Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Several researchers have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, these models are Gaussian, and hence interest rates may become negative. Here, a family of interest rate models is proposed where (i) humped volatility structures are possible; (ii) the interest rate volatility may depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. Proportional models of the Ritchken and Sankarasubramanian type and the Black and Karasinski model are proposed. To ensure computational tractability the embedding of all models in this paper in either the Ritchken and Sankarasubramanian or the Hull and White class of models is demonstrated.
