This paper introduces an HJM-style forward modeling framework for valuing American options. Instead of modeling the dynamics of the underlying asset, we model the maturity-indexed forward drift of the gain process, leading to two no-arbitrage representations of the option value. The first is an additive model, where the American option price equals the current gain plus an integral of forward drifts. This representation embeds the early-exercise premium directly and yields a forward drift characterization of the optimal stopping rule. The second is a multiplicative model that provides an arbitrage-free term structure of option values across maturities via a forward rate, in the spirit of the HJM interest rate theory. While it does not determine the early exercise boundary, it is useful for modeling European option price curves and their evolution. We develop the corresponding drift restrictions, spot consistency conditions, and valuation formulas for both representations and provide numerical examples.
Fernando et al. (Tue,) studied this question.