@ CM2DD: Same thing as my comment on wupper: The couterpart of "one's" is "mine" and "its", not "my" and "its".
This topic shows very well why it makes sense to distinguish clearly between pronouns and determiners.
a) Personal pronouns
Subject case: I, you, he, she, it, we, you, they
Object case: me, you, him, her, it, us, you, them
Possessive case: mine, yours, his, hers, its, ours, yours, theirs
b) Possessive determiners
my, your, his, her, its, our, your, their
In this context, "one" is another personal pronoun with the subject case "one", the object case "one" and the possessive case "one's". Since there is no specific possessive determiner for "one" (as there is no specific possessive determiner for "Peter" either), we use the possessive case of "one" to mark possession. (Wenn man das Deutsche "man" als Gegenstueck sehen will: dort wird ersatzweise der possessive determiner der dritten Person Sg. neutrum verwendet.)