early 14c., wrappen, "to wind (something around something else), cover (something), conceal; bind up, swaddle; fold (something) up or back on itself," of uncertain origin, perhaps via Scandinavian (compare Danish dialectal vravle "to wind"), from PIE *werp- "to turn, wind," from root *wer- (2) "to turn, bend." Or perhaps a variant of lap (v.2). To wrap up "put an end to" is from 1926. Related: Wrapped; wrapping. Wrapping paper is from 1715.
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Harper, D. (n.d.). Etymology of wrap-up. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/wrap-up