Tightness and duality of martingale transport on the Skorokhod space

Abstract

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuoustime martingale transport on the Skorokhod space of càdlàg paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 . 1 During the preparation of the final version of this paper, we knew from Mete Soner about the latest version of Dolinsky & Soner [24] which includes the finitely-many marginal constraints setting.where (X, Y ) denotes the canonical process on R d × R d , i.e. X(x, y) = x and Y (x, y) = y for all (x, y) ∈ R d × R d . Then the optimal transport problem consists in optimizing the expectation of some measurable function ξ : R d × R d → R among all probability measures in P(µ, ν). Various related issues are studied, e.g. the general duality theory and optimality results, we refer to Rachev & Rüschendorf [53] and Villani [55] for a comprehensive account of the literature.Recently, a martingale optimal transport problem was introduced in Beiglböck, Henry-Labordère & Penkner [5] in discrete-time (see Galichon, for the continuous-time case), where a maximization problem is considered over a subset M(µ, ν) := P ∈ P(µ, ν) : E P [Y |X] = X, P-a.s. :Each element of M(µ, ν) is called a transport plan. Similarly to the classical setting, the corresponding dual problem is defined byThe last dual formulation has the interpretation of minimal robust superhedging cost of derivative security defined by the payoff ξ by trading the underlying security and any possible Vanilla option. When d = 1, as observed by Breeden & Litzenberger [11], the marginal distributions of the underlying asset are recovered by the market prices of calls for all strikes, and any Vanilla option has a non-ambiguous price as the integral of its payoff function with respect to the marginal. Therefore, the inequality (1.2) represents a super-replication of ξ, which consist of the trading of the underlying and Vanilla options at different maturities. Since there is no specific model imposed on the process (X, Y ), the dual problem may be interpreted as the robust superhedging cost, i.e. the minimum cost to construct super-replications. Similar to the classical setting, the duality P(µ, ν) = D(µ, ν) holds under quite general conditions. The present paper considers the continuous-time martingale optimal transport problem. Let X := ω = (ω t ) 0≤t≤1 : ω t ∈ R d for all t ∈ [0, 1] , where X is either the space of continuous functions or the Skorokhod space of càdlàg functions. Denote by X = (X t ) 0≤t≤1 the canonical process and by M the set of all martingale measures P, i.e. X is a martingale under P. For a given family of probability measures µ = (µ t ) t∈T , where T...

Robust pricing–hedging dualities in continuous time

“…We explicitly specify several important special cases, including the setting when finitely many options are traded, some dynamically and some statically, and the setting when all European call options for n maturities are traded. The latter gives the martingale optimal transport (MOT) duality with n marginal constraints which was also recently studied in a discontinuous setup by Dolinsky and Soner [26] and, in parallel to our work, by Guo et al [31].…”

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

“…Moreover, our results consider the multiple marginals case, such an extension of their technique seems not obvious, see also the work of Hou & Ob lój [37] and Biagini, Bouchard, Kardaras & Nutz [6]. More recently, an analogous duality is proved in the Skorokhod space under suitable conditions in Dolinsky & Soner [22], where the underlying asset is assumed to take values in some subspace of càdlàg functions (see also [29]). …”

The geometry of multi-marginal Skorokhod Embedding

“…While we do not pursue this direction of research in this article we emphasize that our approach yields a systematic method to address this problem. In particular, the general framework of super-replication results for model-independent framework now includes a number of important contributions, see [20,26,3,38], and most of these papers allow for information at multiple intermediate times.…”