Spanning in Security Markets and The Modigliani-Miller TheoremTwo Essays for the The New Palgrave

The span of a given collection of securities is the set of state-contingent claims to value that can be obtained by trading the securities. In a simple one-period model, this is literally the linear span (the set of all finite linear combinations) of the payoffs of the individual securities. For example, if there are n securities whose payoffs are represented by the random variables X1,…,Xn (OD some probability space), their span is the set of all payoffs of the form a] X1+…+ anxn, where al, - - -, an are constants representing the numbers of units of the respective securities held in a given portfolio._x000B_ In a multi-period model, the opportunity to retrade in each successive period dramatically increases the span of a given set of securities. For example, the option pricing model of Black and Scholes (1973) has two primitive securities, a bond and the stock on which the option is written. Their span, in a dynamic sense, is the set of all contingent claims that can be generated by trading the stock and the bond over time. In this model, under idealistic conditions such as the absence of transactions costs, independently distributed stock returns, continuous price paths, and the ability to trade continually and with unlimited speed, the span of the stock and bond contains any contingent claim whose payoffs can be viewed as a (measurable) function of the path taken by the stock price. A call option on the stock is thus in the span. This fact is of particular interest in finance because it implies, in the absence of arbitrage, that the price C of the call option must be the amount V that one invests in the stock and the bond in order to synthesize the option payoff. If not, say if C > V, then one could create an arbitrage by selling the option for C and investing V in the stock-bond strategy that replicates the option payoff. There would be an initial riskless profit of C - V, and no further cash flows since the payoff of the option replication strategy meets any obligations from the short option position. There is a like arbitrage if C < V, so the modeled option price is V, which is known explicitly under certain assumptions._x000B_ The application of spanning in the Black-Scholes option pricing model has been extended to the point that it is now the most commonly used approach in the business world for pricing securities. Like the Black-Scboles model itself, most of its extensions are partial equilibrium models in which certain “primitive” security prices are taken as given, with the focus placed on pricing other “derivative” securities whose payoffs are in the span of the_x000B_ 2_x000B_