  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 price process is driven by Brownian motion, an associated "master
equation" for the dynamics of the conditional probability density is derived
and expressed in integral form. By a "model" for the conditional density
process we mean a solution to the master equation along with the specification
of (a) the initial density, and (b) the volatility structure of the density.
The volatility structure is assumed at any time and for each value of the
argument of the density to be a functional of the history of the density up to
that time. In practice one specifies the functional modulo sufficient
parametric freedom to allow for the input of additional option data apart from
that implicit in the initial density. The scheme is sufficiently flexible to
allow for the input of various types of data depending on the nature of the
options market and the class of valuation problem being undertaken. Various
examples are studied in detail, with exact solutions provided in some cases.
