Contingent capital with discrete conversion from debt to equity

We consider the problem of valuing contingent capital in the form of debt that converts to equity when a capital ratio falls below a threshold. With continuous monitoring of the conversion trigger and with asset value modeled as geometric Brownian motion, the value admits a closed-form expression. Here we focus on the case of a discretely monitored trigger and the simulation of three potential mechanisms for conversion in discrete time. We show how to use the continuous-time formulas as control variates through exact joint simulation of the discrete- and continuous-time processes. We then investigate continuity corrections to approximate discrete-time results using continuous-time formulas and compare results across alternative conversion mechanisms.