Home Forums Main Forum Basics: Decomposing Butterflies

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  • Curve Advisor
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    #701 |

    You can decompose any equally-weighted butterfly structure (single fly, double fly, etc) into its component 3mo butterflies. For example, ED1-3-5 6mo fly can be broken up into ED1-2-3 fly, plus two ED2-3-4 fly, plus ED3-4-5 fly (see graphic). This is by construction, so if you add up the prices of the four component three month flies, you will get the price of the 6 month fly. Similarly, you can do the same breakdown for a 9 month fly, a 1 year fly, condors, double flies, etc.

    This has a lot of interesting applications in:
    1) the sizing of positions
    2) trading around existing positions
    3) analyzing your risk
    4) determining the quickest way to hedge your curvature risk.

    I will give you some time to think about this, and I will post more later.

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  • Curve Advisor
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    1) Sizing of positions.

    So when you are long a year fly, you are actually long four 6mo flies. That is why you can size a 6mo fly much larger – up to 4x as many of a 1yr fly. However typically it could be a little less, since 4x a 6mo fly will tend to be a little noisier than a 1 year fly.

  • Curve Advisor
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    Actually, I just had an example of this earlier today. I had suggested a trade, and needed something to hedge. so I suggested selling a particular 6mo fly OR 0.25x as many of a nearby 1 year fly. When you do a double fly trade around a particular part of the curve, it may be easier (more liquid) to do one leg in a slightly different form (i.e. using 0.25x a 1 year fly instead of a 6 mo fly, etc).

  • Curve Advisor
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    4) determining the quickest way to hedge your curvature risk.

    So when you have an existing position you want to take profit on (or stop out of), you can sometimes use a larger butterfly to neutralize the curvature risk. So if you were long 1000 of the U8 6mo fly, you can cover buy selling 250 of the H8 1 year fly. This will make you roughly curvatrure neutral. And you can take your time in exiting the remaining position. This is also useful when the U8 6mo fly is lagging, but for some reason the H8 1 year fly is noticeably outperforming.

  • Curve Advisor
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    2) trading around existing positions

    If you are long M7-M8-M9 fly @12, for example, you do not have to wait for the entire structure to get to your take-profit level. You can choose to take the trade off in pieces. Because it sometimes happens that one piece of it could be in more demand than others. For example, if there is large enough selling in Z8, you can consider unwinding the M8-Z8-M9 piece of the flys you are long, leaving you with one M8 6mo fly and long two Z8 6mo flies. This leads me to my final point…

  • draker
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    more basics please

  • Curve Advisor
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    3) analyzing your risk

    If you trade a lot of butterflies up and down the curve, you will eventually have some overlap. If you know that you can decompose a butterfly into 3 month butterflies, you can therefore deconstruct your fly positions into 3 month fly risk “buckets.” So instead of looking at your book as a strange combination of single and double flies up and down the curve, you can break down your positions into how much curvature risk you have have per 3 month fly. This is another way of looking at your curvature risk (and in my opinion a better way) than looking at a list of your FUTEQs.

  • Curve Advisor
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    So I had a situation today where I needed to do a #4 (determining the quickest way to hedge your curvature risk). As a hypothetical example, if you are short 200 ED5-7-9 fly and short 600 ED8-10-12 fly, the fly curve is roughly smooth, and you need to hedge that risk quickly, a good alternative is to just buy 200 ED6-10-14 fly. A year fly is equivalent to FOUR 6mo flies. So by doing the above, you are left with:
    * short 200 ED5 6mo fly vs long 200 ED6 6mo fly (this double fly should have a very tight trading range, as the component flies are only 3 months apart)
    * short 200 ED8 6mo fly vs long 200 ED10 6mo fly (this is just a regular double fly, and this too should trade in a reasonably tight range)

    That’s it. You had a situation where you were short 800 combined 6 month flies, and rather than possibly paying 1/2 bp on up to 800 flies to unwind, you may have paid only 0.5bps on 200 year flies to cover the risk. You can now take your time to unwind the double flies. Also, if you thought there was some value in shorting the ED5 6mo fly and the ED8 6mo fly to begin with (relative to the rest of the curve), you can also pick up a little relative value on the unwind (because you are still short some of those flies).

    There are times when you just should unwind the outrights, but there are other times when doing it this way makes more sense (perhaps if the year fly is offered, the markets aren’t liquid, etc). The key point is not that this is how you should unwind a complex fly position. The point is that you should be aware of all your options, so that you can make the best optimizing decision possible.

  • Curve Advisor
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    I had another example of how decomposing flies helped to manage a position. If you are short, say 40 of the ED13-15-17 6mo fly, that you are not as crazy about any more. You can always just unwind. However, if you saw some value on that part of the curve to begin with, you can try and capture that value by converting it to 10 double double butterflies. All you need to do is buy 10 ED11-15-19 1 year fly. What you would be left with is:
    * +10 ED11-13-15 6mo fly vs -10 ED13-15-17 6mo fly
    * -10 ED13-15-17 6mo fly vs +10 ED15-17-19 6mo fly
    In effect, you unwound 50% of the ED13-15-17 6mo flies you were long, and you converted the rest into a form that should be more “contained”, but still carry value.

  • Curve Advisor
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    Another application of this is if you plan to trade something in a range, and you want to hedge some profits without unwinding the underlying trade. Instead, you can save some brokerage by range trading the hedging 1 year fly, instead of 4x as many 6 mo flies back and forth.

  • Curve Advisor
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    Another thing you can decompose into butterflies are spreads. Consider the ED5-ED6 3 month spread. If you think about it, that is equal to:
    ED5 3mo fly + ED6 3mo fly + ED7 3mo fly + ED8 3mo fly + …
    This is because the (short) back spread of one fly gets cancelled out by the (long) front spread of the next fly.

    I currently have an outstanding trade recommendation that utilizes the above concept. So when everyone gets the trade on, I will let you post here and show you how to use the above information to analyze the trade.

  • Curve Advisor
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    So hypothetically, consider a structure where you buy ED9-11 spread, and sell 3x as many ED11-13-15 flies. Your futures position would look as in column 2. Your “3mo fly buckets” would look as shown in column 3. So this position is net long 3 mo fly buckets. A ton of them. You would generally want to put something like this on when the 3mo flies you are short happen to be the flies that are the highest on the curve, and the fly curve is disproportionately flat further out (as the ones you are long are only worth a fraction of a bp, so there is much limited downside). The premise is that it is very difficult for the markets to price in extremes in the fly curve so far out the curve. So if you put on this type of trade, you are saying the ratio of the steepness of the curve to the level of flies is at an extreme.

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    • Curve Advisor
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      In case you are having problems figuring this out, the easiest way to think about this to break up your fly position into 3 month spreads. In the top table, we show the positions broken up as a sum of various 3 month spreads. You can check that column (a) is equal to the sum of columns (b) thru (g). We then use the property listed in post 897. In particular, 100 of a 3 month spread is going to equal 100 of each of the 3mo flies behind it. If we convert columns (b) thru (g) into 3 month fly buckets, you get the result in the bottom table. If we add those columns, we get our total risk in fly buckets.

      I’ll leave it up to the reader to calculate what the formula to convert a list of futures into fly buckets without the intermediate steps (it is not that long).

      Note that there is a pattern of +200 flies that continues in perpetuity. This is not a sign of excess risk – curvature further out the curve is generally very small and does not move as much. But this is a sign that you have 200 more 3mo steepeners than flatteners in your structure. This may or may not be attractive, depending on your view. To remove this, all you would need to do is to sell 100 6mo spreads (or 200 3mo spreads) further out the curve, where it fits your view. Keep in mind however, that not all parts of the curve are equally volatile (or equally rich/cheap), and the goal does not have to be that you have an equal number of flies in your structure.

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  • Curve Advisor
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    I think it may be difficult to see the usefulness of this thread. I liked trading all types of fly structures up and down the curve, so the concepts in this thread allowed me to decompose my entire list of curvature trades into “fly buckets.” For example, I could sit on the bids on the part(s) of the curve that I thought was cheap, and sit on the offers on the part(s) of the curve I thought was rich – and they did not have to be the same type of fly structure. When managing my risk, I would look at VaR (see the VaR thread), but another thing I could do was to look at the size of my fly buckets. I could see how much positive/negative curvature risk on the various parts of the curve. This is not always going to be clear from looking at a list of flies and double flies, where there may be overlap in contracts. With the risk bucketing by 3 month fly buckets, you see exactly how much curvature risk you have at each point. This way, you don’t have to keep a list of the various versions of the same trade you may have legged into – buying the H8 6mo fly, M9 3mo fly, the H8 6mo condor, the 2x Z7 6mo spread vs M8-M9 spread, etc. And as you unwind, you can just focus on taking off exposure in that part of the curve, based on what structure is the best value, without concerning yourself about unwinding the exact same trade you put on. For example, you may have bought a bunch of 6 month flies and condors, but it may be that when you want to take some profit, selling a nearby year fly provides better value. Fly bucketing allows you to see what your risk is, so that you know how many 1 year flies you need to sell.

    If you only trade a few structures, the above is not important. However, as you start developing a better feel for the various curve structures, you will want to see what your risk looks like in fly buckets.

    [to be continued]

  • Curve Advisor
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    Here is an example of a hedging application of decomposing flies. This assumes that your brokerage (transactions) costs were low, and that your carrying costs (to have outstanding futures contracts) were negligible. Depending on your particular situation, this may or may not be an attractive strategy.

    A “Flip” Trade I had put out is to buy the M9-Z9-M0 6mo fly @ 1 (to sell @ 2). Because the Z9 turn was chronically undervalued, this structure had a hard time trading 2s. However, 4.5s were trading in the Z8-Z9-Z0 year fly. So rather than “taking” profit on the M9-Z9-M0 fly, “hedging” profit on the fly by selling 25% as many Z8-Z9-Z0 fly made sense. At some point, when the Z9 turn reflated, we would be able to unwind the structure at a slightly better level. Most importantly, if the flies around Z9 all went lower, we would be able to buy back the Z8-Z9-Z0 fly at a lower level and profit from the choppiness in this sector of the curve. We would also be range trading this structure with a structure that is lower in brokerage (since you only need to trade 25% as many year flies than 6 month flies).

    This opportunity presented itself from knowing that the 25% of a year fly in the back of the curve was a reasonable proxy for the 6 month fly. Not that you can’t do this as easily around the reds, since there could be more chance for P&L noise.

  • Curve Advisor
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    [duplicate post – reply to post 905] In case you are having problems figuring this out, the easiest way to think about this to break up your fly position into 3 month spreads. In the top table, we show the positions broken up as a sum of various 3 month spreads. You can check that column (a) is equal to the sum of columns (b) thru (g). We then use the property listed in post 897. In particular, 100 of a 3 month spread is going to equal 100 of each of the 3mo flies behind it. If we convert columns (b) thru (g) into 3 month fly buckets, you get the result in the bottom table. If we add those columns, we get our total risk in fly buckets.

    I’ll leave it up to the reader to calculate what the formula to convert a list of futures into fly buckets without the intermediate steps (it is not that long).

    Note that there is a pattern of +200 flies that continues in perpetuity. This is not a sign of excess risk – curvature further out the curve is generally very small and does not move as much. But this is a sign that you have 200 more 3mo steepeners than flatteners in your structure. This may or may not be attractive, depending on your view. To remove this, all you would need to do is to sell 100 6mo spreads (or 200 3mo spreads) further out the curve, where it fits your view. Keep in mind however, that not all parts of the curve are equally volatile (or equally rich/cheap), and the goal does not have to be that you have an equal number of flies in your structure.

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  • Curve Advisor
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    Here’s a real-world example of how knowing how to decompose flies can be helpful. The H7-U7-H8 fly traded as low as -0.5 earlier this week. It was my contention that this made no sense from a model perspective, in light of the U6-U7-U7 fly being +5.5 at the time. As we know from the previous posts:

    U6 1yr fly = U6 6mo fly + 2 x H7 6mo fly + U7 6mo fly

    If you are uncertain, write out the contracts and confirm for yourself. This relationship is true. If the 1 yr fly curve is smooth and “straight” enough, you can use the following approximation:

    2 x H7 6mo fly ~ U6 6mo fly + U7 6mo fly

    The above is just saying the three flies are roughly in a straight line. This does not have to be the case, but if the curve is monotonically decreasing (or increasing), as the 1 year fly curve gives an indication it should be, it may be reasonably close. Obviously the closer the three flies are to a straight line, the more accurate the relationship will be. Combining the two equations, we get:

    U6 1yr fly = (U6 6mo fly + U7 6mo fly) + 2 x H7 6mo fly
    ~ (2 x H7 6mo fly) + 2 x H7 6mo fly
    ~ 4 x H7 6mo fly

    So we basically have a noticeably positive year fly being roughly equivalent to 4x a negative 6mo fly. This won’t be as accurate at the front of the curve, but this is mostly for illustrative purposes… that when the 1 year fly curve is positive over a particular area, you shouldn’t expect any noticeably negative 6mo flies, because you can decompose the 1 year fly into a bunch of 6mo flies.

    But the world is not just a model – it’s possible that there is a timing issue that would make it unlikely for the Fed to aggressively hike in the months leading up to Q3 2017. I just don’t see it now. But the nice thing about being aware of things that could go wrong with the model, is that you can be one of the first ones to get out if the conditions evolve adversely.

    The two big takeaways are:
    * You should think about the interrelationship of the various fly curves (1yr vs 6mo, etc). But in addition, think about how the fly curves relate to the spread curves and the single contract curves.
    * you should know when there is the possibility for some of the “smooth curve” assumptions could break, and the various curves to get severely kinked.

  • Curve Advisor
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    Another reason to think there is value is that the 1 year double flies (3-6 months apart) centered in that part of the curve are noticeably positive. The H7-U7 6mo double fly is -1.5. That is a highly unlikely situation from the model construction perspective.

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