[Paper Review] A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

Céline Labart, Jérôme Lelong|arXiv (Cornell University)|Feb 23, 2011

Stochastic processes and financial applications4 citations

TL;DR

This paper presents a scalable parallel algorithm for solving backward stochastic differential equations (BSDEs) to price and hedge American options in high-dimensional local volatility models. By combining Picard iterations with adaptive control variates and leveraging MPI-based parallelization on a 512-CPU cluster, the method achieves linear speedups up to dimension 10, demonstrating strong scalability for computationally intensive financial derivatives pricing.

ABSTRACT

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementation

Motivation & Objective

• Address the computational challenge of pricing and hedging American options in high-dimensional local volatility models.
• Develop a scalable parallel implementation of a BSDE solver to reduce computation time for high-dimensional problems.
• Enable accurate computation of option prices and deltas (hedges) using a Monte Carlo-based BSDE approach.
• Demonstrate linear speedup and scalability of the algorithm in dimensions up to 10 using a cluster of 512 CPUs.
• Overcome limitations of sequential algorithms that require several hours for high-dimensional computations.

Proposed method

• Adapt the Gobet and Labart (2010) algorithm for solving standard BSDEs using Picard iterations and adaptive control variates.
• Apply penalization techniques to approximate reflected BSDEs, enabling solution of American option pricing problems.
• Implement parallelization via MPI across a cluster, distributing Monte Carlo simulations and polynomial regression tasks.
• Use a domain decomposition strategy where each processor handles a subset of time-space grid points and associated stochastic paths.
• Optimize communication by minimizing synchronization points and balancing computational load across processors.
• Compute corrections using polynomial regression of total degree ≤ η, with complexity proportional to Mpd³ per iteration.

Experimental results

Research questions

• RQ1 Can a parallel implementation of a BSDE solver achieve linear speedup in high-dimensional option pricing?
• RQ2 How does the algorithm perform in terms of scalability and efficiency when applied to American options in dimensions up to 10?
• RQ3 To what extent does the algorithm maintain accuracy in pricing and hedging under high-dimensional local volatility models?
• RQ4 What are the performance bottlenecks in low-dimensional regimes, and how do they affect speedup measurements?
• RQ5 Can the algorithm be efficiently scaled to thousands of processors for ultra-high-dimensional problems?

Key findings

• The algorithm achieves linear speedup in dimension 6 up to 256 processors, with speedups of 1.041 at 256 CPUs, demonstrating strong scalability.
• In dimension 3, speedups degrade beyond 28 processors due to low computational load per processor, highlighting the importance of problem size for parallel efficiency.
• For a 5-asset American put option in dimension 5, the algorithm converges in a few iterations, yielding a price of approximately 6.30 and delta vector of (-0.102159, -0.102893, -0.103237, -0.110546, -0.106442).
• The method computes both option prices and deltas (hedges) simultaneously, offering a comprehensive solution for risk management.
• Despite slightly higher computational times than some alternatives, the algorithm's ability to provide hedging information makes it highly efficient overall.
• The use of a modified speedup metric (CPU time for 8 processors / 8 divided by n × CPU time for n processors) accounts for the lack of a sequential baseline, explaining minor speedups above 1 in some cases.