A Local Linear Regression Method Using a Discrete Kernel Function with Applications to Bond Curve Construction

Abstract

Local linear regression is generally used to estimate the dependency of a random variable Y on another random variable X from a finite sample of data points. Except for the chosen kernel function, no model assumptions on the relationship between X and Y are taken into account in the estimation. In this paper, we show that discretizing the kernel function is consistent with a piecewise linear model assumption on the relationship between X and Y . Under certain boundary conditions, this relationship can be shown to be continuous. We apply the local linear regression with discretized kernel (LLRDK) to a problem from finance: the construction of a bond spread curve from a discrete set of observed bond market prices. The method we present comprises a valuation approach that combines a reference interest rate curve, survival probabilities from CDS markets and the statistical estimation of the bond CDS basis spreads using the LLRDK. This allows for the consistent combination of local linear regression as a statistical estimation method with model assumptions on the bond CDS spread such as a piecewise constant or piecewise linear dependency on time.