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From Discrete to Continuous Financial Models: New Convergence Results For Option Pricing

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  • Nigel J. Cutland
  • Ekkehard Kopp
  • Walter Willinger
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    Abstract

    In this paper we develop a new notion of convergence for discussing the relationship between discrete and continuous financial models, "D"-super-2-convergence. This is stronger than weak convergence, the commonly used mode of convergence in the finance literature. We show that "D"-super-2-convergence, unlike weak convergence, yields a number of important convergence preservation results, including the convergence of contingent claims, derivative asset prices and hedge portfolios in the discrete Cox-Ross-Rubinstein option pricing models to their continuous counterparts in the Black-Scholes model. Our results show that "D"-super-2-convergence is characterized by a natural lifting condition from nonstandard analysis (NSA), and we demonstrate how this condition can be reformulated in standard terms, i.e., in language that only involves notions from standard analysis. From a practical point of view, our approach suggests procedures for constructing good (i.e., convergent) approximate discrete claims, prices, hedge portfolios, etc. This paper builds on earlier work by the authors, who introduced methods from NSA to study problems arising in the theory of option pricing. Copyright 1993 Blackwell Publishers.

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    Bibliographic Info

    Article provided by Wiley Blackwell in its journal Mathematical Finance.

    Volume (Year): 3 (1993)
    Issue (Month): 2 ()
    Pages: 101-123

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    Handle: RePEc:bla:mathfi:v:3:y:1993:i:2:p:101-123

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    Web page: http://www.blackwellpublishing.com/journal.asp?ref=0960-1627

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    Cited by:
    1. Frederik Herzberg, 2008. "On the foundations of Lévy finance: Equilibrium for a single-agent financial market with jumps," Working Papers 406, Bielefeld University, Center for Mathematical Economics.
    2. Johannes Leitner, 2000. "Convergence of Arbitrage-free Discrete Time Markovian Market Models," CoFE Discussion Paper 00-07, Center of Finance and Econometrics, University of Konstanz.

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