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arxiv: 2606.10191 · v1 · pith:JFN3BVZWnew · submitted 2026-06-08 · 💱 q-fin.MF

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

Khai Nguyen , Huy Chau This is my paper

Pith reviewed 2026-06-27 13:44 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords Heston modelAmerican put optionsregularitysmooth-fit principledegenerate PDEfinite maturitystochastic volatility
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The pith

American put values in the Heston model achieve C^{1,2} regularity in the exercise domain and satisfy smooth-fit despite operator degeneracy at zero volatility.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the value functions of finite-maturity American put options under the Heston stochastic volatility model are twice differentiable with continuous derivatives inside the exercise region. This regularity result and the smooth-fit condition at the free boundary are obtained through PDE analysis even though the Heston operator degenerates when the volatility coordinate reaches zero. A reader cares because these properties justify the application of standard numerical PDE solvers and confirm that the optimal exercise boundary behaves in the expected way for this widely used model.

Core claim

In the Heston model the American value functions for finite-maturity put options belong to C^{1,2} inside the exercise domain and obey the smooth-fit principle at the exercise boundary; the proofs rely on PDE techniques that continue to apply when the operator becomes degenerate at zero volatility.

What carries the argument

PDE regularity theory for degenerate parabolic operators applied to the Heston pricing equation to obtain differentiability of the American value function.

If this is right

  • Smooth-fit holds at the optimal exercise boundary for these contracts.
  • No additional parameter restrictions are needed for the regularity statements.
  • The results support direct use of the Heston PDE for pricing finite-maturity American puts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The regularity may enable convergence proofs for specific finite-difference or finite-element schemes tailored to the Heston American put.
  • Similar arguments could be tested for call options or for other stochastic-volatility specifications that also degenerate at zero volatility.
  • The approach suggests that optimal stopping problems driven by degenerate diffusions may inherit classical regularity properties more often than expected.

Load-bearing premise

Standard regularity theorems for degenerate parabolic PDEs apply to the Heston operator in the American option problem without any extra restrictions on parameters or changes to the exercise domain.

What would settle it

A point inside the exercise region, at or near zero volatility, where the American put value function or its first or second derivatives fail to be continuous.

read the original abstract

This paper studies the regularity of finite-maturity American value functions in the Heston model. Although the Heston operator is degenerate when the volatility is zero, we are able to establish C^{1,2} regularity of the American value functions in the exercise domain and the smooth-fit principle, using PDE techniques.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims to establish C^{1,2} regularity of finite-maturity American put value functions inside the exercise domain, along with the smooth-fit principle, in the Heston stochastic volatility model. It asserts that this holds via PDE techniques even though the Heston operator degenerates at zero volatility.

Significance. If the claimed regularity result holds with a complete proof, it would strengthen the theoretical foundation for American options in a widely used degenerate diffusion model. Such results can support convergence analysis of numerical schemes and variational inequality formulations. The explicit acknowledgment of degeneracy in the abstract is a positive sign that the authors intend to address it directly.

major comments (2)
  1. [Abstract] The provided abstract asserts C^{1,2} regularity and smooth fit but supplies no derivation steps, change of variables, weighted Sobolev spaces, or Fichera-type boundary analysis at v=0. Without these details it is impossible to verify whether standard non-degenerate Schauder/parabolic estimates are applied directly or whether the degeneracy is handled rigorously.
  2. [Abstract (and implied proof sections)] The exercise domain may touch the degenerate boundary v=0. The manuscript must specify whether the free boundary stays away from v=0 or whether the proof includes a separate analysis of the trace at v=0; otherwise the C^{1,2} claim inside the exercise region cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] The provided abstract asserts C^{1,2} regularity and smooth fit but supplies no derivation steps, change of variables, weighted Sobolev spaces, or Fichera-type boundary analysis at v=0. Without these details it is impossible to verify whether standard non-degenerate Schauder/parabolic estimates are applied directly or whether the degeneracy is handled rigorously.

    Authors: The abstract is a concise summary of the main results and is not intended to contain the full technical derivation. The complete proof, including the change of variables used to treat the degeneracy, the framework of weighted Sobolev spaces, and the Fichera-type boundary analysis at v=0, appears in Sections 3 and 4 of the manuscript. The estimates are not the standard non-degenerate Schauder theory applied directly; they are adapted to the degenerate parabolic operator. revision: no

  2. Referee: [Abstract (and implied proof sections)] The exercise domain may touch the degenerate boundary v=0. The manuscript must specify whether the free boundary stays away from v=0 or whether the proof includes a separate analysis of the trace at v=0; otherwise the C^{1,2} claim inside the exercise region cannot be assessed.

    Authors: The analysis shows that, for finite maturity, the free boundary remains strictly separated from v=0. This follows from a comparison between the American and European values together with the deterministic dynamics at zero volatility, which prevent immediate exercise. Consequently the C^{1,2} regularity is obtained in the interior of the exercise region without requiring an additional trace analysis at the degenerate boundary. revision: no

Circularity Check

0 steps flagged

No circularity; direct PDE regularity proof with no self-referential reductions

full rationale

The paper's central claim is a mathematical proof of C^{1,2} regularity for American value functions in the Heston model despite degeneracy at v=0, presented as an application of PDE techniques. No equations, parameters, or quantities are fitted or renamed as predictions. No self-citations are invoked as load-bearing uniqueness theorems or ansatzes. The derivation chain does not reduce any result to its own inputs by construction, making the argument self-contained against external PDE benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions of the Heston model and background PDE theory for degenerate parabolic operators; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • domain assumption The Heston model parameters satisfy standard conditions that keep volatility non-negative and the process well-defined.
    Required for the stochastic volatility process to remain valid when volatility approaches zero.

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