9.4.2 Modeling Curves
Poorly designed value-at-risk measures employ numerous key factors in an attempt to reproduce the actual shape of term structures or other curves. Consider the Euro swap curve for March 3, 1999, which is plotted in Exhibit 9.25.
How many key factors are needed to model this term structure over time? In fact, it can be accurately reproduced with just three. The secret is to not model individual swap rates as key factors, but to model changes in individual swap rates.
Exhibit 9.26 indicates the same swap curve as well as the swap curve from the previous week. Over 1 week, the curve shifts upward with a slight tilt. Otherwise, its shape hardly changes.
Exhibit 9.27 plots the change in the swap curve—swap rates for March 3 less corresponding swap rates for February 24. This is a simple curve. We could easily model it with three key factors and interpolate for the rest. In this manner, we can accurately model the swap curve itself—it equals the change in the swap curve plus the previous week’s swap curve. Lets formalize this.
Measure time in weeks. A primary mapping models the Euro swap curve with a 15-dimensional key vector
[9.56]
We would like to implement a variables remapping that reduces the number of key factors. Consider a remapping of form
[9.57]
Let 
represent changes in the 1-, 10-, and 30-year swap rates
[9.58]
We interpolate between these to obtain changes in the swap rates for other maturities y. We might quadratically interpolate, but an interpolation function of the form
[9.59]
tends to provide a better fit. The result is a vector 
of changes in the swap curve at each of the 15 maturities modeled by 1R. We add the current swap curve 0r to to obtain
[9.60]
which approximates 1R.
This approach of modeling changes in a curve is invaluable for modeling yield curves, forward curves, volatility skews, and other curves. It is based upon an observation that changes in financial curves tend to have smoother shapes than the curves themselves. Hence, they can be accurately modeled with fewer key factors.
Exercises
Measure time in trading days. Today is November 14, 2001. A portfolio holds IPE Brent oil futures. A primary mapping has key factors 1R for prices of the first 12 nearby futures. A variables remapping is applied:
[9.61]
where represents changes in the first, second, sixth, and twelfth nearby prices. approximates changes in all twelve nearby prices. It is obtained from by setting 
= 
and interpolating between 
, 
, and 
to obtain the remaining components of . An interpolation function of form
[9.62]
is used for this purpose, where k represents the number of a nearby, from 2 to 12. We do not include the first nearby in the interpolation because it tends to move independently from the rest. 1Q approximates 1R and is defined as
[9.63]
Current values 0r for 1R are
[9.64]
A day goes by, and is realized as
[9.65]
Determine the corresponding realization 1q of 1Q.
Solution