# Why the Inconsistency in the Derivation of BS for Dividend-Paying Underlying?

The basic idea is that we get two expressions for $$\Delta \Pi = ...$$ and equate them.

The thing that does not make sense is that in one we take into account the dividend

$$\Delta \Pi = \frac{d}{dS}V \cdot \Delta S -1 \cdot \Delta V \boxed{+ (D \cdot \Delta t) \cdot \frac{d}{dS}V \cdot S}$$

where as in the other we totally ignore it

$$\Delta \Pi = \Pi \cdot k$$ $$\Delta \Pi = \bigg(\frac{d}{dS}V \cdot S -1 \cdot V \bigg) \cdot \bigg(r \cdot \Delta t \bigg)$$

How come?

• See a discussion in this question. A self-financing portfolio should have no cash in or out. – Gordon Nov 21 '19 at 17:40

In the first $$\Delta \Pi$$ expression we are trying to eliminate risk from the portfolio. It just so happens that to do the delta hedging in this case we need to take into account the dividend.
The second $$\Delta \Pi$$ expression comes from the no-arbitrage principle, which is the same (as is its equation $$\Delta \Pi = r \cdot \Pi \cdot \Delta t$$) be it dividend or no divident. It is based on wishful, albeit logical, thinking that if we manage to create a synthetic asset (a delta-hedged portfolio) that is risk (volatility) free, then it must be equivalent to an interest-bearing bank account (because, well, it too is risk .., I mean volatility, free). And there are certainy no dividends in interest-bearing bank accounts.