The market maker controls the bid and ask price only if the security is illiquid and there are no orders on the order book within the MM's bid/ask range. Any trader who places an order at a better price becomes the market until someone else supersedes him
For example, the NBBO quote is $24.70 x $25.00 and the MM is on both sides. If I place a buy order at $24.71 then I become the bid and the MM remains the ask with the quote then becoming $24.71 x $25.00. This new quote will remain until either someone takes out either side or an order comes in at a better bid or better ask price.
Options are the same as above (the auction determines the quote) except that in addition, if the MM is on both sides, his computerized program will adjust the option price as the price of the underlying changes. If this did not occur, there would be price dislocation and free money.
For example, XYZ is $52 and the soon to expire in-the-money $50 call is $1.90 x $2.10 (the intrinsic value is $2.00). XYZ rises to $53 and the $50 call's intrinsic value is now $3.00. In the absence of other market participants, why would the MM leave a stale ask price quote of $2.10, allowing you to buy a call worth $3 for only $2.10? That would be free money. Prices adjust automatically.
There is a small amount of price dislocation with options when the B/A spread is wide and/or the delta is lower. The option's price isn't going to change penny by penny as the underlying changes penny by penny unless the underlying and its options are heavily traded (for example, the SPY).