# Perpetual American put option with zero interest rate

I want to find an optimal time when we should exercise perpetual American put option.

In other words I want to maximize the following equation:

$$V(S) = \sup_{\tau \in \mathcal{\tau}}\mathbb{E}[e^{-r\tau}(K-S_{\tau})|S_0 = S],$$ where $K$ is the strike price and $S_t = S_0 e^{\left(r-d-\frac{\sigma^2}{2}\right)t + \sigma W_t}$

I know that this problem can be solved using PDE approach.

But I that about the case when $r = 0$.

Then we obtain: $$V(S) = \sup_{\tau \in \mathcal{\tau}}\mathbb{E}[(K-S_{\tau})|S_0 = S],$$ where $K$ is the strike price and $S_t = S_0 e^{\left(-d-\frac{\sigma^2}{2}\right)t + \sigma W_t}$

My question is: Is there an optimal time when we should exercise such option?

Intuitively: I think that if $-d-\frac{\sigma^2}{2}\leq0$, then we should wait as long as possible to get the maximum profit, because a drift of $S_t$ is nonpositive so an optimal time is: $\tau^{*} = + \infty.$

In turn, if $-d-\frac{\sigma^2}{2}>0$, then there is no optimal time to exercise it, so formally we also should assume that $\tau^{*} = + \infty.$

Am I right in my considerations? Although they are not too formal mathematically.

• This is treated very well in the famous Shreve working paper, here efinance.org.cn/cn/FEshuo/stochastic.pdf. Chapter 25. I cannot recommend this document enough for quantitative minded derivatives enthusiasts and practitioners. – Ivan Apr 28 '18 at 11:35
• Thank you! I know this paper. Why you cannot recommend it? I – MMM Apr 29 '18 at 12:42
• I can’t recommend it enough ! It’s a very good general quand finance resource. – Ivan Apr 29 '18 at 18:10
• I am sorry, but I cannot find exactly where in this book a situation with zero interest rate: $r = 0$ is covered. – MMM May 3 '18 at 14:53