Options typically do not move as much as their underlying stock unless they are deep-in-the-money and/or very close to expiration. There are valid mathematical reasons for this.
Delta is the amount you can expect an option premium to change given a one-point move in the underlying stock (all other conditions being equal). We derive Delta from the Black-Scholes formula for pricing options. It represents roughly how much the option behaves like the underlying stock. A Delta of .50, for example, means that an option can be expected (all other things being equal) to move about $0.50 for every one-point move in the underlying product. Delta changes with time to expiration as the option moves more in- or out-of-the-money. Volatility of the underlying stock also affects Delta.