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