Option prices are largely influenced by their intrinsic value: the underlying price, and the strike price. However, what effect do other factors have, including option demand? Is this ever reflected in option pricing?
Change in the underlying's price affects both intrinsic and extrinsic value unless the option is 100 delta and then, it only affects intrinsic value.
Demand for options increases implied volatility which in turn increases extrinsic value.
Carry cost and dividends also affect premium.
Other factors. The movement pattern (volatility) of the underlying would be the biggest factor. Even if an option does not trade, a change in movement pattern (volatility) of the underlying will have an enormous effect on how the option will be priced. Rarely are options priced far from the actual volatility of the underlying (corresponding to the term of the option). This becomes particularly true with longer-term options.
I should mention, obviously intrinsic, as you stated, but I never think about intrinsic. For me, an in the money call is a put, and an in the money put is a call.