A new approach for American option pricing: The Dynamic Chebyshev method

TL;DR

This paper presents a novel Chebyshev interpolation-based method for American option pricing that separates model-dependent computations into an offline phase, enabling fast online pricing and sensitivities with theoretical error bounds.

Contribution

The paper introduces a flexible, efficient approach for American option pricing using Chebyshev polynomials, with offline computation of moments and explicit error analysis.

Findings

Fast convergence of prices and sensitivities demonstrated

Efficiency gain over least-square Monte Carlo method

Applicable to various stock price models

Abstract

We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalised (conditional) moments is computed by an appropriate numerical technique such as a Monte Carlo, PDE or Fourier transform based method. Thanks to this methodological flexibility the approach applies to a large variety of models. Online, the backward induction is solved on a discrete Chebyshev grid, and no (conditional) expectations need to be computed. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. Moreover, the same family of (conditional) moments yield multiple outputs including the option prices for different strikes, maturities and different payoff profiles. We provide a theoretical error analysis and find conditions that imply explicit error bounds for a variety of stock price models. Numerical experiments confirm the fast convergence of prices and sensitivities. An empirical investigation of accuracy and runtime also shows an efficiency gain compared with the least-square Monte-Carlo method introduced by Longstaff and Schwartz (2001).