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Gaussian copula

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 and a half

August 11, 2009 in Finance, Math, Risk

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

One of the comments to part II of this series asked about recovery assumptions, observing that a 40% static recovery assumption would cause a 60-100% tranche to always be worth par, since it would never take damages. That’s correct and raises an issue I will dance around in part three: recoveries constitute another unobserved input to the copula model.

When issuers default, they do not drop completely out of the picture – there is a some recovery level assumed, on the premise that in bankruptcy there will be enough assets to cover some portion of the debt owed. It could be anywhere from 0% to 100%. If the issuer recovers 100%, the CDO takes no damage (as there is no need to pay out insurance).

The most simple way to address recovery assumptions is to take them as exogenous static inputs. They don’t necessarily have to be 40% (widely accepted as the long-run average recovery rate). Recovery assumptions could be based on an analyst’s estimate, implied from CDS, taken from a market quote (recoveries are traded quantities), spit out of a random number generator, etc. It doesn’t matter in an exogenously-specified static model.

The second way of dealing with them is to model them as stochastic quantities, which may or may not be correlated with default intensities. A simple (and hardly stochastic) method would be to lower recoveries as more issuers default, on the assumption that in such a scenario the business environment must be very bad. This would have the effect of compounding losses – not only are issuers defaulting, but the surviving issuer recoveries are falling over time. Alternatively, you could get very fancy and let the recoveries dance as random variables – incorporating a gamma process, for example. However, that’s a little beyond the scope of my series (for the moment).

There is generally enough information about recoveries available through research and market information that an exogenous model provides satisfactory results. One question we must ask is whether it is “bad” that 40% recoveries would cause a senior tranche to never take damages. Honestly, I think it isn’t because that’s a scenario that could easily be reflected in reality: if it really happened, then the tranche wouldn’t take damages. It might trade down, however, as people fear recoveries would drop – and that is where the model specification comes into play. It is the role of the model caretaker, not the model itself, to make sure its input parameters and dynamics are reflective of reality.

There’s nothing wrong with creating a risk free bond – and to take this example to extreme, the 99-100% tranche is most likely risk-free under any recovery assumptions. The only problem is to expect to get paid for holding it. Moreover, even a 60-100% tranche with 40% recoveries could trade under par depending on interest rates and (in some cases) prepayment probabilities. But that’s another chapter.

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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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Monte Carlo: house of cards?

May 8, 2009 in Finance, Math, Risk

The WSJ recently ran apiece on Monte Carlo risk management:

Here is how a typical Monte Carlo retirement-planning tool might work: The user enters information about his age, earnings, assets, retirement-plan contributions, investment mix and other details. The calculator crunches the numbers on hundreds or thousands of potential market scenarios, guided by assumptions about inflation, volatility and other parameters…

These models were supposed to help quantify and manage the risks of mortgage-backed securities, credit-default swaps and other complex instruments. But given the events of the past couple of years, it appears that the models often gave big institutions, as well as small investors, a false sense of security.

I can’t stand idly and let another “models are just the tool” opportunity go by. Monte Carlo simulations are among the most powerful and versatile tools available. As an empiricist, I prefer their discretized approximation of reality to grappling with overly simplified closed form models. There are few substitutes to a Monte Carlo expectation for determing the outcome of a complex process, given a reasonable set of starting assumptions.

Yes: given a reasonable set of starting assumptions.

A Monte Carlo simulation is nothing more than a bunch of random numbers which, taken together, determine a path through time. At each step along the path, more random numbers are generated. The rules by which those numbers are generated – what we refer to as their marginal distributions and pairwise correlations – are not part of the Monte Carlo process, strictly speaking. They are purely at the discretion of the operator. And their specification is critical to the success of the model (which, in their absence, is just a big random number generator).

Monte Carlo methods are really useful because of the existence of path-dependent outcomes. Path-dependent means that the end value depends on intermediate values. For example, a stock price is not path dependent because having a price X on day 1 does not preclude having a price Y on day 2. But what if a bond can be called at par if – and only if – it trades above par for 20 days? Then the price of the security depends heavily on its price in the past. Monte Carlo analysis solves such problems because it actually generates a price path; other methods can not account for such securities.

So why are these risk mangement companies using Monte Carlo methods to value stocks? Beats me – a closed form expression would do just as well (given the same assumptions as the current set up) without the need to waste cycles on each Monte Carlo iteration. I have a sneaking suspicion, however, that it’s because it’s nice to show investors “probable stock paths”, something which Monte Carlo can do and “math” can not.

This author actually comes close to the mark:

Critics emphasize that the problem isn’t Monte Carlo itself, but the assumptions that go into it. Since no standard approach exists, one user might plug in a range of assumptions on interest rates, inflation or volatility that is different from another user.

But this implies that a solution is to have a “standard approach,” which would presumably take the form, “Using 2% interest rates, 2% inflation, 30% volatility gives you the right answer.” Obviously, that doesn’t exist.

Now, some of the assumptions going into these Monte Carlo calculators are wrong because the end-user has mis-specified them (ridiculous interest rates, for example). But the real mis-specification is that the Monte Carlo models described in the article use normal distributions as the heart of their engine. Even the best end-user assumptions will be rendered useless if the distributional assumptions are wrong.

And here’s the rub: a non-path-dependent normally-distributed Monte Carlo simulation can be perfectly replicated by… a Gaussian copula.

Shocker.

“Gaussian copula” has become a dirty word in finance, because we recognize the havoc that these models have wreaked on the CDO industry. We – by which I mean countless journalists – have railed against this model because it was not suited to the task (in my opinion, this is a little like blaming Porsche when you fill your tank with 83 and ruin your car). And yet here it is again in another guise! If I write an article pointing out that normal Monte Carlo is just a discrete Gaussian copula (and I guess I am), can I start a new crusade against Monte Carlo models?

I hope not, because I think Monte Carlo models are wonderful tools. Like the Gaussian copula, they are well suited to answering a variety of questions. Unlike Gaussian copulas, they can be adjusted to account for non-normality (that’s sort of a cheap shot, since a Gaussian copula is by definition normal; a non-normal copula could be substituted instead). It’s sad that at the moment such models are needed most, we choose to blindly rage against the method rather than aknowledge the deficiency of its current implementation and just simply, astoundingly simply, correct it.

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Signs of the apocalypse

February 24, 2009 in Finance, Math, Risk

Wired has published an article attacking the Gaussian copula: Recipe for Disaster: The Formula That Killed Wall Street.

It’s a very typical “hate the game, not the player” article which finds fault with a tool rather than the people who use it. Not that I completely disagree with the critique – but imagine my surprise when an idea rooted in that obscure statistical field in which I claim expertise showed up on the [digital] glossy front page of a pop sci magazine.

It seems like the Gaussian copula is the new VaR – everyone uses it, everyone claims to know its limitations, but when the world blows up everyone is quick to point fingers at the model. From a mathematical perspective, the two are actually incredibly close, which is perhaps unsurprising given their pitch of “easy risk measurement with minimal parameterization.” VaR has dubious use as a relative indicator through time – if day 2’s VaR is higher than day 1’s VaR, maybe risk increased. But as soon as one applies meaning to the VaR, they have overstepped its utility. The Gaussian copula has similarly limited value – under its assumptions, one can gauge the relative magnitude of risk and price shocks.

In truth, we use oversimplifications like this all the time in finance, in places few pop sci magazines dare tread. Case in point: options pricing. Volatility, like correlation in the Gaussian model, is a plug – it’s the number that gets you to the right place. It has no meaning outside a log-normal no-arbitrage zero-drift tax-free world. And yet we use it just fine.

The Gaussian copula may be a poor fit to reality, but it’s good math. It’s simple and gets a reasonable approximation. In the next month the credit world will officially adopt a newly open-source JP Morgan model for all CDS pricing. Like the Gaussian copula and VaR – two other JP Morgan innovations – this model oversimplifies the world in order to provide an easy view of pricing and risk. Does that make it bad? Certainly not. The fault is with the trader, risk manager, or analyst who blindly adopted the model without thought.

Surely if a man with an exotic flamethrower accidentally burnt his village down, we would blame the man and not the unfamiliar machine for causing the damage.

I’m definitely not a supporter of the financial applicability of the Gaussian copula – I spent a good chunk of my academic career ranting against it – but, just as with VaR, much of the the fault lies with the users and not the mathematics.

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