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

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

2011-02-11

arxiv.org/abs/1102.2285v2

Economy

Quantitative Finance

Pricing of Securities

15 pages

Scientific paper

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 speci?c 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 in?nity under suitable conditions.

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