# 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.

• A) which part are you short when you buy such put? B) where would you make money indefinitely? In fact you only make money when the company ceises to exist because early exercise is suboptimal barring dividend consideration. Most importantly such options are purely theoretical academic constructs. Nobody trades them. In my 14 years in the industry have I not come across a single such contract. I would occupy my mind with more fruitful pursuits. But just my 2 cents.
– Matt
Mar 14, 2013 at 17:56
• Please describe your arbitrage strategy properly. Are you long put, short underlying? It's not clear at all from your question. Mar 14, 2013 at 20:31
• nevermind margin? or the fact that the call prices negative infinity while the put prices positive infinity? (from the model)
– user3232
Mar 14, 2013 at 23:34
• Freddy, yes, I'm aware it's a theoretical construct, and my question is a theoretical one. Is that not appropriate for this site? Also, please rest easy that I am occupying the bulk of my time with other thoughts. :) Mar 15, 2013 at 2:17
• In the scenario of holding a short or long position in some fundamental asset, there is no derivative contract involved. Jun 24 at 15:29

shorting the stock for arbitrary lengths of time

Thus what you describe is not an arbitrage strategy.

• That's my point though. If shorting stock for arbitrary lengths of time is not allowed, then how can you delta-hedge this perpetual option? And if you can't delta-hedge the option, how is the price you get under risk-neutral pricing argument the price? Mar 16, 2013 at 0:09
• @Chan-HoSuh You can hedge because you can continuously adjust the position in underlying, i.e. do dynamic hedging. Also note that in general you don't need a hedging strategy to get a price for a contingent claim. Mar 16, 2013 at 10:27
• @Chan-HoSuh, I do not understand your comment here. On one side you say your question is of pure theoretical nature, hence you happily accept the assumptions made in the theoretical framework, but then you say that shorting stocks for arbitrary lengths is not allowed? Which side you want to be on? You have to chose one.
– Matt
Mar 18, 2013 at 1:51
• @Freddy, I didn't say that. I said "if". Mar 18, 2013 at 20:56
• @Alexey, I'm not sure I understand. Dynamically hedging the perpetual put would require shorting the stock for arbitrary lengths of time, would it not? Mar 18, 2013 at 20:57

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.