Sampling of One-Dimensional Probability Measures in the Convex Order and Computation of Robust Option Price Bounds

Alfonsi, Aurélien; Corbetta, Jacopo; Jourdain, Benjamin

doi:10.1142/s021902491950002x

Cited by 38 publications

(38 citation statements)

References 15 publications

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“…Acciaio et al [1] consider an object related to the adapted Wasserstein distance in continuous time in connection with utility maximisation, enlargement of filtrations and optimal stopping. Glanzer et al [26] prove a deviation inequality for the so-called nested distance in a discretetime framework, 3 and consider acceptability pricing over an ambiguity set described through the nested distance. Bion-Nadal and Talay [18] study via PDE arguments a continuous-time optimisation problem which is related to the adapted Wasserstein distance.…”

Section: Literaturementioning

confidence: 99%

“…We fix ε > 0 and assume π is bi-causal ε-optimal for AW p (P, Q) andπ is bi-causal ε-optimal for AW p (Q, R). In the next couple of lines, ω always denotes the first coordinate of a vector in 3 , η the second and γ the last. Let…”

Section: Proof Of Lemma 32 It Is Clear Thatmentioning

confidence: 99%

“…Note that d w p is not symmetric and that as a consequence of Jensen's inequality, we always have d w p ≤ d p . Problems akin to d w p (μ, ν) go under the name of "weak optimal transport" and have been recently introduced by Gozlan et al in [28], but see also Alfonsi et al [3], Alibert et al [4], Backhoff-Veraguas et al [11,9] and Gozlan et al [27]. We have Theorem 3.10 Let P, Q ∈ SM p ( ), let π be a bi-causal coupling between P and Q, let C : → R be Lipschitz with constant L, and let H ∈ H k .…”

Section: Stochastic Integrals and A Contraction Principlementioning

confidence: 99%

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Adapted Wasserstein distances and stability in mathematical finance

et al. 2020

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Assume that an agent models a financial asset through a measure Q with the goal to price/hedge some derivative or optimise some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, the agent is left with the possibility that Q does not provide an exact description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is No. Models which are similar with respect to the Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler (

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Section: Literaturementioning

confidence: 99%

Section: Proof Of Lemma 32 It Is Clear Thatmentioning

confidence: 99%

Section: Stochastic Integrals and A Contraction Principlementioning

confidence: 99%

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Adapted Wasserstein distances and stability in mathematical finance

et al. 2020

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“…Since its introduction in Beiglböck et al (2013), this MOT problem has received recently a great attention in the financial mathematics literature. In particular, the structure of martingale optimal transport couplings has been investigated by Beiglböck and Juillet (2016); Campi et al (2017); De March and Touzi (2019); Ghoussoub et al (2019); Henry-Labordère and Touzi (2016), continuous time formulations by Dolinsky and Soner (2014); Galichon et al (2014); , links with the Skorokhod embedding problem by Beiglböck et al (2017), numerical methods by Alfonsi et al (2020Alfonsi et al ( , 2019; De March (2018); Guo et al (2019); Henry-Labordère (2019) and stability properties by Backhoff-Veraguas and Pammer (2019); Jourdain and Margheriti (2020); Wiesel (2012). The MOT problem is a particular instance where the measurable cost function C : R d × P 1 (R d ) → R is linear in the measure component (C(x, η) = R d c(x, y)η(dy)) of the Weak Martingale Optimal Transport problem inf π∈M(µ,ν) R d C(x, π x )µ(dx).…”

Section: Introductionmentioning

confidence: 99%

“…In two recent papers, Alfonsi et al (2019Alfonsi et al ( , 2020 propose to restore for (µ, ν) ∈ P ≤ × P 1 (R d ) the convex ordering from any finitely supported approximations μ and ν of µ and ν. In dimension d = 1, one may define the increasing (resp.…”

Section: Introductionmentioning

confidence: 99%

Quantization and martingale couplings

Jourdain¹,

Pagès²

2022

ALEA

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Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its quadratic optimal) primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to ∞.

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Approximation of martingale couplings on the line in the adapted weak topology

Beiglböck

Jourdain

Margheriti

et al. 2022

Probab. Theory Relat. Fields

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Our main result is to establish stability of martingale couplings: suppose that $$\pi $$ π is a martingale coupling with marginals $$\mu , \nu $$ μ , ν . Then, given approximating marginal measures $$\tilde{\mu }\approx \mu , \tilde{\nu }\approx \nu $$ μ ~ ≈ μ , ν ~ ≈ ν in convex order, we show that there exists an approximating martingale coupling $$\tilde{\pi }\approx \pi $$ π ~ ≈ π with marginals $$\tilde{\mu }, \tilde{\nu }$$ μ ~ , ν ~ . In mathematical finance, prices of European call/put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call/put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.

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Sampling of One-Dimensional Probability Measures in the Convex Order and Computation of Robust Option Price Bounds

Cited by 38 publications

References 15 publications

Adapted Wasserstein distances and stability in mathematical finance

Adapted Wasserstein distances and stability in mathematical finance

Quantization and martingale couplings

Approximation of martingale couplings on the line in the adapted weak topology

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