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.
This is the option table for a stock that closed Friday at $175.50.
These options expire this coming Friday, i.e. 5 trading days left.

[![enter image description here](https://i.stack.imgur.com/jUltZ.jpg)](https://i.stack.imgur.com/jUltZ.jpg)

I've been trading options for over 35 years, and never took the time to study "the greeks." What I know from this table is that the market is telling us this stock is not likely to move $5 in 5 days. If it did, your $2.15 bet on the $175 strike would go to $5.50, or a $0.95 bet on $177.50 would be a $3 return. The former is a 2.6X return vs 3.2X return for the latter. Stocks can and do move 10% in a week, but it's most often as a result of remarkable news, or earnings. 

In the case of this particular stock, it's not hard to find times where a return of 10X was available for a 50% move in the stock over a longer period, 18 months. 

If the current answers don't really address the question as you asked it, you should study the Black Scholes options model for the effects of time and volatility on the options price. It's all about risk, reward, and the bell curve.