Fitting No Arbitrage Term Structure Models Using a Regularisation Term

Abstract

In this paper we develop a generic, flexible fitting method that can be applied to no arbitrage term structure models. Flexibility in the model fitting methodology is generated by assuming that observed prices are not exact but have a bid/ask spread. A regularisation term is incorporated into the model’s fitting algorithm in order to generate smoother expected interest rates while still enforcing the no arbitrage constraint. The regularisation term acts as a roughness penalty term by measuring the smoothness of the models expected future interest rates. This fitting technique more closely reflects the market expectation of a smooth transition between consecutive points on the yield curve. We demonstrate this model fitting approach by applying it to the single factor Black-Derman-Toy interest rate model. The model is calibrated to observed quarterly term structure data from the US treasury market. The results compare this new fitting method using a regularisation term with the traditional calibration method of matching the observed mid prices. Graphs show how this new approach to model fitting more closely reflects market expectation leading to a smoother evolution of the term structure of interest rates.