In the Longstaff and Schwartz method of pricing American options, what is the point of the regressions at each step?
The goal is to approximate an optimal stopping time for each path. However, why approximate, when you can get an exact stopping time for each path by just ... you know ... using the values you've simulated to figure out when the best stopping time is?
For example, say the maturity is $T$, and at time $T - 1$, we want to estimate the continuation value for each path. But I already know what the continuation value is for each path, I know it exactly, I just use what the stock value for that particular path is, and then calculate what my option payoff will be in the next period. Why do I need to use regression? What is the point?