In this paper we model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the asset is driven by Brownian motion we derive an associated "master equation" for the dynamics of the conditional probability density, and express this equation in integral form. By a "model" for the conditional density process we mean a solution to the master equation along with a specification of the initial density and a specification of the volatility structure for the density. The volatility structure in particular is assumed at any given time and for each value of the argument of the density function to take the form of a functional that depends on the history of density up to that time. The choice of this functional determines the particular model for the conditional density, and in practice one specifies the functional modulo sufficient parametric freedom to allow for the input of additional option data apart from that already implicit in the specification of the initial density. The scheme is sufficiently flexible to allow for the input of various different types of data depending on the nature of the options market under consideration and the class of valuation problem being undertaken. Various specific examples are studied in detail, with exact solutions provided in some cases. (Co-authors: D. Filipovic, Ecole Polytechnique Fédérale de Lausanne, Switzerland, and A. Macrina, King's College London and Kyoto Institute of Economic Research.)