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How can I calculate how much an option would be worth after X days if the underlying stock changed by +/- $Y?
You'd need to know the delta and the theta of the option. You can either calculate them yourself using a model like Black-Scholes (assuming you have a market price and can imply a volatility, and know the other factors that go into the model) or, you can see if your broker quotes "greeks" as well (mine does). The delta is the sensitivity (rate of change in value) to the underlying stock price, and the theta is the sensitivity to time passing (usually expressed in $/day).  So if your option has a delta of .5 and a theta of -.04, when one day passes and the underlying stock goes up $3, the option will gain roughly $1.50 due to the underlying stock price and lose $0.04 due to time passing.