Whose faith prompted the healing of the paralytic man?

And behold, some people brought to him a paralytic, lying on a bed. And when Jesus saw their faith, he said to the paralytic, "Take heart, my son; your sins are forgiven." - Mt 9:2

On seeing this paralysed man, Jesus goes on to forgive his sins. But it appears on first glance there is no mention of the paralysed man's faith. The "their", whose faith Jesus saw, seems to point specifically to the man's helpers rather than to the man himself. This throws up several possible questions: does this mean the friends' faith saved the man? Was the man saved without expressing faith of his own? Did Jesus forgive the paralytic man's sin without addressing the salvation of those who brought him?

There are at least three possible solutions:

1. "Their" actually includes both the man and his friends
2. Although Jesus specifically highlights the faith of the friends, we can presume the man had some faith given he was willing to be taken to Jesus. So, "their" refers to the friends but the man must have had some faith too
3. The friends' faith was in Jesus' ability to physically heal the paralysed man. Jesus was concerned to demonstrate that forgiveness, as with healing, is entirely a work of God. Thus, Jesus forgives the paralysed man, knowing (a) his sin had been forgiven in eternity past and (b) he will exhibit saving faith upon seeing the proof his sins had been forgiven

The key drawback to option 1 is its lack of consistency with the object of the sentence. It seems apparent the friends are the object of "their", not the man. It also begs the question why Jesus pronounced the man's sins as forgiven when he already had possession of saving faith. Similarly, why was the man forgiven when his friends, whose faith Jesus saw, were not?

Option 2 appeals in several respects. It makes the friends the object of "their" whilst allowing for the man's own faith to be the basis of his forgiveness. The major drawback is that the only grounds to presume the man had faith is the forgiveness Jesus grants. We have no grounds to presume they approached Jesus with any expectation other than physical healing (and, for the paralysed man, we don't know even that). Further, if they already had saving faith, why would Jesus need to pronounce the sins of the man forgiven (a) when this was already known and (b) without pronouncing the same of his friends, whose faith prompted Jesus in the first place?

Option 3 is also appealing in several ways. It maintains the friends as the object of "their" and makes sense of the surprise that came with Jesus' pronouncement (it does appear that nobody presumed the man already forgiven). Further, this reading fits nicely with the effectual calling of Matthew immediately following. The major drawback of this reading is that it might imply some are saved without possessing saving faith. As an initial riposte, we might note (a) the ordo salutis expects a work of the Spirit prior to recognition of our own conversion and (b) if election means anything, it is that God knows in eternity past that our sin is forgiven even though we do not.