Yes, it does. It replaces the act of discarding the card to removing it from the game and allowing you to cast it for the madness cost. If it didn't let you cast it at "instant speed" you would never be able to cast it because it tells you to cast the card in the middle of the Madness trigger..
Considering this, can madness be cast at instant speed?
Torment's madness cards inspired a deck known as "Wild Mongrel", or "UG Madness", which, using the signature card Wild Mongrel, enabled instant-speed discard, allowing cards with Madness to be played at instant-speed.
Similarly, is madness a triggered ability? THE OFFICIAL WORD. 502.24a Madness represents two abilities. The first is a static ability of cards that functions while the card is in a player's hand. The second is a triggered ability that functions whenever the first ability is applied.
Considering this, when you cast an instant or sorcery?
Whenever a player casts an instant or sorcery spell, you draw a card. Whenever you cast an instant or sorcery spell, you may put three +1/+1 counters on target land you control. If you do, that land becomes a 0/0 Elemental creature with haste that's still a land. Whenever you cast an instant or sorcery spell, scry 1.
How does bag of holding work with madness?
The ability does trigger on Bag of Holding for cards with madness as it does with any discarded card, it just doesn't exile anything. So madness used to allow you to choose whether or not you want the card to go directly to the graveyard when discarding or go to exile for potential casting.
Related Question Answers
When can you discard a card in MTG?
One is at the end of his or her turn. During the discard phase each player reduces their hand to seven cards if they have more in hand. Otherwise, you discard a card if a card tells you to. If you discard a card with the madness ability you may pay its madness cost to play it instead.Can I play a sorcery in response to an instant?
An instant is just like a sorcery, except you can cast it just about any time you want, even during your opponent's turn or in response to another spell. Like a sorcery, an instant has its effect, then you put it into your graveyard.Can I cast a sorcery during combat?
So no you cannot cast them during combat. 307.1. A player who has priority may cast a sorcery card from his or her hand during a main phase of his or her turn when the stack is empty. Casting a sorcery as a spell uses the stack.When you play a creature draw a card?
Whenever you sacrifice a creature, you may draw a card. Until end of turn, creatures you control gain trample and "Whenever this creature deals combat damage to a player, draw a card." At the beginning of your end step, if you control a creature with power 5 or greater, you may draw a card.What is Storm MTG?
702.39a Storm is a triggered ability that functions on the stack. “Storm” means “When you cast this spell, copy it for each other spell that was cast before it this turn. If the spell has any targets, you may choose new targets for any of the copies.”Are morph creatures colorless?
If you cast a morph creature directly (without paying the morph cost), it isn't colorless. If you cast a morph creature face down for it is colorless.When can you cycle MTG?
Cycling is played as an instant, so you can use it any time that instants are legal. You can even use cycling as a response to another instant. Remember, though, that you draw the replacement card as the batch containing the cycling ability resolves, so you can't play that card until the whole batch has resolved. 
