Boundary Conditions for Perpetual American Options and One-Shot Derivatives

Summary

The document asks about a boundary condition in a textbook treatment of perpetual American options. The setup includes a one-shot derivative that pays a fixed amount when the stock first reaches a specified level. The questioner accepts that its value equals the payoff at that level, but challenges the condition that its value is zero when the stock price is zero, reasoning that the price might later rise to the trigger. They also ask why the one-shot payoff framework is relevant to perpetual options, which have no fixed expiry.

No answer is supplied, so the document does not establish the boundary condition or connect the derivative to the valuation of perpetual calls and puts. The issues concern the assumptions behind the stock-price process and its behavior at zero, as well as how an option’s value can be analyzed through a first-passage payoff. Those assumptions are essential to resolving the questions; the document itself gives no model details, derivation, or evidence.

Key ideas

• The question concerns a derivative paying a fixed amount when the stock first reaches a threshold.
• It challenges a zero-value boundary at a stock price of zero because the price might subsequently recover.
• It asks how a first-passage payoff helps analyze perpetual American options.
• The document contains no answer or model assumptions that resolve either issue.

Tags

# why does "one-shot derivative" boundary condition hold for perpetual American call options?


# why does "one-shot derivative" boundary condition hold for perpetual American call options?












When reading John Hull's book about perpetual options (Section 26.2: "perpetual American call and put options", 10th edition), I am a bit confused about the boundary conditions used here.

The author starts with discussion of the "one-shot derivative", which pays off a fixed amount $Q$ when stock price $S=H$ for the first time. Then author claims the boundary conditions for this derivative are $f=Q$ when $S=H$ and $f=0$ when $S=0$. I understand that $f=Q$ when $S=H$, but does anyone know why $f=0$ when $S=0$ (I expect that although $S=0$ at some time, it could still rise to $H$ at a future time therefore its price should not be exactly 0)?

Another question is, why does this discussion of the "one-shot derivative" apply to perpetual options? In theory, we could wait forever with a perpetual option, so I assume it should not have this sort of "one-shot" behavior.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.