# Options and market completeness

#### Abstract

A competitive equilibrium in an incomplete financial market needs not be Pareto efficient. Assuming a finite contingencies space, Ross (1976) showed that a collection of ordinary call options written on a resolving underlying asset can be constructed to restore completeness in any incomplete securities market. In the same framework, Arditti and John (1980) and John (1981) show that resolving underlying assets are generic, i.e., dense in the space of contingent claims. These results suggest that options are Pareto improving assets. In my work the contingencies space contains infinitely many states of nature. Brown, Huijsmans and de Pagter (1991) proved that options on any continuous and resolving asset span the space of continuous contingent claims, provided that the contingencies space is compact (e.g., the interval [0, 1]). My work shows that if the compactness assumption is relaxed, then options on a continuous and resolving asset uniformly span the space of continuous contingent claims with finite limit at the points of the closure of the contingencies space. Relying on this result, options written on any almost surely continuous and resolving underlying asset are proven to span all Borel measurable and p-integrable random variables, for 1 ≤ p < ∞. Under the assumption that the contingencies space is a Polish topological space, options on any almost surely resolving underlying asset are shown to span all Borel measurable and p-integrable random variables, for 1 ≤ p < ∞. Moreover, almost surely resolving underlying assets are proven to be dense in the space of contingent claims. Hence, the spanning properties of ordinary options enjoy the hallmark of generality of the finite dimensional setting for most of the spaces of states of nature that are commonly encountered in the financial literature.

#### Degree

Ph.D.

#### Advisors

Aliprantis, Purdue University.

#### Subject Area

Finance|Economics

Off-Campus Purdue Users:

To access this dissertation, please **log in to our
proxy server**.