# For equity options, why sometimes ATM vol of shorter expiration is higher than that of longer expiration?

Basically a negative forward vol in the ATM vol term structure. For index options, it's probably rare. But for single name options, I've seen a bunch of examples on Bloomberg. Does this relationship admit some weak form of arbitrage?

Update: I think I have falsely claimed about the forward vol in this situation. Some people might call it "nagative forward vol", but now I don't think it makes much sense (might just be an analogy to interest rate). And as commented in one of the answers, the implied forward vol should be calculated as suggested on Wikipedia, assuming that the two periods are independent of each other. Otherwise, some correlation coefficient is needed in the calculation.

Now I read a bit more on this topic, this kind of phenomenon doesn't admit any arbitrage, but another situation would. If we factor in the time component and calculate the implied variance as $\sigma^2T$, then a "negative forward variance" would admit arbitrage under Black-Scholes dynamics.

• 1. Are you familiar with calendar arbtirage opportunities (both expressed in terms of total variance in the (log-forward-)moneyness x time space and in terms of option prices)? 2. I think that if you gave precise example it would help steer the discussion. What I mean is that it is possible to observe such a phenomenon as an artefact of the method used for deriving the implied forward curve and volatility surface in the first place. – Quantuple Dec 14 '16 at 8:21
• @Quantuple Thank you. Yes I just updated my question to distinguish the total variance from implied vol. I was writing about one but thinking the other. And to your second question, as mentioned in one of the answers, I believe it should be fairly common to observe this phenomenon. – Will Gu Dec 14 '16 at 8:58
• Indeed now I understand what you mean and you are right. – Quantuple Dec 14 '16 at 9:03
• @Quantuple By far the best explanation on here. Seems like very few, have the experience you have. – Ted Taylor of Life Jan 18 '17 at 4:45

## 2 Answers

It doesn't have to imply negative forward vol. for example , 1month 20 Vol and 2 month 19 vol implies (approximately ) that 1 month vol in 1 month time will be 18.

A large part of single stock volatility comes just after and around quarterly earnings announcements. An option with short expiry over one announcement thus may price at higher annualized vol than a longer expiry that also encompasses precisely one announcement.

• This makes a lot of sense. Thank you for your explanation. – Will Gu Dec 14 '16 at 8:01
• @MatsLind Your answer only holds for that specific case.... – Ted Taylor of Life Jan 18 '17 at 4:43