Conditional Density Models for Asset Pricing

We model the dynamics of asset prices and associated derivatives by considerationof the dynamics of the conditional probability density process for the value of an assetat some specied time in the future. In the case where the asset is driven by Brownianmotion, an associated \master equation" for the dynamics of the conditional probabilitydensity is derived and expressed in integral form. By a \model" for the conditionaldensity process we mean a solution to the master equation along with the specicationof (a) the initial density, and (b) the volatility structure of the density. The volatilitystructure is assumed at any time and for each value of the argument of the density tobe a functional of the history of the density up to that time. This functional determinesthe model for the conditional density. In practice one species the functional modulosuffcient parametric freedom to allow for the input of additional option data apartfrom that implicit in the initial density. The scheme is suffciently exible to allowfor the input of various types of data depending on the nature of the options marketand the class of valuation problem being undertaken. Various examples are studied indetail, with exact solutions provided in some cases.[...]