###### 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 ^{1}*R*_{1}:3

^{1}

*P*depends upon a single key factor

^{1}

*R*

_{1}as shown.

Using a derivative approximation, we approximate ^{1}*P* = θ(^{1}*R*_{1}) with linear remapping = (^{1}** R**). 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|0}**μ** = ^{0}*E*(^{1}** R**) 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.