When is time continuous?

Continuous-time stochastic processes are approximations to physically realizable phenomena. We quantify one aspect of the approximation errors by characterizing the asymptotic distribution of the replication errors that arise from delta-hedging derivative securities in discrete time, and introducing the notion of temporal granularity which measures the extent to which discrete-time implementations of continuous-time models can track the payoff of a derivative security. We show that granularity is a particular function of a derivative contract's terms and the parameters of the underlying stochastic process. Explicit expressions for the granularity of geometric Brownian motion and an Ornstein-Uhlenbeck process for call and put options are derived, and we perform Monte Carlo simulations to illustrate the empirical properties of granularity. (C) 2000 Elsevier Science S.A. All rights reserved. JEL classification. G13.