Why did my option go up less than the stock?
Because an option's price moves by its delta, not dollar for dollar. A call near the stock's price has a delta of about 0.5: it moves about 50 cents a share when the stock moves a dollar. Delta grows as the call goes deeper in the money and shrinks as it goes out, and time and implied volatility move the price too.
Try it: a call on a $50 stock, strike $50, 30 days left, 30% implied volatility (made-up numbers). Move the stock and compare 100 shares with one contract.
Simulated example, priced by the StrikePath engine (Black–Scholes, interest 4.3% a year). For learning; not advice.
Why less, and why it changes
A call at the stock's price has a delta near 0.5 (0.53 here), so for a small move its price follows the stock about half as fast. Delta measures how fast the price moves, not the chance the call pays. Deep in the money it behaves almost like the shares (a delta near 1); far out of the money it barely moves (near 0).
Less per dollar, but on less money
One contract covers 100 shares but costs far less than 100 shares, so a move of a few dollars can be a large share of what you paid, either way. The most you can lose on a bought call is what you paid.
What this shows, and what it doesn't
The stock moves at once here, with the days and the expected swing held still. In a market, time passes and implied volatility changes too, which can add to or take from the move.