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Closed formula for options with discrete dividends and its derivatives

Author

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  • Veiga, Carlos
  • Wystup, Uwe

Abstract

We present a closed pricing formula for European options under the Black-Scholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr, together with an unusual change of measure technique that relies on the replacement of the initial condition. The closed formulas are attained for the case where no dividend payment policy is considered. Despite its small practical relevance, a digital dividend policy case is also considered which yields approximation formulas. The results are readily extensible to time dependent volatility models but no so for local-vol type models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis using the formulas here obtained. 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

  • Veiga, Carlos & Wystup, Uwe, 2008. "Closed formula for options with discrete dividends and its derivatives," CPQF Working Paper Series 16, Frankfurt School of Finance and Management, Centre for Practical Quantitative Finance (CPQF).
  • Handle: RePEc:zbw:cpqfwp:16
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    References listed on IDEAS

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    1. Ansgar Belke & Thorsten Polleit, 2005. "(How) Do Stock Market Returns React to Monetary Policy? - An ARDL Cointegration Analysis for Germany," Diskussionspapiere aus dem Institut für Volkswirtschaftslehre der Universität Hohenheim 253/2005, Department of Economics, University of Hohenheim, Germany.
    2. Ansgar Belke & Thorsten Polleit, 2007. "How the ECB and the US Fed set interest rates," Applied Economics, Taylor & Francis Journals, vol. 39(17), pages 2197-2209.
    3. Christina Bannier, 2007. "Heterogeneous multiple bank financing: does it reduce inefficient credit-renegotiation incidences?," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 21(4), pages 445-470, December.
    4. Wollersheim, Jutta & Barthel, Erich, 2008. "Kulturunterschiede bei Mergers & Acquisitions: Entwicklung eines Konzeptes zur Durchführung einer Cultural Due Diligence," Frankfurt School - Working Paper Series 94, Frankfurt School of Finance and Management.
    5. Bannier, Christina E. & Hänsel, Dennis N., 2007. "Determinants of banks' engagement in loan securitization," Frankfurt School - Working Paper Series 85, Frankfurt School of Finance and Management.
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    Cited by:

    1. Yu, Xiaofan, 2011. "A spatial interpretation of the persistency of China's provincial inequality," Frankfurt School - Working Paper Series 171, Frankfurt School of Finance and Management.

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