The project I'm working on requires me to model the share price of a firm through time using the Merton and Black-Cox credit models. The model is used here to induce the leverage effect in the share price.
I was initially simulating the share price path using a GBM for the value of assets, $A_t$, and simply equating the equity value, $E_t$ to $\max(0,A_t-D)$, where $D$ is the debt value. The rationale here is that this respects the accounting equation: $A_t = E_t + D$.
Now, I'm not so sure about this and was wondering if it is more correct to instead use $$ E_t = BScall(V_t,D,...)$$ for $0\leq t \leq T$.
Would this be any different for the Black-Cox model (which allows for default prior to $T$).
Any references would be appreciated.