Question: OTM (out-of-the-money) options are cheap but even decent move can yield dramatic returns that eclipses ATM (at-the-money) or ITM (in-the-money) options, but what I don't understand is that delta and gamma is supposed to be low for OTM so what causes the giant leap in returns when price approaches the strike? Often, it doesn't even need to touch the strike but a large swing can drive up the premium.

Near expiry options (1 week let's say) are also priced cheaper, but does it contribute to the high leverage?

Answer: 
I have a feeling you're looking at large *percentage* increases which is very different that *absolute* increases. Delta and Gamma measure *absolute* changes in value per dollar of change in the underlying, not *percentage* changes. If a option that costs 5 cents goes up another 5 cents, that's a 100% increase, but a $1 option that goes up 5 cents is only a 5% increase. If the 5 cent option goes *down* 2 cents, though, it's a 40% *drop*.

So yes, you can see huge swings in relative returns just because the prices are so small relative to ITM options, but those swings go both ways.

The other problem is that the *reason* those options are so cheap is because there's very little chance that they'll be worth anything at expiration.

Note that this has nothing to do with *leverage*, where returns are multiplied because you borrow money to multiply the amount of money invested, it's just the mathematical phenomenon of larger percentage changes when the denominator is smaller.
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> but what I don't understand is that delta and gamma is supposed to be low for OTM so what causes the giant leap in returns when price approaches the strike
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yeah delta and gamma and absolute returns don't exist in a vacuum

gamma is a measurement of the rate of change of the delta, and delta is the measurement of rate of change of the option price relative to the underlying asset. gamma increases the closer in the money you get, and so does delta (because of gamma) until it gets to a value of 1

I guess thats the answer?