The Projected-Generator Method for Path-Dependent Derivative Pricing

The projected-generator method is a deterministic, simulation-free engine for pricing path-dependent derivatives — American and Bermudan exercise, barriers, autocallables, coupon memory, and structured products. Instead of Monte Carlo paths, regression continuation estimates, or high-dimensional PDE grids, it projects the model's infinitesimal generator onto a finite basis and propagates continuation values by the resulting matrix semigroup; path dependence is carried by finite event-state transitions around the same propagation primitive. The same finite expansion yields Greeks without bump-and-reprice, and the projection residual gives an explicit error budget: pricing is exact when the finite space is invariant under the generator, and for analytic payoffs under spectral bases the error decays exponentially. The closure and error-certificate results are formally verified in Lean 4 (with Python verification scripts). Key results: 7 theorems / propositions. Maturity: Draft. Target venue: Mathematical Finance / SIAM Journal on Financial Mathematics. Part of The Latent research program. The method is strongest for low-to-moderate-dimensional models with a tractable generator; high-dimensional baskets, rough volatility without a controlled lift, and severe discontinuities remain better served by Monte Carlo, finite differences, or adapted bases.