Spanning and Completeness with Options

The role of ordinary options in facilitating the completion of securities markets is examined in the context of a model of contingent claims sufficiently general to accommodate the continuous distributions of asset pricing theory and option pricing theory. In this context, it is shown that call options written on a single security approximately span all contingent claims written on this security and that call options written on portfolios of call options on individual primitive securities approximately span all contingent claims that can be written on these primitive securities. In the case of simple options, explicit formulas are given for the approximating options and portfolios of options. These results are applied to the pricing of contingent claims by arbitrage and to irrelevance propositions in corporate finance.