Isn't Black's approximation for American options inconsistent?

Question

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?

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[EDIT]

Dividends should be discounted using the factor $e^{-rt_D}$, not $e^{-rT}$.

Answer 1

There is a logical fallacy in your argument.

The price of a European call expiring 1 day before a dividend payment may well be greater than that of a call expiring after it.

In other words, claiming that

$$ C_E (S_0,K,t_D-1\text {day}; D, t_D) < C_E (S_0,K,T; D, t_D) $$

is not necessarily true.

Try the above inequality with a huge dividend (e.g. $D = 90\%$ of the current spot price $S_0$) paid at $t_D$ with $T = t_D + 1\text {day} $ to convince yourself.

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[Edit]

Let $BS(S_0,K,T)$ denote the standard Black-Scholes formula.

If the value of an American call with one discrete dividend is given by the Black approximation you refer to: $$C_A^{FB}(S_0,K,T;D,t_D) = \max(BS(S_0-De^{-rt_D},K,T), BS(S_0,K,t_D-1/252))$$ then you should compare it to the value of the European call with 1 discrete dividend which is usually given by: $$C_E(S_0,K,T;D,t_D) = BS(S_0-De^{-rt_D}, K, T)$$ (escrowed model assumed) and not simply $C_E(S_0,K,T)$.

Your problem is thus twofold:

1. you should use $De^{-r t_D}$ and not $De^{-rT}$ for the present value of the capital distribution
2. you are always using $C_E(S_0,K,T)$ as a reference (no dividend accounted for), while you should really use $C_E(S_0,K,T;D,t_D)$

Using the above notations, it is clear that $C_A^{FB}(S_0,K,T;D,t_D)$ will always be greater than $C_E(S_0,K,T;D,t_D)$, because

\begin{align} C_A^{FB}(S_0,K,T;D,t_D) &= \max( C_E(S_0,K,T;D,t_D), BS(S_0,K,t_D-1/252) ) \\ &\geq C_E(S_0,K,T;D, t_D) \end{align}