22-Credit Risk Model Derivs Page 734 Wednesday, February 4, 2004 1:12 PM
734 The Mathematics of Financial Modeling and Investment Managementmakes it more likely for defaults to occur. Extending the default horizon
therefore has the effect of increasing the measured default correlation.
Indeed we must be careful to specify the horizon when we quote a
default correlation. On the other hand, correlation of default times is
independent of the trade horizon (i.e., the tenor of the default swap).
There is also a link between default correlation and the hazard rate.
For a fixed horizon, increasing the hazard rate for all assets makes
default more likely within that horizon. If the assets are correlated, the
measured default correlation must increase. However, the increase in
default probability makes the distribution of default times more
weighted towards earlier defaults. Yet, the default time correlation can
remain unchanged.
The analysis below shows that the default correlation is always
lower than the default time correlation. This can be understood in quali-
tative terms as follows: To have the same basket price we have the same
number of defaults before maturity. As default correlation is a direct
measurement of the likelihood of two assets to default within a fixed
horizon, it is more closely linked with the pricing of a basket default
swap than a correlation of default times. Indeed, as we have shown in the
one-period model above, the value of the basket default swap is a linear
function of the default correlation. Though a correlation of default times
introduces a tendency for assets to default within a given trade horizon,
it is an indirect way to do this. As a result, a simulation of defaults with
a certain default time correlation will always tend to have a lower default
correlation. In other words, less default correlation is required in order
to have the same effect as a correlation of default times.^51SUMMARY
■ There are different forms of credit risk: default risk, spread risk, and
downgrade risk.
■ Credit derivatives are financial instruments designed to transfer credit
risk between two parties.
■ Credit default swaps are the most popular credit risk derivatives.
■ In a credit default swap, the protection buyer pays a fee, the swap pre-
mium, to the protection seller in return for the right to receive a pay-
ment conditional upon a default, also called a credit event.(^51) Numerical examples for pricing credit default swap baskets in the single-period
and multi-period cases are provided in Chapter 10 in Anson, Fabozzi, Choudhry,
and Chen, Credit Derivatives: Instruments, Applications, and Pricing.