9.4.2 Modeling Curves

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.

Exhibit 9.25: The Euro swap curve for March 3, 1999, is depicted with values for 15 maturities from 1 year to 30 years.

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 9.26: Euro swap curves for February 24 and March 3, 1999.

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.

Exhibit 9.27: The change in the Euro swap curve going from February 24 to March 3, 1999.

Measure time in weeks. A primary mapping models the Euro swap curve with a 15-dimensional key vector


We would like to implement a variables remapping that reduces the number of key factors. Consider a remapping of form


Let  represent changes in the 1-, 10-, and 30-year swap rates


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


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


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.


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:


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


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


Current values 0r for 1R are


A day goes by, and  is realized as


Determine the corresponding realization 1q of 1Q.