This is early work that explores approximations of the kernel ecxy where c is a constant and x and y are variables ranging from -1 to 1.
I consider approximating the behavior of a portfolio across a set of scenarios. No asset pricing model is assumed; accuracy is measured by worst-case approximation error. The novel approximation I provide here is the first to have verifiable optimality properties for portfolios of bonds, interest rate swaps, credit default swaps, and equities.
The numerically-optimal approximation earns its name by achieving, to within rounding error, a lower bound that I derive and compute for the worst-case error of any approximation. It can be interpreted as assembling a portfolio using a small set of simple approximating cashflow streams to approximate any given cashflow stream’s behavior.
The approximation introduced here can calculate scenario valuations more than 1,000,000 times faster than a direct method. My approximation can be 20 times (rank-three), 70 times (rank-four), 300 times (rank-five), or 350,000 times (rank-twelve) more accurate in worst-case error than standard Taylor-series approximations.
Approximation of Sign-regular Kernels
This paper introduces, for any sign-regular kernel, a new method to efficiently compute a lower bound on any rank-n approximation’s uniform error. It also provides a novel method to construct a rank-n approximation that numerically achieves this lower bound in every example that has been examined.