Perpetual American Calls, Warrants, and Common Stock Analogies
Summary
The document considers whether perpetual American call options trade on exchanges and describes two related instruments: perpetual warrants and common equity. It reports that perpetual warrants are generally issued by companies and traded over the counter or through a registered broker-dealer, while common stock can serve as a conceptual analogue to a perpetual call with a zero strike.
Using the Black–Scholes call formula, the answer argues that as maturity tends to infinity, the value of a call on a non-dividend-paying stock approaches the stock price. This creates an apparent puzzle: a positive-strike perpetual warrant might seem no more valuable than the underlying stock. The explanation offered is that real stock prices may not follow geometric Brownian motion and that equity value reflects expected dividends. The discussion is theoretical and does not establish that perpetual calls are exchange-listed products; it also cautions against applying the model without considering dividends and actual price dynamics.
Key ideas
- The answer reports no known exchange trading for perpetual call options.
- Perpetual warrants are described as company-issued instruments commonly traded over the counter.
- Under Black–Scholes assumptions, a very long-dated call on a non-dividend-paying stock approaches the stock price.
- Common equity resembles a zero-strike perpetual warrant in this limiting argument.
- Dividend expectations and departures from geometric Brownian motion limit the analogy.
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Full text
# Are perpetual american options traded on real stock exchanges?
# Are perpetual american options traded on real stock exchanges?
I am looking for any information about perpetual american options from practical point of view? Are they traded on stock exchanges? Do investment banks deal with such products?
## Answer by David Addison (score 2)
https://quant.stackexchange.com/a/38387
I am not aware of perpetual call options being traded on any exchanges, but there are two very close analogues.
Perpetual warrants are issued directly by companies so trading is done OTC or through a registered B/D.
Common equity is also a very close analogue to a perpetual American call option.
The theoretical value of a perpetual warrant (or option) for a non-dividend paying stock is equal to the value of the stock itself (Samuelson, 1965; Merton, 1990).
If we have standard Black-Scholes, we can see that as $t \to \infty$, $\varPhi[d_1] \to 1$, and $\varPhi[d_2] \to 0$:
$$ V_t[S_t,K,\sigma_S,r,t] = S_t \varPhi[d_1] - K e^{-r (T-t)} \varPhi[d_2]$$
where: $d_1 = \frac{\ln\left(\frac{S_t}{K}\right)+{(r+\sigma^2/2)(T-t)} }{\sigma \sqrt{T-t}}$; $d_2 =d_1 - \sigma \sqrt{T-t}$; and $\varPhi[X]$ is cumulative distribution function (of the normal distribution).
Therefore, common stock may be viewed as a perpetual warrant on itself with an exercise price of zero. However, this results in a paradox because we would expect a lower price for a warrant with a strike price greater than zero.
The reason for this paradox is two-fold:
- Actual underlying prices may not actually follow a random walk, a-la GBM;
- Stocks' values are premised on future dividend payments; there is no such thing as an equity (with $S_t\gt0$) which has the perpetual expectation of zero dividend payments.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.