They have huge leeway. Read the prospectus. They also largely trade amongst themselves. +3x trades with -3x.

Read the prospectus.

collapse can be assured in EV neutral situations due to the fact that one significant enough outlier will take it to actual 0 (as opposed to taking it to a number that is simply approaching 0).

Too lazy. Whether they hold 1x the underlying because they choose to or because they're required to is inconsequential to me.

Feel free to post whatever parts you think are relevant to the discussion though.

This is all pretty funny because it's exactly why your double short martingale strategy will eventually break you.

You're risking 100% of your portfolio because a collapse that gets you at most 100% of your initial position size "can be assured" on an infinite timeline. If you don't go broke first.

I have to imagine CDL is trolling at this point. He offered a 25k bet on a 10 year time horizon, and is now claiming he was only ever talking about an infinite time horizon, when his whole 'infinite timeline' point is completely irrelevant to the topic of this thread, since it applying to all investments.

Anyways:

Lets add some parameters, straight off SPY to start

Daily drift = 0.00034487351697150048
Daily vol = 0.011808957609098045

Formulating mean reversion as the likelihood of the return to make the price trend towards its long term mean:

Let us see how the expected value over 10 years changes, depending on our assumption of mean reversion.

The unleveraged product will always have a mean of 2383, since mean reversion does not impact its mean, by definition.

The leveraged product should have exactly double the ev with no mean reversion, and effectively double the ev with 100% mean reversion (no real volatility), with some sort of curve in between.

My results are suspicious, so I'll have to review the code. It's possible that even a small drag compounds heavily over time, meaning that if mean reversion is >0 it doesn't matter how much there is it will drag all the very positive scenarios over 10 years, and therefore we benefit from higher mean reversion which decreases volatility. That doesn't seem likely to be true.

I was talking about infinite horizons when arguing that they MUST collapse. I was talking about decades when discussing that the will probably collapse (meaning >50% of them going down on the aggregate, but not that each individual one will go down) if you're allowed to invest in a basket of levered ones of your choice.

Then Pete wanted ridiculous constraints like upside capture and stuff. I was never arguing about the asymmetric returns which obviously have skew. I was demonstrating that the median of actual returns of intermediate term timeframes of a broad basket of hypothetical short ETFs would be negative.

I'm not sure if you actually believe what you are saying here


In this conversation you were clearly okay with proportional payouts, with your only concern being escrows. You were not trying to bet on the median and everybody had long ago agreed that the median loses money.

You're either goalpost shifting or re-imagining your previous thoughts, not sure which.

ibavly, i think your results are correct but the way you define mean reversion is suspect.
if i understand it, when you have mean_reversion = 100% it just guarantees a constant return on the underlying every day? which would mean no volatility.

i would rather define mean reversion as a tendency to return to past price levels than to drift toward an expected return. in any case, however you define it, it shouldn't reduce volatility.

i guess i did know that they did move from some outright buying to mostly swaps or maybe they were always mostly swaps.

but if the market if up 1.5% today. and you have $10B double leverage fund and it's contracted at end of day to increase 3% equity exposure ($300MM) at the close to rebalance. wouldn't their counterparty have to be buying that stock to hedge............. isn't the buying pressure still there in an indirect way?

much more extreme example is something i've wondered about: what happens if you go to dealer and want to do $1B high yield bond index swap? so someone offers you swap but then they need to do hedging trade in something that has hundreds of underlying components that never trade...

basically all this stuff needs to be hedged - at least partially - so i'd think buying pressure for s&p 500 would be there at the other end........ i agree it's different than knowing they'll be some gigantic hyper-inflated print at the end of the day that you can easily front-run.

shun (shum?) at york university did a good paper on this.

they offer you a price that compensates them for all the risks. if they don't want or can't take on the risk and can't hedge it in the open market, they might try to find someone to take the other side of the trade.

he seemed to mean proportional at 1 to 1 when he started talking about paying over the cap. I imagined the capped payout being triggered at like 10% gain or loss and anything below that being proportional. I stopped debating it when it was clear he wouldn't engage in a binary bet or something close to it. He wants to take advantage of the skew when my argument about math is not about EV and skew, but rather median outcomes in the intermediate and the leverage taking these to 0 on an infinite timeline.

I am still willing to bet 25k if I get to pick the basket and the max payout is obtained at a relatively low return without excess return mattering.

If I wanted proportional payouts as he seems to want I would just buy a structured note.

Refreshing my memory, what I meant to go for is something like this

dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}

so the mean reversion parameter only affects the drift

Thats the challenge of institutional trading, if you're into that sort of thing. You can probably try to clear out some of the exposure by seperating the risk factors and delta hedging the EQ(idio), IR01 and CS01 with different products.

The ETF providers that have levered ETFs also have inverse ETFs and the exposure across different underlyings also has correlation so they have correlated swaps elsewhere to mitigate risk too.

Daily drift = 0.00034487351697150048
Daily vol = 0.011808957609098045

New mean reversion formula:

With these historical parameters it seems like a 2x return product is excellent even with severe mean reversion. Fwiw you need to get over 50 years before the 2x starts underperforming, even with 100% mean reversion.



leverage starts getting dangerous quickly, at 3.5x you don't have much room for error (keep in mind a trending market would have benefit in opposite direction):



If we get rid of the drift it shows the 2x leverage clearly hurting with mean reversion:

Thanks for posting those. That's starting to look a lot more like what I'd expect from the results.

Just so I'm understanding what you're doing, your expected_value is just the drift * time? So if mean_reversion = 1 your process just fluctuates around the y = drift * time by the daily volatility?

If I'm understanding this correctly, a mean_reversion term of even 0.1 would be incredibly strong? So if you plotted the path with mean_reversion = 0.1 you'd see almost a straight line over say, a 10 year period?

Interesting question. You have my mean reversion formula correct. Do you have a better way of formulating it? It does seem like any non zero mean reversion has to have a big impact. If you look at the paper I posted above they make an argument for mean reversion not actually existing. I think thats especially likely in the long run.

mr=0



mr=.1



mr=1

Stepping away from a conceptual discussion of mean reversion for a bit, here is what would have happened if you had bought SPY with daily leverage of returns in 1993 and held until today:



Changing the y axis to log to see more clearly how the excess leverage can start hurting:



I can run this easily for any index/time period with publicly available data if anyone wants

I think your formula is probably okay, you should just be using much smaller mean_reversion values for realistic results

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