Closed-form expression for the pinch-off time t* in GBM double-boundary American options

Derive a closed-form expression for the pinch-off time t* at which the lower and upper exercise boundaries X*(t) and X**(t) intersect in the geometric Brownian motion model with constant coefficients under the double-boundary regime (for American perpetual options with r < q < 0 for calls or q < r < 0 for puts) when the volatility satisfies σ > σ* = |√(-2r) − √(-2q)|. The goal is to express t* explicitly in terms of the model parameters (r, q, σ) to characterize the finite time at which the exercise interval collapses.

Background

In regimes with negative interest rates and/or convenience yields, American options can exhibit two exercise boundaries. For the Black–Scholes/GBM model with constant parameters, Andersen and Lake (2021) define a critical volatility threshold σ* = |√(-2r) − √(-2q)| and prove that, for American perpetual options in the double-boundary regime (r < q < 0 for calls or q < r < 0 for puts), if σ exceeds σ, the exercise boundaries intersect at a finite time t, causing the exercise interval to pinch off.

While the existence of such a finite pinch-off time t* is known under the stated conditions, an explicit closed-form expression for t* has not been established. Determining t* in closed form would provide a precise characterization of when early exercise becomes non-optimal before maturity and would enhance analytical and numerical understanding of boundary dynamics in negative-rate settings.