Sharp asymptotics for a free boundary touching a degenerate boundary: the perpetual Heston put
Abstract
We study the exercise boundary of the perpetual American put in the Heston model near zero variance. There the pricing operator degenerates, and the exercise boundary, a free boundary in the stock--variance half-plane, touches the boundary of the half-plane. No Feller condition is imposed. The boundary is analytic at positive variance and extends as a regular C1 curve tangent to the zero-variance axis. Its approach to the axis follows one of three laws, according to whether the Feller ratio 2κθ/σ2 is larger than, equal to, or smaller than one: a logarithmic correction to linear order, an exponential correction in the square root of the logarithm, or a fractional power. In the first two cases the leading coefficients are explicit and the tangent is not Hölder continuous at the axis. In the third case the exponent is selected by a possibly nonsymmetric stable operator on the axis, and the amplitude is finite and positive.
Related
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
- Stabilization of a degenerate beam on an elastic foundation with axial force under fractional derivative-type delayed boundary feedback
- On the Large Λ Asymptotics of the One Phase Bernoulli Free Boundary Problem
- Long-time asymptotics of nonlinear Fokker-Planck equations with inhomogeneous diffusion and natural boundary conditions
- Asymptotics of Laplace and Stokes solutions in the gap between a sharp tip and a smooth boundary
BibTeX
@misc{kim2026sharp,
title = {{Sharp asymptotics for a free boundary touching a degenerate boundary: the perpetual Heston put}},
author = {Takwon Kim and Junkee Jeon and Jehan Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29253},
eprint = {2610.29253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29253},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp asymptotics for a free boundary touching a degenerate boundary: the perpetual Heston put
AU - Takwon Kim
AU - Junkee Jeon
AU - Jehan Oh
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29253
UR - https://arxiv.org/abs/2610.29253
AB - We study the exercise boundary of the perpetual American put in the Heston model near zero variance. There the pricing operator degenerates, and the exercise boundary, a free boundary in the stock--variance half-plane, touches the boundary of the half-plane. No Feller condition is imposed. The boundary is analytic at positive variance and extends as a regular C^1 curve tangent to the zero-variance axis. Its approach to the axis follows one of three laws, according to whether the Feller ratio 2κθ/σ^2 is larger than, equal to, or smaller than one: a logarithmic correction to linear order, an exponential correction in the square root of the logarithm, or a fractional power. In the first two cases the leading coefficients are explicit and the tangent is not Hölder continuous at the axis. In the third case the exponent is selected by a possibly nonsymmetric stable operator on the axis, and the amplitude is finite and positive.
ER -