American Options and Early Exercise

Most option theory starts with European options, which you can exercise only on the expiry date. Real exchange-traded options are usually American: you can exercise them any day up to expiry. That extra freedom can only help you, you keep every choice a European holder has, plus more, so an American option is always worth at least as much as its European twin. The gap between them is the early-exercise premium, and figuring out when exercising early is actually worth it is one of the classic quant puzzles.

Value = max of exercising now versus waiting

At any moment, an American option holder faces a simple fork: exercise now and collect the intrinsic value (what you'd get today), or hold on and keep the option alive. So the option's value at every point is

$$V = \max\big(\text{intrinsic value now},\; \text{value of continuing to hold}\big).$$

The "value of continuing to hold" is what a pricing model computes, the discounted expected payoff from waiting. Whenever the intrinsic value climbs above that, it's time to exercise. Recall the intrinsic value is just the option payoff, drag the sliders below to see the payoff (green) that an American holder can lock in by exercising, versus the profit after premium (amber):

When is early exercise actually worth it?

The surprising answers:

• American call on a non-dividend stock: never exercise early. It's always better to sell the call than exercise it. Exercising throws away the option's remaining time value and forces you to pay the strike early, giving up the interest you could have earned on that cash. So a call on a non-dividend-paying stock is worth exactly the same American or European, its early-exercise premium is zero.
• American put: often worth exercising early. A put deep in the money is different. If the stock has crashed near zero, there's almost nothing left to gain by waiting (the stock can't fall below zero), but by exercising now you collect the strike $K$ in cash today and start earning interest on it. That interest incentive makes early exercise of a deep-in-the-money put genuinely optimal.
• Call on a dividend-paying stock: maybe, just before a dividend. Since holding the option doesn't earn the dividend but exercising to own the stock does, it can pay to exercise a call the instant before the stock goes ex-dividend.

The exercise boundary

For an American put there's a critical stock price, the exercise boundary $S^*(t)$. Below it, the put is deep enough in the money that exercising beats waiting; above it, you hold. As expiry approaches, the boundary rises toward the strike, because there's less and less time value left to protect.

Worked example: an American put on a tree

Nothing makes early exercise click like a binomial tree. Take a two-step tree with $S_0 = 100$, up factor $u = 1.2$, down factor $d = 0.8$, and a per-step growth of $e^{r\Delta t} = 1.05$. Price an American put with strike $K = 100$. First the risk-neutral probability:

$$p = \frac{1.05 - 0.8}{1.2 - 0.8} = \frac{0.25}{0.4} = 0.625.$$

The stock tree runs $100 \to \{120, 80\} \to \{144, 96, 64\}$. Put payoffs at expiry ($\max(100 - S, 0)$) are $0$, $4$, and $36$. Now roll back, checking early exercise at each node:

• Up node ($S = 120$): holding is worth $\tfrac{1}{1.05}[0.625(0) + 0.375(4)] = 1.43$; exercising gives $\max(100-120,0) = 0$. Keep holding, value $1.43$.
• Down node ($S = 80$): holding is worth $\tfrac{1}{1.05}[0.625(4) + 0.375(36)] = \tfrac{16}{1.05} = 15.24$; but exercising gives $\max(100-80,0) = 20$. Since $20 > 15.24$, exercise early, value $20$.
• Today ($S = 100$): $\tfrac{1}{1.05}[0.625(1.43) + 0.375(20)] = \tfrac{8.39}{1.05} = 7.99$.

So the American put is worth $7.99. If we had ignored early exercise (the European value), the down node would be worth only $15.24, and rolling back gives $6.29. The difference, $7.99 - 6.29 = 1.70$, is the early-exercise premium. That extra $1.70 is precisely the value of the right to cash in early at the down node.

Why parity only becomes an inequality

Because American puts can be exercised early, the clean equality of Put-Call Parity breaks. For European options $C - P = S_0 - Ke^{-rT}$ holds exactly; for American options it weakens to a pair of inequalities, because the put now carries that early-exercise premium the equation doesn't account for. Interviewers love this: "does put-call parity hold for American options?" The answer is no, only bounds.

Common pitfalls

• Exercising a call early to "capture" a gain. On a non-dividend stock this destroys value, you're better off selling. Only dividends can justify early call exercise.
• Pricing an American option with a plain European formula. Black-Scholes gives the European value; it underprices American puts. Use a tree or finite differences that checks exercise at every node.
• Forgetting the boundary moves. The optimal exercise price isn't fixed; it drifts toward the strike as expiry nears. A rule calibrated for a long-dated option is wrong near expiry.
• Assuming parity still holds. For American options it's an inequality, not an equation. Don't back out a "riskless" arbitrage from it.