Your bear market comment is irrelevant. Mr Chanos is short several biillion dollar of stock, and he gets paid on outperformance vs inverse S&P. He is long a boatlad of SPY ETF to hedge (he could use ES if he wanted)
I also wonder how these things work. If you're short stock, but long SPY ETF, there must still be a directional bet here cause one position is bigger than the other. How does the math work with hedging? If your stocks go down 10 million, how much SPY ETF long do you need and what will the payout be? If you make 5 million on the SPY, but lose 10 million, you're still down 5 million, but why not just lighten up your short stock position? I really don't get how this hedging works. Perhaps if its more like insurance in that its really cheap to buy insurance, but if the market crashes they maybe you have some crazy PUTS that were worth nothing but now gain massively, this I could understand. But short stock and long SPY ETF doesn't make sense to me for the above reason.
You can be short a stock (or portfolio of stocks) and long the market in general. There's no contradiction and you can profit from all your positions even if the market rises sharply. To answer your 2nd question, if you thought there was short-term market risk (e.g poor US farm payroll data) and you wanted to hedge, you would need to calculate your SPY ETF position and deduct your short stock portfolio position x its correlation with the S&P Index. Once that is done you now know your net correlated position that needs to be hedged by selling S&P Index futures.
Yes, but the idea is that you're using one of those trades as a hedge, correct? So you know that one will cost you money, but the second should make money, and that's why you do it. Its just like insurance for shipping lets say. You got a million bucks worth of stuff on the boat that you hope crosses the ocean and gets delivered. You pay 20k to insure it, and if there is a big storm and the boat sinks, that 20k cost will pay out the million dollars worth of goods that were lost. But in the case of trading, if you think your short stocks are actually at risk, and so you hedge with the SPY, sure you make some money on the long in case your short is wrong, but its not like you can ever fully protect your position, right? If your're worried that your short will cost you 100k, and hence you hedge with the SPY long to make back this short loss, then if you are right on the original short, then your hedge losses money in the equal way, no? In other words, I can see buying insurance in the form of deep out of money calls or puts in case there is a severe reaction, but I don't see how protecting from small market moves does anything because either your original position or your hedge loses money, or vice versa.
My guess is the hedgers would be people especially large corporations who usually holding assets, such as bonds or interest rate futures, that have negative correlation against stocks. That would be an indirect hedge, taking advantage for most likely calls options on SPX are relatively cheaper than puts.