There's something I've been trying to understand for a while now, and I just can't quite understand, with regards to skew. In the same month, why can't you buy a option that has low implied vol on the skew and sell another option that has a high implied vol and dynamically hedge? For instance, let's say XYZ is trading at $30 and has a normal skew (smirk). If you buy Aug 45 calls at an implied vol of 24 and sell Aug 20 calls at an implied vol of 40 and dynamically hedge, wouldn't you be guaranteed to make money? If stock moved on a 50 vol you would make 26 vol points of profit on your long call and only lose 10 vol points of profit on your short call. Any scenario you construct you would make money in terms of implied vol vs realized vol. Obviously this can't be true, can someone please explain why?
It also depends on at what levels of the spot the higher vol gets realized. In your example:
if you buy an option on a 40 vol expiring in a month and over the next month stock the average vol of the stock is 60 and you dynamically hedge, are you guaranteed to make money? If not could you please give me a simple example perhaps where you'd wind up losing money?
If the high volatility materializes in a region of the spot where your option has very little gamma, and the stock realizes high vol when your option has high gamma. Then even though on average realized > implied you might not have made any money. So your gamma scalped pnl is dependent on the path the stock takes.
So to make the above point more clear. Let's use your own numerical example:
For instance, let's say XYZ is trading at $30 and has a normal skew (smirk) you buy Aug 45 calls at an implied vol of 24 and sell Aug 20 calls at an implied vol of 40 and dynamically hedge
So we know that you are short the K = 20 at vol of 40; and you are long K =45 at vol=24. Now start your blackscholes pricer and look up what the gamma is of your long and your short. You will notice that both options have very little gamma. Therefore no matter what realized is currently your not making much money from gamma hedging, but you are losing your theta. So not good. You need the realized vol to happen near the strikes that you are long (or short).
lucky for you the stock drops like a brick and now trades around S=20. You re very happy cause you are short vol at 40. However, unlucky for you the realized turns out to be 50. So you lose on your short. Maybe you will now say...."hang on, i am also long vol at 24. So therefore i make 26vol points on my long call". Unfortunately for you your long call K = 45 is so far out that it has 0.0 gamma left. So you just got creamed. Had the ultra high vol of 50 realized while the stock was at S = 45, then yes you d be making a lot.
So. lucky for you the stock shoots up again and it is trading now at S=45. You remember that the vol used to be 50 while it was trading low. You hope to get a few high vol days at the strike that you are long. Too bad for you. Vol realizes to be 2. And you lose on your long call and your short call has 0.0 gamma at these levels and you're screwed again.
Long story short it matters where vol realizes relative to the strikes where you have gamma.
The market does not follow Black-Scholes assumptions, as you clearly know : there is a skew and vol levels change. Neither does it follow any other particular known model. So when you say "dynamically hedge" you have to understand this as an approximate hedge that still leaves some significant risk. Vols will move, and not always together and in the way you predict. The skew may flatten or steepen; the spot may be up or down. In the end you have some outcome of profit or loss and no guarantee it is one or the other.
In an interview on Tastytrade, NYU's Phil Maymin talks a bit about this exact question:
A full example of this is also covered in Maymin's book, Financial Hacking.