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

We prove a fundamental result concerning the connection between discrete-time models of financial markets and the celebrated Black–Scholes–Merton continuous-time model in which “markets are complete.†Specifically, we prove that if (a) the probability law of a sequence of discrete-time models converges (in the functional sense) to the probability law of the Black–Scholes–Merton 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 with bounded risk: For any ^{E} > 0, a consumer in the discrete-time economy far enough out in the sequence can synthesize a claim that is no more than ^{E} different from the target contingent claim x with probability at least 1 - ^{E}, and which, with probability 1, has norm less or equal to the norm of the target claim. This shows that, in terms of important economic properties, the Black-Scholes-Merton model, with its complete markets, idealizes many more discrete-time models than models based on binomial random walks.