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A Note on Constructing 50-50 Step Probability Binomial Lattices to Replicate Wiener Diffusion


  • Allen Abrahamson


Binomial lattices are sequences of discrete distributions commonly used to approximate the future value states of a financial claim, such as a stock price, when the instantaneous rate of return is assumed to be governed by a Wiener diffusion process. In that case, both pedagogical and professional conventions generally follow the lattice construction methodology used by Cox, Ross, and Rubinstein ("CRR") in their classical article. In some applications, it is more convenient to replace the "implied" branching probabilities of that construction with a more natural and tractable alternative: that is, with the probability of "up" and "down" branching being exactly one-half, or, vernacularly, with a "50-50 step" probability. This elementary note reviews such an alternative formulation for constructing a binomial lattice, which can be viewed as simply entailing multiplicative shifts of every state value on a CRR-constructed binomial lattice. This transformation maintains (in fact, improves) the equivalence of the lattice values' moments to those arising from the replicated diffusion. The expression of that transform is derived, and the effect on the lattice values' moments and orders of convergence to the limit imposed by the continuous process are given. To show the absence of numerical effect, the values of some simple European options obtained from the two alternative binomial lattice constructions are compared against the limiting Black Scholes values.

Suggested Citation

  • Allen Abrahamson, 2003. "A Note on Constructing 50-50 Step Probability Binomial Lattices to Replicate Wiener Diffusion," Finance 0305004, EconWPA, revised 17 May 2003.
  • Handle: RePEc:wpa:wuwpfi:0305004 Note: Type of Document - PDFLatex; prepared on IBM PC; to print on Laser Printers; pages: 10; figures: none. PDF with hyperlinks

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    References listed on IDEAS

    1. Duffee, Gregory R, 1999. "Estimating the Price of Default Risk," Review of Financial Studies, Society for Financial Studies, vol. 12(1), pages 197-226.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    3. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-664, May.
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    5. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    6. Robert A. Jarrow & David Lando & Stuart M. Turnbull, 2008. "A Markov Model for the Term Structure of Credit Risk Spreads," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 18, pages 411-453 World Scientific Publishing Co. Pte. Ltd..
    7. Robert A. Jarrow & Stuart M. Turnbull, 2008. "Pricing Derivatives on Financial Securities Subject to Credit Risk," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 17, pages 377-409 World Scientific Publishing Co. Pte. Ltd..
    8. Li Chen & Damir Filipovic, 2003. "Pricing Credit Default Swaps Under Default Correlations and Counterparty Risk," Finance 0303009, EconWPA.
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    Cited by:

    1. Allen Abrahamson, 2003. "Efficient Path-Dependent Valuation Using Lattices: Fixed and Floating Strike Asian Options," Finance 0305005, EconWPA.

    More about this item


    Binomial Lattices; Wiener Processes; Option Valuation Methods;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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