Toward A Convergence Theory For Continuous Stochastic Securities Market Models1

Abstract

This article reviews some recently developed approximation schemes for financial markets with continuous trading. Two methods for approximating continuous‐time stochastic securities market models whose exogenously given prices have continuous sample paths are described and compared One method approximates both the paths and the information structure; the other is an approximation in distribution with a Markovian structure. In both cases, the approximating models have a finite state space, discrete time, and possess the same “structural” properties (e.g., “no arbitrage” and “completeness”) as the continuous model. the latter characteristic is an important criterion for judging the merits of the approximations. Taking advantage of the “structure‐preserving” characteristic, one can formulate a convergence theory for frictionless markets with continuous trading. the theory provides convergence results for objects such as contingent claim prices, replicating portfolio strategies (hedging policies), optimal consumption policies, and cumulative financial gains (i.e., stochastic integrals), which are constructed along the approximation. the convergence theory enables one to combine the intuitive appeal of discrete models and the analytic tractability of continuous models to provide new insight into the theory of modern financial markets. We survey the current state of such a convergence theory and illustrate the results with some examples of well‐known continuous securities market models.

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