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

TL;DR

This paper extends semi-analytical pricing methods for American options in time-dependent models, focusing on solving nonlinear Volterra integral equations efficiently, including exploring machine learning approaches for numerical solutions.

Contribution

It fills the gap in applying the generalized integral transform method to American options and evaluates numerical and machine learning methods for solving the associated integral equations.

Findings

Efficient semi-analytical pricing for American options in time-dependent models.

Extension of the generalized integral transform method to American options.

Assessment of machine learning techniques for solving Volterra integral equations.

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.