9.3  Function Remappings

Function remappings directly approximate a mapping function θ with some simpler function of a specified form. Key vector ¹R is unaffected. This is illustrated schematically as

[9.19]

The function  can take many forms. Linear or quadratic polynomials are most common, in which case the remapping is called a linear or quadratic remapping. Other terminology refers to how the mapping function  is constructed. We may speak of interpolated, least squares, gradient, or gradient-Hessian remappings.

If  is to have a nonlinear polynomial form and the dimensionality of ¹R is high, constructing  can be computationally expensive. Two ways to streamline computations are:

1. precede the polynomial remapping with a holdings remapping, and
2. limit the number of quadratic or higher-order coefficients in the functional form of .

Compared to holdings remappings, which tend to provide excellent approximations, function remappings offer more varied results. Considerable care should be exercised when incorporating them into a mapping procedure. Regardless, linear and quadratic remappings are popular because they facilitate linear and quadratic transformation procedures.

9.3.1 Selecting a polynomial form

Consider portfolio mapping ¹P = θ(¹R) where ¹R is n-dimensional. Linear and quadratic remappings have forms

[9.20]

and

[9.21]

where a is a scalar, b is an n-dimensional row vector, and c is an n×n symmetric matrix. Linear remappings are used with portfolios comprising “linear” instruments such as forwards. Quadratic remappings are used with portfolios that hold options or other “nonlinear” instruments.

In the quadratic case, c has

[9.22]

unique components. Assigning all of these values can be computationally expensive if n is large, so it is worth limiting the generality of a quadratic form where possible. This is done by fixing certain components c_i, j and b_i equal to zero in advance.

Selecting which components to assign nonzero values requires experience and some experimentation. It is often reasonable to model all b_i terms and the single a term. Deciding which diagonal terms c_i,i to model is a question of which key factors contribute significantly to nonlinearity. In an options portfolio, these are the key factors for the options’ underliers. You might also permit nonzero diagonal coefficients c_i,i for key factors representing implied volatilities, although these are less significant. Usually, most or all off-diagonal terms c_i, j can be set equal to 0.