When Is Time Continuous?
Massachusetts Institute of Technology (MIT) - Sloan School of Management
Massachusetts Institute of Technology (MIT) - Sloan School of Management; National Bureau of Economic Research (NBER)
Andrew W. Lo
Massachusetts Institute of Technology (MIT) - Sloan School of Management; Massachusetts Institute of Technology (MIT) - Computer Science and Artificial Intelligence Laboratory (CSAIL); National Bureau of Economic Research (NBER)
August 3, 1998
MIT Laboratory of Financial Engineering (LFE) Working Paper No. LFE-1033-98
Continuous-time stochastic processes have become central to many disciplines, yet the fact that they are approximations to physically realizable phenomena is often overlooked. We quantify one aspect of the approximation errors of continuous-time models by investigating the replication errors that arise from delta hedging derivative securities in discrete time. We characterize the asymptotic distribution of these replication errors and their joint distribution with other assets as the number of discrete time periods increases. We introduce the notion of "temporal granularity" for continuous-time stochastic processes, which allows us to quantify 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 function of the contract specifications of the derivative security, and of the degree of market completeness. We derive closed form expressions for the granularity of geometric Brownian motion and of an Ornstein-Uhlenbeck process for call and put options, and perform Monte Carlo simulations that illustrate the practical relevance of granularity.
Number of Pages in PDF File: 51
JEL Classification: C22, G12, G13working papers series
Date posted: September 2, 1998
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