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

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.

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

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

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 $$t, and $$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):

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