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Approximating functionals of local martingales under lack of uniqueness of the Black-Scholes PDE solution

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  • Qingshuo Song
  • Pengfei Yang

Abstract

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, the Black-Scholes PDE associated with a European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knock-out barrier moves to infinity under suitable conditions.

Suggested Citation

  • Qingshuo Song & Pengfei Yang, 2015. "Approximating functionals of local martingales under lack of uniqueness of the Black-Scholes PDE solution," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 901-908, May.
    • Handle: RePEc:taf:quantf:v:15:y:2015:i:5:p:901-908
    DOI: 10.1080/14697688.2013.838634
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    Cited by:

    1. Yukihiro Tsuzuki, 2024. "Boundary conditions at infinity for Black-Scholes equations," Papers 2401.05549, arXiv.org, revised Sep 2024.
    2. Yukihiro Tsuzuki, 2023. "Pitman's Theorem, Black-Scholes Equation, and Derivative Pricing for Fundraisers," Papers 2303.13956, arXiv.org.

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