Calibration of Diffusion Models

In the models of the two preceding chapters, the derived prices and yields are functions of the state variables and the few parameters entering the dynamics of the state variables. Consequently, the resulting term structure of interest rates will typically not fit the currently observed term structure perfectly. This seems inappropriate when the models are to be applied to the pricing of derivative securities — if the model cannot price the underlying bonds correctly, why trust the model prices of derivatives? This chapter shows how to calibrate the model to the observed yield curve by replacing one of the constant parameters with a carefully chosen time-dependent. Special attention is given to the one-factor Vasicek and Cox–Ingersoll–Ross models. The models can also be calibrated to additional market information such as the term structure of interest rate volatilities by replacing other parameters with time-dependent functions. However, a direct dependence on calendar time in the drift and volatility of the short rate or other state variables is unrealistic, and care has to be taken not to introduce unrealistic time dependencies that are likely to affect the prices of the derivative securities.