Compact Securities Markets for Pareto Optimal Reallocation of Risk

The securities market is the fundamental theo­ retical framework in economics and finance for resource allocation under uncertainty. Securi­ ties serve both to reallocate risk and to dissem­ inate probabilistic information. Complete securi­ ties markets-which contain one security for ev­ ery possible state of nature-support Pareto op­ timal allocations of risk. Complete markets suf­ fer from the same exponential dependence on the number of underlying events as do joint probabil­ ity 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 inde­ pendencies encoded in the market structure, then the market is operationally complete: risk is still Pareto optimally allocated, yet the number of se­ curities can be exponentially smaller. For col­ lections of agents of a certain type, agreement on Markov independencies is sufficient to admit compact and operationally complete markets.