There is a simple piece of financial mathematics that goes a long way to explaining why we have experienced the volatility that we have seen recently. It is based on the idea of complete markets versus incomplete markets. Readers familiar with the binomial option pricing model can verify my remarks using elementary calculations. The ideas are embedded in all of the pricing calculations behind modeling derivatives and they become particularly relevant to pricing derivatives on illiquid assets. The simplest definition of a complete market is one where every risk or state is insurable. In this situation, we have a wonderful theory where we get a unique riskneutral measure that leads to a unique derivative price: this is the basis of Black- Scholes pricing.
In an incomplete market, where risks are not insurable, we have many risk-neutral
measures that lead to many possible derivative prices. This is the general case and
describes the situation prevalent in many credit markets.
This raises the immediate question: which risk-neutral measure was being used
by our friends, the derivative pricers? On this point, the answer is by no means clear.
In many cases, the modelers simply assumed a complete market, even when there
was obviously not one. In others, they presented a distribution, which they claimed
to be the true representation of the world (“nature’s measure” as it is known), and
undertook transformations of it to render it risk neutral. In either case, this presents
interesting challenges to the validation of the models, which may extend beyond
simply comparing the output with the forecast. A close examination of the assumed
structure may well find “black swans” hiding behind the model’s foliage.
This Winter 2009/10 issue contains four papers, the first of which, by Miu and
Ozdemir, looks at stress testing for some of the components of Basel II requirements
and presents a framework for stress testing. The emphasis is on implementation so
that issues, such as a lack of internal data, for example, are included in the analysis.
This should be of interest to the many risk groups who have Basel II responsibilities
within lender firms.
The second paper, by Wong et al looks at a class of structural credit risk models
that are built around the idea of a default boundary. These are further sub-divided
into exogenous and endogenous boundary cases. Although both types of models
are subjected to rigorous analysis, there does not appear to be much discernible
difference in performance. In such a situation, one should wield Occam’s razor!
The third paper, by Morone and Cornaglia, provides a general framework in
a ratings system for decomposing asset and default dynamics. This allows some
macroeconomic aspects of the structure to be measured; in particular, a measure
of cyclicality is presented.
The final paper, by Tashman, presents a regime-switching approach to stresstesting
risk models. I have used regime-switching models in the past for forecasting
and found them to be most frustrating. However, I had speculated, that, perhaps,
their most important use would be in a more backward-looking activity. This paper
illustrates that they are, indeed, appropriate for stress testing. They provide a natural
framework for including scenarios in an integrated way.
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