Prove the critical-yield asymptotic for the Gibson–Schwartz model

Prove the heuristic near-zero-window asymptotic for the critical initial convenience yield in the Gibson–Schwartz model, namely the relation involving the volatility term \(\big(\sigma_\delta^2+2\rho\sigma\sigma_\delta\delta_c+\sigma^2\delta_c^2\big)^{1/2}\), the universal constant \(C\), and the short-rate level \(r\).

Background

For the Gibson–Schwartz model with stochastic convenience yield, the paper characterizes monotonicity and the critical yield below which the futures price equals spot. Numerical results suggest a near-expiry approximation for that critical yield based on the square-root boundary asymptotic established for geometric Brownian motion.

The displayed approximation is described as heuristic, and the authors explicitly state that it has not been proved. Establishing it would extend the near-expiry asymptotic analysis to the two-factor stochastic-convenience-yield setting.

References

The heuristic analogue of Theorem~\ref{thm:parabolic} for this model (Remark~\ref{rem:parabolic}(a), with the volatility of the convenience flow $\delta X$ in place of $\sigma\bar x$), $\delta_c+C\sqrt T\,\big(\sigma_\delta2+2\rho\sigma\sigma_\delta\delta_c+\sigma2\delta_c2\big){1/2}=r$, gives $2.6\%$ and $-0.8\%$; we have not proved it, and it is only an asymptotic statement.

— The end-of-month option in futures contracts and the Gittins index of holding the underlying  (2610.21196 - Bayraktar, 15 Oct 2026) in Section 4.2, subsection “Gibson–Schwartz”