# Lever Options: Gamma Decay & Smiles

Levered ETFs are fascinating financial instruments. Derivatives on levered ETFs, termed *lever options* here (or, *levers* for shorthand), are even more interesting. As lever options are likely to be the first exotic to gain substantial exchange-traded volume, volatility traders should be salivating!

One of the most interesting aspects of levers is *gamma decay* (liberally inventing terminology, borrowing from the classic greek gamma):

Gamma decay is the decay in option value due to the leverage decay of the underlying levered instrument.

In other words, leverage decay in the underlying results in a direct effect on the derivative captured by gamma decay. Gamma decay is a non-trivial departure from the standard BSM assumption of a Weiner process underlying. Note gamma decay differs, although is similarly derived, from the linear scaling of the drift and volatility.

Most fundamentally, gamma decay should cause pause to evaluate the classic BSM gamma-theta tradeoff, now that both gamma and theta decay: .

Quick scan of the smiles for August expiry SSO and UPRO indicate MMs may still be pricing using classic BSM. Multi-order polynomial smile for SSO (fitted in blue, unfitted in gray), assuming 1.24% cont yield and 0% risk-free:

Similar smile for UPRO, assuming 0% cont yield and risk-free:

Subsequent posts will continue analysis of levers given they are such a rich vein.

Hey Q,

I’ve spent a lot of time researching some of the same stuff you have here (I was enthralled by the 3x levered ETFs when I started to see how they played out in the real world), but admittedly, I’m having trouble with some of the concepts that you launched here. I’m not a seasoned quant, but I dabble. Hopefully my ignorance will be useful in one way or another.

Let me key in on what I found confusing:

In other words,leverage decayin the underlying results in a direct effect on the derivative captured by gamma decay. Gamma decay is a non-trivial departure from the standard BSM assumption of a Weiner process underlying. Note gamma decay differs, although is similarly derived, from thelinear scaling of the drift and volatility.“Most fundamentally, gamma decay should cause pause to evaluate the classic BSM gamma-theta tradeoff, now that both gamma and theta decay: (½σ2S2Γ + θ).First, you mention the concept of leverage decay. I am confused as to what you want this to mean. There is no necessary downward movement in these ETFs, although it seems mathematically likely. In fact, if interested, I can send you over some spreadsheets that I put together that simulates these things on the Dow Jones, S&P500 and Nikkei going back as far as the Yahoo data would allow for me to. The appearance of trending markets makes stacking a more common occurrence than what would be implied by an assumption of brownian motion. A trending market can make one of these things stack (and effectively produce greater than 3x leverage). I have done more detailed simulated analysis on empirical data, if you’re interested in taking a look (so let me know).

Second, I have no idea what this means, “linear scaling of the drift and volatility.”

Third, I don’t follow how this system subsequently produces gamma decay, hopefully you can help to clarify.

My best guess to tie the two together is that as time to expiry comes closer, the implied probability that allows for ‘drift’ (or divergence between the underlying and lever that is caused by volatility) diminishes, and subsequently this causes the internal volatility of the levers to simply ‘leak out.’ In other words, the longer that they get to operate the more havoc they get to wreak.

The last two are complete ignorance, the first is possible ignorance, I’m just hoping that the potential usefulness of the first offsets the, um …quality of the second and third. Perhaps, if were not too much of a hassle, I would ask that you spend a bit more time building up through the functioning of the levers, as the concepts that you’ve introduced here are a bit heavy.

Finally, maybe I’m missing something, but MM don’t price on BSM, which is why there are volatility smiles. By BSM they would be poker faces; a flat line (as far as I understand it). My understanding is that options pricing follows something much closer to a log-normal distribution (fatter tails for my fellow quasi-quants), which is how you end up with the smiles.

@Bill: thanks for your interest and question; I am myself very early exploring levers (and avid to do more), so I have

manymore questions than answers. I am excited to discuss with those having performed deeper analysis, and definitely interested to see xls.With that preface, allow me to elaborate briefly on my (unintentionally shorthand) prose and thinking: “leverage decay” is intended to refer to the asymmetry in equal inverse percentage returns on a daily basis: x% down day followed by x% up day does not net to zero (conflicting with intuition built from classic binomial pricing trees, in which up/down nodes measured in absolute values do net to zero). QuantumFading, among others, has covered this in some non-technical detail.

“Linear scaling of drift and volatility” is intended as reference to

xμ andxσ from the Wiener process in equation (23) of Cheng and Madhavan article, where x is the linear scale factor.Given those, gamma decay is my poor attempt to capture the intuition that in mean-reverting markets, asymmetry of equal percentage returns will result in a decay of the value of levered ETF (and thus lever options)

with respect to the underlying index(due to the leverage decay). In other words, the delta of the delta (i.e.gamma) is asymmetric as it does not net to zero based upon equal inverse underlying volatility. Although I am slowly working through the stochastic calculus, this apparently asymmetric gamma is what keyed me in. Admittedly, I intended to be a bit obtuse here in the post–as there are some interesting volarb trades which I am underway exploring. If you are interested, happy to discuss this privately.Re MM BSM: totally agree, smiles are due to implied vol distortion in the wings reflecting hedge risk aversion to long-tail events (among other effects); my intent in critiquing BSM in this context referred to its classic assumption of inverse relationship between Γ and θ.

Do you mind elaborating on your ‘leak out’ concept?

Q,

A buddy of mine sent me the link to this article, and as a Level 1 (2 if I focus) quant, it’s great to see a blog so rich with information and ideas.

I would like to add my $0.02 to the discussion here and ask that it be taken for what it’s worth only, as it’s simply my thoughts based on the ideas you’ve posed here combined with work I have also been doing on these ETFs. BGI and JP Morgan have done some good work on the mechanics of these levered ETFs if you’re interested (I would be happy to pass them along). Borrowing from their work, I believe the gamma decay you’re describing is a function of how these ETFs are constructed.

Forgive me if the following seems random, but I think the issue of gamma decay/volatility leakage is a function of the fact that the ETFs are ‘rebalanced’ daily. As you pointed out, there is a path dependency to the ‘payoff’ of the ETF. BGI/JPM describe the divergence in value of these ETFs as the inclusion of a path dependent option in the price. I think this directly relates to the two issues that have been highlighted in the following way:

Gamma Decay: The security being used as the basis for pricing is not the same security that is delivered when the option matures, nor is it the same from any time after t=0. Because the leverage is adjusted daily, the amount of ‘equity vs debt’ embedded in the ETFs changes on a daily basis relative to a given starting point (if that makes sense).

Volatility Leakage: Because there is a path dependency in the ETF ‘payoff’ over time, the relative sensitivity of the lever options to changes in the underlying ‘misbehaves’ because some of the value of volatility is being transferred to this embedded path dependent option. This accumulated value can be ‘realized’ at those points in time where there is more vol to the upside than downside and the ETFs outperform their intended return, otherwise when volatility is balanced or more to the downside you lose value that is transferred to the option.

In purchasing an option on a levered ETF you’re actually purchasing an option on a dynamic security and a path dependent option. I think the answer to this ‘problem’ lies in identifying a stable methodology for deconstructing these ETFs into their component parts.

Q and Futrbllnr,

First, Futrbllnr, thinking about this as a new security every day is probably the healthiest way to think about these things, first of all. For those of us who are foaming at the mouth to try to out math the market makers….this is only that much more fun.

When I did my long distance simulation of the 3x etfs, what I found was that the character of the market was more important to determining which qualities would be the most important. Generally, there are narrow trending markets (bull markets) where the bull 3x will compound greater than 3x, and the bear will compound less than 3x; range bound flat markets, where there will be slight degradation over a long enough horizon; and bear markets where these things simply get whipped to hell. Knowing the form of market that you’re in is the most important factor over the long run.

@Bill / Futrbllnr: do either of you have link to the Despande, et al. paper from Barclays cited by Cheng and Madhavan?

@ Futrbllnr: totally agree; daily rebalancing is the construct driving this. I am digesting your deconstruction argument, which amplifies my open question re both replication and non-wing hedging (vis-a-vis volarb). Def interested in BGI/JPM work on this topic (I very briefly covered Cheng and Madhavan in Leveraged ETFs and Market Close post), if you are game to share.

@Bill: digesting your comments. Concur on the focus vis-a-vis the underlying index; this screams multi-instrument volarb, pending sorting out the path dependencies per @Futrbllnr. Re market regime: fully agree; coincidentally, my recent post on backtesting was essentially aimed at a very similar thesis: market regime really matters (along with other similar market context), a lot more than often given credit in the quant world. Towards this end, some recent research on variant-scale market regime detection might be relevant. I will email reply privately to continue discussion.

@ Q/Bill : It appears you already have the BGI paper (Cheng and Madhavan) – the JMP work is similar in nature and gets more into mean-reversion. Forgive me for asking, but how might I e-mail the two of you the JPM paper? I can’t seem to figure out how to access your e-mail addy from the comments. As for the referenced paper on Ultrashorts, I can’t seem to find it on the web and will have to see if I can get access to it via other means.

@ Bill: That’s very interesting and it highlight what investors experienced last year with both the UL and US ETFs on same index performing about the same. One might posit that this in- line performance, underperformance and outperformance underscore the fact that you are implicitly taking a vol position by purchasing these securities and that correct characterization of the market you’re in is important in taking positions.

@ Q/Bill: I was inspired by Bill’s work on backtesting and think I will try to put something together myself as well. I think the key is dissecting daily prices in such a way that you can separate the evolution of your original position and the effect of releveraging daily in dollar terms.

Another starting point would be to characterize the path dependent option so that it can be priced to see if the implied value in the ETF aligns with the numerical approach.

Q,

If you could send an email to both of us, we can have a little off-blog discussion.

Futrbllnr,

You are essentially taking a volatility decision when you try to arb these things. And it is all about the character of the market.

When we get the email set up I’ll send you my big xls (which I’ll probably have to break up to send). It has all of the data that I’ve mentioned, and generally provides a decent sense of how these things would work in a series of different markets.

No matter what, trying to accurately value these things is more about predicting the future (there’s no unidirectional arb if you allow for different market regimes), it’s just that there are a couple of more interesting caveats that you see here, rather than in value based ETFs. Besides that there may be a math contest between you and the MM, but you’d have to develop some pretty interesting models to figure out what good prices would be.

Q,

Okay, now I understand what you mean by leverage decay. I looked back in my xls work, and I had called it levered degradation – I haven’t seen many others within the field look into the 3x levered quite this far at the time (mid april), or well for that matter. Although I can say I’ve seen and looked through the Cheng and Madhavan article. So the concepts are the same thing, but the reference to the other blog certainly helped to get me on the same page. My concept of leaking out is the same as the levered degradation, or leverage decay, so I guess ignore that for now.

However, I still am having issues following the concept of gamma decay. I generally understand gamma and how it’s useful, and basically how to gamma scalp. Let me parse your statement, and see if I can’t at least ask some questions to help me understand this.

Given those, gamma decay is my poor attempt to capture the intuition that inmean-reverting markets, asymmetry of equal percentage returns will result in a decay of the value of levered ETF (and thus lever options) with respect to the underlying index (due to the leverage decay). In other words, the delta of the delta (i.e. gamma) is asymmetric as it does not net to zero based upon equal inverse underlying volatility. Although I am slowly working through the stochastic calculus, this apparently asymmetric gamma is what keyed me in.What I’m focusing most on is that I’d like to see how you define mean reverting markets as these subsequently guide everything else. If by this you mean day after day inverse movements, then I can wholly understand what you mean by gamma decay. There is a partial natural movement lower in the ETFs, and subsequently, they lose their potency very quickly. The main issue with that of course is the issue of trending in markets, or the more formal sense of ‘long dependence,’ oh, sweet little H. With long dependence you can see trending that would make a levered ETF exhibit qualities that would either rev up, or simmer down their longer term performance (effective long term leverage can outpace 3x in a trending market, while the bear would exhibit less than 3x effective leverage). Although leverage decay is a factor, that seems like it’s more an observation than an innate quality directly effecting gamma. Or, perhaps there is another factor based on compounding that has to also be built into a model. The one thing I just realized (after thinking about this since I woke up today) is that when you mention gamma decay, it is more an issue of decay

relative to the underlying it is seeking to trackrather than of the actual lever. Gamma decay would be a reality in high volatility markets (fast mean reversion), but in low volatility, trending markets (slow mean reversion) you instead see gamma expand, again only in reference to the underlying.Right now, that’s my best guess.

Oh, also feel free to email me so I can send you my main xls.

Lu, Wang, and Zhang just recently published their working paper on long-term performance of leveraged ETFs, as originally cited in Cheng and Madhavan (as working draft). Introduction makes a few interesting claims re quadratic variation and auto-variation (

e.g.p. 3), but the relevant analysis (in the Appendix) seems quite lacking.If you could clarify what they mean by auto-variation and quadratic variation, I would be very grateful. My quick intuitive understanding is that auto-variation is kind of the mean reverting (degrading/decaying) effect, and quadratic variation is the trending/compounding effect. This – again – is a market regime issue, as the appearance of one form of variation being more dominant than the other was focused on a topsy/bear market, and subsequently did not allow them to see any more.

Nice conversion… I hope you guys keep it out of email and on this post 🙂

In case you guys are interested, I wrote a tool that allows simulation of Nx ETFs via historical data of the underlying index. The results confirm what Bill pointed out, that the characteristics of the leveraged ETFs are significantly dependant on the volatility vs trend of the underlying index. Like he said, during low volatility and high trend of a bull market, the 3x scream to the upside due to the compounding outweighing the decay.

And on another note, (you may find this interesting), I recently came up with a ‘Decay Indicator’ that calculates the amount of decay over a given period, much like RSI is calculated over a period.

Take a look at a FAS simulation:

The yellow line on the second chart down is the 30 day decay indicator. Notice how during the past crisis, in a 30 trading day window, the decay reached a rate of 40% for that period. Compare that to a maximum 30 day rate of decay of 7.6% in the 2000-2003 bear market. Not only that, but during the bull market the rate of decay is only 0.3 to 1% per 30 trading days. In other words, during this bear market, FAS has periods where it was decaying over 100 times the rate of a bull market. Personally I find all of this fascinating.

Kevin

I’m a bit late to the party but interested in joining the discussion and seeing what has been covered so far. Would someone be kind enough to e-mail me?

Thanks

Q,

Mind sending Justin an email?

My e-mail address is (my first name) dot harper at gmail.

Hopefully that should be good enough to keep the spam bots away.