American options and early exercise
1. The idea
An American option can be exercised at any time before expiry, so at every node of the tree its holder has a choice: exercise now, or keep the option. The value is the better of the two:
\[ V = \max\Big(\text{intrinsic value},\; e^{-r\Delta t}\big[\,p\, V_u + (1-p)\, V_d\,\big]\Big), \]
where the first term is the option’s intrinsic value, the payoff from exercising now: \(\max(S - K, 0)\) for a call and \(\max(K - S, 0)\) for a put, with \(S\) the share price at the node and \(K\) the strike price. The second term is the value of keeping the option, worked out as for a European option: \(V\) is the option’s value at a node, \(V_u\) and \(V_d\) its values at the two nodes one step later, \(p\) the risk-neutral probability of an up move, \(r\) the risk-free rate and \(\Delta t\) the time step, as in the two-step tree. The check is made at every node, working back from expiry, so a node’s value already includes the chance to exercise later.
The intrinsic value is never negative, since no one exercises when exercising would cost money. An option with a positive intrinsic value is in the money. The rest of the option’s value is its time value, as in the two-step tree:
\[ \text{time value} = \text{option value} - \text{intrinsic value}. \]
For an American option the time value is never negative, since exercising now is always one of the choices. A European put deep in the money can have a negative time value: its intrinsic value is more than its value, but it cannot be exercised before expiry.
When does exercising early pay?
- A put, deep in the money. Exercising brings in the strike price now instead of later, and the interest on it can be worth more than the chance that the share falls further. Sometimes this holds even today, and the put is then worth exactly its intrinsic value.
- A call on a share without dividends: never. Keeping the call is worth at least \(S - Ke^{-r(T-t)}\), where \(t\) is the node’s time and \(T\) the time to expiry, both in years from today, and never less than zero. When \(r > 0\) that is more than \(S - K\), so keeping a call in the money beats exercising it; when it is not in the money, exercising pays nothing. So an American call on such a share is worth the same as a European one.
- A call on a share with dividends: sometimes, just before an ex-dividend date. The ex-dividend date is the first day on which a buyer of the share does not get the coming dividend, and on that day the share price falls by about the dividend. Exercising before it gives the share with the right to the dividend, which the option holder does not receive.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The tree, node by node
A new example, with three steps: an American put with a strike price of $88 on a share at $80, expiring in 9 months, so each step is 3 months; \(u = 1.15\), \(d = 0.85\) and the risk-free rate is 8%. First, what is the put’s intrinsic value today?
Now step through the tree. A solid orange frame marks a node where the put is exercised early. Click a node to see the comparison made there.
Move the strike price. The gap between the two lines is the American put’s time value. Deep in the money it shrinks to zero: there, exercising at once is best.
4. One worked example, then your turn
The worked example is a put with an early exercise one step before expiry.
Now one with hints: a call on a share with a high dividend yield.
And one on your own. Check today’s node too.
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 put and one for a call. 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: