In the umbrella example, that's more or less what it is.

そのかさは私のです。 = そのかさは私の物（もの）です。

I think the real answer is a little more complicated, in that particles such as の and は establish the grammatical function of the word they're attached to. の is mostly used in constructions of the form ＡのＢ, but it can also work on its own.
In this case you can just think that the 物 was dropped though.

大きいのはいくらですか。

This case is a bit different, as you wouldn't say 大きいの物は….
Contrast it with this though: 大きいはいくらですか。 "How much is big?"
Well, how much is big?
It doesn't make any sense without clarifying particle, and の is the one to use in this case. You may also say 大きいやつは…. "The big guy..." or "The big thing...". の basically fulfills the function of やつ here, and since it's purely grammatical it doesn't have any connotations (that やつ can have).