Category / Financial Models

Discounting in the presence of rate uncertainty July 28, 2011 at 6:21 am

We all know how to discount future cashflows – they teach you that in Econ 101. But what do you do if you aren’t certain of the rate at which to discount? Mark Buchanan, in a fascinating blog post discussing work by Farmer and Geanakoplos (HT Naked Capitalism) answers the question. While you do what most people would expect (OK, most people with a quant background), namely take the probability-weighted average over paths of the effective discount factor on each path, what you end up with is more surprising. According to Farmer and Geanakoplos, in this setting discount factors follow a power law:

D(T) = (1 + aT)^-b

where a and b are constants.

The crucial observation is that this falls off much slower than the usual exp(-rT), and hence typically gives much more value to cashflows in the distant future. If one wanted to be hyperbolic about this (yes, yes, pun intended), then one would say that the cure for short termism is simply to use the right discounting function.

And here comes Hurst, they think it’s all over! It is now! July 14, 2010 at 6:06 am

There has been some comment recently about a paper by Reginald Smith on the impact of high frequency trading (HFT) on market dynamics. I want to spend a little timing explaining what the paper says, roughly, and why it matters.

We can clearly demonstrate that HFT is having an increasingly large impact on the microstructure of equity trading dynamics… the Hurst exponent H of traded value in short time scales (15 minutes or less) is increasing over time from its previous Gaussian white noise values of 0.5. Second, this increase becomes most marked, especially in the NYSE stocks, following the implementation of Reg NMS by the SEC which led to the boom in HFT. Finally, H > 0.5 traded value activity is clearly linked with small share trades which are the trades dominated by HFT traffic. In addition, this small share trade activity has grown rapidly as a proportion of all trades.

So first, what is a Hurst exponent?

Roughly speaking, Hurst exponents measure autocorrelation or, even more loosely, predictability. If H is close to 0.5, the series is a random walk, or what we were told equity prices did in Finance 101. In particular, if H = 0.5, the idea of volatility makes sense, and we can quantify risk using volatility.

If H is bigger than 0.5, though, the series shows positive autocorrelation: roughly, it has very busy periods when volatility is high, and quieter low volatility periods. It switches regimes between these with no warning. Thus we might try to calibrate a simple risk model but if we are unlucky we will calibrate it to a low vol period and then when the high vol hits, our risk estimates are wrong.

So, what the paper seems to have proved (and I have not checked all the details) is that HFT has changed the nature of stock price returns from being a random walk (H = 0.5) to having significant positive autocorrelation. Increasingly we see quiet periods when not much happens followed by periods of intense volatility, and the change between these is unpredictable. Now notice the time period cited, 15 minutes or less. What is happening, then, is that HFT appears to be creating islands of high volatility amid an ocean of more stable prices. Something sets off a price change, which creates a flurry of HFT activity, exacerbating volatility; this then dies away over a period of minutes or hours.

Why does this matter to the ordinary investor? Simply that their trading might hit one of those flurries of activity, and they might well get a significantly worse price than average if it does. Moreover of course simple risk models such as VAR will be less and less accurate risk gauges the higher the autocorrelation. I suspect on the typical VAR one day holding period this does not matter much, but it might.

Finally, there is the issue that HFT might be increasing the risk of flash crashes. If autocorrelation is too high then the probability of very large deviations from the mean over short timescales increases dramatically. I have no idea if this research supports the idea that we have got to that point yet. But I do think that someone should find out.

Quants, Lightbulbs and the Demise of the Financial System January 28, 2010 at 6:48 am

From Naked Capitalism reader Matthew G:

How many quants does it take to screw in a lightbulb?

Using ten racks of co-located blade servers, one quant can detect a janitorial inefficiency, step in between janitor and light fixture, and screw in 49,500 bulbs in less than a millisecond, keeping five hundred lightbulbs of profit.

Two quants competing with each other can screw in 99,998 bulbs in a millisecond, with each quant retaining a profit of one lightbulb.

When ten quant firms try to screw in a light bulb, the bulb explodes, the light fixture gets ripped from the ceiling, the building falls down, the entire electrical grid of the city of Greenwich shuts down, innocent civilians all over the world have their retirement accounts electrocuted, and the Federal Reserve has to give the counterparties of each quant firm five hundred million light bulbs to maintain the stability of the system.

Update. FT alphaville saves me from having an entirely frivolous post by referrring to this article at Trader’s Magazine. They say:

Bryan Harkins, an executive with the Direct Edge ECN, noted the market is “saturated” with high-frequency shops. He doesn’t expect overall industry volume to increase substantially in the next few years.

Volume, in the past three years, has doubled due to a large extent to the activities of high-frequency traders. Average daily volume is about 10 billion shares today. That compares to 5 billion shares in early 2007.

“Someone leaves a high-frequency trading shop to start a new one,” Harkins said. “You do a meeting [with them] and they say ‘We’re going to do 100 million shares a day.’ You get all excited with the next big account and then six months later they’re struggling to stay in business.” About half of Direct Edge’s volume comes from high-frequency trading firms, Harkins said.

[NYSE Euronext's Paul] Adcock noted the changes in volume at NYSE Arca’s top five high-frequency accounts mirror those of the VIX “almost perfectly.” And because most high-frequency strategies are similar, he adds, only the “biggest and fastest will make those strategies work.”

(Emphasis mine.) The comments above refer to the good times, too. Imagine what would happen if one of those big guys liquidates, or if we have a very high volatility episode with extreme decorrelations, as might happen for instance if there is a sovereign crisis. It won’t be pretty but we cannot say that we have not been warned.

I’ll take an alpha please Bob* December 1, 2009 at 2:50 pm

Dealbreaker says:

Robert Litterman is head of quantitative resources at Goldman Sachs Asset Management… And as he sees it, … quantitative hedge funds have to do a better job of making money for their clients. And in Litterman’s considered opinion, they need to find new ways of making money. New and non-quantitative, apparently.

We’re putting together data that’s not machine-readable.

I see. Any other pearls of wisdom?

You have to adapt your process. What we’re going to have to do to be successful is to be more dynamic and more opportunistic.

Totally worth the price of admission to the Quant Invest 2009 conference (flight to Paris not included). Thank you, Bob.

Now that is quite amusing, but perhaps a little unfair. What is clear is that you can make money for extended periods of time by being long liquidity premiums and short volatility. Many hedge fund ‘strategies’ are just versions of this strategy: get exposure to illiquid assets, leverage up, and hope there is not a flight to quality before you have got paid your 2 and 20. If you can guarantee your leverage through good times and bad (or are not leveraged at all and can lock investors in for long enough), this strategy is often successful even through a crisis. But if you have to sell into the storm, things will go rather less well.

One thing that might be interesting, then, is somehow measure alpha relative to probability of having to deleverage. That is, we ought to level the playing field between funds that generate high alpha at the expense of running the risk of having to sell into a crisis and those funds which generate less excess return, but which never have to deleverage.

*OK, some of you might not remember Blockbuster. It was a classic, in the sense of classically, heroically awful.

Physics envy, History envy June 29, 2009 at 8:51 am

Physics is in some ways the geekiest science. It’s fundamental, it has hard maths in it, and it has had enormous success at explaining the phenomena it tries to study. What other subject can successfully predict something to twelve decimal places?

As a result, some practioners in other fields have physics envy. This is a notable problem for finance quants, many of whom didn’t make it as academic physicists (or did make but didn’t like the salaries). Indeed in retrospect one can make a case that one of the causes of the Credit Crunch was the collapse of the Soviet Union – the argument would go that the collapse freed up lots of highly trained mathematians and physicists, some of whom came to work for investment banks – no bulge bracket firm was without its Academy of Sciences prize winner; the geeks used used the maths that they knew, which was mostly stochastic calculus, to model things; these models were dangerous but not easy to falsify (because they were only really wrong in a crisis); so the industry used them and was subsequently screwed. In one way at least communism brought capitalism down with it.

Anyway, the desire to build highly mathematical models has in practice lead finance down a dangerous path. Perhaps the aspiration was good, but the implementation has been deeply flawed.

Let me instead propose a different aspiration. History envy. History is a lovely subject. There are lots of facts, but most historians ignore many of the relevant ones. They are interested in motivations, in causes, in the evolution of ideas. They want to understand the why as well as the what. A good history text is carefully argued and insightful. It provokes discussion, and casts fresh light on the present. It’s not clearly wrong, given the evidence, but it can never be said to be right, either.

How much better would finance be if it took these desiderata? Abandon the spurious and misleading quest for quantification. Just try to make an interesting argument about why things happen.

Rebuilding May 22, 2009 at 9:59 am

There is a lot of comment around at the moment about how broken finance is: here, for instance, is a piece by Pablo Triana. And certainly there are many, many issues that we have no idea how to deal with in practice, including fat tails, autocorrelation, correlation smiles, and hidden systematic risks. These phenomena challenge option pricing models, CDO pricing, basket option pricing, ABS pricing and all sorts of quantitative risk management model.

But, but, but. There are some things that work. The huge push in the 1990s on the behaviour of the yield curve has at least left us with a good idea how to manage single currency swaps books. Vanilla puts and calls can mostly be hedged effectively. Credit derivatives – despite stident claims otherwise – have not caused the end of the world as we know it.

We need then to return to the things we do actually know, and to be very critical about what has worked well, what has worked acceptably, and what has turned out to be unhelpful. Saying the whole edifice of mathematical finance is rotten is just as counterproductive as saying that none of it is. For once, finance theorists need to be disinterested and critical observers of reality rather than cheerleaders (or hooligans). Let’s see what we need to tear down and what is still standing now that the tumult is dying down.

Copula counterfactual April 28, 2009 at 6:30 am

How different would the world be if David Li had written about a variety of different copulas rather than just the Gaussian one? (Do read the excellent Sam Jones piece that the link points to.)

More on models April 22, 2009 at 8:16 am

From Daniel Kahneman, via portfolio.com:

A group of Swiss soldiers who set out on a long navigation exercise in the Alps. The weather was severe and they got lost. After several days, with their desperation mounting, one of the men suddenly realized he had a map of the region.

They followed the map and managed to reach a town. When they returned to base and their commanding officer asked how they had made their way back, they replied, “We suddenly found a map.” The officer looked at the map and said, “You found a map, all right, but it’s not of the Alps, it’s of the Pyrenees.”

Correlation is not causality April 3, 2009 at 8:51 am

From the social science statistics blog via Naked Capitalism, an amusing illustration of this truth:

Lemons vs. fatalities

Sociologists do models, kinda February 13, 2009 at 6:59 am

From Reflexive Modeling: The Social Calculus of the Arbitrageur by Daniel Beunza and David Stark:

Modeling entails fundamental assumptions about the probability distribution faced by the actor, but this knowledge is absent when the future cannot be safely extrapolated from the past…

By privileging certain scenarios over others, by selecting a few variables to the detriment of others, and in short, by framing the situation in a given way, models and artifacts shape the final outcome of decision-making. This … is the fundamental way in which the economics discipline shapes the economy, for it is economists who create the models in the first place…

…models can lead to a different form of entanglement. In effect, models can lock their users into a certain perspective on the world, even past the point in which such perspective applies to the case at hand. In other words, models disentangle their users from their personal relationship with the actor at the other side of the transaction, but only at the cognitive cost of entangling them in a certain interpretation.

Despite the focus on relatively uninteresting models (merger arb), this is an interesting paper for anyone interested in how traders really use models.

Tarring Taleb January 21, 2009 at 12:44 pm

I have always been a little suspicious of Nassim Taleb. He seems to take too much pleasure in discussion of crises. And his first book — a very conventional account of hedging — isn’t actually very useful for actually running portfolios of options. Now a post on Models and Agents (an excellent blog I have only found recently) gives a more focussed critique:

the current crisis is not a black swan. Alas, the world’s economic history has offered a slew of (very consequential) credit and banking crises … So not only aren’t credit crises highly remote; they can be a no-brainer, particularly if they involve extending huge loans to people with no income, no jobs and no assets.

Taleb also recommends that we buy insurance against good black swans—that is, investments with a tremendous (though still highly remote) upside but limited downside. For example, you could buy insurance against the (unlikely?) disappearance of Botox due to the discovery of the nectar of eternal youth. And make tons of money if it happens.

And that surely is the point. Yes, the unexpected happens with considerable frequency. But knowing which black swan is more likely than the market is charging for is the hard part. Buying protection in the wings on everything is far too expensive to be a good trading strategy. If all Taleb’s observations amount to is the claim that being long gamma can sometimes be profitable, then they are hardly prophetic. What would be much more useful would be his analysis of when, exactly, black swan insurance is worth buying.

No arbitrage requires arbitrageurs November 20, 2008 at 6:14 am

No arbitrage conditions are not natural laws. You can only rely on them if there are enough arbitrageurs around to keep the markets in line. At the moment, that isn’t true in many settings. John Dizard points out an example from the Tips market:

seven-year Tips bonds are asset swapping at 130 basis points over Libor

As Dizard says, this is partly because the Tips are illiquid and hard to finance (and thus to leverage), and partly because there is not enough risk capital around:

The dealers can’t afford to make efficient markets, given their decapitalisation, downsizing, and outright disappearance. That means anomalies sit there for weeks and months, where they would have disappeared in minutes or seconds. The arbs, well, they thought they had risk-free books with perfectly offsetting positions. These turned out to be long-term, illiquid investments that first bled out negative carry, and then were sold off by merciless prime brokers.

What are the dynamics of risky bond prices? November 4, 2008 at 7:11 am

The Basel Committee’s two papers on incremental risk in the trading book (incremental to that captured by VAR, that is) – here and here – led me to muse on what the real dynamics of risky bond prices are.

Firstly clearly there is an interest rate risk component. Let’s ignore that as it is the best understood.

Second there is jump to default risk. The phrase itself is slightly misleading in that bond prices often fall a long way in the period before default, and indeed recoveries are sometimes higher than pre-default bond prices would suggest. Skip to default might a better term. Still, the idea that there is a jump process which can cause non-continuous changes in risky bond prices is reasonable.

Then there are ‘everyday’ movements in credit spreads. Now, here’s the six hundred and forty billion dollar question (OK, OK, not the size of the corporate bond market I know) – if you take out the jumps, is what you are left with even vaguely normal? My guess is that it isn’t, and that autocorrelation is significant even after jumps have been taken out. The hard part is that you need a lot of credit spread data to look at this kind of thing, and it isn’t easy to come by. CDS data won’t do in this instance simply because single name CDS have only been liquid for ten years or so, and you’d at least want data going back well before the ’98 LTCM/Russian crisis. I’ll get around to this sometime soon…

Spread dynamics are the flipside to my earlier post on what CDS spreads mean: that was about what causes the spread to move; this is about how you can model those movements.

Swaps spreads and other lunch toppings October 26, 2008 at 10:34 am

Why, sometimes I’ve believed as many as six impossible things before breakfast said Alice. This quotation came to mind in the discussion of the 30y dollar swap spread in the FT recently:

“Negative swap spreads have been considered by many to be a mathematical impossibility, just like negative probabilities or negative interest rates,” said Fidelio Tata, head of interest rate derivatives strategy at RBS Greenwich Capital Markets.

Oh dear me. A mathematical impossibility is 2 and 2 adding to 5, or the sudden discovery of a third square root of 4. A physical impossibility is something that we think is impossible according to our current understanding of science: accelerating from rest to go faster than the speed of light, say.

Negative swap spreads are neither of those. They simply represent an arbitrage. An arbitrage is when you can make free money without taking risk. Ignoring for a moment the risk de nos jours – counterparty risk – swap spreads allow one to lock in a positive P/L if one can fund at Libor flat. Free lunches do not often exist in finance, but they do happen in particular when there are no arbitrageurs left standing. No arbitrage relies not on the theoretical possibility of a free lunch, but on enough people actually wanting to dine for nothing that prices move to stop the feast. At the moment there is such a shortage of risk capital that one can indeed find free food. So `impossible’ things are happening not just before breakfast but all through the day. Bon appetit.

What is a derivatives pricing model anyway? September 4, 2008 at 12:29 pm

I had a conversation about this last night and thought it was worth writing some of it down and extending it a little. So…

Let’s begin with the market. For our purposes there are some known current market variables which we assume are correct. This could be a stock price, interest rates, a dividend yield — and perhaps one or more implied volatilities.

Secondly we have a model. The model is often, but not always, standard, i.e. shared between most market participants. Let’s start with standard models. Here the model is first calibrated to the known market variables.

At this point we are ready to use the model. There is a safe form of use and a less safe one. In the safe one we use the model as an interpolator. For instance we know the coupons of the current 2, 3, 5, 7 and 10 year par swaps (plus the interest rate futures prices and deposits) and we want to find the fair value coupon for a 4.3 year swap. Or we know the prices of 1000, 1050 and 1100 strike index options and we want to price a 1040 strike OTC of the same maturity.

The less safe use is when we use the model as an extrapolator. We want a 12 year swap rate, for instance, or the price of a 1200 strike option. That’s not too bad provided we don’t go too far beyond the available market data, but it is definitely a leap.

(Both of these, by the way, count as FAS 157 level 2.)

Note that there are two ways that we realise P/L in derivatives. Either we trade them or we hedge them. If we are in the flow business then trading is important. We need to use the same model as everyone else simply because we are in the oranges business and we need to kInow what everyone else thinks an orange is worth. We take a spread just like traders of other assets, buying for a dollar and selling for a dollar ten, or whatever. The book might well be hedged while we are waiting to trade, but basically we are in the moving business. Swaps books, index options, short term single stock, FX, interest rate and commodity options, and much plain vanilla options trading falls into this camp.

In the hedging business in contrast we trade things that we do not expect to have flow in. Most exotic option businesses are an example here, as are many long dated OTC options. There is no active market here so instead we have to hedge the product to maturity. Thus here the model hedge ratios are just as important as the model prices. Valuation should reflect the P/L we can capture by hedging using the model greeks over the life of the trade. Thus standard models are more questionable in the hedging business than in the moving business since it is not just their prices — which are correct by construction — but also their greeks that matter.

Things start to get really hairy when we move away from standard models. Now we are almost certainly dealing with products where there is no active market (some kinds of FX exotics are a counterexample) and we do not even know that the model prices are correct. There is genuine disagreement across the market as to what some of these things are worth. Different models also produce radically different hedge ratios. How can we judge the correctness of such a model? The answer is evident from the previous paragraph: it is correct if the valuation predicted can genuinely be captured by hedging using the model hedge ratios. [Note that this does not necessarily give a unique 'correct' model.]

In summary then: for flow businesses we need interpolators between known prices and, to a lesser extent, extrapolators. For storage businesses we need models which produce good hedge ratios.

Where are the ‘risk free’ curves in dollars now? August 25, 2008 at 7:43 am

It is a serious question: the Treasury curve is being moved by concerns about the cost of the GSE bailout, with some commentators saying that the US could lose its AAA.

And the Libor curve, according to a money manager quoted by Bloomberg,

aren’t reflective of the entire banking system but of three or four major banks that continue to have pressure on liquidity

So where is the ‘risk free’ curve in dollars exactly?

Undercover equilibrium: Holiday reading 2 August 11, 2008 at 4:17 pm

The other popular economics book I read on holiday was Tim Harford’s Undercover Economist. It’s an interesting if quick read, and I can entirely see how it stimulated admissions to undergraduate economics programmes. It strikes me, though, that Harford’s examples work best when the notion of price is unproblematic. He is presenting classical economics, so he assumes that prices are known and that all agents have a view as to the correct price for a good or service. Things get a lot more interesting once prices are not observable or when agents don’t know what the ‘right’ price is. As, umm, in the debt markets at the moment. The Big Picture has a related discussion concerning the failure of equilibrium economics.

I have several favorite examples of where markets simply get it wrong. When I spoke with the reporter on this, I used the credit crunch as exhibit A. It began in August 2007 (though some had been warning about it long before that). Despite all of the obvious problems that were forthcoming, after a minor wobble, stock markets raced ahead. By October 2007, both the Dow Industrials and the S&P500 had set all time highs. So much for that discounting mechanism.

We’ve seen that sort of extreme mispricing on a fairly regular basis. In March 2000, the market was essentially pricing stocks as if earnings didn’t matter, growth could continue far above historical levels indefinitely, and value was irrelevant.

My own view is that finance is not an equilibrium discipline, mostly, so while classical economics might work well in explaining the price of coffee – one of Harford’s examples – it does rather less well in asset allocation or explaining the return distribution of financial assets. Rather new news arrives faster than the market can restore equilibrium after the last perturbation, meaning that most of the time equilibrium is not a useful concept.

Soros and Equilibrium: Holiday Reading 1 at 4:02 pm

While I was away, perhaps slightly masochistically*, I read the new Soros book, The New Paradigm for Financial Markets: The Credit Crisis of 2008 and What it Means. It is not a particularly good summary of what happened, nor a detailed analysis of why it happened, but it does make an interesting point. Soros claims, I think very plausibly, that finance is reflexive, that is that the very study of it changes the object being studied. I have written about this before, but it is interesting to see Soros making much of the role of reflexity in the formation of asset price bubbles.

Brera Library

Of course, this feature of finance renders the received wisdom of classical economics rather suspect. In particular, models in finance are not time-stable in the same way that a good piece of science is, simply because the way market practitioners behave changes. The S&P return distribution with over half of all trades done by machine (2008) is unlikely to be the as that when most of the market went via floor traders (1988).

* ‘We read popular finance books so you don’t have to dot com’ has not, funnily enough, been registered as a domain name…

Renaissance man December 20, 2007 at 7:47 am

Alea reports a talk that James Simons, founder of Renaissance Technologies, gave at NYU recently. Simons is a highly successful quant investor so his remarks are interesting. The part of the Alea article that really piqued my interest was:


[...] perhaps the most interesting observation came in response to a question posed by the moderator, Nobel Prize-winner Robert Engle: “Why don’t you publish your research, the theory behind your trading methods? If not while you are active in the markets, perhaps later on.”

Simons’ reply – there is nothing to publish. Quantitative investment is not physics. The markets have no fundamental, set-in-stone truths, no immutable laws. Financial “truth” changes constantly, so that a new paper would be needed almost every week.

The implication is that there is no eternal theorem of finance that could serve as an infallible guide through all the ages. Indeed, there can be no Einstein or Newton of finance. Even the math genius raking in $1 billion and consistently generating 30%-plus annual returns wouldn’t qualify. The terrain is just too lawless.

Simon’s view seems to me to be obviously true, although I don’t quite agree with the Alea spin. It isn’t that there is no law, it is that the law changes as the behaviour of market participants change. Yesterday’s arb is today’s theorem is tomorrow’s unrealistic simplification. As I said over a year ago, mostly the market trades based on the current orthodoxy. But big news changes that orthodoxy – as is happening at the moment in the liquidity markets, and so to make a lot of money you need to be willing to keep changing your theory of asset prices.

This neatly brings me to a related topic, the non-equilibrium nature of financial markets. In retrospect, Walras’ idea of an auctioneer groping towards equilibrium (word of the week – tâtonnement) is really unhelpful because it suggests that there is enough time for this process to be completed and equilibrium reached before the next piece of news hits the market. I don’t think this is true. Rather I conjecture that the process is much more like a game of tetherball, with each new news item changing people’s opinions and hence moving the market long before equilibrium is reached from the previous piece. The ball almost never hangs by the pole, so any theory which analyses where it will come to rest isn’t much use in determining who is going to win the game.

The last piece of the puzzle is the primary role of transactions. There are no prices without transactions to establish them – lots of transactions. So it is only opinions about asset prices which lead to trading that matter. You can be right for a long period about the fundamentals, but if your assumption about how fundamentals lead to trading is wrong then you will lose money. For example I called the weakness of Japan completely correctly through the 2nd half of the 1990s and first half of 2000s, but I was wrong for extended periods on dollar/yen because I hadn’t accounted for the actions of the BoJ and other market participants beliefs about the BoJ. To make a lot of money you need to predict what most other market participants trading-related beliefs will be and get your position on before they do. Predicting fundamentals is only useful if they will influence future trading: on the flip side, predicting wrong beliefs is just as good as predicting right ones if they pertain for long enough for you to make money.

Level 3 and default correlation November 12, 2007 at 8:46 am

Suppose you take a collection of assets of known price and put them into an SPV. The SPV issues tranched debt. Obviously the sum of the values of the tranches equals the sum of the values of the original assets (unless there is some external credit enhancement for the SPV which adds value, or some other external influence).

Clearly too in a conventional tranching structure the senior tranche bears the least default risk and the junior tranche the most. But what is the fair value of each silo individually?
This is the problem many banks are facing at the moment. They have either retained tranches in their own CDOs or purchased tranches in other people’s, and these securities have not traded for some time. Therefore level 1 (mark to an observable market price) valuation is impossible. The assets have to be valued, though, so they have to do something, albeit in level 3.

The basic modeling problem is to assign value between tranches. This assignment depends on something that is popularly known as default correlation, but should properly be called default comovement: if the occurrence of one default in the CDO assets makes another much more likely, then the senior tranche is less valuable. If one default is more or less unrelated to another, then the senior is safer and hence more valuable.

Default comovement is idiosyncratic. There is no reason to believe it will be the same between one pool of mortgages and another, let along between one diverse pool of ABS and another. Some limited information on it in particular cases can be inferred from the prices of the few liquid securities – the iTraxx tranches and the ABX, for instance – but there is no compelling reason to believe this carries over to other asset pools. This means that more or less any CDO tranche at the moment is a level 3 asset and any valuation necessarily has a measure of model risk and/or model parameter risk.