# Prove that the value of a perpetual American put is time-independent

I know that the value of a perpetual American put is time-independent. I think it is very intuitive property and it results from the fact that we do not have any expiry date.

My question is: Is it possible to prove this fact in a more formal mathematical language?

• There is no expiry, so the price function is not a function of time $t$, it depends only on the level of the underlying, and market conditions... Mathematically, you can view it as the limit of the price of a regular american put when the expiry $T \rightarrow \infty$. Also, if you take two different times $t_1$ and $t_2 \neq t_1$ for which the underlying level and market conditions are the same, the price is necessarily the same, otherwise the price function wouldn't be a function (two different prices for the same parameters ($S$, $r$, etc.) – byouness May 9 '18 at 8:28
• I would put it this way "if a formula to value the perpetual American put exists, that formula is time independent". And the argument is the intuitive one that you gave. However it might still be that the put value is undefined, or infinite, diverges, etc. – Alex C May 10 '18 at 21:05
• Thank you. I have another question: As you wrote, if I will obtain that the put value is undefined, what can I say about this option? Can I say that there is no optimal strategy to exercise such option? – MathMen May 11 '18 at 8:44