American Options: When to Exercise Early and Why It Matters

European options are simple: you wait until expiry, check if they are in the money, and either collect the payoff or walk away. American options give you more power and more headaches. You can exercise at any time before expiry, which sounds great but raises a hard question: when exactly should you do it? Chapter 9 of Wilmott’s book tackles this problem, and the ideas that come out of it show up again and again throughout the rest of quantitative finance.

The Basic Idea

An American option can be exercised at any time up to (and including) expiry. Most traded stock and futures options are American. Most index options are European.

The right to exercise early is clearly valuable. An American option is always worth at least as much as an equivalent European option. But the extra value comes with a puzzle: part of the pricing problem is figuring out the optimal exercise strategy. When should you pull the trigger?

Starting Simple: The Perpetual American Put

Wilmott begins with the easiest possible case. A perpetual American put has no expiry date. You can exercise it for the payoff max(E - S, 0) whenever you want. Because there is no time limit and the contract details do not depend on time, the option value depends only on the stock price S, not on t. This makes the math much simpler.

Two key principles apply:

1. The option value can never fall below the payoff. If it did, you could buy the option, immediately exercise, and make a riskless profit. No-arbitrage says this cannot happen:

V(S) >= max(E - S, 0)

2. While the option value is above the payoff, it must satisfy the Black-Scholes equation. Since there is no time dependence, this reduces to an ordinary differential equation:

(1/2)sigma^2S^2V’’ + rSV’ - rV = 0

The general solution is V = AS + BS^(alpha), where alpha = -2r/sigma^2. For a put, A must be zero (the value goes to zero as S goes to infinity), leaving V = B*S^(alpha).

The Optimal Exercise Point

Now comes the interesting part. We have an unknown constant B and an unknown exercise boundary S*. When S reaches S* from above, we exercise the put. This gives one equation (the value must equal the payoff at S*):

B(S)^alpha = E - S***

But we have two unknowns. Where does the second equation come from?

Wilmott shows that we should choose S* to maximize the option value. If we can exercise whenever we want, we should pick the strategy that makes our option worth the most. Differentiating with respect to S* and setting the result to zero gives:

S = alpha*E / (alpha - 1)*

This is always less than E (you exercise before the stock falls all the way to the strike).

Something elegant happens at this optimal exercise point: the slope of the option value curve matches the slope of the payoff. The option value and the payoff touch smoothly, without any kink. This is called the smooth-pasting condition (or high-contact condition).

Think about what happens if the slopes do not match. If you exercise too early (S* too high), the option curve meets the payoff with a kink pointing down, and you are leaving money on the table. If you exercise too late (S* too low), there is an arbitrage opportunity. The smooth-pasting condition is the sweet spot.

The Key Result: American Calls Without Dividends

When there are no dividends, the American call option should never be exercised early. Its value is exactly the same as the European call.

This might seem counterintuitive. You have a right you are not using, and yet the option is not worth more? The reason: exercising a call early means paying the strike price now instead of later. You lose the interest on that money. And since a call’s upside is unlimited, there is always a chance the stock will go higher. The combination of lost interest and unlimited upside makes early exercise suboptimal.

But add dividends and everything changes. With a continuous dividend yield D, the stock “leaks” value to the holder. If you hold the option instead of the stock, you miss those dividends. At some point, the dividend income is worth more than the remaining time value, and you should exercise.

For the perpetual call with dividends, the exercise boundary is:

S = alpha+ * E / (alpha+ - 1)*

where alpha+ is the positive root of a specific equation involving r, D, and sigma. When D = 0, alpha+ becomes infinity, confirming that you never exercise.

The General American Option Problem

The perpetual case is nice because it reduces to an ODE. The real-world case with a finite expiry is much harder because the option value depends on both S and t, and the exercise boundary S*(t) moves over time.

For the general case, we get an inequality instead of an equation:

(1/2)sigma^2S^2V_SS + rSV_S + V_t - rV <= 0

Combined with the constraint V >= P(S,t) (the payoff function), this defines what is called a free boundary problem. We do not know in advance where the boundary between “hold” and “exercise” regions is. It must be found as part of the solution.

This makes the problem non-linear. You cannot add two American option solutions together to get a third one. Wilmott illustrates this with the perpetual American straddle: an American straddle exercised as a single contract is not the same as the sum of an American call and an American put, because there is only one exercise opportunity for the straddle but two separate opportunities for the individual options.

Close to Expiry Behavior

For an American call on a dividend-paying stock:

When r > D (the usual case for equities), close to expiry the optimal exercise boundary behaves like:

S(t) ~ E*r/D - const * sigma * sqrt(T-t)*

As expiry approaches, the boundary comes down from infinity toward E*r/D. When D = 0, there is no boundary at all.

When r < D, the boundary starts at E at expiry, and the local analysis is more subtle.

Dividends and Jump Conditions

Discrete dividends create interesting interactions with American options. When the stock drops by the dividend amount, the jump condition for the option applies:

V(S, t_before) = V(S - D_i, t_after)

But for American options, this jump could push the option value below the payoff just before the dividend date. This is not allowed by the no-arbitrage constraint, so we impose:

V(S, t_before) = max(V(S - D_i, t_after), P(S))

The option value may actually become discontinuous at the dividend date, but only if the holder should have exercised before the dividend was paid. This is not a flaw; it is correct behavior.

One-Touch Options

The one-touch option is a clever American-style contract. It pays $1 if the asset ever reaches a specified level. Once the level is reached, you exercise immediately (there is no benefit in waiting), so the free boundary problem simplifies into a fixed boundary problem.

For a one-touch call (pays $1 when S reaches S_u from below), you solve the Black-Scholes equation with V(S_u, t) = 1 and V(S, T) = 0, only for S < S_u. Wilmott gives an explicit formula involving the cumulative normal distribution.

The double one-touch has both an upper and lower level. Interestingly, its value is not the sum of a one-touch call and a one-touch put (another example of non-linearity in problems with boundaries).

Bermudan Options and Other Variants

A Bermudan option allows exercise only at specific times (say, every Thursday). Mathematically, the no-arbitrage constraint is only “switched on” at those dates. Between exercise dates, you just solve the regular Black-Scholes equation. Numerically, this is no harder than the American case.

The make-your-mind-up option requires advance notice before exercise. You declare your intention to exercise, and the actual exercise happens some fixed time later. You cannot change your mind (unless the contract specifically allows it). The valuation requires solving for two functions: the option value given that you have declared exercise, and the option value given that you have not.

The instalment option requires the holder to keep paying premiums to keep the contract alive. If a premium is missed, the contract lapses. The holder must decide at each payment date whether it is worth continuing to pay. This is an optimal stopping problem similar to the American option, but with the decision reversed: you must optimally choose when to stop paying, rather than when to exercise.

Why This Chapter Matters So Much

The concepts in this chapter extend far beyond American options:

• Free boundary problems appear whenever an optimal decision must be made as part of the valuation
• Non-linearity from free boundaries means you cannot simply add solutions together
• Smooth-pasting provides the extra condition needed to locate the boundary
• Numerical methods are essentially the only practical way to solve these problems

Wilmott emphasizes that even if a contract is not explicitly labeled “American,” it may contain early-exercise features or optimal-strategy elements that create the same mathematical structure. Being able to recognize and handle free boundary problems is one of the most important practical skills in quantitative finance.

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