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Discrete dividends after maturity adjust the stock and strike prices

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  • Kevin W. Lu

Abstract

The standard method to price European calls on a discrete dividend-paying stock is to subtract the present value of the dividends from the initial stock price in the Black-Scholes formula. However, when there are dividends after maturity, this is inconsistent with the model. Within the escrowed dividend model, we highlight an extension of the Black-Scholes formula in which those dividends after maturity adjust both the stock price and strike price, allowing for calls over all maturities to be priced in a model-consistent way. As a related application to American calls with a single dividend before maturity, we establish a neglected case where it is always optimal to early exercise and derive an extension of the Roll-Geske-Whaley formula when there are dividends after maturity, including fully characterising the optimal exercise policy.

Suggested Citation

  • Kevin W. Lu, 2026. "Discrete dividends after maturity adjust the stock and strike prices," Papers 2607.24973, arXiv.org.
    • Handle: RePEc:arx:papers:2607.24973
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    References listed on IDEAS

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    1. M. H. Vellekoop & J. W. Nieuwenhuis, 2006. "Efficient Pricing of Derivatives on Assets with Discrete Dividends," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(3), pages 265-284.
    2. Whaley, Robert E., 1981. "On the valuation of American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 9(2), pages 207-211, June.
    3. Roll, Richard, 1977. "An analytic valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 5(2), pages 251-258, November.
    4. Alexander Buryak & Ivan Guo, 2012. "New Analytic Approach to Address Put--Call Parity Violation due to Discrete Dividends," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 37-58, May.
    5. Tian-Shyr Dai & Yuh-Dauh Lyuu, 2009. "Accurate approximation formulas for stock options with discrete dividends," Applied Economics Letters, Taylor & Francis Journals, vol. 16(16), pages 1657-1663.
    6. Geske, Robert, 1979. "A note on an analytical valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 7(4), pages 375-380, December.
    7. Robert A Jarrow & Arkadev Chatterjea, 2024. "An Introduction to Derivative Securities, Financial Markets, and Risk Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 13797, March.
    8. 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.
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