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Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE

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  • Qingshuo Song

Abstract

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an 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 of 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 knocked-out barrier moves to infinity under suitable conditions.

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  • Qingshuo Song, 2011. "Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE," Papers 1102.2285, arXiv.org, revised Sep 2012.
    • Handle: RePEc:arx:papers:1102.2285
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    References listed on IDEAS

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    1. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    2. Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086, arXiv.org, revised Sep 2009.
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