Option pricing in the multidimensional Black-Scholes-Merton market with Gaussian Heath-Jarrow-Morton interest rates: the parsimonious and consistent Hull-White models of Vasicek and Nelson-Siegel type

Abstract

An explicit state-price deflator for the multidimensional Black-Scholes-Merton market with a multiple factor Gaussian bond price dynamics is constructed. It immediately yields an extension of the Margrabe formula in this multiple risk economy. Restricting further the attention to those Gaussian Heath-Jarrow-Morton interest rate models with time-homogeneous sensitivities that share the Markov diffusion property, one is led to consider models of the Hull-White type only. For practical reasons, only consistent families of yield curves are retained. That is, if an initial forward rate curve has a given form, then its future evolution should remain of the same form. Given two simple consistent forms and their most parsimonious parameterizations with two respectively four parameters, as well as their corresponding Hull-White models, we derive an explicit generalized Black-Scholes formula that takes into account the Hull-White term structure.