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Convergence of strategies: An approach using Clark-Haussmann's formula

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Listed:
  • Jan Pedersen

    () (Departments of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark Manuscript)

Abstract

We consider a binomial model that converges towards a Black-Scholes model as the number of trading dates increases to infinity. The models considered are complete and hence every claim is generated by an appropriate trading strategy. Fixing a path dependent claim the paper treats weak and pathwise convergence of the corresponding strategy. It is well known that in a binomial model the generating strategy is easily expressed in terms of stock prices and prices of the claim. In contrast, the Black-Scholes model essentially only allows an explicit representation when the underlying claim is differentiable (in some sense), in which case the strategy is defined in terms of Clark-Haussmann's Formula. Hence, attention is restricted to the case when the claim is differentiable. The strategy is then shown to be convergent and a (very simple) version of Clark-Haussmann's Formula is established.

Suggested Citation

  • Jan Pedersen, 1999. "Convergence of strategies: An approach using Clark-Haussmann's formula," Finance and Stochastics, Springer, vol. 3(3), pages 323-344.
  • Handle: RePEc:spr:finsto:v:3:y:1999:i:3:p:323-344
    Note: received: October 1997; final version received: August 1998
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    Keywords

    Binomial and Black-Scholes models; complete markets; Clark-Haussmann's formula; convergence of strategies;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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