On regularity of finite-maturity American put options in the Heston model

Battle Count: 95

This paper provides foundational theoretical results for American option pricing under stochastic volatility. While not directly a trading strategy paper, the regularity results are essential prerequisites for: (1) rigorous derivation of early-exercise premium formulas, (2) justification of numerical pricing algorithms used in practice, (3) proper characterization of optimal exercise boundaries for American-style derivatives. The Heston model is widely used in quantitative finance for options with stochastic volatility, making these results practically relevant for derivatives desks and risk management.

This is a highly theoretical paper requiring deep expertise in degenerate parabolic PDE theory, viscosity solutions, Sobolev spaces, and stochastic analysis. The proof techniques involve: penalty method construction, covering arguments for interior estimates, Arzela-Ascoli compactness, Sobolev embedding theorems, and comparison principles for viscosity solutions. There is no code or numerical implementation. Understanding and verifying the proofs requires graduate-level knowledge in PDEs and probability theory.

This is a purely theoretical mathematics paper with complete proofs. All results are self-contained with detailed derivations. The proofs can be verified by following the logical chain: viscosity solution uniqueness (Prop 4.3-4.5), penalized PDE existence (Prop 5.7), uniform bounds (Prop 5.9), Sobolev estimates (Prop 5.10, 5.13), and smooth-fit derivation (Prop 5.15-5.16). No numerical experiments are needed for verification.