If all puts and calls on an underlying are related via put-call parity, boxes, and rolls, how can different IV's exist on different strikes and expirations? Why aren't these immediately arbed out?
They are related in exactly that way. You're probably forgetting that one side has slightly different IV due to the cost of carry involved in performing the conversion. McMillan has detailed information on this.
Except for the put-call parity those relationships do not imply single volatility number. So, there's no way to arb out volatility skew/smile.
I'd be surprised if the cost of carry is the reason for the subject IV discrepancy. Are you sure that McMillan says that? The cost of carry is part of the pricing formulas and should not have to be fudged by the IV figure which sometimes is a catchall for many unknowns. For dividend paying stocks I aways suspected that this discrepancy was due to the dividend not accurately input into many of the web site IV calculations. For non-dividend stocks I never noticed that the IV discrepancy was significant, often swamped out by the B/A spread. Unless, of course, the underlying stock was not shortable. Edit: My comments apply to IV discrepancies between Put/calls at the same strike which I think is what Damon was addressing. Don