9.4 Variables Remappings

9.4  Variables Remappings

In value-at-risk modeling, we frequently work with changes of variables. Every mapping is a change of variables. Consider a mapping that expresses a vector of discount factors 1Q as a function φ of corresponding interest rates 1R. We describe it schematically as


Every mapping is a change of variables.

Here, mapping function φ reflects an exact relationship between 1Q and 1R. A change of variables can also be an approximation—and hence a remapping. Suppose futures prices  are used to approximate forward prices 1R. Based upon our convention that horizontal arrows denote exact relationships and vertical arrows denote approximations, we describe this relationship schematically as


Change of variables remappings are approximations.

A variables remapping is a remapping in which a key vector 1R is approximated with some other risk vector. This can happen in two ways. Sometimes, a new key vector   directly approximates 1R:


A change of variables remapping.

Other times, an approximating risk vector 1Q is constructed as a function  of a new key vector  :


Another form of change of variables remapping.

In schematic [9.50], key vector  replaces key vector 1R, but portfolio mapping function θ is unaffected. In schematic [9.51], both key vector 1R and portfolio mapping function θ change.