Abstract
This paper presents a method for interpolating yield curve data in a manner that ensures positive and continuous forward curves. As shown by Hagan and West (2006), traditional interpolation methods suffer from problems: they posit unreasonable expectations, or are not necessarily arbitrage-free. The method presented in this paper, which we refer to as the “monotone preserving r(t)t method", stems from the work done in the field of shape preserving cubic Hermite interpolation, by authors such as Akima (1970), de Boor and Swartz (1977), and Fritsch and Carlson (1980). In particular, the monotone preserving r(t)t method applies shape preserving cubic Hermite interpolation to the log capitalisation function. We present some examples of South African swap and bond curves obtained under the monotone preserving r(t)t method.
Highlights
A yield curve is a plot depicting the spot rate of interest for a continuum of maturities, in some time interval
A model is required to interpolate between adjacent maturities of observable securities, and to extract spot rates from more complicated securities such as coupon bonds, swaps, and Forward Rate Agreements (FRAs)
The entire yield curve is explained through a single parametric function, with the parameters typically estimated through the use of some least-squares regression technique
Summary
A yield curve is a plot depicting the spot rate of interest for a continuum of maturities, in some time interval. As noted by Andersen (2007) only a limited number of fixed income securities trade in practice, very few of which are zero-coupon bonds. The entire yield curve is explained through a single parametric function, with the parameters typically estimated through the use of some least-squares regression technique. Important contributions in this field have come from Nelson and Siegel (1987) and Svensson (1992). As noted by Andersen (2007) the resulting fit of such parametric functions to observed security prices is typically too loose for mark-to-market purposes, and may result in highly unstable term structure estimates. Financial institutions involved in the trading of fixed income securities rarely rely on parametric models
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