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The stochastic balance equation for the American option value function and its gradient

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  • Shashiashvili, Malkhaz

Abstract

In the paper we consider the problem of valuation and hedging of American options written on dividend-paying assets whose price dynamics follow a multidimensional diffusion model. We derive a stochastic balance equation for the American option value function and its gradient. We prove that the latter pair is the unique solution of the stochastic balance equation as a result of the uniqueness in the related adapted future-supremum problem. The latter problem has an attractive interpretation: the given adapted stochastic process can be adjusted by a martingale in such a manner, that the observer will gain the perfect foresight of the resulting future-supremum process via the Snell envelope of the given stochastic process.

Suggested Citation

  • Shashiashvili, Malkhaz, 2023. "The stochastic balance equation for the American option value function and its gradient," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:spapps:v:166:y:2023:i:c:s0304414923001886
    DOI: 10.1016/j.spa.2023.09.011
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    References listed on IDEAS

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    1. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    2. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
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