Options and Efficiency in Multiperiod Security Markets

We extend the result of Ross (1976) that European options generate complete markets from the single-period to a multiperiod setting. We find that multiperiod European options on a trading strategy generate dynamic completeness for every arbitrage-free price process, provided that the trading strategy has non-negative terminal dividends and separates states at the terminal date. Furthermore, we show that if the uncertainty and information structure in an economy are such that the number of immediate successors of every non-terminal event is non-decreasing over time, then multiperiod European options on a trading strategy generate dynamic completeness for almost every arbitrage-free price process under a significantly weaker condition on the trading strategy's terminal dividends. This condition requires the trading strategy to have non- negative terminal dividends and to separate states at the terminal date conditional on the information available at the previous date. Finally, we examine the minimum number of options generating dynamic completeness for almost every arbitrage-free price process.