I have came across a formula suggested by Fisher Black (Fact and fantasy in the use of options, FAJ, July–August 1975, pp.36) for approximating the price of an American call written on a dividend-paying stock. See Wikipedia's article Black's approximation for further details.
Consider a stock with current price $S_0$, which pays a dividend $D$ at date $t_D$; both the dividend amount and the payment date are known at time $t=0$. Consider also an American call option with strike $K$ and maturity date $T$; we can write its Black-Scholes price $C_A$ as a function: $C_A = C_A(S_0,K,T)$. Let $C_E$ be the price of an European call option; then Black's approximation $C_A^{FB}$for $C_A$ is:
$C_A(S_0,K,T) \approx C_A^{FB}(S_0,K,T) = max[C_E(S_0-De^{-rT},K,T),C_E(S_0,K,t_D-1)]$
I understand the financial logic behind this approximation; however what strikes me is that it is inconsistent with Black-Scholes theory. Indeed, we must have:
$C_E(S_0-De^{-rT},K,T)<C_E(S_0,K,T)$
$C_E(S_0,K,t_D-1)<C_E(S_0,K,T)$
Thus:
$C_A^{FB}(S_0,K,T)<C_E(S_0,K,T)$
So the American call Black's approximated price would always be less than its European counterpart, whereas American options are at least as valuable as their European counterparts.
Is there anything wrong in my reasoning? Does anyone understand the logic of this approximation, given this inconsistent feature?
[EDIT]
Dividends should be discounted using the factor $e^{-rt_D}$, not $e^{-rT}$.