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More derivative witch hunts

November 17, 2009 in Finance

Going through the FT’s original post on exchange traded currency notes, I saw a couple of sentences that really bothered me. One thing we do not need right now are witch hunt statements without basis (a point especially compounded by the fact that the FT completely misunderstood how these products worked, even as they wrote a piece describing them):

First:

In quick conclusion, the ETCs appear to be another fine example of how exchange-traded products are mutating from their transparent replication-based beginnings into ever more complex instruments.

Granted, an ETC isn’t going to be as easy to understand as SPY. But that doesn’t mean it’s “out to get” investors. Remember when Seeking Alpha tried to help people lose money even faster with FAZ and FAS? Now those were truly frightening derivatives – leveraged, options and/or swaps based plays with complex end of day delta-balancing schemes. These ETC’s are nothing compared to that. In fact, they’re no more terrifyingly complex – and much less manipulable – than the commodity ETFs Alphaville covers so frequently. Yes, they’re the first currency ETFs (sorry, FT insists that it’s wrong to confuse these for ETFs, even though they don’t say why). Get over it.

Second:

The type of financial whizz-kidery that brought us CDOs, meanwhile, appears to be thriving well in ETFs.

I don’t even know where to start with this one. Maybe the author wrote this because the word “collateral” appears frequently in the prospectus. I wonder if it’s the same financial whizz-kidery that brought us secured loans and mortgages, too? What is this sentence doing here, besides making people associate these products with those that ruined the financial system? We’re talking about total return indices on the deepest, most liquid market in the world – not distressed CDO tranches.

Third (regarding Morgan Stanley, the derivative counterparty):

Morgan uses the proceeds it receives to hedge its total-return-swap exposure — but essentially can do whatever it pleases with the money.

What does this really mean? Morgan Stanley might be fooling investors, opting to take their cash elsewhere rather than hedging their exposures? Well in that case, Morgan Stanley is taking on currency risk equal and opposite to those investors; so it’s not exactly a free trade. One of two things must be true:

MS doesn’t want currency exposure. In this case, they have two options: they use the ETC proceeds to hedge their currency risk OR they “steal” the ETC proceeds and use cash from elsewhere to hedge the exposure. The economic outcome is identical.
MS wants currency exposure. Again, two options: they use the ETC proceeds to hedge their currency risk, after which they put on the desired currency exposure OR they “steal” the ETC proceeds and pray that the ETC investors in aggregate have taken the exact opposite viewpoint from the one MS wants to take. Since option two is extraordinarily unreasonable and volatile, there’s really only one option here: hedge the currency risk with the ETC proceeds.

Fourth:

Bank of New York Mellon has the responsibility of monitoring the eligibility of the collateral, but to all extents and purposes, from what we can make out, Morgan Stanley determines the valuation on a daily mark-to-market basis.

Another piece of conspiracy-bait. Fortunately, FT lays out what that collateral can consist of: “AA-rated G20 government bond, AAA-rated shares of government or treasury money market funds, AAA-rated supranational bonds, unsubordinated bonds issued by Ginnie Mae and any equity listed on ’specified indices’ anywhere in the world” (and I’m going to hazard that adding “anywhere in the world” is yet more bait, since the “specified indices” are major ones in developed economies). Let’s call it like it is: Morgan Stanley is not going to be able to make up their own arbitrary marks, thereby cheating the investor out of their collateral backing, on these deep and liquid securities. In any case, they are disincentivized to do so (as FT reveals in the next paragraph) by a set of over-collateralization rules.

Fifth:

As for the investor — remembering the products were launched as a response to investor demand for “secure, transparent and liquid currency package”– it means a potential upside scenario of receiving all of the performance of a currency index, for relatively low management fees, but without any interest or dividend (no carry trade here then) and downside scenarios that include credit-exposure to Morgan Stanley, covered by a claim on potentially illiquid securities, as valued by Morgan Stanley. Compulsory redemptions at inopportune moments due to a myriad of different triggers. And in the event of counterparty default, a position third-in-line for repayment.

Right off, the carry trade claim is simply wrong. Moreover, these are total return indices, so there ARE all the benefits of interest and dividends – they are just reinvested rather than distributed. The downside scenario is correct that this gives some credit exposure to MS, but the “potentially illiquid” line goes a little too far. I know that the prospectus says that these securities might not have a deep secondary market, because it has to, but in default they are extraordinarily liquid – they represent claims on FX derivatives! There’s no uncertainty about what they are worth in default.

Sixth:

ETF Securities’ ETCs are based on Morgan Stanley’s MSFX Total-Return Currency Indices. The way they achieve that performance, however, is not by replicating the components of those indices, but by taking out a total return swap with a counterparty that assures the performance of that index.

In ETF Securities’ case that counterparty happens to be Morgan Stanley (and only Morgan Stanley for the time being).

Again, misplaced suspicion. Here’s a scenario: to get exposure to the S&P 500 I can either 1) use cash to buy all 500 stocks and actively manage their exposure every day, making sure to reinvest dividends or 2) enter a total return swap which tracks the level of the actual S&P 500, plus dividends, perfectly. (There’s a third option, which is to buy SPY – effectively paying someone to do option one on my behalf.) Which one has a lesser chance of error? (Hint: it’s the swap.) Yes, a TRS is a derivative – but it’s not evil by that virtue. It’s exactly the same as a vanilla interest rate swap, the most liquid derivative in the world, only it reference the level of the S&P 500 instead of Libor. So let’s not get all suspicious of these crazy methods for replicating payoffs.

I’m not going to pretend that these ETCs are vanilla securities. They carry risks – perhaps large ones – and will likely experience liquidity difficulties until (and if) their market attracts traders, just like any security. No, I haven’t read the prospectus, and my comments are based purely on the FT post; moreover, my concern regards the FT’s attitude rather than the securities themselves. I can’t endorse any sort of derivative witch hunt of this sort – its unfounded, based if anything in popular fears that themselves were borne out of ignorance (on the part of both retail investors and institutions). It may be in journalistic vogue, but it’s hardly appropriate here.

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Deconstructing the Gaussian copula, part III

August 11, 2009 in Finance, Math, Risk

(Parts I, II and II and a half of this series are also available.)

In the first two parts of this series, I respectively addressed some misperceptions about the Gaussian copula and described its common use in CDO pricing. Part III focuses more on the model components and the intuition driving them.

I am a staunch supporter of a “models are just the tool” viewpoint, an opinion more elaborately and memorably stated by George Box as, “All models are wrong, but some are useful.” With that in mind, what you will find here is not a campaign against the Gaussian copula itself; merely its blind application to certain problems in finance. I find it as difficult to blame this model alone for the 2008 recession as I find it hard to blame the sinking of the Titanic on its hull design (new research actually suggests the rivets were more at fault) – while it certainly contributed to a general sense of invincibility and well-more-than-advisable risk taking, it is naive to think that in the absence of this notorious model, 2008 would have turned out just fine.

As I recently (and strangely, given my campaigns against it) stated about VaR, the Gaussian copula does exactly what it is supposed to do – the error lies in its interpretation and its application in the first place. I join Paul Wilmott in his crusade for less equations and more common sense among quantitative financiers: getting the number is good but explaining it is better.

Gaussian Dependance

Copulas are nothing more than descriptions of how two or more random variables relate to each other. To be more specific, copulas refer to the co-behaviors of uniform random variables only; but any distribution may be transformed to the uniform case via its CDF, and that is the appeal of copula models: they describe dependance without concern of the marginal distributions. The Gaussian copula, we may conclude, doesn’t necessarily have anything to do with normal distributions as we typically think of them (i.e. in the “normal distributions are useless in finance” sense)! Rather, it describes the sort of dependance that arises when a bunch of normally-distributed variables are correlated with each other.

Gaussian dependance isn’t easy to describe like a Gaussian distribution is. For the latter case, just think of a bell curve. The former is more difficult to identify, so here’s a picture of a two uniform random variables with a Gaussian dependance structure (click to zoom):

A first observation is that the dependence is regular (meaning even) and smooth. It lacks any significant clustering. More importantly, it lacks a property called tail dependence. Tail dependance is the probability of observing extreme observations in all random variables at once. Strictly speaking, it measures the probability of observing joint tail events. As you move further out in the tail, that probability converges to 1 in the limit for structures exhibiting tail dependence. It is extremely surprising and counter-intuitive to learn that the Gaussian copula lacks tail dependence. In plain English, this means that tail events in the Gaussian copula are asymptotically independent of each other – and that is the chief problem with using Gaussian dependence in finance.

In finance, extreme events co-occur all the time, as recent memory bears witness. If risk management is the process of ascertaining, measuring, and avoiding those situations, then doesn’t it seem a little odd to use a model which is explicitly unable to account for them? Tail dependence is a necessary condition for a dependence model in finance. The Student t copula exhibits it and is only marginally more difficult to implement than a Gaussian copula; but simplicity is king and there was obviously a decision made at some point that tail events didn’t require consideration, anyway. It brings to mind my favorite VaR metaphor as an airbag that always works, except in a crash.

Correlation

Another element of the Gaussian model which does not carry well to finance is the idea that linear correlation is a sufficient statistic for the dependence distribution. Consider these two plots, each of which shows two variables that, by construction, have a correlation of 0.7 to each other. First, a Gaussian dependence structure (this looks different than the above plot because the former was the copula itself, as indicated by the uniform marginals, whereas this is a full copula-derived multivariate distribution):

Next, a dependance structure exhibiting lower tail dependance (this is from a Clayton copula and is a stylized depiction of behaviors more characteristic of finance). You can plainly see the impact of the tail dependance, in contrast to the Gaussian plot above:

The two distributions are very obviously different, and yet if you merely measured their correlation you’d describe them in exactly the same way. Correlation alone is insufficient to describe more complex dependence structures such as those observed in finance. And yet, it is the only descriptive statistic of a multivariate Gaussian distribution.

Financial covariates tend to resemble the second plot – when a large negative event occurs in one, it more than likely will occur in the other. This, by the way, accounts for some of the skewness in financial distributions – it is possible to have two perfectly normal distributions whose combination is nonetheless skewed if the dependence structure exhibits tail effects like this.

Again, we have a call for clarity: it is imperative for the underlying dynamic of any model to resemble the behaviors of the system in question.

The Single Correlation Factor

In a CDO pricing framework based on the Gaussian copula, not only is correlation the sole determinant of the dependence structure, it is assumed to be the same for every name in the basket. This has caused much alarm. Certainly, using more factors would provide a more accurate model – allowing different industries to have different correlations, for example. Unfortunately, this comes at the cost of model accuracy.

It is very important, where possible, for a model to have no more than one unobserved input for every output. Think of a Black-Scholes option: future volatility can not be known, so we plug in whatever value gets the model to spit out the current market price of the option (a “the market is always right” approach). If there were two volatilities (say, a short term value and a long term value), we would be unable to create a consistent model, for there would likely be an infinite number of volatility pairs that would satisfy the market price. For every additional parameter, we need one more output metric to match. If we could match an option’s price and also it’s delta, just for arguments sake, then there is probably a unique combination of two volatilities for that output space.

This is why using multiple correlations is problematic not just from a fitting standpoint, but from a model integrity standpoint – if you take the thousands of necessary pairwise correlations and estimate just a handful of them incorrectly, the model could deliver completely spurious results.

(For a very concrete example of this, consider pricing a mezzanine or senior CDO tranche, which requires two correlation inputs. Without knowledge of the corresponding equity tranche price – and consequently the attachment point correlation – this becomes a very difficult puzzle indeed).

However, in my mind this is one of the more minor problems. That’s not to say it isn’t an issue, but I’d much rather have a single-parameter tail-dependent model than a multi factor Gaussian one. Why? Because it’s more important to me that a model captures downside risk in some regard than that it captures the distribution’s central dynamics more faithfully.

Correlation (again)

We’ve discussed why correlation is insufficient to describe the CDO dynamics, and also why a single-factor model may lack fidelity. But in some ways, the entire discussion is slightly off base. Correlation (as I’ve alluded before) is an implied measure – it is whatever plug gets the model to output the “right” price.

There is a raging debate about how similar correlation is to Black-Scholes volatility, but I think for the purposes of this exercise we can highlight their similarities (though I will not necessarily agree with that under more rigorous terms). both are plug values; both have intuitively “correct” ranges but can not be directly measured or observed; both are the single unobserved input in the most simple pricing models of their respective derivatives.

Because of this, a lot of our reasoning on the problems with correlation goes backwards, since we begin with the premise that correlation is arbitrary and/or unmeasurable, and therefore conclude that a correlation-based model must fail. However, in practice we actually start with a tranche price, and work out the implied correlation value from that price. So I don’t really care if my correlation comes out to 60% or 70% because I’m not going to read too much into that figure – it’s just a parameter that will keep my model ticking consistently with the market, all else equal.

“But wait,” you say, “that’s the dumbest thing I’ve ever heard!” What if the market price is arbitrarily high and implies a correlation greater than 1 (or just 1, since the input is bounded)? Then that’s great, you get the price right in that instant, but the second you try to measure any sort of risk or even price it the next day, you’ll fail because a correlation of 1 doesn’t reflect reality at all. Moreover, take this to its logical conclusion: why not have a model whose sole input and output is just the price. In this scenario, you would see a tranche trading at 20, and set your “model” to 20 (the implied price). Tomorrow, your model still says 20 – so when the actual tranche trades for 19, you need to adjust your “model parameter” (i.e. price) down. Obviously, a ridiculous situation and it speaks to the critical need for any model to balance a reasonable representation (even if a simple one) of reality with an acceptable range of input parameters.

To reiterate, this is why I would prefer a simplistic one-factor tail dependent model to a multifactor Gaussian one.

Other Copula Models

All of this must raise the question, why are we stuck using the Gaussian copula?

And like so much else, the answer is: because its easy.

As mentioned, the Student t copula exhibits tail dependence and is only slightly harder to build than the Gaussian variety. So why not use it? The dark secret (unless you read part II) is that single factor Gaussian copula models are really just massive simplifications of copula-derived mathematics. The engine itself relies on arithmetic and an integral – nothing that would suggest a copula model on the surface. It is the mathematically friendly properties of the Gaussian distribution that make this possible (though frankly, it seems to me a t implementation shouldn’t be much farther off). More obscure copulas, like those in the Archimedean family, don’t necessarily follow “real world” behaviors in high dimensions, as it pertains to finance.

Moreover, like all problems of this ilk, CDOs suffer from a massive curse of dimensionality. In such situations, familiarity is key – in fact, it is sometimes the only hope of finding answers in the massive cosmos of sparse data.

Finally, Gaussian copulas have a nice property – they are easy to explain (keep in mind, lately such explanations aren’t much at all). In particular, the error rates are easy to quantify – we can be 99.975% sure of an outcome. Knowing a concrete chance of failure, even if that probability is completely bogus, makes the model easy to accept. More complicated copula structures, by contrast, are harder to work with (read: make it harder for risk managers to promise certain error rates within certain error bounds).

Finally, more complicated does not necessarily mean better. Even after all I’ve written, a pinch of common sense applied to a single factor Gaussian model might do more wonders than a more advanced model in the hands of a naive user.

Here endeth the lesson.

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Deconstructing the Gaussian copula, part II

July 9, 2009 in Finance, Math, Risk

(Parts I, II and a half and III of this series are also available.)

Recently, I addressed a great deal of misinformation regarding the Gaussian copula and it’s role in the 2008 crisis. I would like to try and follow that up with a succinct description of the copula and its use in CDO pricing. (This may seem a defense of the math behind the process, but you know I’m just setting it up for a fall.)

Introduction

David Li’s contribution to quantitative finance was the rapidly-standardized “single factor Gaussian copula” CDO pricing framework. The real crux of the problem was the “single factor” part – not the Gaussian copula itself (though we won’t pull any punches here). In an extraordinarily broad sense, a copula is a mathematical function that describes how two or more random variables interact. “Correlation” is a simple way of describing the copula, which should give the function some intuitive grounding. But let’s back up a second and figure out why we even need a copula in the first place.

Aside: Why Copulas?

If you try to model the behavior of many random variables, you need a multivariate distribution. The most mathematically friendly distributions are from the Gaussian family, including the familiar bell (or normal) curve. This is why such models are prevalent in all manners of statistics. For most purposes, the model is not only easy to work with but asymptotically correct (which is a nice feature, to put it mildly). However, there are some areas where the model choice is more for pragmatic reasons than justified ones – finance being prime among them. Indeed, financial distributions do not behave normally, but only recently have tools been developed that can describe them – and even there large joint distributions are daunting.

So, it is unsurprising that the Gaussian copula arose as a natural choice for modeling the joint distribution inherent to CDOs – which are essentially just collections of many intercorrelated credits.

But I’m getting ahead of myself. (This is much easier to discuss than to write about, I think, because you can guage your audience’s comfort which each boldfaced section before moving on. I hope, brave reader, that you are still there.) Lets talk about CDOs.

CDOs

A CDO is nothing more than a collection of various bonds, all held together in a basket. The principal risk of a CDO is default: the chance that one or more of the bonds will not survive to maturity. To isolate this risk, it is instructive to think of the CDO as a basket of sold CDS contracts, rather than a basket of purchased bonds (and indeed, “synthetic CDOs” are nothing more than CDS portfolios and have rapidly gained market share from bond portfolios). Thus, the buyer of a CDO needs to draw two conclusions regarding the basket:

Will any of the credits default?
When will all of those defaults occur?

The first point is obvious; the second gets at the heart of the problem. Both the timing and the correlation of defaults matter. If the CDO basket is comprised disproportionately of financial companies, then default by one may imply a greater likelihood of default for the others; a more diversified basket may not exhibit such dependencies.

This issue is compounded by the introduction of tranches – a staple of the CDO industry. Again, it is helpful to consider a CDO as a basket of sold CDS. The most junior (or “equity”) tranche has, by definition, sold insurance on the first few issuers to default – say, the first 3. The next tranche does not experience a loss until the 4th issuer defaults. The key here is that when a portfolio is tranched, investors have not sold CDS on specific issuers by name, but rather by time of default. They can not know ahead of time which issuers they are effectively responsible or on the hook for.

Bathtub Correlation

To understand why tranching compounds the correlation problem, think of the CDO as a rectangular bathtub interspaced with mines that represent each issuer’s default. The CDO investors are aboard a boat on one side of the bathtub, and need to cross to the other side. If the boat hits a mine, that issuer defaults, and the explosion of the mine will damage the boat. The equity tranche has an extremely thin hull and will sink quickly; the senior tranche has a thick hull and can withstand many blasts without taking damage. Finally, the boat moves across the bathtub via geometric brownian motion – which is to say, randomly.

In a low-correlation world, the mines are dispersed uniform randomly across the bathtub; hitting one mine does not imply or necessitate hitting any other. With high correlation, the mines cluster somewhere in the water; hitting one mine makes it relatively certain that another will be hit.

As a consequence, equity investors prefer high correlation. They are indifferent to hitting just a few mines or many, as they are wiped out in both situations. Therefore, they prefer the mines to be clustered, as this leaves more clear paths across the bathtub. In contrast, senior investors prefer low correlation – they can withstand glancing off a few mines, but hitting a cluster would wipe them out.

From this intuitive example, it should be clear that not only the timing of the defaults, but also their expected clustering (i.e. correlation) is important when valuing a CDO tranche.

Correlation in the Guassian Copula

Let us first draw the connection I’ve sketched out already: CDOs are composed of many issuers that may interact with each other; and a multivarite normal distribution is a common method of describing such behavior. So far, so good.

Like any Gaussian multivariate model, the Gaussian copula takes as parameters the correlation of every pair of variables under consideration. (In other words, to make the model work, you need to “explain” to it how every issuer interacts with every other issuer – these are the parameters.) Thus, the number of parameters increases with the square of the number of variables being considered – specifically, there are $\frac{N(N-1)}{2}$ parameters. If you had a CDO of 100 names, you would need to compute 4,950 parameters to describe their behavior! It doesn’t take a statistical degree to appreciate the flimsiness of a model which relies on such assumptions – it’s just too many to estimate reliably. Clearly, the traditional model simply won’t do.

Enter David Li, whose principal contribution to this field is to boil 4,950 parameters down to just one.

Shocking! Dastardly! The decision that caused the 2008 crisis! Well, not really. Though I am full of doubts about the validity of the Gaussian copula for this task in the first place, I do not think that the compression of its parameter space is the chief culprit by any means.

What Li was suggesting amounted to this: instead of modeling the intricate inter-corporate correlation structure, in which financials are highly correlated to each other but bear little semblance to utilities, which themselves are very similar, he said why not just model everything at the average correlation of the CDO names? Actually, he just said that one correlation level will be enough to describe the CDO price – he did not say it was the average (I just added that to make the notion more tolerable at first glance). He didn’t care if you chose a higher or lower correlation than any pair in the whole CDO exhibited; his claim was that there was some single number that would get the model to output a price that matched the market.

Before we get up in arms about this let’s remember that most financial instruments are priced this way. One or more variables of the equation are left free to change, such that for some level the model will output the “correct” (or market-observed) price. With options, this is called volatility; with swaps this is the fixed rate; with bonds this is the yield – I particularly like the last example because most people assume this is limited to derivatives. It’s not, “real” securities exhibit this problem too — for stocks, it’s called a P/E ratio.

So, we’ve boiled correlation down to one parameter which can take any value, but forces all issuers to have the same correlation to each other AND (this is a much more important caveat) exhibit a Gaussian dependance structure.

Now What? This Is Getting Boring.

Ok, let’s price a CDO.

If I have CDS prices for all the issuers in my CDO, I can back out the probability of each issuer defaulting. (That’s a whole other lecture, but please take my word that if we have the price of default insurance, we can calculate the probability of default. Otherwise I’ll go on for another 2000 words…) This answers my first question: will defaults occur? Combine that with a correlation number and I can answer the second question: when will all the defaults occur? So now I can price the CDO, right? Unfortunately, no.

The default probabilities backed out of the CDS data are conditional default probabilities, meaning they have the market’s 4,950 correlation factors baked into them. Company A may be doing fine, but it’s very correlated to company B which is not so healthy. The result is that company A’s CDS will exhibit a relatively high default probability even though that’s more B’s fault than A’s.

In statistics, we like to deal with independent or unconditional probabilities, because the math becomes dramatically easier. So the conditional probabilities extracted from the CDS are not so useful, and must be transformed into independent probabilities. To achieve this goal, we do something that I think is very clever:

We set up a model in which defaults are driven by a shared “market factor” and an idiosyncratic factor, similar to a regression with one dependent variable and an error term, hence the name “single factor model.” Now, I know I just said there are two factors, but one is specific to each individual issuer, so it doesn’t count as one of the model factors — if this troubles you, chalk it up to statistical nuance. Anyway, the two drivers are weighted by a correlation term; as correlation increases the market factor dominates, and as it decreases the idiosyncratic factor dominates.

Now, suppose for a moment we knew the value of the [random] market factor. In this case, default would be driven solely by the idiosyncratic factor (since the market factor is fixed, and we have chosen it such that all names either are – or are not – in default). The idiosyncratic factor is, by definition, independent across all issuers. Therefore, we have artificially created a scenario in which defaults are independent for each issuer by conditioning the market factor on a certain level. More specifically, we have generated a set of conditionally-independent default probabilities. Now, repeat the process for every issuer and every market factor level. The result is a complete picture of how every issuer behaves in every possible situation. From this, the unconditionally independent probabilities can be extracted.

(If that isn’t quite clear, suffice to say there’s a bit of math behind it. Interestingly, the math is surprisingly simple, but with the exception of the number of factors in a Gaussian model I have promised not to write out any equations in this post, so in the absence of symbols I hope you will accept my reasoning.)

So now, we have the probability of every issuer independently defaulting at any given time – with that information, it is relatively straightforward to figure out the expected loss on the portfolio. In fact, it’s mainly arithmetic at this point: the value of the portfolio is just the probability-weighted average payoff of all the issuers.

And that’s really it – that’s how the Gaussian copula is used to price a CDO, or a collection of sold CDS on many issuers. We calculate the default probabilities from the CDS, then we use the Gaussian copula to tell us how they relate to each other. You’ll notice that I never actually mentioned the copula when discussing the probability model – that’s because you don’t really need it. It happens that the copula math simplifies nicely into something that is almost, but not quite, entirely unlike a copula (hey! a Douglas Adams reference!). However, the copula-based approach is more informative, even if copula-specific math per se doesn’t enter the picture.

And why is this so bad?

A few of the modeling decisions I’ve described above are unquestionably poor ones, though it may not be obvious how to improve them. Here is my brief rundown:

The Gaussian dependence structure – what’s wrong with it? What alternatives are there? Why are they better?
The single factor – is it really sufficient to describe the behavior?
The single correlation number – is it sufficient to describe the behavior? Can we reliably estimate more relationships? Is correlation the right metric in the first place?

I’ll attempt to answer all these and more in part III…

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Deconstructing the Gaussian copula, part I

June 5, 2009 in Finance, Math

(Parts II, II and a half, and III of this series are also available.)

Newsweek has a new article about Paul Wilmott called “Revenge of the Nerd” which I really enjoyed, with two caveats.

In its opening the article compares quants to aeronautical engineers who design faulty planes (CDOs). The author observes:

Yet while aeronautical engineers who willfully designed a faulty plane might be on trial for criminal negligence, Wall Street’s math gurus are, for the most part, still employed. Strangely, the banks need quants more than ever right now.

But where is the logical conclusion of the aeronautical metaphor, that other engineers would be needed to fix the planes? In that framework, there’s no contradiction.

But that point is minor. Here’s what really bothered me (my comments are inserted in bold):

In 2000, the CDO market was jump-started by David X. Li, who, while working at JPMorgan, created the Gaussian copula function (no, he didn’t), a formula for determining the correlation between the default rates of different securities (no, it’s not). In theory, the model tells you the odds that, if one CDO goes bad, others will too (no, it doesn’t). The apparent genius of the Gaussian copula is its abstraction (true, but not in the way the author means). Rather than relying on the immense amount of data used to figure the odds that a CDO might default (there is no such data; issuers default, not CDOs), Li appeared to have discovered a law of correlation (no, he didn’t). That is, you didn’t need the data; the correlation was just there. Armed with it, quants could price CDOs much faster, and traders could buy and sell them at record speeds. Gaussian was rocket fuel for the CDO market (“Gaussian” is an adjective, not a noun). The global volume of CDO deals went from $157 billion in 2004 to $520 billion in 2006. As more banks got in on the game, the once large profit margins started to shrink. In order for banks to make the same kind of returns, they had to pack more and more loans into a CDO, essentially making bigger bombs. Li was on his way to a Nobel Prize when the world blew up (no, he wasn’t).

I have no problem with the simplification of difficult topics (in fact, I encourage it). I also have no problem with bashing Gaussian copulas as applied to CDO’s (the argument was featured in my thesis years ago). But I have severe issues and frustrations with poor reporting of false information. Will 99% of this paragraph’s readers realize it’s incorrect or even care? Of course not! But it doesn’t make it all right.

Deep breath. Ready to go deeper?

This paragraph seems to be lifted largely from a recent Wired article (my response to that article would evolve into my “models are just the tool” tirade). Anyway:

“David X. Li…created the Gaussian copula function.” The Gaussian copula is rooted in research from more than 250 years ago. In fact, Gauss – a prodigal mathematician whose influence extends far beyond the bell curve – died in 1855! It’s unclear when the first bivariate extensions were arrived at, but Wikipedia notes that it must have been developed by 1872. The copula itself would not be described until 1959, but almost immediately mathematicians used it to decompose the multivariate normal distribution into a pairing of Gaussian marginals and something the new vocabulary termed a Gaussian copula. All David Li did was pair the copula and CDO pricing for the first time.

“…a formula for determining the correlation between the default rates of different securities.” Copulas describe the dependence structure of random variables. Correlation is a way of condensing the information contained in the copula down to a single number. The sentence as written suggests that the copula is used to measure correlation when in fact it is the other way around. In fact, you can not even create a Gaussian copula until after you decide what correlation to use.

“In theory, the model tells you the odds that, if one CDO goes bad, others will too.” I assume the author meant to write “if one issuer goes bad” rather than “if one CDO goes bad”, because the Gaussian copula as applied to CDO’s describes the issuers within the CDO, not CDOs to each other. In this framework, the sentence is correct: copulas describe the dependence structure, which essentially means “how one issuer relates to other issuers.” In this case, the thing being measured is default probability.

“The apparent genius of the Gaussian copula is its abstraction.” This is a true statement as it stands: the brilliance of the copula function is that it abstracts the dependence structure from the marginal distributions, meaning the dependence of, for example, two dice numbered 1-6 has the same copula as the behavior of two dice numbered 2-7. Before the development of the copula, the 1-6 dice would have a completely different function than the 2-7 dice, because one would have to account for the marginal differences while defining their dependence.

However, the abstraction the author is referring to is that “you don’t need the data, you only need the correlation” (see below). The Gaussian copula as Li implemented it boils all correlation down to a single number, enabling such an abstraction.

“Rather than relying on the immense amount of data used to figure the odds that a CDO might default…” The available data consists of CDS spreads and bond z-spreads, which may be used to imply a default probability for each issuer. However, to figure out if a CDO will default, one must evaluate the probability of multiple firms defaulting within a given time frame. This is the correlation parameter. Thus, the data alone does not tell you about the likelihood of CDO default.

The default probabilities extracted from historical data are not independent, and so can not simply be added (or multiplied, to be more precise) together. Moreover, the correlation which may be measured in CDS is the correlation of changes in default probability, and the jump to correlations of actual bankruptcy events is much more difficult, not in the least because there are relatively few historical defaults, compared to the number of issuers.

This isn’t to say that the data can’t be used – in fact the data must be used – but the key realization is that without a model (I struggle to think of one capable of handling such data that isn’t a copula), the data yields no worthwhile insights. Merely having the data is not enough to price a CDO.

“Li appeared to have discovered a law of correlation.” As I’ve mentioned, Li did not “discover” anything. He merely applied an existing model to a new dataset.

“You didn’t need the data; the correlation was just there.” Of course you need the data – the correlation is meaningless without the default probabilities extracted from the data. What the author presumably means is that your correlation number does not have to represent the “true” level of correlation observed in your data (which, as I’ve stated, is a nearly impossible thing to observe in the first place).

But having said that, this is probably the one thing the author has correct. After some futile efforts, researchers stopped measuring correlation and started holding a finger in the air to determine the “right” level. Similar to implied volatility in option pricing, correlation was unobservable and the “right” correlation was whatever level made the model price come out the same as the market price. Unfortunately, in a space where traders became so dependent on their models, the chain was circular: markets were informed solely by correlation-based models, which were themselves calibrated to the market.

The critique is not limited to the use of a Gaussian copula, however.

“Gaussian was rocket fuel for the CDO market.” Another true statement, but one which reveals the author’s unfamiliarity: “Gaussian” is an adjective used to describe a type of model. It’s a person’s name. This is like saying “Newtonian revolutionized the world of physics” when you want to talk about a model of gravitational acceleration or “Darwinian turned the study of biology upside down.”

“Li was on his way to a Nobel Prize when the world blew up.” No, he most emphatically was not. This is a repeat of a one of Felix’s statements from the Wired article. Even if the model had been perfectly accurate, do today’s financial journalists think pricing a financial derivative is worthy of a Nobel prize? Black/Scholes/Merton didn’t win a Nobel prize for their option pricing model, they won it for the research they did into the economics of asset pricing. The option model was just a nice benefit on the side.

A fundamental issue with this paragraph, on top of all these highlights, is that not once does it explain the actual problem. If you read the paragraph, and I asked you why did they blow up, could you tell me? I’m sure you’d say something about the correlation not being reflective of the data. And I’d respond, well then why didn’t we just start using the data, or start using the right correlation?

I’ll try to answer these questions soon in part II.

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