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When Is Time Continuous?

In: Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume II)

Author

Listed:
  • DIMITRIS BERTSIMAS

    (Sloan School of Management, E40-147, Massachusetts Institute of Technology, Cambridge, MA 02139, United States)

  • LEONID KOGAN

    (Sloan School of Management, E40-147, Massachusetts Institute of Technology, Cambridge, MA 02139, United States)

  • ANDREW W. LO

    (50 Memorial Drive, E52-432, Cambridge, MA 02142–1347, United States)

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.

Suggested Citation

  • Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 2001. "When Is Time Continuous?," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume II), chapter 3, pages 71-102, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789812810663_0003
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    JEL classification:

    • B16 - Schools of Economic Thought and Methodology - - History of Economic Thought through 1925 - - - Quantitative and Mathematical
    • B23 - Schools of Economic Thought and Methodology - - History of Economic Thought since 1925 - - - Econometrics; Quantitative and Mathematical Studies
    • F65 - International Economics - - Economic Impacts of Globalization - - - Finance
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets

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