Suppose X write a 1yr ATM plain vanilla call on S&P500 and Y write a 1 yr out of money call S&P500. And the market implies the ATM implied vol is 16% and 10% for OTM option Y is writing. Both X and Y delta hedge their position using implied vol. Suppose the realized vol is 14% next year (for simplicity, assuming daily realized vol is the same across the year). Regardless where the S&P finishes, X makes money and Y loss money. The daily P&L is the difference between implied vol and realized vol weighted by gamma. Then, why Y is willing to accept 10% implied vol for an out of money option and X requires higher vol. Let’s say all parties on the market writing ATM option is call group X and all parties writing OTM is called group Y. Why group Y is willing to accept lower implied vol than group X. They will be exposed to the same realized vol next year. Is it because gamma is different for ATM and OTM option? Or I miss something here?

there is not an easy answer to this. i suggest you look at the faq. there are different ways of looking at this. one, paper flow sells otm calls as a buy-write which depresses prices. this is done regardless of vol to "enhance" returns. i'm not saying this is necessarily rational, but it happens. two, there is a negative correlation of vol with spot. that is, Y may expect that although vol is 14% now, that if the index climbs implieds and realized will fall. many option traders are not "write and hold." they may roll their position after 6 months or if vol drops.HTH a little.

The reason is that you are trying to look at real life options with Ideal Geometric Brown. mothion. Yes, in GBM you would loose money, but no underlying behaves like one. It may be well mean-reversion model (like commodities), or just non GBM motion. Then, in GBM you consider inplied volatility as known input- wrong again. Stricktly speaking hedging using local vol+Black-Scholes is incorrect- you should use "volatility smile" tree.

Thanks apine and PlainVanilla.apine, I can understand your first point. Price of OTM Call is depressed due to call overwriting. I also understand the negative correlation of vol and spot. But even there is negative correlation, dealers who wrote ATM and OTM will be exposed to the same realized vol: we have only one market, right? If spot is up and both implied vol and realized vol is down, dealers can roll the oringinal ATM and OTM options, both will generate gains. In other words, will the existence of negative correlation impact ATM and OTM differently when spot is up? PlainVanilla, I know using implied vol to hedge is not correct and the index is not really GBM. My question is even the underlying is not GBM and vol is not constant: if I write the option with 16% implied vol and delta hedge using implied vol, the realized vol turn out to be 14% next year, I still have gains. I know I use the wrong formula(BS), wrong delta and wrong vol in my hedging. But if the implied vol gives me the correct intial price and the realized vol is less than implied vol, I would have gains if I use BS to hedge regardless the true underlying process and vol smile. If I can somehow get the TRUE model and hedge correctly, my P/L in risk reward profile (or not risk at all) will be better than using BS. But it won't change the sign of the results(still positiv gain), right?

QuoteOriginally posted by: wonderingPlainVanilla, I know using implied vol to hedge is not correct and the index is not really GBM. My question is even the underlying is not GBM and vol is not constant: if I write the option with 16% implied vol and delta hedge using implied vol, the realized vol turn out to be 14% next year, I still have gains. I know I use the wrong formula(BS), wrong delta and wrong vol in my hedging. But if the implied vol gives me the correct intial price and the realized vol is less than implied vol, I would have gains if I use BS to hedge regardless the true underlying process and vol smile. If I can somehow get the TRUE model and hedge correctly, my P/L in risk reward profile (or not risk at all) will be better than using BS. But it won't change the sign of the results(still positiv gain), right?I suspect the 'flaw' is that you can't really show that, in a completely model-independent way, that your statement istrue. But I am willing to be convinced. Can you post a proof with the fewest possible assumptions about the price process?regards,

QuoteOriginally posted by: wondering But it won't change the sign of the results(still positiv gain), right?No, do not think this is the correct statement. Just tried to do hedging strategy on a constantly rising underlying- you loose. Also about smiles- in equities you have a smile due to the punters want to buy 1p puts and wait until crush (using BS crush can happen only once in 10.000 years, it happens every now and then), for indices- almost the same thing, instead call side has less volatility because there is such a thing as crush in index, but in can not jump to positive side with the same force. Yes, sellers of high vol usually win, but only 9 years in a row. If you want to back-test historically you need to incorporate crashes as well, but not just "normal behavior", which is VERY very similar to BS

Thanks Alan and PlainVanilla. Alan, you are right. I didn't think about my statement carefully enough. If the underlying is GMB with vol varies by time and spot, is my statement correct? I remember the P&L is the difference between realized vol and implied vol weighted by gamma if using implied vol and BS to hedge. But that conclusion is based on GMB with vol varies by time and spot (Am I rigght?).PlainVanilla, I think if the index is constant rising, as long as the realized vol is constant and smaller than implied vol, there is no loss when using BS to hedge. If I write a call for 16% implied vol, and market is up 1% every day for the entire year, I won't lose any money when I use BS to hedge my position. I understand your statement about market crash. You are saying the likelyhood of downside jump is greater than what is in the log normal dist assumption in BS. And dealers writing put option with strike lower than spot will require a higher vol than ATM. Under this theory, the equity vol smile is coming from put side, not call side. But to be consistent with market and not being arbitraged against, dealers have to write OTM call with less vol even though it is the put option that makes vol of higher strike less expensive than vol with lower strike. Did I understand you right?

QuoteOriginally posted by: wonderingThanks Alan and PlainVanilla. Alan, you are right. I didn't think about my statement carefully enough. If the underlying is GMB with vol varies by time and spot, is my statement correct? I remember the P&L is the difference between realized vol and implied vol weighted by gamma if using implied vol and BS to hedge. But that conclusion is based on GMB with vol varies by time and spot (Am I rigght?).Yes, I have seen proofs for that special case -- which means it doesn't necessarily work if wehave stochastic volatility, jumps, or other stochastic factors, right?