[Paper Review] Pricing American options using martingale bases

Jérôme Lelong|arXiv (Cornell University)|Apr 12, 2016

Stochastic processes and financial applications24 references3 citations

TL;DR

This paper proposes a novel dual method for pricing American options using Wiener chaos expansions to approximate uniformly square integrable martingales, enabling efficient, scalable, and path-dependent option pricing via sample average approximation. The approach achieves high performance and parallel scalability—reducing computation time from over an hour to under 11 seconds on 512 cores—while accurately pricing 40-asset options within 3% relative error in under 3 minutes.

ABSTRACT

In this work, we propose an algorithm to price American options by directly solving the dual minimization problem introduced by Rogers. Our approach relies on approximating the set of uniformly square integrable martingales by a finite dimensional Wiener chaos expansion. Then, we use a sample average approximation technique to efficiently solve the optimization problem. Unlike all the regression based methods, our method can transparently deal with path dependent options without extra computations and a parallel implementation writes easily with very little communication and no centralized work. We test our approach on several multi--dimensional options with up to 40 assets and show the impressive scalability of the parallel implementation.

Motivation & Objective

• To develop a dual-based algorithm for pricing American options that avoids the curse of dimensionality and regression-based limitations.
• To enable transparent handling of path-dependent options without additional computational cost.
• To achieve high scalability in parallel computing environments for high-dimensional problems.
• To provide a finite-dimensional, convex, and differentiable optimization framework for the dual problem.
• To demonstrate the method's efficiency and accuracy on multi-asset, path-dependent options up to 40 dimensions.

Proposed method

• The method approximates the space of uniformly square integrable martingales using truncated Wiener chaos expansions, which regularize the optimization and allow exact conditional expectation computation.
• The dual optimization problem is reformulated as a finite-dimensional, convex, and differentiable problem over the coefficients of the Wiener chaos expansion.
• A sample average approximation (SAA) technique is employed to solve the optimization problem efficiently using Monte Carlo sampling.
• The algorithm is implemented in a pure MPI environment, enabling fine-grained parallelism with minimal communication and no centralized bottleneck.
• The approach leverages the fact that Wiener chaos expansions allow exact computation of conditional expectations, avoiding regression approximations.
• The method is tested on geometric basket put options with up to 40 underlying assets and various volatilities and correlations.

Experimental results

Research questions

• RQ1 Can a dual approach to American option pricing be made scalable and efficient for high-dimensional problems using finite-dimensional martingale approximations?
• RQ2 How does the use of Wiener chaos expansions improve the conditioning and implementability of the dual optimization problem?
• RQ3 To what extent can the method handle complex path-dependent payoffs without increasing computational cost?
• RQ4 What is the scalability of the parallel implementation across thousands of CPU cores for high-dimensional options?
• RQ5 How does the number of samples in the SAA affect convergence and accuracy, especially in high-dimensional settings?

Key findings

• The algorithm prices a 40-asset geometric basket put option with 3% relative error in under 3 minutes, demonstrating strong performance on high-dimensional problems.
• On a 190-node supercomputer with 3204 cores, the computation time dropped from 4365 seconds (1 hour 13 minutes) to 10.7 seconds using 512 cores, achieving 59% efficiency.
• With 128 cores, the runtime was reduced to 52 seconds, achieving 73% efficiency, showing excellent scalability.
• For a 10-asset option, a second-order Wiener chaos approximation achieved accurate results within seconds, outperforming traditional dual methods.
• Overfitting was observed when the number of samples was too small relative to the number of chaos expansion coefficients (e.g., m=40 for d=40, p=2), leading to a slightly underestimated price.
• The method naturally handles path-dependent options without additional computational overhead, unlike regression-based approaches.