Interior smoothness and zero-variance geometry for finite-maturity Heston American puts

Paper: arXiv 2610.20000

Authors: Junkee Jeon, Takwon Kim, Jehan Oh

Abstract

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on rigorous mathematical proofs for the properties of American options in the Heston model. Its novelty lies in extending the understanding of boundary behavior, especially at zero variance, without the Feller condition. However, it lacks empirical validation or practical implementation details.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze American Puts in Heston Model] --> B{Key Methodology: Analytical & PDE Analysis};
    B --> C[Inputs: Finite-Maturity Heston Model Parameters, No Feller Condition];
    C --> D{Computational Process: Derivation of Boundary Properties & PDEs};
    D --> E[Outcome 1: Boundary Smoothness & Behavior at Positive/Zero Variance];
    E --> F[Outcome 2: Smooth Fit Failure in Variance Direction];
    F --> G[Outcome 3: Quantitative Boundary Gap Bounds & Vertical Identities];