Dividends
Cash dividends drop the spot on a date. How to price around them, and why they are the only reason to exercise a call early.
Yield versus cash
Everywhere else in this app dividends are a continuous yield \(q\): the stock leaks value at a steady rate, which is a fine description of an index. A single stock pays cash amounts on ex-dividend dates. On such a date the price drops by the dividend, all at once. For a short-dated option that difference matters, and for an American call it changes the answer qualitatively.
The escrowed-dividend model
Split the spot into two parts: the present value of the dividends that will be paid before expiry, which is riskless and simply accrues at the rate, and the remainder \(S^* = S - \mathrm{PV}(D)\), which is assumed to follow geometric Brownian motion. The option is then a Black-Scholes option on \(S^*\) with no yield:
$$ C = S^* N(d_1) - K e^{-rT} N(d_2), \qquad S^* = S - \sum_{t_i \lt T} D_i e^{-r t_i} $$Put-call parity holds with \(S^*\) in place of \(S\). The model is simple and widely used, though it slightly understates volatility early in the option's life, because it treats part of the spot as riskless. Setting \(q = -\ln(1 - \mathrm{PV}(D)/S)/T\) gives a continuous yield with the same European price, which is the usual way to fold known dividends into a yield.
Early exercise of calls
Without dividends an American call is never exercised early: waiting keeps the insurance and earns interest on the strike. A dividend breaks this. Just before the ex-date the holder can exercise, collect the stock and the dividend, and give up the remaining time value. If the dividend is larger than the time value left, exercise wins. So the exercise region for a call is a set of instants, one before each ex-date, and only when the option is deep enough in the money. On a tree the exercise decision at each node compares the payoff on the cum-dividend spot, \(S^* + \mathrm{PV}_t(\text{remaining } D)\), with the value of waiting.
Puts
Dividends make puts more valuable, since the spot will drop, and they discourage early exercise of a put just before an ex-date: better to wait and let the price fall first. The tree captures both effects without any special cases, which is one reason trees remain the workhorse for single-stock options.
Check your understanding
4 questions. Pick an answer for each, then check. Nothing leaves your browser session.
1. In the escrowed-dividend model, the option is priced on
2. When can it be optimal to exercise an American call early?
3. A cash dividend makes a European put
4. The equivalent continuous yield q satisfies