Do market makers price inverse leveraged ETF options simply by using the IV of the un-leveraged version and multiplying that IV by the leverage factor? I've assumed that was the case (here's an old discussion). But it seems that's not always true... Here is a comparison of SPXU (-3x ETF) put IV and SPY call IV. This is using data from the middle of the day on 11/2/18 for options expiring 1/17/20. The IVs seem roughly aligned, as I would expect. BUT the following is from 10/25/19 for options expiring 1/15/21. Why is the SPXU IV noticeably higher? (Approximately 1.5 vol points higher.) When analyzed this way, the inverse leveraged ETFs seem more expensive lately - not just a one day fluke. And across a number of such products (e.g. SQQQ, SDOW). Am I over simplifying by assuming the pricing should be the same with just the IV adjusted by the leverage factor? (By the way, the IVs were calculated by using implied forwards for both products to account for borrow rates and dividends, and the Black76 formula. Then SPXU IV and %-OTM is divided by 3. Result shows IV of bid/ask/mid.)
One way to think about this is to try to buy a portfolio of 3X long and 3X short options that will exactly balance each other out, assuming the ETF did exactly what it was supposed to, namely moved 3X and -3X the daily percentage move of the underlying. Set up a spreadsheet with 5 days of random moves of the index, calculate the value of each ETF at the end of each day (remembering that it's 3X the daily percentage move!). Then calculate just the intrinsic value of the options you chose, as if they expired on that 5th day and were european options. You'll find that while they mirrored each other the first day, from the second day on they are way out of whack with each other. Because after more than 1 day the 3X and -3X ETFs aren't going to mirror each other, so the options won't either. And because delta between the strike and each ETFs price are now so much different, options only amplify that issue. The answer to your question is really in the fact that the ETFs mirror the daily percentage change but your math is assuming they just equal the SPX times 3 or -3 at the end of a period lasting more than one day. But if you do the little exercise I describe above it will be much more intuitively clear.
Market makers need to price options based on supply and demand which is reflected in IV. So basically if someone figures out that options are too cheap then they may keep buying them while MMs will need to keep increasing their price (and vice-versa). And they need to keep wide bid/ask spreads to not fall pray to various models that may price them differently and beat them. Along the line they all analyze data and improve their models over time. So for example if you’re using a model that shows that those options should be cheaper then you should sell tons of them, but then you may also affect their price. Unless your model is so clearly wrong that MMs can hedge using the underlying or other/different instruments. Though in this case I suspect the high IV may be due to compounding effect where a leveraged ETF may compound its gains or losses quicker than the underlying. If the underlying goes up by 10% from $100 to $110 in a day then 2x etf would go up by 20% from $100 to $120 (extreme example). But if this happens again the next day then the stock would end up again 10% higher at $121 while the ETF at $144. So the stock went up by 21% total while the ETF by 44% over 2 days (compounded daily), more than twice the stock’s move. They get even more out of sync when moving in both directions.
Have not done the exercise but if you use take the binomial tree of the underlying vs the 3x leveraged, should be able to work out the payoffs and then you can price the option from there.
Thank you all for your comments. I'm familiar with the effect some of you refer to (and very elegantly explained!) - where the levered/inverse products diverge over time. But if that is reason to boost the IV of SPXU, what were "they" thinking back in 2018 when it was not more expensive? Did the market makers just not incorporate that into their thinking? I thought this effect has been well understood by the pros for some time.