graph from one in which the bank has “planted” the dense subgraph (a.k.a. boobytrap). Formally, the two kinds of graphs are believed to be computationally indistinguishable for polynomial-time algorithms and this was the basis of a recent cryptosystem proposed by Applebaum et al. 3

The conjecture is that even quite large boobytraps may be undetectable: the expected yield of the entire portfolio could be much less than say V – n1.1 and yet the buyer may not be able to distinguish it from a truly random (i.e., honestly constructed) portfolio, whose yield is V – o(n).

We conclude that if buyers are computationally bounded, then introducing derivatives into the picture not only fails to reduce the lemon wedge, but paradoxically, amplifies it even beyond the total value 2n of all lemon assets. Though the above example is highly simplified, it can be embedded in settings that are closer to real life and similar results are obtained.

4. 1. Can the cost of complexity be mitigated? In Akerlof’s classic analysis, the no-trade outcome dictated by lemon costs can be mitigated by appropriate signaling mechanism—e.g., car dealers offering warranties to increase confidence that the car being sold is not a lemon. In the above setting, however, there seems to be no direct way for seller to prove that the financial product is untampered i.e., free of boobytraps. (It is believed that there is no simple way to prove the absence of a dense subgraph; this is related to the NP ¹ coNP conjecture.) Furthermore, we can show that for suitable parameter choices the tampering is undetectable by the buyer even ex post. The buyer realizes at the end that the financial products had a much lower yield than expected, but would be unable to prove that this was due to the seller’s tampering. Nevertheless, we do show in our paper5 that one could use ideas from computer science in designing derivatives that are tam-perproof in our simple setting.

4. 2. Complexity ranking

Recently, Brunnermeier and Oehmke9 suggested that traders have an intuitive notion of complexity for derivatives. Real-life markets tend to view derivatives such as CDO2 (a CDO whose underlying assets are CDOs like the one described earlier) as complex and derivatives like CDO3 (a CDO whose underlying assets are CDO2) as even more so. One might think that the number of layers of removal from a simple underlying real asset could be a natural measure of complexity. However, as Brunnermeier and Oehmke9 point out, such a definition might not be appropriate, since it would rank e.g. highly liquid stocks of investment banks, which hold CDO2s and other complex assets, as one of the most complex securities. Our paper5 proposes an alternative complexity ranking which is based on the above discussed notion of lemon cost due to complexity. This ranking also confirms the standard intuition that CDO2s are more complex than CDOs. Roughly speaking, the cherry-picking possibilities for sellers of CDOs described in this paper become even more serious for derivatives such as CDO2 and CDO3.

5. DIsCussIon

The notion that derivatives need careful handling has been extensively discussed before. Coval et al. 10 show that pricing (or rating) a structured finance product like a CDO is extremely fragile to modest imprecision in evaluating underlying risks, including systematic risks. The high level idea is that these everyday derivatives are based upon the threshold function, which is highly sensitive to small perturbations of the input distribution. Indeed, empirical studies suggest that valuations for a given financial product by different sophisticated investment banks can be easily 17% apart6 and that even a single bank’s evaluations of different “tranches” of the same derivative may be mutually inconsistent. 12 Thus one imagines that banks are using different models and assumptions in evaluating derivatives.

The question studied in our work is: Is there a problem with derivatives even if one assumes away the above possibilities, in other words the yield of the underlying asset exactly fits the stochastic model assumed by the buyer? Economic theory suggests the answer is “No”: informed and rational buyers need not fear derivatives. (Recall our discussion of DeMarzo’s theorem.)

The main contribution of our work has been to formal-
ize settings in which this prediction of economic theory
may fall short (or even be falsified), and manipulation is
possible and undetectable by all real-life (i.e., computation-
ally bounded) buyers. We have worked within existing con-
ceptual frameworks for asymmetric information. It turns
out that the seller can benefit from his secret information
(viz., which assets are lemons) by using the well-known fact
that a random election involving n voters can be swung with
significant probability by making voters vote the same
way; this was the basis of the boobytrap described earlier.
The surprising fact is that a computationally limited buyer
may not have any way to distinguish such a tampered CDO
from untampered CDOs. Formally, the indistinguishabil-
ity relies upon the conjectured intractability of the planted
dense subgraph problem.h
The model in our more detailed paper has several nota-
ble features:

 

1. The largeness of the market—specifically, the fact that sellers are constructing thousands of financial products rather than a single product as was the case in the model of DeMarzo11—allows sellers to cherry pick in such a way that cannot be detected by feasible rational (computationally bounded) buyers—i.e., all real-world buyers—while it can be detected by fully rational ( computationally unbounded) buyers.

2. The possibility of cherry picking by sellers creates an Akerlof-like wedge between buyer’s and seller’s valuations of the financial product. We call this the lemon cost due to computational complexity. In our detailed paper we can quantify this wedge for several classes of derivatives popular in securitization. This allows a par- h Note that debt-rating agencies such as Moody’s or S&P currently use simple simulation-based approaches to evaluate derivatives, which certainly do not attempt to solve something as complicated as densest subgraph.