Difference in value - American call and a European call - stock pays a dividend

Question

For a stock paying a single dividend prior to expiration, I would like to estimate the difference in value between an American call and a European call with the same expiration, strike and underlier.

Pointers to literature addressing this topic are appreciated.

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Proposed method:
The value of an American call $C_{\textrm{a}}$ is identical to the value of a European call $C_{\textrm{e}}$ after the dividend ex-date $t_{\textrm{d}}$, $$ C_{\textrm{a}}\left(S,K,T-t_{\textrm{d}}^+\right) = C_{\textrm{e}}\left(S,K,T-t_{\textrm{d}}^+\right) ~. $$

The difference in value between $C_{\textrm{a}}$ and $C_{\textrm{e}}$ arises from the opportunity to "catch" the dividend.

Isolating this distinguishing event with European options written at time $t_0$ for the period from $t_0$ to $t_{\textrm{d}}$, the difference in value between an American call and a European call for a stock paying a single discrete dividend prior to expiry $T$ is attributable to

$$ C_{\textrm{e}}\left(S,K \textrm{e}^{r \left(T-t_{\textrm{d}}\right)},t_{\textrm{d}}^-\right) - C_{\textrm{e}}\left(S - D \textrm{e}^{-r t_{\textrm{d}}}, K \textrm{e}^{r \left(T-t_{\textrm{d}}\right)},t_{\textrm{d}}^+\right) ~. $$

Result:

The difference in value between an American call and a European call for a dividend paying stock may be estimated using European calls

$$ C_{\textrm{a}}\left(S,K,T\right) - C_{\textrm{e}}\left(S,K,T\right) \approx C_{\textrm{e}}\left(S,K \textrm{e}^{r \left(T-t_{\textrm{d}}\right)},t_{\textrm{d}}\right) - C_{\textrm{e}}\left(S - D \textrm{e}^{-r t_{\textrm{d}}}, K \textrm{e}^{r \left(T-t_{\textrm{d}}\right)},t_{\textrm{d}}\right) ~. $$

Answer 1

Whaley's "On the valuation of American call options of stocks with known dividends" shows that there is a boundary $S_{t_d}^*$ on the range of the ex-dividend stock price $S_t$ (where $S_t:= P_t-D\exp(-r(t_d-t))$ if $t<t_d$, and $S_t:= P_t$ if $t_d\leq t\leq T$, where $P_t$ is the cum-dividend stock price), defined by equation: $$C_E(t_d, S_{t_d}^*;T,K) = S_{t_d}^* + D - K.$$ One then exercises getting $S_{t_d} + D - K$, if $S_{t_d} > S_{t_d}^*$, and continues holding, if $S_{t_d} \leq S_{t_d}^*$.

Once we have this fixed boundary, and given that we have a single dividend, we can construct a replicating portfolio:

a) a long position paying $(S_T - K)^+$ at $T$,

b) a long position paying $(S_{t_d} - S_{t_d}^*)^+$ at $t_d$, and

c) a short position paying $$ \left( C_E(t_d, S_{t_d};T,K) -(S_{t_d}^* + D - K) \right)^+ = \left( C_E(t_d, S_{t_d};T,K) - C_E(t_d, S_{t_d}^*;T,K)\right)^+ $$ at $t_d$.

If $S_{t_d} \leq S_{t_d}^*$, b) and c) vanish (due to the call price monotony in $S$), leaving a) to be paid at $T$, that is, we decided to hold.

If $S_{t_d} > S_{t_d}^*$, the b) and c) payouts give $$ (S_{t_d} - S_{t_d}^*) + (S_{t_d}^* + D - K) = S_{t_d} + D - K$$ at $t_d$, and $$ -(S_T-K) $$ at $T$. The latter payout gets cancelled by the the payout from c). Hence, under this scenario, we decided to exercise.

These payouts, with c) being a compound option (call on call), have a closed-form pricer (uses only normal and binormal cdfs) under Black-Scholes dynamics, as shown by Whaley.

So, a parallel equation to your equation is (for $t<t_d$):

$$ C_A(t, S_{t};T,K) - C_E(t, S_t;T,K) $$ $$ = C_E(t, S_t;t_d,S_{t_d}^*) - \exp(-r(t_d-t))\mathbb{E}_t\left[ \left( C_E(t_d, S_{t_d};T,K) -(S_{t_d}^* + D - K) \right)^+ \right]$$.