Intepretation of $\begin{Bmatrix} \tau(\omega) \leq t \end{Bmatrix} \in \Im_t$

Question

I've read a lot of posts on the subject but I can't figure it out yet.

How do I interpret the definition of stopping time as from title? If $\Im_t$ are the information that I have in $t$ (now) e $\tau$ is the instant I decide to exercise my American option based on the information in $t$, how can I exercise the option in a time previous to $t$? Trivially, the time $\tau$ is passed...

Answer 1

In plain words, $\tau$ is a stopping time (or rule). For example, $\tau$ will be the first time when something has happened. Then $\{\tau\leq t\}$ is the event that such time (or rule or event) has happened before $t$. Being the event $\{\tau\leq t\}\in \mathcal{F}_t$ is saying that by time $t$ and with the information $\mathcal{F}_t$ at hand, it will be possible for us to know whether $\tau$ happened or not on the time interval $[0,t]$. It's simply that :)

Again:

1. We are at time 0 now.
2. We make up a rule for exercising some option (or doing something) if some even happens. If such event happens we exercise our option, and this will happen at a random time $\tau$.
3. We fix some future time, say $t>0$. Being $\tau$ a stopping time, that is being $\{\tau\leq t\}\in \mathcal{F}_t$ means that it is possible for us to now know, whether we will know whether we will have exercised (or not) the option by time $t$.

3') It can also be said as follows: at any time $t$, it is possible to know whether the event has happened, hence, whether we have exercised the option at time $\tau$.

Examples in the context of finance:

i) Sell when the stock price reaches $100\$$ (it is possible for us to know at any time whether this has happened.

ii) Sell when the stock price is at its maximum value. This is not a stopping time, because we can not know when the maximum is reached before the price goes down again, in which case, it is too late to exercise the option.

Maybe it was too much overexplained but I hope it helpes :)

Answer 2

Example. We flip a coin repeatedly. $\Omega = \{\omega : \omega = (s_1, s_2,\cdots), s_j \in \{H,T\}\}$ Our "time" set is $\{0,1,2,\dots\}$. Time $0$ is before the first toss, time $1$ is after the first toss but before the second, etc.

Betty's stopping rule is: she will stop the first time tails appears. Exercise: prove that Betty's stopping rule is a stopping time. $\tau(\omega) = \min\{k : s_k = T\}$.

Alfred's stopping rule is: he will stop just before the first tail. Exercise: prove that Alfred's stopping rule is not a stopping time. $\tau(\omega) = \min\{k : s_k = T\} - 1$.

There is some mathematical work on "prophet theorems", where we picture a gambler who knows the future compared to a normal gambler who does not, and analyzes how much advantage the prophet has.

Example:

Harten, Friedrich; Meyerthole, Andreas; Schmitz, Norbert, Prophet theory. Prophet inequalities, prophet regions, games against a prophet, Teubner Skripten zur Mathematischen Stochastik. Stuttgart: B. G. Teubner. viii, 210 S. (1997). ZBL0886.60033.