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?

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    $\begingroup$ No, I believe they are a theoretical construct by P. Samuelson nd R. C. Merton. There are none actually traded to my knowledge. $\endgroup$ – Alex C Feb 21 '18 at 20:13

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:

  1. Actual underlying prices may not actually follow a random walk, a-la GBM;
  2. 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.

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