Why are options on Leveraged ETFs cheaper than ETFs — on the same underlying index and expiration?

Question

I had always reckoned that IVs on Leveraged ETFs (LETF) are "increased by the same amount as the leverage i.e. if it is 2x then IV will be 2x. This will essentially double your cost (in my example), while likely paying more bid/offer.". But like others, I have noticed the opposite.

We shall compare LETFs with ETFs with the SAME expiration and underlying index. I know that the strike prices of LETFs will differ from the ETFs'.

1. So how can we fairly and reasonably compare their different strike prices?

2. Are options on Leveraged ETFs cheaper?

3. If true, why? Simply because options on LETF are less liquid? Because options on LETF are demanded less?

Answer 1

It's probably because the leveraged ETF tracks the daily simple return rather than the daily log return. What it means is: suppose the log return of unleveraged version is $\ln(S_{t+1}/S_t)=r$, that's equivalent to its simple return $S_{t+1}/S_t - 1 = e^r - 1$. Now the simple return of the 2x leveraged ETF would be twice of that, hence $P_{t+1}/P_t - 1 = 2(e^r - 1)$. Therefore the log return of the 2x leveraged ETF would be $\ln(P_{t+1}/P_t)=\ln(2e^{r}-1)$ instead of $2r$.

I thought the difference in standard deviation between $\ln(2e^{r}-1)$ and $2r$ is the reason why the leveraged iv is smaller than twice of the original. But in fact there is not much difference. So my guess of why the leveraged iv is smaller is due to the fact that when the daily decrease of the unleveraged version is greater than $50\%$, there's still some residual value in the 2x leveraged ETF from its actual contract construction, leading to a smaller than expected daily move.

This does not seem to be a very convincing answer, please accept other better answers if they appear.