Options are always in the black
First of all, you need to define what options you are talking about. Consider a digital option for example which is really fairly vanilla since you can proxy it as a combination of two European callswhich pays 1 of the stock is beyond a certain level at maturity and nothing otherwise.
The intrinsic value of the option is just 1 or 0, but would you pay more than 1 to buy such an option? In fact, there is a general point here, a option with a limited upside can have a price lower than the intrinsic value.
And this may even apply to European vanilla options, although in very specific circumstances: take a European put option with 0.
Also it can not happen for American options there would definitely be an arbitrage here, since you could by the option and exercise it right away.
You buy the stock and a European put option with maturity T and strike K. To avoid arbitrage, you need this cost to be greater than the minimum of 1 and 2 above, which is K. This is the actual arbitrage condition, which is close to saying the intrinsic value is a lower bound of the put option but not quite.