American options under tempered space-fractional diffusion: Riesz scaling, risk-neutral pricing, and exercise boundary analysis
Résumé
Pure stable log-price models with stability index γ < 2 have no finite positive exponential moment and therefore cannot be used directly as risk-neutral exponential stock models. This paper reformulates the space-fractional American put problem under a symmetric tempered-stable log-price process, whose finite moment generating function gives an explicit martingale drift correction. The Riesz operator is retained as the short-maturity principal part of the tempered generator. After removing drift and discounting, this principal equation admits the Riesz heat scaling x ~τ 1/γ , which is broken globally by tempering and by the American obstacle but remains a useful asymptotic guide near expiry. We state the resulting exercise-boundary scale as a conjectural ansatz with a slowly varying factor, rather than as a proved expansion, and assess it against a Fourier-collocation projected successive overrelaxation benchmark for the tempered obstacle problem. The benchmark indicates small absolute pricing errors near the exercise and at-the-money regions, while also showing the boundary displacement caused by omitting the slowly varying logarithmic correction.
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