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
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:
- precede the polynomial remapping with a holdings remapping, and
- 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
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
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 ci, j and bi equal to zero in advance.
Selecting which components to assign nonzero values requires experience and some experimentation. It is often reasonable to model all bi terms and the single a term. Deciding which diagonal terms ci,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 ci,i for key factors representing implied volatilities, although these are less significant. Usually, most or all off-diagonal terms ci, j can be set equal to 0.