In Matthew's account of the Last Supper, Jesus speaks about the significance of His blood:
for this is my blood of the covenant, which is poured out for many for the forgiveness of sins. (Matthew 26:28 ESV)
τοῦτο γάρ ἐστιν τὸ αἷμά μου τῆς διαθήκης τὸ περὶ πολλῶν ἐκχυννόμενον εἰς ἄφεσιν ἁμαρτιῶν
"Forgiveness" is ἄφεσις. It could also be translated as "release" (cf. LXX-Leviticus 16:26 NETS). Paul uses this word in the letters to the Ephesians and the Colossians:
In him we have redemption through his blood, the forgiveness of our trespasses, according to the riches of his grace (Ephesians 1:7)
ἐν ᾧ ἔχομεν τὴν ἀπολύτρωσιν διὰ τοῦ αἵματος αὐτοῦ τὴν ἄφεσιν τῶν παραπτωμάτων κατὰ τὸ πλοῦτος τῆς χάριτος αὐτοῦ
In him we have redemption through his blood, the forgiveness of our trespasses, according to the riches of his grace (Colossians 1:14)
ἐν ᾧ ἔχομεν τὴν ἀπολύτρωσιν τὴν ἄφεσιν τῶν ἁμαρτιῶν
What I find surprising is in Romans, where the Gospel is so thoroughly laid out, Paul never uses ἄφεσις. He does use ἀφίημι, the verb from which ἄφεσις is derived (cf. 1:27, 4:7). The second is aquoted from Psalm 32:
Blessed are those whose lawless deeds are forgiven, and whose sins are covered
(Romans 4:7)
Blessed is the one whose transgression is forgiven, whose sin is covered. (Psalm 32:1)
In what way does Paul explain ἄφεσις in Romans? And why does he avoid the direct use of a term which seems to be straight forward and basic to the Gospel?