American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support

ArXiv ID: 2307.13870 “View on arXiv”

Authors: Unknown

Abstract

Semi-analytical pricing of American options in a time-dependent Ornstein-Uhlenbeck model was presented in [“Carr, Itkin, 2020”]. It was shown that to obtain these prices one needs to solve (numerically) a nonlinear Volterra integral equation of the second kind to find the exercise boundary (which is a function of the time only). Once this is done, the option prices follow. It was also shown that computationally this method is as efficient as the forward finite difference solver while providing better accuracy and stability. Later this approach called “the Generalized Integral transform” method has been significantly extended by the authors (also, in cooperation with Peter Carr and Alex Lipton) to various time-dependent one factor, and stochastic volatility models as applied to pricing barrier options. However, for American options, despite possible, this was not explicitly reported anywhere. In this paper our goal is to fill this gap and also discuss which numerical method (including those in machine learning) could be efficient to solve the corresponding Volterra integral equations.

Keywords: American Options, Ornstein-Uhlenbeck Model, Volterra Integral Equation, Semi-Analytical Pricing, Machine Learning, Options

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats
  • Why: The paper is heavily theoretical, deriving nonlinear Volterra integral equations for exercise boundaries across multiple models (OU, Hull-White, CEV, CIR) and discussing analytical transforms and numerical solvers, indicating high mathematical density. However, it contains no empirical backtests, code, or data; the focus is on algorithmic formulation and numerical efficiency, placing it in the ‘Lab Rats’ quadrant.
  flowchart TD
    A["Research Goal<br>Extend semi-analytical pricing<br>for American options in time-dependent<br>Ornstein-Uhlenbeck models<br>and evaluate efficient numerical solvers"] --> B["Methodology<br>Generalized Integral Transform"]
    B --> C["Mathematical Formulation<br>Derive Volterra Integral Equation<br>for the exercise boundary S(t)"]
    C --> D["Data & Model Inputs<br>• Model parameters (OU process)<br>• Option specs (T, K, r)<br>• Initial asset price S0"]
    D --> E["Computational Process<br>Solve Volterra Equation for S(t)<br><br>Numerical Methods:<br>• ML-supported solvers<br>• Numerical quadrature<br>• Finite difference schemes"]
    E --> F["Outcomes<br>• American option price P(t,S)<br>• Exercise boundary S(t)<br>• Computational efficiency &<br>accuracy comparison vs. PDE solvers"]