Can an Option Have Negative Intrinsic Value?

By definition, an option cannot have a negative intrinsic value. The intrinsic value is calculated as the greater of (favorable difference) or zero.

If the calculation results in a negative number ▴ meaning the option is out-of-the-money ▴ the intrinsic value is simply zero. The option holder has the right, but not the obligation, to exercise.

They would never exercise an option at a loss, thus the minimum intrinsic value is always zero.