Sharp asymptotics for a free boundary touching a degenerate boundary: the perpetual Heston put

Source 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 $C^1$ 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.