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Closed Formula for Options with Discrete Dividends and Its Derivatives


  • Carlos Veiga
  • Uwe Wystup


We present a closed pricing formula for European options under the Black-Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and a proposition that relates expectations of partial derivatives with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis, covering calls and puts, together with results on their partial derivatives. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.

Suggested Citation

  • Carlos Veiga & Uwe Wystup, 2009. "Closed Formula for Options with Discrete Dividends and Its Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 517-531.
  • Handle: RePEc:taf:apmtfi:v:16:y:2009:i:6:p:517-531
    DOI: 10.1080/13504860903075498

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    Cited by:

    1. Carlos Veiga & Uwe Wystup & Manuel Esquível, 2012. "Unifying exotic option closed formulas," Review of Derivatives Research, Springer, vol. 15(2), pages 99-128, July.
    2. Martina Nardon & Paolo Pianca, 2012. "Extracting information on implied volatilities and discrete dividends from American options prices," Working Papers 2012_25, Department of Economics, University of Venice "Ca' Foscari".


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