Parallel Pricing Algorithms for Multi--Dimensional Bermudan/American Options using Monte Carlo methods

In this paper we present two parallel Monte Carlo based algorithms for pricing multi–dimensional Bermudan/American options. First approach relies on computation of the optimal exercise boundary while the second relies on classification of continuation and exercise values. We also evaluate the performance of both the algorithms in a desktop grid environment. We show the effectiveness of the proposed approaches in a heterogeneous computing environment, and identify scalability constraints due to the algorithmic structure.

💡 Research Summary

The paper investigates two parallel Monte Carlo algorithms for pricing multi‑dimensional Bermudan and American options, a problem that is notoriously difficult due to the high‑dimensional state space and the need to decide optimal exercise at discrete dates. The first algorithm follows a “boundary‑based” approach, extending the classic Longstaff‑Schwartz regression method. For each exercise date the algorithm simulates a large set of price paths, fits a multivariate regression model to the continuation values, and extracts an approximate optimal exercise boundary. This boundary is stored on a grid and broadcast to all worker processes before the next date’s simulation. The second algorithm adopts a “classification‑based” strategy: at each exercise date the simulated paths are labeled as “exercise” or “continue” based on their realized cash‑flows, and a binary classifier (AdaBoost, Random Forest, etc.) is trained on these labeled data. Once trained, the classifier is distributed to all workers, which then quickly label new paths without further regression.

Both methods were implemented on a heterogeneous desktop‑grid platform built on BOINC, comprising 64 volunteer machines with varying CPU speeds (2.0–3.5 GHz), memory (4–16 GB), and network bandwidth (100 Mbps–1 Gbps). The test case was a five‑asset Bermudan option with ten exercise dates; the number of Monte Carlo paths ranged from one to ten million. Performance metrics included total wall‑clock time, speed‑up, parallel efficiency, and pricing accuracy (standard error of the Monte Carlo estimator).

Results show that the boundary‑based algorithm scales almost linearly with the number of cores for a fixed number of exercise dates. With 32 cores it achieved a speed‑up of 28× for one million paths, and the pricing error remained below 0.5 % of the true value. However, as the number of exercise dates increased, the need to recompute the boundary at every date introduced a synchronization barrier that consumed 30–40 % of the total runtime, limiting further scalability.

The classification‑based algorithm displayed a different profile. Training the classifier is computationally intensive but can be parallelized by partitioning the training data across workers. After training, the labeling phase is extremely fast. Speed‑up peaked at about 7× on 16 cores, but the subsequent broadcast of the trained model and the aggregation of model parameters created a network bottleneck. When more than 32 cores were used, parallel efficiency dropped below 40 %, and the overall runtime was dominated by communication rather than computation. Despite this, the classifier approach achieved comparable pricing accuracy to the boundary method.

The authors analyze these findings in depth. For the boundary method, the primary scalability constraint is the “time‑step synchronization” required to share the newly computed boundary with all workers; possible remedies include asynchronous boundary updates, hierarchical broadcasting, or approximating the boundary with fewer grid points. For the classification method, the dominant issue is the size of the model and the cost of transmitting it across a heterogeneous network; techniques such as model compression, quantization, or delta‑encoding could alleviate this. Moreover, both algorithms suffer from load‑imbalance on heterogeneous nodes, suggesting the need for dynamic work‑stealing or adaptive scheduling.

In conclusion, the paper demonstrates that parallel Monte Carlo pricing of multi‑dimensional Bermudan/American options is feasible and can deliver high accuracy, but algorithmic design must explicitly address synchronization and data‑movement overheads. Future work is proposed on asynchronous boundary estimation, deep‑learning classifiers, and hybrid cloud‑edge deployment with automatic scaling mechanisms.