Finite-Rank Overshoot and Mellin–Volterra Pricing of American Options with Jumps

This paper revisits a Mellin-transform approach to American option pricing under exponential Lévy dynamics originally developed in 2005. We first identify a nonlocal boundary-crossing term omitted from the original finite-maturity exercise source. For a general two-sided Lévy process, positive jumps from the stopping region generate an overshoot operator depending on the entire continuation-value function. We then show that, under a hyper-exponential positive jump law with m exponential components, this operator has rank at most m. The continuation dependence is summarized exactly by m weighted continuation-gap moments, and the finite-maturity American put problem reduces to a closed (m+ 1)-dimensional nonlinear Volterra system for the exercise boundary and these moments. Kou's double-exponential model is the rank-one case. A verification result links the Volterra representation to the original stopping problem. We further clarify what multivariate Mellin transforms achieve for American basket puts and what remains genuinely multidimensional. A direct rank-one Mellin–Volterra solver is then benchmarked against an independent highresolution PIDE. A systematic convergence study in space and time shows close agreement for prices, the free boundary and the overshoot moment, while documenting the associated wall-clock response times. A four-level Nτ = 100, 200, 400, 800 Volterra refinement and generalized Richardson extrapolation provide a method-internal asymptotic benchmark for the exercise boundary. In the Kou benchmark, the jump-overshoot correction represents more than twenty percent of the local exercise source near the boundary. A stylized desk-level jump-risk stress test further shows how the representation can be used for model validation and model-risk control: under a combined two-sided jump-regime stress the at-the-money value rises by about 9.6%, the exercise boundary moves by about 1.54 stock-price units, and the overshoot share of the local exercise source increases from about 20.9% to 41.8%, while the independent PIDE and Mellin–Volterra engines remain close. The paper thereby separates the historical Mellin contribution from the present correction, the new finite-rank structural result, and its computational and risk-management consequences.