Calibrating Term Structure Models to an Initial Yield Curve

Abstract

The modelling of the short rate offers many advantages, with the models explored in this dissertation all offering closed-form, analytic formulae for bond prices and for options on bonds. Often, a vital primary condition is for a model to be calibrated to the initial term structure and to recover the bond prices observed in the market – that is, to be calibrated to the initial yield curve. Under the two exogenous models explored in this dissertation, the Hull-White and the CIR++, the effect of increasing the volatility parameter of the SDE increases the mean of the short rate. Increasing volatility of an SDE is a common approach to stress testing a model, as such, the consequences of bumping volatility in a calibrated model is a vital concern. The Hull-White model and CIR++ model were calibrated to market data, with the former being able to match the observed cap prices, while the latter failed, displaying an upper bound on cap prices. Investigating this, under CIR++ model, bond option prices are shown to not be straightforward increasing functions of the volatility parameter. In fact, for high volatility, bond option prices display an upper limit before decreasing, thus providing a limit to the level of cap prices too. This dissertation points to the reason residing in the underlying CIR model from which the CIR++ is based on, and the manner in which the model is extended