x, z). The cost function is given by c(x, y, z) = c1(x, y) + c2(y, z) + c3(x, z) where c1(x, y) = |y − x|, c2(y, z) = |z − y|,… view at source ↗

**Figure 2.** Zoomed view around ε = 1 for first-order approximation vs. tree-like method for the third moment of a sum (sum of cross term only). via the tree-like method. We observe that, for ε = 1, all three values from the tree-like method with p = 1, 2, 3 lie within the first-order approximation bounds Ql(1) and Qu(1), supporting the idea that the first-order expansion provides a good approximation for the price bound [PITH_… view at source ↗

**Figure 3.** Comparison of first-order approximation and tree-like method for the [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

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Works this paper leans on

23 extracted references · 21 canonical work pages

[1]

Martingale optimal transport in the discrete case via simple linear programming techniques

Nicole B¨ auerle and Daniel Schmithals. Martingale optimal transport in the discrete case via simple linear programming techniques. Mathematical Methods of Operations Research, 90:453–476, 2019

work page 2019
[2]

Model- independent bounds for option prices—a mass transport approach

Mathias Beiglb¨ ock, Pierre Henry-Labordere, and Friedrich Penkner. Model- independent bounds for option prices—a mass transport approach. Finance and Stochastics, 17:477–501, 2013. 24

work page 2013
[3]

On a problem of optimal transport under marginal martingale constraints

Mathias Beiglb¨ ock and Nicolas Juillet. On a problem of optimal transport under marginal martingale constraints. The Annals of Probability, 44(1):42 – 106, 2016

work page 2016
[4]

Prices of state-contingent claims implicit in option prices

Douglas T Breeden and Robert H Litzenberger. Prices of state-contingent claims implicit in option prices. Journal of Business , pages 621–651, 1978

work page 1978
[5]

Instability of martingale optimal transport in dimension d ≥ 2

Martin Br¨ uckerhoff and Nicolas Juillet. Instability of martingale optimal transport in dimension d ≥ 2. Electronic Communications in Probability , 27:1–10, 2022

work page 2022
[6]

The cboe skew index

CBOE. The cboe skew index. White paper, 2011

work page 2011
[7]

Dittmar, and Eric Ghysels

Jennifer Conrad, Robert F. Dittmar, and Eric Ghysels. Ex ante skewness and expected stock returns. The Journal of Finance , 68(1):85–124, 2013

work page 2013
[8]

Exotic options trading

Frans De Weert. Exotic options trading . John Wiley & Sons, 2011

work page 2011

Show all 23 references

[9]

Optimal transport methods in economics

Alfred Galichon. Optimal transport methods in economics . Princeton Uni- versity Press, Princeton, NJ, 2016

2016
[10]

Model-free hedging: A martingale optimal trans- port viewpoint

Pierre Henry-Labord` ere. Model-free hedging: A martingale optimal trans- port viewpoint. CRC Press, 2017

2017
[11]

An explicit martingale version of the one-dimensional Brenier theorem

Pierre Henry-Labord` ere and Nizar Touzi. An explicit martingale version of the one-dimensional Brenier theorem. Finance Stoch., 20(3):635–668, 2016

2016
[12]

An ordinary differ- ential equation for entropic optimal transport and its linearly constrained variants

Joshua Zoen-Git Hiew, Luca Nenna, and Brendan Pass. An ordinary differ- ential equation for entropic optimal transport and its linearly constrained variants. arXiv preprint arXiv:2403.20238 , 2024

2024 arXiv
[13]

Robust price bounds for the forward starting straddle

David Hobson and Martin Klimmek. Robust price bounds for the forward starting straddle. Finance and Stochastics , 19(1):189–214, 2015

2015
[14]

Robust bounds for forward start options

David Hobson and Anthony Neuberger. Robust bounds for forward start options. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics , 22(1):31–56, 2012

2012
[15]

Options, futures, and other derivatives

John C Hull and Sankarshan Basu. Options, futures, and other derivatives . Pearson Education India, 2016

2016
[16]

Recovering probability dis- tributions from option prices

Jens Carsten Jackwerth and Mark Rubinstein. Recovering probability dis- tributions from option prices. The Journal of Finance , 51(5):1611–1631, 1996

1996
[17]

Conditional convex orders and measurable martingale couplings

Leskel¨ a Lasse and Matti Vihola. Conditional convex orders and measurable martingale couplings. Bernoulli, 23(4A):2784 – 2807, 2017

2017
[18]

Multiperiod martingale transport

Marcel Nutz, Florian Stebegg, and Xiaowei Tan. Multiperiod martingale transport. Stochastic Processes and their Applications , 130(3):1568–1615, 2020. 25

2020
[19]

Optimal transport for applied mathematicians

Filippo Santambrogio. Optimal transport for applied mathematicians. Birk¨ auser, NY, 55(58-63):94, 2015

2015
[20]

The existence of probability measures with given marginals

Volker Strassen. The existence of probability measures with given marginals. The Annals of Mathematical Statistics , 36(2):423–439, 1965

1965
[21]

Note on multidimensional breeden– litzenberger representation for state price densities

Jarno Talponen and Lauri Viitasaari. Note on multidimensional breeden– litzenberger representation for state price densities. Mathematics and Fi- nancial Economics, 8:153–157, 2014

2014
[22]

Springer, 2009

C´ edric Villani et al.Optimal transport: old and new , volume 338. Springer, 2009

2009
[23]

On the Monge–Kantorovich problem with additional linear constraints

Danila A Zaev. On the Monge–Kantorovich problem with additional linear constraints. Mathematical Notes, 98:725–741, 2015. 26

2015

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Martingale optimal transport in the discrete case via simple linear programming techniques

Nicole B¨ auerle and Daniel Schmithals. Martingale optimal transport in the discrete case via simple linear programming techniques. Mathematical Methods of Operations Research, 90:453–476, 2019

work page 2019

[2] [2]

Model- independent bounds for option prices—a mass transport approach

Mathias Beiglb¨ ock, Pierre Henry-Labordere, and Friedrich Penkner. Model- independent bounds for option prices—a mass transport approach. Finance and Stochastics, 17:477–501, 2013. 24

work page 2013

[3] [3]

On a problem of optimal transport under marginal martingale constraints

Mathias Beiglb¨ ock and Nicolas Juillet. On a problem of optimal transport under marginal martingale constraints. The Annals of Probability, 44(1):42 – 106, 2016

work page 2016

[4] [4]

Prices of state-contingent claims implicit in option prices

Douglas T Breeden and Robert H Litzenberger. Prices of state-contingent claims implicit in option prices. Journal of Business , pages 621–651, 1978

work page 1978

[5] [5]

Instability of martingale optimal transport in dimension d ≥ 2

Martin Br¨ uckerhoff and Nicolas Juillet. Instability of martingale optimal transport in dimension d ≥ 2. Electronic Communications in Probability , 27:1–10, 2022

work page 2022

[6] [6]

The cboe skew index

CBOE. The cboe skew index. White paper, 2011

work page 2011

[7] [7]

Dittmar, and Eric Ghysels

Jennifer Conrad, Robert F. Dittmar, and Eric Ghysels. Ex ante skewness and expected stock returns. The Journal of Finance , 68(1):85–124, 2013

work page 2013

[8] [8]

Exotic options trading

Frans De Weert. Exotic options trading . John Wiley & Sons, 2011

work page 2011