Theoretical proof of validity under unbounded short‑rate settings

Prove that the CTMC approximation results for convertible bond pricing remain valid when Assumption 3 (existence of a uniform lower bound for the short rate) is violated, specifically under the Black–Scholes–Hull–White model, by identifying parameter regimes ensuring theoretical convergence.

Background

Assumption 3 imposes a lower bound on the short‑rate process to facilitate certain convergence arguments. The numerical experiments suggest that the CTMC methodology still performs well for models such as Black–Scholes–Hull–White where this assumption does not hold.

The authors indicate that establishing a formal theoretical proof of convergence in this unbounded short‑rate setting is a pending task.