A recent study shows that American‑style options with irregular payoff patterns can be priced precisely using a universal super‑hedging formula.

The authors prove that any strong optional supermartingale under all such martingale measures can be broken down explicitly: it equals its initial value plus a stochastic integral with a fixed integrand plus two non‑decreasing parts—one continuous and one purely discontinuous that captures the right jumps of the process. Importantly, this decomposition works uniformly for all probability measures and does not require any smoothness of the payoff.

The result relies on a uniform optional decomposition theorem: given a supermartingale as above, there exist a predictable integrand, a right‑continuous non‑decreasing process, and a purely discontinuous non‑decreasing process that together reconstruct the supermartingale. The stochastic integral part is uniformly integrable under every martingale measure used in the pricing.

However, the decomposition only holds when the full set of equivalent local martingale measures is considered. If a restricted family of measures is used, the integral part may fail to be representable as a stochastic integral of the underlying asset, and the neat decomposition can break down.