In an illiquid market, the market maker sets the prices. His computerized programs will automatically adjust the options price when the underlying changes enough to warrant a change in the price of the option (delta of option times the underlying's price change).
In a liquid market where market participants are the market, there are various types of algorithmic orders in play. Similarly, they automatically adjust order price as well. Some of these might be:
Various pegged orders
One type of pegged order would be the Pegged-to-Stock Order. Like the market maker explanation above, it continuously adjusts the price of the option order by the product of a user defined delta and the change of the option's underlying stock price.
Your broker must offer such order algorithms in order to be able to utilize them.