Well, you're conflating it a bit.
You are buying 1 option, so that REPRESENTS 100 shares.
Value of shares = 100 bux.
So hypothetically you would pay in your example, 10 bux for the 1 contract option to buy 100 shares at $1. A call option.
If the stock goes up to 2 bux per share, you would exercise your option to buy it at 1 per share.
So you paid 10 bux for the call option. You exercise the option, and you now own those 100 shares with a value of 200 bux, minus the 10 paid for the option, so you have 190 bux in your pocket if you can then immediately sell those 100 shares at 2 bux per.