A Family of Humped Volatility Structures

Abstract

Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Mercurio and Moraleda (1996) and Moraleda and Vorst (1996a) 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, both models are Gaussian, and hence interest rates may become negative. In this paper we propose a family of interest rate models where (i) humped volatility structures are possible; (ii) the interest rate volatility can 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. We propose, for instance, models of the type of the proportional Ritchken and Sankarasubramanian (1995) and the Black and Karasinski (1991) model. To gain the computational tractability (iii), we show how to embed all models in this paper in either the Ritchken and Sankarasubramanian (1995) or the Hull and White (1990, 1994) class of models.

JEL Classification: G13