Here is a simple version. Let's say you have two options, one is with the strike of 100% and IV of 20%, the other one is struck at 90% and has IV of 23%. So that's a skew of 3% which means that the market expects that if the underlying goes from 100% to 90% over the life of these options, it's going to realize an average vol of 21.5%. So, if you are delta hedging and it follows that path, if volatility is higher then 21.5% you will lose money and if volatility is lower, you will make money. In real life, however, this all would be complicated by a variety of facts (your delta model, hedging frequency, hedging vol etc).
I don't understand why in this example you lose money if realized vol is higher than 21.5% and make money if its lower. I see how that would be the case if you had sold both options, but since you are buying the 100% strike and selling the 90% strike shouldn't you be neutral to realized vol? Say realized vol ends up at 21.5%. Why is it not the case that you would profit (23-21.5) = 1.5 from the put you sold and (21.5-20) = 1.5 from the call you bought? I get that it has to do with the underlying moving from 100% to 90%, but am struggling to understand why that matters.
Because your delta-hedged P&L is the change of underlying weighted by the gamma on any given day. At the inception, long gamma from the ATM option dominates your P&L, while as the underlying is moving toward the other strike, at some point short gamma from the wing will be dominant.
Ok I think I am beginning to understand. Would another way of saying this be: the IVs from the call and the put are not fungible. In an extreme case where the underlying falls significantly the overall position is reduced to being short the 90% put and delta-hedging, which is why you lose money if realized vol ends up higher than 21.5% (lower than 23% to account for the premium you paid for the 100% call). EDIT: Disregard that last sentence in parentheses "lower than 23% to account for the premium you paid for the 100% call". It's 21.5% because thats just what the skew is saying about realized vol in the 90%-100% price area
I am not sure, to be honest, I need to think about your analogy a bit. Best way to think of it is that once you introduce multiple strikes, your are going to be mainly accruing the implied/realized difference for the strike you are closest to at the moment.