Dividends, stock indices and currencies
1. The idea
The holder of an option does not receive what the asset pays before expiry. As in put-call parity and the bounds of Level 5, the formula then counts the asset without those payments.
A cash dividend. On the ex-dividend date (Level 4), the first day on which a buyer of the share does not get the dividend, the share price falls by about the dividend. As the textbook does, we take the dividend to be paid on that date. So the formula uses the share price less the dividend’s present value, \(S_0 - D\), in place of \(S_0\):
\[ c = (S_0 - D)\,N(d_1) - K e^{-rT} N(d_2), \qquad d_1 = \frac{\ln\big((S_0 - D)/K\big) + (r + \sigma^2/2)\,T}{\sigma\sqrt{T}}, \]
where \(D\) is the present value of the dividends whose ex-dividend dates are before expiry, and the other symbols are those of the formula, with \(d_2 = d_1 - \sigma\sqrt{T}\) and the put’s price \(p = Ke^{-rT}N(-d_2) - (S_0 - D)\,N(-d_1)\). Strictly, \(\sigma\) is then the volatility of \(S_0 - D\), the share price without the dividends; using the share’s own volatility for it, as we do, is an approximation, close when the dividends are a small part of the share price.
A dividend yield. A stock index pays dividends all the time, at a yearly rate \(q\), its dividend yield (Level 2). Its options use \(S_0 e^{-qT}\) in place of \(S_0\), and \(r - q\) in place of \(r\) in \(d_1\):
\[ c = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2), \qquad p = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1), \]
\[ d_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)\,T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}, \]
where \(S_0\) is the index level today and \(q\) the dividend yield with continuous compounding, as a decimal, and the other symbols are those of the formula. The prices are in index points; an index option contract is usually on 100 times the index.
A currency. Holding a foreign currency earns the foreign risk-free rate \(r_f\), as an index earns its dividend yield (Level 2). Options on one unit of a currency use the same formulas with \(r_f\) in place of \(q\), and \(S_0\) the spot exchange rate, in dollars.
An example. A stock index stands at 4,800, with a dividend yield of 2%; the risk-free rate is 4% and the index’s volatility 18%. For a three-month call with a strike price of 4,900,
\[ d_1 = \frac{\ln(4{,}800/4{,}900) + (0.04 - 0.02 + 0.18^2/2) \times 0.25}{0.18 \times \sqrt{0.25}} = -0.1285, \qquad d_2 = -0.2185, \]
and \(c = 4{,}800\,e^{-0.02 \times 0.25} \times 0.448877 - 4{,}900\,e^{-0.04 \times 0.25} \times 0.413520 = 137.78\) index points. Without the dividend yield the call would cost 148.79: dividends that the index pays out lower a call’s price and raise a put’s.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The dividend on a chart
First, the dividend’s present value.
Now move the size of the dividend. A larger dividend lowers the share that the formula uses, so the call’s price falls and the put’s rises.
4. One worked example, then your turn
In the worked example, we price a put on a stock index.
Now a call on a currency with hints, one step at a time.
And one on your own: a put on a share that pays a dividend.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a share with a dividend, one for a stock index and one for a currency. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: