American Options
A European option
can be exercised only at its expiry date. An American option can be
exercised at any time up to expiry. That extra freedom sounds like a small footnote,
but it transforms the problem: the holder must now decide not just whether the option is
valuable, but when to pull the trigger — turning pricing into an
optimal stopping problem.
Pricing as optimal stopping
Because the holder may exercise at any
stopping
time \tau \le T, the fair (risk-neutral) value is the
best such choice — the supremum over all stopping rules of the expected discounted payoff:
V_0 = \sup_{\tau \le T} \mathbb{E}^{\mathbb{Q}}\!\left[ e^{-r\tau}\, \text{payoff}(S_\tau) \right].
Crucially, \tau must be a genuine stopping time — the decision to exercise
can depend only on information seen so far, never on a peek at tomorrow's price. There is no formula
as tidy as Black–Scholes; instead the value is computed by backward induction (the
"Snell envelope"): at each moment, the option is worth the larger of exercising now (its
intrinsic value) and holding on (its continuation value). That comparison traces out an
early-exercise boundary — a free boundary in the pricing PDE.
- Exercisable at any time up to expiry, so pricing is an optimal-stopping
problem: V_0 = \sup_{\tau \le T} \mathbb{E}^{\mathbb{Q}}[e^{-r\tau}\,\text{payoff}].
- An American option is worth at least the matching European one — the right to
exercise early can only add value.
- At each instant its value is \max(\text{exercise now},\ \text{continue}).
When early exercise does — and doesn't — pay
The most famous result is a surprise: for an American call on a stock that pays
no dividends, it is never optimal to exercise early. Holding the option is
always at least as good as exercising, so an American call equals its European twin. But an American
put genuinely can be worth exercising early — when it's deep in the money, taking
the cash now (and earning interest on it) can beat waiting. So the American put is strictly more
valuable than the European put, while (dividend-free) the calls agree.
Exercising a call early throws away two things. First, you pay the strike sooner than you must, so
you lose the interest you could have earned on that cash in the meantime. Second,
you surrender the option's insurance: if the stock later crashes below the strike,
an unexercised call simply expires worthless, but an exercised one has already locked in the shares
at a loss. Holding keeps the upside and the downside protection and your cash
earning interest — so, with no dividend tempting you to grab the shares, you always wait. (Add a
juicy dividend and the calculus flips: capturing it can make early exercise worthwhile.)
-
An American option is always ≥ the European one — never cheaper. The gap is the
early-exercise premium, which can be zero (the dividend-free call). "American ≥ European"
does not mean "always strictly greater".
-
The exercise decision must be a stopping time: "exercise at the price's peak" is
not allowed — it requires seeing the future. Optimality is about the best rule using only past
information.
-
At a node, compare intrinsic
\max(K-S,0) with continuation (the discounted
risk-neutral average of the children). The American value is the larger of the two — not their
average, and not automatically the intrinsic.
A two-step American put, by hand
Take a put with strike K = 100 on a stock sitting at
S = 100. Each step the stock is multiplied by
u = 1.2 or d = 0.8, and £1 in the bank grows
to £1.10 — a loud 10% per step, on purpose, so the early-exercise gap shouts. The risk-neutral
up-probability is
q = \frac{1.10 - 0.8}{1.2 - 0.8} = 0.75.
Work backwards from expiry. At a live node the continuation value is the discounted risk-neutral
average of the two children,
(1/1.10)\,[q\,V^u + (1-q)\,V^d],
and the American value is
\max(\text{intrinsic},\ \text{continuation}).
| node |
S |
intrinsic (K-S)^+ |
continuation |
American |
decision |
| expiry, uu |
144 |
0 |
— |
0 |
expire |
| expiry, ud |
96 |
4 |
— |
4 |
expire |
| expiry, dd |
64 |
36 |
— |
36 |
expire |
| t = 1, up |
120 |
0 |
0.91 |
0.91 |
hold |
| t = 1, down |
80 |
20 |
10.91 |
20 |
exercise |
| t = 0 |
100 |
0 |
5.17 |
5.17 |
hold |
The down node is the whole lesson in one comparison. The stock is 80, so exercising now pays 20.
Holding on is worth only
(1/1.10)\,[0.75\cdot 4 + 0.25\cdot 36] \approx 10.91.
Cash in hand, earning interest, beats waiting — early exercise is strictly optimal. (With
r = 0 those two numbers would have tied at 20; any positive rate tips
the put toward exercising.) The matching European put, forced to wait, is worth only about 3.10 at
t = 0, so the early-exercise premium is about 2.07.
Rollback on the tree
The same numbers, drawn as a tree. Terminal payoffs are locked in; then each earlier node takes
\max(K-S,\ \text{continuation}). The down node (highlighted) is where
exercise beats holding.
The schematic below is not a pricing formula — there isn't a tidy closed form for the American put.
It is the inequality the tree is computing:
\text{American} \ge \max(\text{European},\ \text{intrinsic}).
The hockey-stick is the put payoff (K-S)^+; the smooth curve sitting on
and above it is what an American put's value looks like.
See it explained