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

TL;DR

This paper investigates approximating the smallest hedging price of European options when the underlying is a strict local martingale, using rebate barrier options to address PDE non-uniqueness issues.

Contribution

It introduces a rebate barrier option approach to approximate the minimal hedging price in cases of PDE non-uniqueness due to local martingale behavior.

Findings

Rebate barrier options can approximate the smallest hedging price.

A specific rebate option with a continuous rebate function corresponds to the unique PDE solution.

The asymptotic convergence rate is established as the barrier moves to infinity.

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.