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I'm trying to better understand leveraged etfs, and specifically how they have convexity and volatility decay similar to options.

An older post on this site asked a similar question and one of the respondents and the article they linked talked about how if you pair trade 2 leveraged etfs, where you either short 2 related leveraged etfs or go long two leveraged etfs. The idea being, that by doing so you're creating a position similar to a straddle, so if you go long say SPXL and long SPXS you're long a straddle and you're long gamma (convexity) and short theta. But where does that show up? I created a simple example in excel where I tried to simulate something like this, but all I see is 0 PnL and no gamma and no theta.

I created a simple simulation. I assume you have 2 triple leveraged etfs, one is a triple long, the other is a triple short. I assumed the underlying index moves randomly anywhere between -15 and 15%, and the triples obviously move 3x each day.

I assume that both indices start off at $100, and we purchase 1,000 units each, and then systematically re-balance a the end of each day to maintain a 50-50 exposure.

When I do this, my portfolio value, unsurprisingly remains flat at $200k.

As an example, the first day we come in with a position of +1000 units in the 3x Long etf, and +1000 in the 3x Short etf. The index moves down 7%, so the long etf declines to 79 dollars and the short etf declines to 121. Portfolio value remains flat at $200k

Then I rebalance, increasing long index exposure to 1.26k and decreasing short index exposure to 826. Same result. I only included 10 days of data, but I tested this multiple times and nothing changes, this isn't surprising after all.

If we assume r is the return of the underlying index our portfolio value is this for any given day:

On the first day we have:

100k *(1+3R) + 100k(1-3R) = 200k

. So it never changes.

I must be missing something, and I can't figure it out. Where is the convexity, where's the theta? Can someone please explain?

Simulation

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Both products actually have positive convexity, they will buy more underlying (SP500) when the price goes up and sell it when it goes down.

However, if you hedge every day, you will just cancel out that gamma convexity. You have to let the position run a few days if you want to trade the gamma, because it is generated by the daily hedging of the 3x etf, not intraday.

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  • $\begingroup$ Ah I see. Thank you, this makes sense on the one day horizon the return is linear, but beyond that it compounds hence the convexity. I'll play around in my sheet some more. Thank you $\endgroup$ – user49866 Sep 4 at 2:03
  • $\begingroup$ Yes, but if you hedge at the end of the day it becomes linear again, you actually do the exact opposite that the ETF is doing. $\endgroup$ – Lliane Sep 4 at 2:22
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I disagree that these products are convex*.

At any point in time, the ETF exposure to the underlying is linear, it's just that it changes through time. A 2x ETF will just have 2x exposure to the underlying - where the exposure is based on the nav at the point of rebalancing.

Say the nav is \$100 per share, then it will hold \$200 of exposure to the underlying (for the sake of the example, we'll ignore the drag coming from management fees). If the underlying is worth \$4, and it moves up to \$4.25, then in the example above we'd have been holding 50 of the underlying, which would mean the NAV increases by 50x0.25 = \$12.5. so now the nav per share is \$112.5, the exposure before a rebalance though is still 50 units - i.e. it will be 50x4.25=\$212.5. the ETF will rebalance though, such that the exposure becomes 2x again, so it will need to get to $225, which means it needs to buy \$225-\$212.5=\$12.5 of the underlying in order to get back to being 2x levered. But at all points in time, the ETF is only holding the underlying - which is linear.

The ETF at all points is only holding linear products, thus it itself is linear.

*The exception here is when the etf level is low enough that it could hit zero, as you cannot owe the fund anything, which means that you essentially have an option at zero. Well constructed ETFs though will normally have some kind of provision to delever in situations where this becomes likely (as the fund does not want to be in a position where the fund can go negative, as they'll be short gap risk).

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  • $\begingroup$ For the holder of the ETF, it is linear over a daily period, but for longer periods it is no longer linear to the holder. The rebalancing process you describe is from the point of view of the sponsor/hedger of the ETF, not the investor holding stakes. $\endgroup$ – Daneel Olivaw Sep 5 at 15:24
  • $\begingroup$ the product is always linear though, which means that there should be no decay associated with a convexity. $\endgroup$ – will Sep 6 at 13:45
  • $\begingroup$ I was going to write the exact same thing. The products are 100% linear until the last second when the rebalance is complete. The amount that the fund manager needs to trade to rebalance moves around, but that's not related at all to the exposure of the fund. $\endgroup$ – JoshK Sep 8 at 3:39
  • $\begingroup$ @will I am not sure what you mean. For example, if you trade an inverse ETF in which you buy on January 1st and sell on December 31st, assuming the underlying closes the year at the same level than it started, the corresponding ETF will close lower than it started (ignoring fees etc.). I think this is a pretty acknowledged fact and for me this means that the ETF is not linear. What do you mean by "the product is always linear"? $\endgroup$ – Daneel Olivaw Sep 8 at 15:04
  • $\begingroup$ The way I see it, the end value of the leveraged/inverse ETF depends on realized volatility, hence I see this as non-linearity. $\endgroup$ – Daneel Olivaw Sep 8 at 15:10
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As @Lliane explains, you are actually describing a position in which the underlying is rebalanced everyday, hence the compounding effect of the leveraged ETF vanishes.

Maybe a bit of modelling can be helpful to illustrate the relationship between leveraged ETFs and volatility. Let $S_t$ be the value of the underlying and $V_t$ the value of a leveraged ETF with leverage being a positive or negative integer $\alpha\in\mathbb{Z}/\{0\}$. The dynamics of the ETF value are determined by the constraint: $$\frac{dV_t}{V_t}\triangleq\alpha\frac{dS_t}{S_t}$$ If we assume the familiar Geometric Brownian Motion dynamics for the underlying, we get: $$dV_t=\alpha\left(\mu V_tdt+\sigma V_tdW_t\right)$$ That is: $$\begin{align} V_t&=V_0\exp\left\{\alpha\left(\mu-\frac{\alpha\sigma^2}{2}\right)t+\alpha\sigma W_t\right\} \\ &=V_0\exp\left\{\alpha\left(\mu-\frac{\sigma^2}{2}\right)t+\alpha\sigma W_t\right\} \exp\left\{\alpha(1-\alpha)\frac{\sigma^2}{2}t\right\} \\ &=V_0\left(\frac{S_t}{S_0}\right)^\alpha \exp\left\{\alpha(1-\alpha)\frac{\sigma^2}{2}t\right\} \end{align}$$ Unless there is no leverage, i.e. $\alpha=1$, we observe that the value of the ETF will depend upon the volatility experienced by the underlying. In particular, notice that given $\alpha\in\mathbb{Z}/\{0\}$, the term $\alpha(1-\alpha)$ will always be negative, thus the exponential will have a value lower than 1 and therefore the higher the volatility, the higher the drag on the ETF value.

For example, for both a x2 leveraged ETF or an inverse ETF, we have $\alpha(1-\alpha)=-2$. Assuming a one-year period $t=1$ and that the volatility is not too high, then by the approximation: $$\exp\{-\sigma^2\}\underset{0}{\sim}1-\sigma^2,$$ you can expect these ETFs to experience a drag approximately equal to the annual variance, e.g. if the annual vol is 30% then you can expect to lose 9% of the value due to volatility.

All this is really a consequence of volatility drag and the concavity of the logarithm. Maybe my answer to this question can be helpful to understand further.

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  • $\begingroup$ Hi thank you for your response. This makes a lot of sense. The thing that's a bit confusing now is, if being long a leveraged etf means you are short volatility/short gamma, this should mean that you are long theta somewhere right, at least that's how options work. Where would that show up? I've run more simulations in excel after your post and I've observed how increasing vol decreases leveraged etf returns, but I'm not sure where the positive theta is then. $\endgroup$ – user49866 Sep 15 at 16:06
  • $\begingroup$ Also my apologies for such a late response, I was out of commission for a while. I really appreciate such a thorough answer. $\endgroup$ – user49866 Sep 15 at 16:10
  • $\begingroup$ I’m thinking that maybe, instead of framing this in terms of gamma/theta, it is more appropriate to think of it in terms of carry, more specifically negative carry. Normally there is gamma/theta when there’s optionality, but what we have here is actually path-dependency. Basically, beyond fees, borrow costs, etc., the above analysis would point towards an inherent cost in holding a daily leveraged position. $\endgroup$ – Daneel Olivaw Sep 15 at 17:19

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