Enhancing accuracy for solving American CEV model with high-order compact scheme and adaptive time stepping

ArXiv ID: 2309.03984 “View on arXiv”

Authors: Unknown

Abstract

In this research work, we propose a high-order time adapted scheme for pricing a coupled system of fixed-free boundary constant elasticity of variance (CEV) model on both equidistant and locally refined space-grid. The performance of our method is substantially enhanced to improve irregularities in the model which are both inherent and induced. Furthermore, the system of coupled PDEs is strongly nonlinear and involves several time-dependent coefficients that include the first-order derivative of the early exercise boundary. These coefficients are approximated from a fourth-order analytical approximation which is derived using a regularized square-root function. The semi-discrete equation for the option value and delta sensitivity is obtained from a non-uniform fourth-order compact finite difference scheme. Fifth-order 5(4) Dormand-Prince time integration method is used to solve the coupled system of discrete equations. Enhancing the performance of our proposed method with local mesh refinement and adaptive strategies enables us to obtain highly accurate solution with very coarse space grids, hence reducing computational runtime substantially. We further verify the performance of our methodology as compared with some of the well-known and better-performing existing methods.

Keywords: Constant Elasticity of Variance (CEV), Fixed-free boundary, Compact finite difference scheme, Dormand-Prince, Local mesh refinement, Equity Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced numerical analysis techniques including high-order compact finite difference schemes, adaptive time-stepping (Dormand-Prince), and non-linear PDE transformations, indicating high mathematical density. While it includes numerical experiments and comparisons, the focus remains on methodological derivation and implementation specifics rather than extensive backtesting or real-market data validation.

  flowchart TD
    A["Research Goal: <br>Improve CEV pricing accuracy & efficiency"] --> B
    subgraph B ["Key Methodology"]
        B1["Adaptive Time Stepping<br>(Dormand-Prince 5(4))"]
        B2["Local Mesh Refinement<br>(Compact Finite Difference)"]
        B3["Analytical Approximations<br>for PDE Coefficients"]
    end
    B --> C{"System of Coupled PDEs<br>Fixed-Free Boundary CEV"}
    C --> D["Computational Process:<br>High-Order Time-Space Discretization"]
    D --> E["Outcome:<br>High Accuracy on Coarse Grids & Reduced Runtime"]
    E --> F["Validation:<br>Comparison with Existing Methods"]