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.
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
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-
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.