Abstract
The payoffs of the barrier options depend on the time path of the underlying price as opposed to just the price at expiry. It implies that both the boundary conditions and the initial condition are imposed on the Black-Scholes partial differential equation. Therefore, the valuation of the barrier options is a boundary value problem. Since the Black-Scholes equation can be converted into a homogeneous linear equation, the boundary integral method will be the most efficient numerical method to calculate the numerical solution for the barrier options. This paper demonstrates the valuation of an up-and-out call by applying the boundary integral method, and explains its risk characteristics.
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