This book has been published at a most awkward time. It discusses a market that has changed dramatically, due to the ongoing financial crisis, since the book was finished. Many commentators blame the market for exacerbating the crisis with toxic assets (see, for example, Scott & Taylor’s editorial in the *Wall Street Journal* from July 20, 2009.) This is a controversial area, so I have divided the review into two parts. The first part describes the book and tries to give a sense of what you might learn from it. It also explains what a synthetic CDO is, and evaluates the modeling approach used in the book. In the second part, I will consider whether synthetic CDOs, or the mathematical modeling of synthetic CDOs, contributed to the current financial crisis.

**Part I: Modeling SCDOs**

It is evident from its title that this book is highly specialized. You will need some familiarity with financial practice and a working knowledge of stochastic processes (computations, but no measure theory) to get anything out of it. The author pledges to avoid financial jargon, but for the uninitiated this pledge is broken by page twelve. In terms of financial background, note that he considers a credit-default swap to be a “plain vanilla” transaction (for which he nonetheless gives a detailed description and analysis.) The author gives comprehensive coverage of synthetic collateralized debt obligations (SCDOs). This sounds quite restrictive, but there is in fact a large variety of securities that fall under this heading.

The book is written for practitioners, so the focus is on effective methods for numerical computation of prices and risk profiles. The exposition is workmanlike. Mathematicians may notice a few errors, but these are incidental to the development. Numerical analysts and applied mathematicians may find fertile ground here to improve on current techniques. For others, you will find a candid discussion of the mathematical machinery being used by professionals, if you have the patience to read it.

Mounfield describes the market’s standard model for SCDOs, as well as the shortcomings of that model. He also describes models that have been introduced to address some of these shortcomings. These newer models typically address a single shortcoming, and some of the shortcomings are not addressed at all. In light of the financial crisis, it seems that the scope of the modeling must be broadened.

Here are the main theoretical issues: (i) quantifying the interdependency of firms with respect to their likelihood of default, (ii) risk analysis and rare events, and (iii) explaining and justifying the economic function of the securities in question. Based on Mounfield’s survey, point (i) is being actively investigated by academics and practitioners, point (ii) is being investigated somewhat, by academics, and (iii) is largely untouched. Each point listed seems to require moving further away from current practice.

*What are SCDOs? *

Before we discuss the points given in the previous paragraph, I will describe briefly what synthetic CDOs are. First, we must describe a credit default swap (CDS). A CDS is a bit like an insurance contract on the credit-worthiness of a corporation. One person in the contract makes regular payments to the second person, and the second person pays a prescribed, larger sum to the first if the corporation named in the contract defaults. (Default can correspond to going bankrupt, missing payments on loans, or not following the rules prescribed by a lender. Note that the corporation named in the contract is neither of the two persons who entered into the contract, which is where the analogy with insurance breaks down.) The contract lasts a fixed number of years, but can end sooner in the case of default.

A synthetic collateralized-debt obligation (SCDO) is a collection of CDSs, all (in principle at least) referring to different corporations. A person can buy part of the SCDO, and then is entitled to receive some of the regular payments from the CDSs. This buyer will also be responsible for making “payments” in the case of a default (actually, the “payment” is realized as a loss in the value of the CDO purchased.) The catch is that the responsibility for making payments in case of default is not spread evenly among the people who buy the SCDOs. The SCDO is divided into pieces called tranches, some of which are have more protection from defaults than others. Purchasers of junior or subordinate tranches must make payments on the first defaults when they occur, and purchasers of senior tranches must make payments only when there have been enough defaults to wipe out the value of the junior tranches.

There are also what Mounfield calls cash flow collateralized-debt obligations, which I will abbreviate CDO. A general CDO is backed by a collection of securities, say corporate loans. A purchaser of the CDO receives payments from the collection of securities, e.g. from corporations’ loan payments. If a corporation does not make its loan payments, the CDO holder does not receive them. Again, tranches are used so that non-payment has different effects on different tranches. The securities behind the CDO (i.e., the collateral) can consist of any kind of debt or loan. Indeed, the tranche structure used in CDOs originated some thirty years ago in the mortgage markets, enabling banks to sell bundles of mortgages they had originated. A SCDO emulates a corporate-loan CDO, but without being involved with corporate loans in any way; losses are determined by a prescribed computation, rather than by non-payment by the corporations.

*Modeling Interdependence*

Since a SCDO refers to many corporations, losses depend on the pattern of defaults among all of them. It is easy to believe that the financial distress of one firm ought to be correlated with the financial viability of other firms, be they suppliers, clients, partners, or competitors of the distressed company. This interdependence must be taken into account to evaluate risk. Mounfield describes several ways in which the interdependency can be modeled. The industry standard seeks to describe the joint probability distribution of default among the corporations, rather than computing only the correlation coefficient between each pair of firms. The joint probability density is not given directly. Instead, one decomposes the density into its marginal distributions and a copula function. A copula gives a deterministic, functional dependence between random variables without prescribing what the (marginal) distributions of those random variables are. This is a neat way of reducing the work of multidimensional probability density estimation, provided the choice of copula is a good one. The joint density is then used to simulate future default behavior.

Modeling the interdependence is a demanding statistics problem, and an incorrect choice of copula function makes the pricing models unreliable. Since default patterns can be viewed from an actuarial or an epidemiological point of view, however, there is a lot of additional statistical machinery available to fit the joint density. Before the financial crisis, this kind of modeling was applied exclusively to the dependence between the corporations referred to in SCDO contracts. It appears that similar analysis should be applied to financial firms who have entered into the contracts (i.e. the “counterparties”). It is clear that the market is moving in this direction. There has been consolidation of contracts in the CDS market, and index of the credit-worthiness of CDS counterparties is now available. Both trends should be of use for synthetic CDOs. To the extent the risk from counterparties is independent from those referred to in the SCDO contracts, it should be easy to adjust SCDO prices accordingly. Unfortunately, it is not clear whether such independence holds. This uncertainty is one of the reasons I believe that the wider economy needs a more detailed treatment in these models.

There is another issue in modeling the default correlations that is especially troubling. One can invert the market’s standard model, and compute, using market prices, the implied correlation of default between different firms. The problem is that there is a clear and persistent inconsistency in these correlations, the so-called “correlation skew”. In particular, the correlation between two firms defaulting varies depending on the kind of tranche being priced, i.e. where it stands among the junior and senior tranches. This is a red flag. Ordinarily, the operations of computing the value of cash flows and combining cash flows (e.g., splitting them into tranches) commute. All financial analysis, elementary or sophisticated, relies on this point. In this particular case, they do not commute. It should be possible to fit statistical models to reflect the correlation skew; but this is not the issue. The inconsistency is troubling because the very meaning of “price” depends on this commutative property. Consequently, the use of these prices in risk models, balance sheets, and regulations may be inappropriate.

*Extreme Risk Analysis*

The second issue is how to conduct risk analysis and model rare events. The most popular measures of risk in this context are based on the idea of “value at risk”. This is essentially a low percentile for the value of an investment over a fixed time period. The probability distribution for the loss is usually simulated based on historical data. One then might compute, say, the first percentile for the value of a given CDO during the next week. The figure would give “the maximum amount we could expect to lose during the next week with 99% confidence”. This is a kind of large deviations analysis, although (as far as I can tell) one works from the distribution of possible profits and losses, not the distribution of the maximum losses per simulation. In any case, the principal problem is that this kind of analysis assumes a normal market, where the historically observed probabilities apply, where one can buy and sell at will, and one’s own actions do not change prices. The financial crises in 2007 and 2008 provide poignant examples where these conditions did not hold, but in fact these assumptions can fail in less extreme circumstances. There seems to be a need to develop models that would reflect these market flaws, although this would entail the introduction of many more parameters, which is not welcome from a practical point of view. In any case, it seems that additional macroeconomic variables would be useful for capturing time-dependent changes in the probability distributions, e.g. to capture the effect of two variables moving together that in ordinary circumstances might be independent. Another approach, more specific to CDOs, would be to model defaults as contagious disease, as outlined in a report on the SIAM Conference on Financial Mathematics and Engineering given in the *SIAM News* from January/February 2009.

*Financial and Economic Grounding*

The status of synthetic CDOs in financial and economic theory is somewhat different than some other derivative securities. In the first place, a perfect hedging mechanism for SCDOs is not possible, even in principle. Second, as I mentioned above, the commutative property for cash flow valuation does not seem to hold. Thus, different valuation methods can lead to different prices. These two phenomena are linked: consistent evaluation methods would follow from a perfect hedge. The main reason for this inability to hedge is that defaults occur discretely and introduce discontinuities to the stochastic analysis. Mounfield devotes a chapter to hedging methods for SCDO portfolios, which he admits is exploratory in nature. There is probably room to introduce some ideas from stochastic control theory here, such as “relaxed controls” for hedges. I suspect that the inability to hedge is not driven entirely by the discontinuities, but also from the economic function of the SCDOs. Because default behavior is linked with the state of the economy overall, it should not be possible to hedge against this factor. Consequently, the securities cannot be treated in an actuarial fashion.

Let us consider what good SCDOs may contribute to the economy. The securities considered by this book are, in a fundamental way, based on the modeling of catastrophic events. Thus, one important economic function they can have is as insurance contracts. Financial firms will also argue that they improve “market liquidity.” This is supposed to have two benefits. The first is that the new securities provide an additional source of information that can be used to cross-check the value of securities in other markets. The second is that SCDOs allow for more kinds of trades, which should allow traders to eliminate inconsistencies between the prices of different securities. By seeking out inconsistencies, traders have an opportunity to arbitrage them, that is, earn safe profits with little effort by buying what is relatively too cheap and selling what is relatively too expensive. The belief is that this will ensure that inconsistencies in prices are small and transient, which is considered good from the standpoints of efficiency and fairness. There is no guarantee that a new security will serve either function, however. First, it does not always make economic sense to insure things. Second, the impetus for new kinds of securities can be to increase leverage and to avoid regulation, which may rely on the continuance of inconsistencies in the market, rather than eliminating them. The point here is that the insurance and liquidity effects should be quantified or modeled somehow. This is not an easy task, but it seems to me that it might be done.

The original motivation for the CDO market, according to Mounfield, was moving liabilities off balance sheet, so that banks could engage in additional lending. However, as the book demonstrates in its later chapters, the hedging and risk control of these contracts is not fully developed, so to some extent the use of SCDOs for this purpose is a way of getting around banking regulations. Further, SCDOs are based on credit default swaps. Thus, an SCDO is a kind of re-insurance. However, since these securities are exposed to systemic risk due to co-movements during recessions and other factors, they are inherently speculative. Opinions differ as to whether such speculation is good or bad. This must be evaluated at a system-wide level, rather than in terms of individual transactions.

The fact that SCDOs cannot be hedged perfectly means that the markets are “incomplete”. Incompleteness of the market can have a precise meaning in different contexts, but it refers to the fact that desirable trades cannot be made. Completeness, in its various guises, is a topological condition ensuring that possible investment strategies lie in a convex set, preferably a simplex of a linear space. It makes the optimization problems facing investors easier to solve, and often one can avoid the optimization problem altogether because there are nice dual problems and equivalent forms that can be used. Each different formulation of the problem gives a price, but these prices agree. In particular, for some derivative securities, one can talk of “risk neutral probabilities” that all parties can agree upon, even if some parties are more risk-averse than others. Once we have incompleteness, much of this neatness goes away. This may have practical implications, but the question is unexamined. For example, the price that an investor may pay for an SCDO tranche is not necessarily the same as the price that a regulator should use when setting reserve requirements.

Completeness of markets is also a concept in general equilibrium models of the economy, where it affects the various kinds of efficiency the economy may attain. These models try to capture the economy as a whole. Incomplete market analysis in finance is more instrumental. It gives constraints on possible investment behavior, but there is no link with the wider economy or financial markets; there is no big picture. From a theoretical perspective, and perhaps from a practical perspective, it seems that an ability to articulate the form of an incomplete market is essential for further progress. It seems plausible that markets can be incomplete in different ways and to different degrees, and that this must be modeled and quantified. No one, I believe, has a pursued a modeling framework for incomplete markets; this is quite challenging. On the other hand, I do not think it is unreasonable. For example, Chichilinsky has formulated the general equilibrium model for the economy in terms of algebraic topology. Although this formulation may not lend itself to empirical use, it does raise the question of whether one could compute homotopy groups for an actual market, or more crudely some measure of connectedness. The practicality of such results, or even their interpretation, is not clear. However, the potential gains in understanding and in risk control are great.

**Part II: Are SCDOs at Fault?**

I will not condemn Wall Street and financial innovation outright, since I am sympathetic to the difficulties financial analysts face. Nonetheless, SCDOs did contribute to the financial crisis.

The financial crisis seems to have hit just as the author was putting the finishing touches on his book. He refers several times to the credit crunch in the summer of 2007, but does not discuss subsequent developments. It seems clear the discussion of the credit crunch was added to the text without revising its core. In some respects, the author cannot be blamed for this: he was describing an evolving state of the art when the liquidity crisis undermined many of the assumptions being used to assign prices to CDOs.

On the other hand, he argues SCDOs are not problematic, and that toxic assets are mainly drawn from non-synthetic CDOs and asset-backed securities. Although he has a (historical) point, the distinction between these different securities is less sharp and less meaningful than he might wish. It seems strange to single out non-synthetic CDOs for being problematic because (at one remove) they are secured by real assets. The main issue is that the CDOs and SCDOs are popular in part because of they allow banks to increase their leverage. Leverage refers to borrowing money to invest with; if the value of the investment goes up, you may keep the entire amount of profit but without committing much of your own funds. High leverage is what made so many financial firms vulnerable to bankruptcy as short-term borrowing became difficult during the crisis. Thus, SCDOs did not cause the crisis but they played their part in magnifying its effects. Further, uncertainty about the accounting for derivatives such as these contributed to the lack of trust that developed in the short-term markets. Neither SCDOs nor derivative securities in general are unique in exacerbating the crisis, but they were hardly a help to the firms that held them at the time. Interestingly, they may now be yielding profits again, although their market now exists on a reduced scale.

There is also the question of whether the mathematical modeling of SCDOs contributed to the financial crisis. I do not think it did, directly. The market standard model worked fairly well for several years, and was flexible enough to handle some surprises, such as during what Mounfield calls the “correlation crisis,” in 2005, when American automobile companies’ debt was dropped to junk-bond status. On the other hand, the more recent financial crisis showed that the models were myopic, and that not much thought had gone into what to do, practically speaking, if the assumptions of the models no longer held. In ignoring certain risks, the models may have emboldened financial firms to take riskier investments. On the other hand, the push for higher returns (and, consequently, riskier investments) was endemic throughout finance, and perhaps a better risk model would not have changed behavior as much as one would hope.

Whether this book will be a stepping stone for the future or a record of arcane and obscure practices in financial history remains to be seen. I expect that it will be a stepping stone, but this depends on the research community rising to the occasion it presents.

John Curran is Assistant Professor of Mathematics at Eastern Michigan University, where he coordinates the actuarial science program. Curran worked for a Wall Street firm for several years before obtaining his Ph.D. in applied mathematics from Brown University.