The General Hull–White Model and Supercalibration

Term-structure models are widely used to price interest rate derivatives, such as swap options and bonds with embedded options. We describe how a general one-factor model of the short rate can be implemented as a recombining trinomial tree and calibrated to market prices of actively traded instruments. The general model encompasses most popular one-factor Markov models as special cases. The implementation and the calibration procedures are sufficiently general that they can select the functional form of the model that best fits the market prices. This characteristic allows the model to fit the prices of in- and out-of-the-money options when there is a volatility skew. It also allows the model to work well with economies characterized by very low interest rates, such as Japan, for which other models often fail. Term-structure models are widely used to price interest rate derivatives, such as the options embedded in swaps and bonds. We describe how a general one-factor model of the short rate can be implemented as a recombining trinomial tree and calibrated to the market prices of actively traded options. The general model encompasses most popular one-factor Markov models as special cases. The implementation and the calibration procedures are sufficiently general that they can select the functional form of the model that best fits the market prices. This characteristic allows the model to fit the prices of in- and out-of-the-money options when there is a volatility skew.The calibration of the model involves choosing volatility parameters so that the model matches as closely as possible the market prices of actively traded instruments, such as caps and European swap options. The volatility parameters are usually parameterized as piecewise linear or step functions, and an algorithm is used to minimize the sum of the squared differences between the market prices and model prices of the actively traded instruments. What we term “supercalibration” is a procedure for extending the calibration to obtain a best-fit functional form of the short-term interest rate. Supercalibration is possible because brokers now provide quotes on not-at-the-money caps as well as at-the-money caps. The supercalibration methodology is analogous to the implied tree methodology that is popular for valuing exotic equity and foreign currency options. It enables the model to work well in economies with very low interest rates, such as Japan, where other models often fail.We illustrate each numerical procedure with a detailed example. A key issue is the extent to which one should add parameters to the model to fit market data. Academics have quite different views from practitioners on this point. Academics prefer a stationary volatility environment, whereas practitioners consider fitting all observed option prices important. As a result, practitioners often use a highly nonstationary model whose future behavior may be quite different from its current behavior. Our view is that a moderate approach should be taken in fitting a model to observed option prices. Usually, modest nonstationarity does not seriously affect the future behavior of a model and allows a good fit to today's prices.