When is Time Continuous?

MIT Laboratory of Financial Engineering (LFE) Working Paper No. LFE-1033-98

51 Pages Posted: 2 Sep 1998

See all articles by Dimitris Bertsimas

Dimitris Bertsimas

Massachusetts Institute of Technology (MIT) - Sloan School of Management

Leonid Kogan

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; National Bureau of Economic Research (NBER); Massachusetts Institute of Technology (MIT) - Computer Science and Artificial Intelligence Laboratory (CSAIL)

Date Written: August 3, 1998

Abstract

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.

JEL Classification: C22, G12, G13

Suggested Citation

Bertsimas, Dimitris and Kogan, Leonid and Lo, Andrew W., When is Time Continuous? (August 3, 1998). MIT Laboratory of Financial Engineering (LFE) Working Paper No. LFE-1033-98. Available at SSRN: https://ssrn.com/abstract=116688 or http://dx.doi.org/10.2139/ssrn.116688

Dimitris Bertsimas

Massachusetts Institute of Technology (MIT) - Sloan School of Management ( email )

E53-359
Cambridge, MA 02142
United States
617-253-4223 (Phone)
617-258-7579 (Fax)

Leonid Kogan

Massachusetts Institute of Technology (MIT) - Sloan School of Management ( email )

100 Main Street
E62-636
Cambridge, MA 02142
United States
617-253-2289 (Phone)
617-258-6855 (Fax)

HOME PAGE: http://web.mit.edu/lkogan2/www/

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Andrew W. Lo (Contact Author)

Massachusetts Institute of Technology (MIT) - Sloan School of Management ( email )

100 Main Street
E62-618
Cambridge, MA 02142
United States
617-253-0920 (Phone)
781 891-9783 (Fax)

HOME PAGE: http://web.mit.edu/alo/www

National Bureau of Economic Research (NBER) ( email )

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Massachusetts Institute of Technology (MIT) - Computer Science and Artificial Intelligence Laboratory (CSAIL)

Stata Center
Cambridge, MA 02142
United States

Register to save articles to
your library

Register

Paper statistics

Downloads
1,834
Abstract Views
6,316
rank
8,390
PlumX Metrics