American options: backward induction on a binomial tree
Early exercise turns option pricing from a one-shot expectation into a backward dynamic program. The binomial tree is the simplest model where you can see the early-exercise boundary appear.
European options can only be exercised at maturity, so their price is the discounted risk-neutral expectation of the payoff. American options can be exercised at any time before maturity, which means at every node the holder compares the intrinsic value (exercise now) against the continuation value (hold and re-decide next step). The optimal stopping problem turns pricing into a dynamic program; the binomial tree is the discrete-time setting where each step of that program is explicit.
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Worked example
Price an American put with $S_0 = 100$, $K = 105$, risk-free rate $r = 5\%$, volatility $\sigma = 20\%$, maturity $T = 1$ year, using a 3-step Cox-Ross-Rubinstein binomial tree. Use $u = e^{\sigma\sqrt{T/N}}$, $d = 1/u$, and risk-neutral probability $p = (e^{r T/N} - d) / (u - d)$. Compare to the European put on the same parameters.
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Reflection
When does early exercise become optimal in this binomial example, and what would change if the underlying paid a dividend? Why do American calls on non-dividend stocks have the same value as European calls?