Asymptotic Synthesis of Contingent Claims with Controlled Risk in a Sequence of Discrete-Time Markets

Abstract

We examine the connection between discrete-time models of financial markets and the celebrated Black--Scholes--Merton (BSM) continuous-time model in which "markets are complete." We prove that if (a) the probability law of a sequence of discrete-time models converges to the law of the BSM model, and (b) the largest possible one-period step in the discrete-time models converges to zero, then every bounded and continuous contingent claim can be asymptotically synthesized in a manner that controls for the risks taken in a manner that implies, for instance, that an expected-utility-maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete-time economies as she can at the continuous-time limit. This implies that, in economically significant ways, many discrete-time models with frequent trading "resemble" the complete-markets model of BSM.