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Rookie_Quant


Total Posts: 742
Joined: Jun 2004
 
Posted: 2016-06-29 20:22
Does anyone have a good primer with the mathematics of credit correlation/dispersion trading?

I'm thinking things like buying protection on CDX constituents and selling index protection (either as a tranche or a vertical slice). I'm trying to find an analogue to the risks of doing this in equity options space.

"These metaphors and similes aint similar to them, not at all." -Eminem

Cheng


Total Posts: 2838
Joined: Feb 2005
 
Posted: 2016-06-30 15:34
No primer at hand, but a few hints.

Single names and CDX index used to have different restructuring clauses. One was without, one was ModR. Since restructuring is a rather common credit event this should make some difference regarding pricing.

Index vs tranches is trickier since the higher tranches usually don't trade a lot if at all (for example some people used iTraxx 9-12 as proxy for the 12-22 since the tail risk is similar... until Lehman, that is).

There was a lenghty discussion of dispersion trading here on NP. I think someone even wrote a primer, sort of.

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

Nonius
Founding Member
Nonius Unbound
Total Posts: 12702
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Posted: 2016-06-30 21:33
I think Bruno Iksil might have some decent notes

Chiral is Tyler Durden

Cheng


Total Posts: 2838
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Posted: 2016-07-02 15:50
I thought he is more into breeding whales.

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

Rookie_Quant


Total Posts: 742
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Posted: 2016-07-27 18:17
Ok let me ask a more granular question....

Let's say I wanted to devise a theoretical trading strategy (in the academic, not practitioner sense) to determine whether or not fixed income correlation is "priced".

If I were to do something like calculate compound correlations and base correlations for each tranche of iTraxx or CDX or whatever, and then go long/short assets which have the lowest/highest implied correlations....does that effectively isolate the risk factor I care about?

Other than counterparty risk, which I'm doubling, what other sensitivities am I exposed to, provided the longs and shorts occur in the same series/reference portfolio?

For reference, Driessen, Maenhout and Vilkov (2009) do an exercise using equity dispersion trading. I'm trying to do the analogue in credit. They buy/sell constituent straddles and sell/buy index straddles as youd imagine.

"These metaphors and similes aint similar to them, not at all." -Eminem

Rookie_Quant


Total Posts: 742
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Posted: 2016-08-01 20:07
Apologies...assume delta-neutrality in the above.

"These metaphors and similes aint similar to them, not at all." -Eminem

Cheng


Total Posts: 2838
Joined: Feb 2005
 
Posted: 2016-08-07 11:23
If I were to do something like calculate compound correlations and base correlations for each tranche of iTraxx or CDX or whatever, and then go long/short assets which have the lowest/highest implied correlations....does that effectively isolate the risk factor I care about?

No. Index tranches trade based on one-factor implied correlation. The best you can get for single names is historical correlation. Which is an entirely different beast. Either way, use base correlation.

Other than counterparty risk, which I'm doubling, what other sensitivities am I exposed to, provided the longs and shorts occur in the same series/reference portfolio?

Potentially different restructuring conventions. Dunno how it is nowadays, in the past CDX index restructuring clauses were different from single names ones.

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

gill


Total Posts: 190
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Posted: 2016-08-08 22:20
Till may 2005 that was a popular strategy to trade equity vs lower mezz to monetize "misspriced" correlation hehe
Nowadays it's purely theoretical exercise -- there is no tranche trading as such. No flow from real money and bid/ask is prohibitively expensive

Rookie_Quant


Total Posts: 742
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Posted: 2016-08-09 06:12
My reading of the economic catastrophe bonds paper (coval, jurek, stafford) is that if you are betting against a catastrophe, there are better places to do it than by selling super senior CDX protection. Insofar as the state of the world where 50% OTM puts become ITM is the same state whereby enough IG defaults occur with sufficient clustering to cause losses in the super-senior, might as well sell puts.

I hear you on the real-world frictions that act as limits to "arbitrage", to use the term loosely. But whether or not the gap between correlations can be monetized, in the theoretical sense it shouldnt exist. Before things fell apart, did desks calculate/care about sensitivities to correlations in a strict sense? Is there some partial derivative with respect to correlation in the asset pool that was monitored?

I guess it just bugs me that the exact same asset pool will price the joint default density so differently for, say, 0-3 and 3-7 tranches. But equity vol smile/smirk/skew stuff bugs me too.

"These metaphors and similes aint similar to them, not at all." -Eminem

gill


Total Posts: 190
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Posted: 2016-08-09 11:20
Obviously the sensitivity to correlation was calculated and itrax/cdx were used to hedge bespokes, but hedges were model dependent. What do you mean by "partial derivative with respect to correlation in the asset pool"?

From theoretical point of view I think that was perfectly normal that tranches were priced at different correlation. Think about it this way, you have two states of the world normal and recessionary.

In normal state of the world you have low credit spreads and low gaussian copula correlation.
In recessionary state of the world you have high credit spreads and high gaussian copula correlation.

Now you need to combine those two in one stochastic model and have just one parameter of gaussian copula correlation which would give you market prices for all tranches.

Thats theory in practice its even much more simple than that. The street saw a bid for mezz pieces from the real money, therefore implied losses were realocated to equity senior pieces. Then leveraged super senior was invented impllied loss was realocated to equity. For equity despite zero coupon equity and combo notes that was not easy to find a buyer therefore equity piece was traded at low correlation.

So the dynamics of an inplied correlation surface was pretty much flow driven, and just one parameter was reponsible for loss realocation accross entire capital structure-- correlation.

Hopefully thats clarifies.

Rookie_Quant


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Posted: 2016-08-09 21:41
What I meant by the partial derivative comment was whether or not there was a closed-form solution for solving first-order sensitivities of the tranche price or spread to changes in the measure of correlation.

But if I understand you correctly, using my words/vernacular:

The mezz tranche was essentially the most liquid and therefore a sort of "fulcrum" security. It was in some sense a more reliable pricing of correlation risk but the only thing driving the spreads in the illiquid corners of the market was supply/demand dynamics.

So, to back it all the way to the simplest of examples, if I sell protection on super senior tranches, I benefit from low realized default correlation, and in a vacuum want to sell them at the highest implied correlation possible so that I maximize my cash inflows from the protection buyer.

Then, do I understand this part of what you said correctly:

"The street saw a bid for mezz pieces from the real money, therefore implied losses were realocated to equity senior pieces. Then leveraged super senior was invented impllied loss was realocated to equity. For equity despite zero coupon equity and combo notes that was not easy to find a buyer therefore equity piece was traded at low correlation."

In practice, markets took mezz trades/liquidity as a stronger or more high quality signal of true correlation risk and implied losses. This new loss expectation became an input into the pricing of less liquid tranches, then?

So let's say the mezz market implies a certain cumulative loss distribution. It's hard to stylize an example with fake numbers that have any meaning, but lets say the mezz is telling you there is very little default correlation and therefore low likelihood of losses in the mezz tranche. Obviously, then, there is even a lower likelihood of worry in the senior tranche. This should cause spreads at the top to be extremely low, correct? Any deviation from this is most likely because there werent many buyers/sellers, and so quotes at the super senior part of the capital structure were not really bets on asset correlation at all.

I think that part makes sense to me...it's the equity piece that I still struggle with. My understanding is that equity tranche prices are increasing in correlation, correct? So then to attract buyers, the equity tranche would have to trade at low correlations. Where I'm confused is what you mean by loss reallocation from super senior to equity...I dont follow how the advent of, say a 70-100 tranche or whatever should affect the equity piece in any way at all. Whether the most senior tranche is 15-30, 30-70, or 70-100 or whatever shouldnt impact the way defaults hit the equity piece.

Apologies if these are stupid or redundant questions, but I'm just learning :)

"These metaphors and similes aint similar to them, not at all." -Eminem

Cheng


Total Posts: 2838
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Posted: 2016-08-10 11:03
The mezz tranche was essentially the most liquid and therefore a sort of "fulcrum" security. It was in some sense a more reliable pricing of correlation risk but the only thing driving the spreads in the illiquid corners of the market was supply/demand dynamics.

Yes and no... think in terms of flow. Everybody and their grandmas were selling mezz protection. It is not as evil as equity (who wants first losses anyway?) and had decent spreads... at least during 2004/2005. Banks were thus long mezz protection and hedged themselves selling index mezz protection. Which brought down prices.

Now the overall amount of losses for the underlying index is known. What is not known is the shape of the loss distribution. Think of it like a toothpaste tube. You can squeeze the contents back and forth, the overall quantity remains constant though. This "squeeze back and forth" is expressed using correlations. If prices for index mezz go down losses are "squeezed" into other parts of the capital structure, namely equity and the more senior tranches.

Which led to tons of research telling you that selling equity protection and buying mezz protection is the next big thing (think about what this means for banks' books to understand why it made sense for them). This worked until May 2005... afterwards new ways to offload those unwanted tranches were invented and outlined by gill.

HTH.

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

gill


Total Posts: 190
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Posted: 2016-08-10 15:38
ping me offline. will be easier to explain that way.

sigma


Total Posts: 105
Joined: Mar 2009
 
Posted: 2016-08-10 16:26
It’s a very interesting discussion.

I worked as a quant for a credit correlation desk pre and during the crisis so here are my thoughts.

In my opinion, on the modeling side, the key failure was trying to apply copula models, which are static, to price risks which are dynamic in the nature. I mean that if you sell protection on senior or mezz tranches with maturities of 10y, you can only model realized losses at 10y assuming some statistical measure for realized defaults and their clustering. As a result, only a real-money investor, who is able to "sell-and-forget" for next 10y, can apply the concept of copula to get the (statistical) value of such trades at 10y horizon.

However, if we are subject to MTM risk computed daily, then the joint dynamics of credit spreads is key for modeling purposes. All unwinds of CPDOs and leveraged super senior tranches in 2008 were triggered because of large MTM losses due to joint widening of spreads, not because of realized defaults.

The copula intends to link the spreads and implied loss dynamics of tranches with the default probabilities implied by credit spreads and credit correlation of default times (a static concept). As a result, the copula mixes static statistical concepts (default probabilities and default time correlations) with the market data for credit and tranche spreads and the risk-premiums implied in these spreads. Thus, in a copula model, there is no account for spread volatilities and joint widening but some implicit structure of dependent default events fitted to whatever liquidity was available in the market.

Anyway, back in the days pre 2008, the desk was making trades of 100mm notional every day with tasty fees, so why would they care?

In the end, I found that for a proper (econometric and fundamental) modeling of credit spreads we indeed need a two- (three-) state model and incorporate risk-premiums. In the recessionary state, the spreads widen because of the risk-premium across all credits which will produce joint spikes in spreads. This widening will then translate into the fundamental problem where the question becomes whether a company can re-finance itself or not (Bear Stearns, Lehman) and, as a result, we get the clustering of defaults if liquidity conditions become tight. Also, the realized recovery rates are highly dependent on the cycle…



Rookie_Quant


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Posted: 2016-08-10 17:29
Thank you gill (msg sent), sigma, and Cheng...

My little brain is stuck on the toothpaste analogy, so I must be thinking of it in some wrong sense. If I own/have sold protection on a security in a CDO or CDO-type index with a known waterfall/priority structure, how those losses actually hit each tranche is a very simple function, no?

I guess my confusion is that you are saying the overall amount of "toothpaste" is known. I understand if there are market segmentation or microstructure reasons for the index mezz spreads being "too low" in some sense, but I'm stuck here:

I see how we can "squeeze" losses down the stack, sorta. I dont see how if we "know" the overall amount of losses, we can squeeze losses up the stack. I must not be certain what you mean by knowing the overall amount of losses.

I'm thinking of it like this: we have 100 credits in our pool. if correlations are really low, I'll have a small number of defaults, say 2. If correlations are really high, let's say I have 10. The price at which I can sell mezz protection will in some sense characterize, perhaps imperfectly, the prevailing view of what this correlation will be. At expiration, my senior suffers no impairment unless my mezz is wiped out, so if markets are liquid, there should be at least some arbitrage forces which keep the relative spreads of these two things connected. Of course, senior isnt liquid and maybe it's not even driven by the same factors/frictions as the mezz market. But in any case, how and why would the market take either of the default cases above (2 or 10) and re-allocate those losses to a tranche more senior?

I can see a case where an illiquid senior market implies 10 defaults at the same time a liquid or artificially compressed mezz market implies 2 defaults, but the way I read and interpret the analogies thus far is like "well, we know there are gonna be 10 defaults and mezz spreads are really low...i guess we should price in the defaults in the senior"

(thinking in real time)....or do you just mean that if I compare a vertical slice of the index to the mezz prices, the only way I can keep the "arbitrage" relationship between the two/match the index to the tranche spreads is to use the senior or equity piece as my free parameter of sorts?


"These metaphors and similes aint similar to them, not at all." -Eminem

deeds


Total Posts: 348
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Posted: 2016-08-10 20:59
maybe something like

how losses actually hit each tranche is arithmetic.

how losses may hit each tranche in the future is probability, subject to estimation.

the impact of changes in estimates causes updates across the stack, subject to constraints...not unlike a tube full of paste.

EDIT: removed duplicate word

Rookie_Quant


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Posted: 2016-08-10 23:36
As the losses probabilistically impact the stack, why would toothpaste be squeezed to the senior and away from the mezz, unless it's not actually updating about future losses that's going on; I can visualize it if we are saying it looks like losses may hit the senior in more or less quantity than before (due to liquidity flows).

"These metaphors and similes aint similar to them, not at all." -Eminem

Cheng


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Posted: 2016-08-11 11:50
You can run a simple experiment. Implement the Large Pool Approximation in Excel, take some reasonable parameters for spread, recovery (thus implying the PD) and correlation and calculate losses for three tranches: 0-100%, 0-5%, 5-100%. Play with the correlation and see what happens to tranche and pool losses. This is basically the toothpaste analogy in an Excel nutshell... Smiley

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

pj


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Posted: 2016-08-11 11:57
great thread
Currently trying to wrap my head around
the concept of the Excel toothpaste in a nutshell.

I saw a dead fish on the pavement and thought 'what did you expect? There's no water 'round here stupid, shoulda stayed where it was wet.'

gentinex


Total Posts: 82
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Posted: 2016-08-11 19:08
Setting aside the fact that the credit correlation market has been dead for years...

Here's a simple way of thinking about the toothpaste analogy w/o setting up large pool approximation: Say you have two events A and B (e.g., the defaults of two companies), and say e.g. the probability of A is P(A) = 1/3, and P(B) = 1/3.

Let's calc the distribution and expectation of the number of events that occur (either 0, 1 or 2) under different assumptions for the correlation between A and B:

Correlation = 0%:
P(exactly 0 events occurring) = (2/3) * (2/3) = 4/9
P(exactly 1 events occurring) = (2/3) * (1/3) + (1/3) * (2/3) = 4/9
P(exactly 2 events occurring) = (1/3) * (1/3) = 1/9
Expected number of events occurring = 0 * (4/9) + 1 * (4/9) + 2 * (1/9) = 2/3

Correlation = 100%:
P(exactly 0 events occurring) = 2/3
P(exactly 1 events occurring) = 0
P(exactly 2 events occurring) = 1/3
Expected number of events occurring = 0 * (2/3) + 2 * (1/3) = 2/3

So we see here that regardless of correlation, we have the same number of expected events (this is what Cheng means by "the overall amount of losses for the underlying index is known"), but the distribution of the number of events varies with the correlation - in particular, note the probability of exactly 2 events occurring shooting up with correlation (this is the driver of the senior tranche pricing).

Rookie_Quant


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Posted: 2016-08-11 19:47
Cheng and gentinex, thank you both for more detail. I surely never will look at toothpaste the same way. I definitely understand the analogy as written above.

I was getting stuck on the idea that the default probabilities were somehow jointly determined with the correlation measure....i/e the number of defaults in the basket was not what I was drawing...I was drawing from a measure of correlation, and then trying to make *that* measure imply the number of defaults that were likely in the asset pool.

"These metaphors and similes aint similar to them, not at all." -Eminem

Rookie_Quant


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Posted: 2016-08-11 19:47
Cheng and gentinex, thank you both for more detail. I surely never will look at toothpaste the same way. I definitely understand the analogy as written above.

I was getting stuck on the idea that the default probabilities were somehow jointly determined with the correlation measure....i/e the number of defaults in the basket was not what I was drawing...I was drawing from a measure of correlation, and then trying to make *that* measure imply the number of defaults that were likely in the asset pool.

"These metaphors and similes aint similar to them, not at all." -Eminem

Cheng


Total Posts: 2838
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Posted: 2016-08-12 13:53
I surely never will look at toothpaste the same way.

Big Smile

"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man)

radikal


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Posted: 2016-08-12 22:14
@gentinex --

This is an AMAZING example to demonstrate this idea. Really, really, thanks!

Am I dumb or is this really counterintuitive to anyone else? I'm less thinking from the credit perspective and more just the general dispersion side, but this kind of leads to the idea of "fucked up distributions can be mathematically reasonable"

There are no surprising facts, only models that are surprised by facts

gentinex


Total Posts: 82
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Posted: 2016-08-16 23:20
I suppose it depends on your definition of "reasonable", which perhaps amounts to what order moment you care about.

The other amusing (non-correlation) example I think about in this vein is the classic trading floor game where everyone puts $20 into a pot, names are selected at random and the last name remaining gets the full pot. While this is popular, you very rarely see the inverse of this game - in which the last name remaining has to pay $20 to everyone else. (Though I guess one does see this occasionally with dinner tabs, but there you typically have a smaller number of people involved.) Most likely because while both the original game and its inverse have the same mean and standard deviation, they have very different skewness :)
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