Boundary conditions at infinity for Black-Scholes equations

Yukihiro Tsuzuki

Papers from arXiv.org

Abstract: We propose a numerical procedure for computing the prices of forward contracts, in which the underlying asset price is a Markovian local martingale. If the underlying process is a strict local martingale, multiple solutions exist for the corresponding Black-Scholes equations, and the derivative prices are characterized as the minimal solutions. When applying numerical methods that are set up on a finite grid, such as finite difference methods, additional spatial boundary conditions are required, which causes the solution to uniquely exist. In this case, the problem is reduced to specifying the boundary conditions. Our boundary condition is based on an expression of the derivative price at infinity in terms of those at finite values, and incorporates volatility behaviors beyond the boundary. The proposed procedure is demonstrated through numerical tests, which show that it outperforms the methods proposed in the literature.

Date: 2024-01, Revised 2024-09
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