Dual Attainment in Multi-Period Multi-Asset Martingale Optimal Transport and Its Computation

We establish dual attainment for the multimarginal, multi-asset martingale optimal transport (MOT) problem, a fundamental question in the mathematical theory of model-independent pricing and hedging in quantitative finance. Our main result proves the existence of dual optimizers under mild regularity and irreducibility conditions, extending previous duality and attainment results from the classical and two-marginal settings to arbitrary numbers of assets and time periods. This theoretical advance provides a rigorous foundation for robust pricing and hedging of complex, path-dependent financial derivatives. To support our analysis, we present numerical experiments that demonstrate the practical solvability of large-scale discrete MOT problems using the state-of-the-art primal-dual linear programming (PDLP) algorithm. In particular, we solve multi-dimensional (or vectorial) MOT instances arising from the robust pricing of worst-of autocallable options, confirming the accuracy and feasibility of our theoretical results. Our work advances the mathematical understanding of MOT and highlights its relevance for robust financial engineering in high-dimensional and model-uncertain environments.

💡 Research Summary

This paper addresses a fundamental problem in model‑independent finance: establishing the existence of dual optimizers for the multi‑period, multi‑asset martingale optimal transport (MOT) problem, and demonstrating that such problems can be solved efficiently in practice. The authors consider a market with d underlying assets observed at N discrete maturities. For each asset i and time t, the marginal distribution μ_{t,i} is assumed known from market data, and the sequence of marginals satisfies the convex order condition μ_{t,i} ≼c μ{t+1,i}, which guarantees the existence of a martingale coupling (Strassen’s theorem). The set VMT(μ) consists of all probability measures on ℝ^{N·d} whose one‑step conditional expectations reproduce the martingale property and whose one‑dimensional marginals match the observed μ_{t,i}.

The primal MOT problem (VMOT) is to minimize (or maximize) the expected value of a measurable payoff c(X₁,…,X_N) over π∈VMT(μ). This yields the tightest arbitrage‑free lower and upper price bounds for a path‑dependent derivative whose payoff is c. The dual formulation involves a static component ϕ = (ϕ_{t,i}) representing European options on each asset at each date, and a dynamic trading strategy h = (h_{t,i}) that trades the underlying assets between dates. Feasibility of (ϕ,h) requires the pathwise inequality

 ∑{t,i} ϕ{t,i}(x_{t,i}) + ∑{t,i} h{t,i}(x_{1},…,x_{t})·(x_{t+1,i}−x_{t,i}) ≤ c(x)

for all paths x∈ℝ^{N·d}. The dual value μ(ϕ) = ∑{t,i}∫ϕ{t,i} dμ_{t,i} is the cost of the semi‑static hedge. Classical results guarantee strong duality (P(c)=D(c)) under mild regularity, but the existence of an optimal dual pair (ϕ*,h*)—dual attainment—has been proved only for very restricted settings (single asset, two periods).

The main theoretical contribution is Theorem 3.1, which proves dual attainment for arbitrary d and N under two key assumptions: (i) each marginal has a finite first moment, and (ii) the family of marginals is “irreducible,” meaning that the supports of successive marginals overlap in a way that prevents mass from being trapped at the boundary. Irreducibility yields uniform L¹‑bounds for candidate dual functions and ensures tightness of the associated measures. The proof proceeds by first obtaining a primal optimizer π* (compactness of VMT(μ) via Prokhorov’s theorem), then constructing a sequence of admissible dual pairs (ϕ^k,h^k) that approximate the primal value from below. Using Banach–Alaoglu and measurable selection arguments, the authors extract a weak‑* convergent subsequence whose limit (ϕ̂,ĥ) satisfies the feasibility inequality everywhere, thanks to the irreducibility condition. Finally, they verify μ(ϕ̂)=P(c), establishing that (ϕ̂,ĥ) is an optimal dual solution. This result extends dual attainment from the classical two‑period, single‑asset case to the fully vectorial, multi‑period setting, providing a rigorous foundation for robust hedging strategies in high‑dimensional markets.

On the computational side, the authors discretize the continuous VMOT problem on a finite grid, turning it into a large linear program with variables representing the transport plan and constraints encoding marginal and martingale conditions. They solve these instances with the Primal‑Dual Linear Programming (PDLP) algorithm, a state‑of‑the‑art interior‑point method designed for massive LPs. Numerical experiments include a worst‑of auto‑callable option, whose payoff depends on the minimum of several assets over multiple observation dates. The experiments demonstrate that PDLP can handle problems with millions of variables and constraints, achieving convergence within reasonable time and memory limits. Moreover, the computed dual variables correspond to explicit static option positions and dynamic trading rules that replicate the optimal sub‑ or super‑hedge, confirming the practical relevance of the dual attainment theory.

In summary, the paper makes three substantial contributions: (1) it proves the existence of dual optimizers for the general multi‑asset, multi‑period MOT problem under realistic market assumptions; (2) it introduces the irreducibility condition to relax the stringent regularity requirements previously needed for dual attainment; and (3) it validates the theoretical findings with large‑scale numerical experiments using PDLP, illustrating that robust price bounds and optimal hedging strategies can be computed for complex, path‑dependent derivatives. These results bridge a critical gap between the abstract theory of martingale optimal transport and its concrete application in quantitative finance.