# Early exercise premium with discrete cash dividends using integral approximation

From my understanding, we have to integrate $$N(d1(S_x-D,B,t))$$ on both asset-price and time-space to derive the Early Exercise Premium $$EEP(B,t)$$ on each $$t$$ before the ex-date to get current early exercise premium $$EEP(S,0)$$. Where $$S_x = S \exp((r-\sigma^2/2) t + x \sigma \sqrt{t})$$ Is this correct?

When I use carry and vol backed out from European option prices, continuous div EEP is at bid of American quotes, which is correct. But when I use discrete cash div version EEP is a fraction of the continuous one, which takes American price below payoff which is wrong. I apply the same method on backing out carry with discrete div which only returns implied borrow/lending fee that I use to plug in as q.

Thanks.