Boundary conditions at infinity for Black-Scholes equations

ArXiv ID: 2401.05549 “View on arXiv”

Authors: Unknown

Abstract

We propose a numerical procedure for computing the prices of European options, in which the underlying asset price is a Markovian strict local martingale. If the underlying process is a strict local martingale and the payoff is of linear growth, multiple solutions exist for the corresponding Black-Scholes equations. When numerical schemes such as finite difference methods are applied, a boundary condition at infinity must be specified, which determines a solution among the candidates. The minimal solution, which is considered as the derivative price, is obtained by our boundary condition. The stability of our procedure is supported by the fact that our numerical solution satisfies a discrete maximum principle. In addition, its accuracy is demonstrated through numerical experiments in comparison with the methods proposed in the literature.

Keywords: Strict Local Martingale, Finite Difference Methods, Boundary Condition at Infinity, Discrete Maximum Principle, European Options, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, focusing on existence, uniqueness, and boundary conditions for PDEs in the context of strict local martingales, involving advanced stochastic calculus and functional analysis; empirical validation is limited to numerical experiments on a finite difference scheme without live data or backtesting on real markets.

  flowchart TD
    A["Research Goal<br>Compute European option prices<br>when the underlying is a strict local martingale"] --> B["Methodology<br>Numerical procedure with boundary condition at infinity"]
    B --> C["Input Data<br>Payoff function parameters<br>and initial asset price S₀"]
    C --> D["Computational Process<br>Finite Difference Method solving<br>Black-Scholes PDE"]
    D --> E{"Key Finding<br>Our boundary condition selects the minimal solution<br>which represents the true derivative price"}
    E --> F["Validation<br>Stability via discrete maximum principle<br>Accuracy via numerical experiments"]