Finite-Difference Solution Ansatz approach in Least-Squares Monte Carlo

ArXiv ID: 2305.09166 “View on arXiv”

Authors: Unknown

Abstract

This article presents a simple but effective and efficient approach to improve the accuracy and stability of Least-Squares Monte Carlo. The key idea is to construct the ansatz of conditional expected continuation payoff using the finite-difference solution from one dimension, to be used in linear regression. This approach bridges between solving backward partial differential equations and Monte Carlo simulation, aiming at achieving the best of both worlds. In a general setting encompassing both local and stochastic volatility models, the ansatz is proven to act as a control variate, reducing the mean squared error, thereby leading to a reduction of the final pricing error. We illustrate the technique with realistic examples including Bermudan options, worst of issuer callable notes and expected positive exposure on European options under valuation adjustments.

Keywords: Least-Squares Monte Carlo, Bermudan Options, Control Variate, Partial Differential Equations (PDE), Valuation Adjustments (XVA), Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical techniques like finite-difference PDE solutions, control variates, and rigorous error analysis, indicating high math complexity. It also demonstrates empirical rigor through extensive backtesting on realistic examples (Bermudan options, worst-of issuer callable notes, CVA/EPE calculations) with performance metrics and computation time comparisons.

  flowchart TD
    A["Research Goal: Enhance LSMC accuracy & stability"] --> B{"Key Methodology: Finite-Difference Ansatz"}
    B --> C["Construct Ansatz: Finite-Difference solution from PDE"]
    C --> D["Use Ansatz as Control Variate in Regression"]
    D --> E["Data/Inputs: Underlying models & instrument parameters"]
    E --> F["Computational Process: MC Simulation with Reduced Variance"]
    F --> G["Outcome: Lower Pricing Error & Improved Stability"]
    G --> H["Applications: Bermudan Options, XVA, Callable Notes"]