Perpetual American Puts, Black–Scholes Assumptions, and Admissible Arbitrage
Summary
The document discusses whether the perpetual American put, a standard Black–Scholes example with an optimal lower exercise barrier, conflicts with the model’s assumptions because hedging may require a stock short held indefinitely. One answer focuses on admissibility: describing an indefinitely maintained short position is not enough to establish an arbitrage strategy. An arbitrage claim requires a sufficiently specified strategy, including its admissibility under the model.
A second answer frames perpetual options as idealized models and notes a limitation in applying option theory to firm equity: firm value is not directly tradable, which weakens the replication argument underlying risk-neutral pricing. The discussion also questions whether firm value follows the assumed stochastic process and whether a finite economic horizon can be identified. It offers conceptual cautions rather than a formal proof or a detailed derivation of perpetual put pricing. The examples therefore clarify the scope of the model, but do not establish that every real-world perpetual claim can be hedged or priced using the basic framework.
Key ideas
- A perpetual American put is presented as having an optimal lower exercise barrier in the standard model.
- An indefinitely held short position alone does not specify an admissible arbitrage strategy.
- Replication arguments rely on the underlying being tradable within the model.
- Using firm value as an option underlying can be problematic because it is not directly tradable.
- Perpetual option formulas are idealizations whose usefulness depends on how well their assumptions fit the application.
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# Doesn't a perpetual option contradict the Black-Scholes framework? # Doesn't a perpetual option contradict the Black-Scholes framework? A standard example when learning to price American options is the perpetual American put. This is a put that has no expiry (or you can consider T = infinity). The standard solution prices this using the basic Black-Scholes assumptions (including no restrictions on short-selling) and ends up concluding the optimal exercise strategy is finding the right lower barrier that effectively knocks out your put with a rebate. But my question is this: doesn't the idea of such a perpetual option contradict the Black-Scholes assumptions? In order for this pricing exercise (and yes, I'm aware it's just an exercise and not a real pricing problem) to be valid, you must be able to delta-hedge, correct? And since this will involve shorting the stock, this would seem to allow shorting the stock for arbitrary lengths of time. But doesn't this allow arbitrage? After all, someone (who doesn't care about the put) can just short the stock indefinitely and never close it out. Isn't this an arbitrage? If not, what am I missing? Is there some adjustment one needs to make to the idea of arbitrage in this infinite time case? Note my question is about the standard Black-Scholes theory. Someone asked about margin, but I don't think that's part of the standard theory. ## Answer by Alexey Kalmykov (score 1) https://quant.stackexchange.com/a/7537 The description of your arbitrage strategy is very vague. Thus it's hard to answer your question. However, this alone makes your "arbitrage" strategy not admissible: > shorting the stock for arbitrary lengths of time Thus what you describe is not an arbitrage strategy. ## Answer by David Addison (score 0) https://quant.stackexchange.com/a/32561 The use case for a perpetual option that I familiar with is equity. In the analogy, equity is a call on the value of a firm. The problem is that the underlying (firm value) is not directly "trade-able", which is a requirement for the no-arbitrage principle, which is turn a fundamental precept of risk-neutral hedging. You can infer the value of a firm by summing the value of equity, debt, capital/operating leases, and non-controlling interests. But even then, the analogy is not perfect since it assumes that the stochasticity of the underlying can be described by a Levy Flight (i.e., following Geometric Brownian Motion in continuous time). This may not be the case. If you can somehow: a) isolate the risky part of the business which has properties of a Levy Flight; and, b) determine a terminal time value at which the underlying cash flow reach an economic limit (independent of the uncertainty), you can then use the Black and Scholes as is. It is when you cannot determine the terminal time that one must adjust the formula for infinite expiry. I think it's key to keep in mind that a perpetual option is just a model which is useful in as far it describes reality; is not necessarily a perfect representation of reality. I hope that helps. If you want more, I would suggest the following paper as a reference: http://people.stern.nyu.edu/plakner/papers/perpetual.pdf.
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