 # 9.3.2 Linear Remappings

###### 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 Exhibit 9.13: A portfolio’s value 1P depends upon a single key factor 1R1 as shown.

Using a derivative approximation, we approximate 1P = θ(1R1) with linear remapping = (1R). The new mapping function is graphed in Exhibit 9.14. Exhibit 9.14: This linear remapping erroneously suggests an absence of market risk.

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.