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

Qingshuo Song and Pengfei Yang

Quantitative Finance, 2015, vol. 15, issue 5, 901-908

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.

Date: 2015
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DOI: 10.1080/14697688.2013.838634

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