Let's say I have a black box stock price model I run Monte Carlo on to estimate European call prices. For a given strike $K$ and expiration $T$, I then back out the Black-Scholes implied volatility $\sigma(K, T)$ from the Monte Carlo price $C_{MC}$, and this assumes the model is lognormal.
I now want to price a digital option using this black box model at the same $K$ and $T$ for which I computed the implied vol. I use Monte Carlo for this and obtain a price $D_{MC}$
Let's say I also compute the price of this digital using the closed-form Black-Scholes price and use $\sigma(K,T)$ as my vol. I obtain a price of $D_{BS}$.
Now, as I understand, $\sigma(K, T)$ is exactly that volatility I had to plug into my Black-Scholes price to obtain $C_{MC}$. I am now pricing another type of option (digital) for this $K$ and $T$ and use $\sigma(K, T)$ in my closed-form. My question is, what is the relation between $D_{MC}$ and $D_{BS}$? Would one expect these to be equal? Would they be equal only if the black box model was actually just the lognormal model?