21-Bond Portfolio Man Page 675 Wednesday, February 4, 2004 1:12 PM
Bond Portfolio Management 675tems made of thousands of scenarios can now be solved on desk-top
machines. Two well-known scenario systems in practical use are SPAN, a
16-scenario system developed by the Chicago Mercantile Exchange and
New York 7, a 7-scenario system use by New York insurance regulators
(National Association of Insurance Commissioner scenarios).
As a general requirement, scenarios must be both “complete” and
“coherent.” Completeness means that scenarios must capture the business-
as-usual situations as well the extremes. Coherence means that scenarios
must respect the conditions typical of many financial variables. For
instance, some financial variables are perfectly anti-correlated, a condition
that must be respected by scenarios. Financial and economic scenarios
must also be free from anticipation of information. A natural way to make
nonanticipative scenarios is the use of information structures as described
in Chapter 5. Information structures require that scenarios are indistin-
guishable up to a given date and then part in a treelike structure.
Consider the generation of interest rates scenarios. This is a prob-
lem that can be solved starting from a model of the term structure of
interest rates. Continuous-time models of interest rates were introduced
in Chapter 15. To create scenarios, these models need to be discretized
as discussed in Chapter 15. Recall that there are different ways of dis-
cretizing a continuous-time model. For example, a Brownian motion
can be simulated as a random walk whose increments are random draws
from a normal distribution. Alternatively, one can adopt a binomial
approximation to the Brownian motion. The first procedure creates a
random sampling from a continuous distribution while the second pro-
duces a discrete-time, discrete-state model.
If we consider only risk-free bonds, the information contained in the
interest rate processes is sufficient to create scenarios. A large number of
scenarios can be created either by sampling or with discrete models. If,
in contrast, we want to consider bonds with default risk, then we need
to generate scenarios according to a specified model of credit risk (see
Chapter 22). For example, if we use a rating process, we need to simu-
late a rating process for each bond taking into consideration correla-
tions. It is clear that we immediately run into computational difficulties,
because the number of scenarios explodes even for a modest number of
bonds. Drastic simplifications need to be made to make problems tracta-
ble. Simplifications are problem-dependent.Multistage Stochastic Programming
After creating scenarios one can effectively optimize, taking into account
that after initial decisions there will be recourses (i.e., new decisions even-