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How do you determine the strike price and duration of the puts you're interested in, say for the S&P 500?

I generally do a black mass facing east at midnight on Hallowe'en.

Really, there is no particularly good answer.
What I do is pick a lot of different strikes, at least 2 at each.
The time value of each one is maximized when the index is equal to the strike,
so I close 1 of each strike as the market passes down through the strike and goes into the money.
At that point the other one at that strike is "free" insurance.

Then, when I think the market has bottomed (or bottomed enough for me), I close the rest.
It's not worthwhile trying this unless you have a strict rule for closing them.
They gain value during a crash, but they lose value VERY fast during the rebound.

Remember that any money spent on puts will probably be a dead loss,
and it's pretty unlikely that the few that make money will make up for those losses.
The entire strategy is likely to cost money.
Spend an amount on them that is appropriate to that foreknowledge.
The only thing going for them is that, when they do make money,
it is at an extraordinarily handy moment to have some spare cash arrive.

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