Advanced American Options: Optimal Exercise and Profit
Here is something that should make every options trader stop and think. The “optimal” time to exercise an American option depends on who you are. The textbook answer assumes the holder is delta hedging. But if the holder were delta hedging, why would they buy the option in the first place? Chapter 63 of Wilmott’s book, based on a 1998 paper with Dr. Hyungsok Ahn, digs into this question and reaches a conclusion that is great news for option writers.
The Standard Story
Let us start with what everyone learns. An American option can be exercised at any time before expiry. It is worth at least as much as a European option because it gives you more rights. To price it correctly, you assume the holder exercises at the worst possible time for the writer. This gives the option its highest theoretical value.
The writer collects this maximum premium and delta hedges. If the holder exercises at the “optimal” time, the writer breaks even. If the holder exercises at any other time, the writer profits.
The writer can never lose. At worst he feels nothing. At best he is very happy. That is straight from Wilmott’s own words.
But Wait, Who Is This Optimal Holder?
Here is the problem with the standard story. The price-maximizing exercise time assumes the holder is a risk-neutral, delta-hedging machine. But think about it. If the holder could delta hedge perfectly, why would they buy the option? They could replicate the payoff themselves. The market is complete. Buying the option would be pointless.
Real option holders are not delta hedging. They buy options as investments. They have views on market direction. They have different levels of risk tolerance. They might follow a buy-and-hold strategy, exercising when it feels right rather than when the math says so.
This is not a zero-sum game either. Both the writer and the holder can trade the underlying asset with other market participants. The worst case for the writer is not the best case for the holder. Their optimal strategies come from different places.
Three Important Assumptions
Before we go further, Wilmott wants you to keep three things in mind.
First, the writer lives in the Black-Scholes world with no transaction costs. Second, the option holder cannot delta hedge (and would not want to). Third, the option cannot be sold on a secondary market. Think of it as over-the-counter. The only way to close the position is to exercise.
Under these conditions, the question becomes: given that the holder follows some strategy other than the textbook optimal one, how much extra profit does the writer make?
The Utility-Maximizing Holder
Instead of assuming the holder exercises to maximize the option’s theoretical value, let us assume the holder exercises to maximize their own expected utility. Remember utility theory from the previous chapter? Each investor has their own utility function that captures their risk preferences.
The holder faces an optimal stopping problem. At each moment, they decide whether to exercise now or wait. But their decision uses the physical (real-world) probability measure, not the risk-neutral one. And it depends on their personal utility function.
This creates a different exercise boundary. The textbook boundary comes from the Black-Scholes variational inequalities. The holder’s boundary comes from their own utility maximization. These two boundaries do not coincide.
The difference between them is where the writer’s profit lives.
How the Exercise Boundary Shifts
Wilmott and Ahn prove several results about how the holder’s exercise boundary depends on drift and risk aversion.
If the holder is sufficiently risk averse, they exercise calls early. This is striking because in the standard theory, you never exercise an American call early on a non-dividend-paying stock. But a very risk-averse person might think: “The stock could crash tomorrow. Let me take my profit now.” From a utility standpoint, that can be perfectly rational.
For call options, the exercise time increases with the physical drift. If you believe the stock is going up fast, you hold the call longer. If you think the stock might decline, you exercise sooner. This makes intuitive sense but is completely absent from the standard pricing theory, which ignores drift.
For put options, the relationship reverses. Higher expected drift means you exercise the put sooner, because the stock is likely moving away from the profitable zone.
Specific Utility Functions
The paper works through several families of utility functions to see how the exercise boundary changes.
CARA (Constant Absolute Risk Aversion): Early exercise happens for both calls and puts, regardless of the risk aversion parameter. The holder always exercises before the textbook optimal time. The exact boundary depends on the risk aversion level, but it always exists.
HARA (Hyperbolic Absolute Risk Aversion): This is the most interesting case because certain HARA utilities produce two separate exercise boundaries. The holder exercises if the stock goes too high or too low. This creates a corridor in the middle where they hold the option.
Expected Return (Linear Utility): A risk-neutral holder only exercises a call early when the drift is below the risk-free rate. This is the closest to the textbook result because a risk-neutral person does not care about variance.
The Writer’s Profit
Here is the punchline and the practical takeaway of the whole chapter.
When the holder exercises at a time that differs from the price-maximizing boundary, the writer earns a profit. This profit accumulates in the region between the two exercise boundaries. The longer the stock price spends in this zone, the more the writer makes.
The profit has two components. First, the time value of the option erodes in the writer’s favor while the stock is between the boundaries. Second, when the holder finally exercises, the option is worth more than the exercise value, and the writer captures that difference.
The numerical results show this profit can be significant. More risk-averse holders generate more profit for the writer. Higher physical drift (for calls) also increases the writer’s expected gain because it pushes the holder’s boundary further from the textbook boundary.
The Practical Lesson
Wilmott and Ahn end with a beautiful observation that was also published in Derivatives Week magazine. Since the writer can never lose from early exercise, and any non-textbook exercise generates a profit, the advice is simple.
If early exercise does not add too much to the theoretical value of an option, always sell it as American rather than European. You charge a small premium for the American feature. And then you sit back and wait for the holder to exercise at a “non-optimal” time. When they do, you collect a surprise windfall.
This is not about tricking anyone. The holder is rationally maximizing their own utility. They are making the right decision for themselves. But because their personal optimal strategy differs from the theoretical worst case, the writer benefits.
Everyone wins. The holder gets an option that matches their risk preferences. The writer gets expected profits above and beyond the theoretical zero. The only thing that changes is our understanding of what “optimal” really means.
Key Takeaways
Three things from this chapter. First, the textbook optimal exercise boundary is optimal only for a delta-hedging holder, which is a contradiction because such a holder would not need the option. Second, real holders with utility functions exercise differently, and their boundaries depend on drift and risk aversion, two factors that standard pricing theory ignores. Third, option writers should love this because any deviation from textbook exercise generates profit. Sell American when you can, hedge it properly, and let human nature do the rest.
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