[Paper Review] Optimal Skorokhod embedding given full marginals and Azema-Yor peacocks

Sigrid Källblad, Xiaolu Tan+1 more|arXiv (Cornell University)|Mar 2, 2015

Markov Chains and Monte Carlo Methods18 references3 citations

TL;DR

This paper establishes a general duality result for the optimal Skorokhod embedding problem under full marginal constraints, extending prior results from finite to continuous-time marginals. It characterizes optimal embeddings and their duals when the reward depends on the maximum of the martingale, using limiting arguments from the Azéma-Yor embedding and martingale transport theory, yielding explicit solutions for extremal peacock processes.

ABSTRACT

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock ("process increasing in convex order", by Hirsch, Profeta, Roynette and Yor). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordere, Obloj, Spoida and Touzi. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.

Motivation & Objective

• To extend duality results from finite-marginal optimal Skorokhod embedding to the full-marginal case over [0,1].
• To characterize optimal martingale peacocks associated with a given peacock process under maximum-dependent reward functions.
• To derive explicit representations of the optimal primal and dual solutions using limiting arguments from the iterated Azéma-Yor embedding.
• To establish model-independent bounds on exotic option prices via extremal martingales with given marginals.
• To generalize the martingale transport approach to continuous-time marginal constraints and maximum-based rewards.

Proposed method

• Derives a general duality result for optimal Skorokhod embedding under full marginals by taking limits of duality results from Guo, Tan, and Touzi (2015) for finite marginals.
• Applies convergence techniques to extend the iterated Azéma-Yor embedding from finite to continuous-time marginal constraints.
• Uses the martingale Spence-Mirrlees condition and limiting arguments to characterize the optimal value and optimizers.
• Relies on Kellerer's theorem to guarantee existence of martingales with given marginals and Monroe's embedding theorem to ensure existence of stopping times.
• Constructs the primal and dual optimizers via the limit of solutions to finite-marginal problems, leveraging right-continuity and uniform integrability.
• Employs the canonical space with random time changes and proves measurability of key functionals using Lévy metric and Borel sigma-fields on path spaces.

Experimental results

Research questions

• RQ1 How can the duality framework for optimal Skorokhod embedding be extended from finitely many marginals to full continuous-time marginal constraints on [0,1]?
• RQ2 What is the explicit form of the optimal embedding when the reward depends on the maximum of the martingale process?
• RQ3 How do the primal and dual optimizers for the full-marginal problem relate to the iterated Azéma-Yor embedding in the finite-marginal case?
• RQ4 What are the model-independent bounds on exotic option prices derived from extremal peacock processes with given marginals?
• RQ5 How can the convergence of finite-marginal solutions to the full-marginal solution be rigorously justified under technical conditions?

Key findings

• A general duality result for the optimal Skorokhod embedding problem with full marginals is established via convergence of finite-marginal duality results.
• The optimal solution for maximum-dependent rewards is characterized as the limit of iterated Azéma-Yor embeddings from the finite-marginal case.
• The optimal value and primal/dual optimizers are explicitly described in terms of the cumulative distribution and maximum process of the peacock.
• The convergence of the finite-marginal solutions to the full-marginal solution is proven using dominated convergence and the Lévy metric on path spaces.
• The dual problem's solution is shown to be related to the convex minorant of the reward function, extending known results from the finite-marginal setting.
• The paper confirms that the optimal martingale is a peacock and provides a constructive method for its pathwise realization via time-changed Brownian motion.