Martingale Constraints for Payoffs on Different Assets

Summary

This question examines how martingale optimal transport should be formulated when a payoff depends on two different assets, such as an equity and a foreign exchange rate. In the usual setup described, two time points belong to the same asset, and the conditional martingale constraint links its later value to its earlier value. Applying that constraint directly between different assets would not express the usual no-drift condition for either asset.

The author asks whether each asset should instead be represented by its own initial and later values, with a separate martingale condition, and whether the chosen initial values matter. The document poses this modeling issue but supplies no answer, derivation, or empirical evidence. It therefore identifies a useful distinction between cross-asset dependence and each asset’s own time evolution, while leaving the correct transport formulation and the role of initial values unresolved.

Key ideas

• A martingale constraint between two observations of one asset differs from a constraint between two distinct assets.
• The question proposes imposing a separate time-based martingale condition for each asset.
• The document does not resolve how to formulate the multi-asset transport problem or choose initial values.

Tags

# Martingale optimal transport with two different nature of assets


# Martingale optimal transport with two different nature of assets












In most of the litterature , for solving the optimal transport problem $sup_{Q\in \mathcal{M}}E^{Q}[c(S_{1},S_{2})]$ where $\mathcal{M}$ is the set of probability coupling such that the marginals of Q verify $Q^{1} \sim S_{1}$ and $Q^{2} \sim S_{2}$ and the martingale condition $E^{P}[S_{2}|S_{1}] = S_{1}$ which means we consider the asset S at two different times as $S_{1}$ and $S_{2}$

My question raises when it is not the same asset , like a payoff on a equity $S_{1}$ and FX $S_{2}$ , it would have no sens to consider the martingale condition defined as previously , we would rather want that each asset verifies it. How do we account for this ? Do we introduce artificially $S^{1}$ as $(S^{1}_{1} , S^{1}_{0})$ and $S^{2}$ as $(S^{2}_{2},S^{2}_{0})$ and say that $E^[S^{1}_{1}|S^{1}_{0}]=S^{1}_{0}$ ? How would the choice of $S_{0}$ be relevant ?

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