  The emph{securities market} is the fundamental theoretical framework in
economics and finance for resource allocation under uncertainty. Securities
serve both to reallocate risk and to disseminate probabilistic information.
emph{Complete} securities markets - which contain one security for every
possible state of nature - support Pareto optimal allocations of risk. Complete
markets suffer from the same exponential dependence on the number of underlying
events as do joint probability distributions. We examine whether markets can be
structured and "compacted" in the same manner as Bayesian network
representations of joint distributions. We show that, if all agents'
risk-neutral independencies agree with the independencies encoded in the market
structure, then the market is emph{operationally complete}: risk is still
Pareto optimally allocated, yet the number of securities can be exponentially
smaller. For collections of agents of a certain type, agreement on Markov
independencies is sufficient to admit compact and operationally complete
markets.
