Why Leveraged ETF Options May Not Scale Implied Volatility
Summary
The note asks why options on leveraged exchange-traded funds can appear cheaper than options on an unleveraged fund tracking the same index and expiry, despite an expectation that leverage should multiply implied volatility. It explains that a daily leveraged fund targets a multiple of the index's simple return, not its log return. Consequently, its log return is a nonlinear transformation of the underlying log return and does not simply equal the leverage multiple times that return.
The answer initially considers whether the difference between this nonlinear return and a scaled log return explains the volatility gap, then observes that this difference appears small. It offers a tentative alternative: the fund's construction may retain residual value after an extreme daily index loss, limiting the size of its move relative to a naive leverage calculation. The response labels this explanation unconvincing and leaves the comparison unresolved. It supplies no empirical option-price analysis or controls for strike, liquidity, or demand.
Key ideas
- Daily leveraged funds target multiples of simple returns, which do not translate into proportionally scaled log returns.
- The resulting log-return transformation is nonlinear.
- The answer speculates that residual value after a very large daily decline may limit leveraged fund moves.
- The proposed explanation is tentative and does not establish why option implied volatility differs.
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Full text
# Why are options on Leveraged ETFs cheaper than ETFs — on the same underlying index and expiration?
# Why are options on Leveraged ETFs cheaper than ETFs — on the same underlying index and expiration?
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 by MainCom (score 0, accepted)
https://quant.stackexchange.com/a/70471
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.