If you want to know what the price of an option will be based on the change in the underlying price, you can use the greeks to get an estimate. For example, if SPY goes up 5%, what will be the future value of a SPY $440 call with 30 days to expiration, SPYprice $437.92 a delta of 0.481 and a gamma of 0.017?
Delta is the change in the price of the option given a change in the underlying price.
Gamma is the change in delta given a change in the underlying price.
Start by getting a good average delta for the entire move in the underlying. You can average the starting delta with the ending delta to get this estimate. We know the starting delta. The ending delta is the gamma * underlyingChange + delta.
Ending delta = 0.481 + gammaEffect 0.372 = 0.853
Average delta = (0.481 + endingDelta 0.853) = 0.667
deltaEffect = stockChange $21.90 * avgDelta 0.667 = 14.61
Ending price of call = startingPrice $7.53 + deltaEffect $14.61 = $22.14
Another way to think about changes in options prices when your prediction is not precise is adjusting the future distribution. If you construct a distribution of future returns based on the time to options expiration that has a zero expected value for the stock, ie summing the percents * stock moves = 0, you can shift the future expectation to whatever your outlook is.
For example, if you think SPY will be up 5% in 30 days, the current distribution D% values the 16-May $440 call at $7.53. Up 5% values it at $22.99.
You can compare the traditional approach to using delta and gamma to forecast a change in options prices to shifting the distribution. The result is close $22.14 vs D% of $22.99.
$22.14 = avgDelta .667 * stockChg 21.90 + origOption 7.53
avgDelta .667 = gamma 0.017 * stockChg 21.9