9.3.2 Linear Remappings
Linear remappings are widely used with portfolios composed exclusively of “linear” instruments—futures, forwards, spot or physical commodities positions, swaps, most non-callable bonds, foreign exchange, and equities. For such portfolios, linear remappings afford excellent approximations. They facilitate the use of linear transformation procedures.
If a portfolio holds even a single “nonlinear” instrument—such as an option—a linear remapping should not be used. Exhibits 9.13 and 9.14 offer a simple but dramatic example of why this is so. Exhibit 9.13 illustrates a portfolio mapping function θ of a portfolio with a delta-hedged, negative-gamma exposure to a single underlier, whose value is represented by key factor 1R1:3
Using a derivative approximation, we approximate 1P = θ(1R1) with linear remapping 
= 
(1R). The new mapping function is graphed in Exhibit 9.14.
The new mapping function is a constant function, suggesting the portfolio has no market risk whatsoever. This ignores the—possibly substantial—risk due to the portfolio’s negative gamma. In this example, a linear remapping does not provide a crude approximation. It is simply wrong. This example is not contrived. Derivatives dealers routinely delta hedge negative gamma positions.
9.3.3 Linear Remappings With Gradient Approximations
Linear remappings are usually constructed with gradient approximations. A gradient approximation can be constructed about any point, but the conditional expectation 1|0μ = 0E(1R) is a reasonable choice. The remapping then has form
[9.23]
Because pricing formulas for many financial instruments are easily differentiated, the gradient can usually be valued analytically. Otherwise, it may be valued with finite differences. This will require n + 1 valuations of θ.