When is time continuous?

Continuous-time stochastic processes are approximations to physically reali zable phenomena. We quantify one aspect of the approximation errors by char acterizing 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 disc rete-time implementations of continuous-time models can track the payoff of a derivative security. We show that granularity is a particular function o f a derivative contract's terms and the parameters of the underlying stocha stic process. Explicit expressions for the granularity of geometric Brownia n motion and an Ornstein-Uhlenbeck process for call and put options are der ived, and we perform Monte Carlo simulations to illustrate the empirical pr operties of granularity. (C) 2000 Elsevier Science S.A. All rights reserved . JEL classification. G13.