Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate (2412.11257v3)

Abstract: For many complex simulation tasks spanning areas such as healthcare, engineering, and finance, Monte Carlo (MC) methods are invaluable due to their unbiased estimates and precise error quantification. Nevertheless, Monte Carlo simulations often become computationally prohibitive, especially for nested, multi-level, or path-dependent evaluations lacking effective variance reduction techniques. While ML surrogates appear as natural alternatives, naive replacements typically introduce unquantifiable biases. We address this challenge by introducing Prediction-Enhanced Monte Carlo (PEMC), a framework that leverages modern ML models as learned predictors, using cheap and parallelizable simulation as features, to output unbiased evaluation with reduced variance and runtime. PEMC can also be viewed as a "modernized" view of control variates, where we consider the overall computation-cost-aware variance reduction instead of per-replication reduction, while bypassing the closed-form mean function requirement and maintaining the advantageous unbiasedness and uncertainty quantifiability of Monte Carlo. We illustrate PEMC's broader efficacy and versatility through three examples: first, equity derivatives such as variance swaps under stochastic local volatility models; second, interest rate derivatives such as swaption pricing under the Heath-Jarrow-Morton (HJM) interest-rate model. Finally, we showcase PEMC in a socially significant context - ambulance dispatch and hospital load balancing - where accurate mortality rate estimates are key for ethically sensitive decision-making. Across these diverse scenarios, PEMC consistently reduces variance while preserving unbiasedness, highlighting its potential as a powerful enhancement to standard Monte Carlo baselines.

• The paper introduces PEMC, integrating machine learning-based control variates with Monte Carlo simulation for unbiased variance reduction.
• It employs neural networks to approximate conditional expectations, significantly lowering RMSE in pricing complex derivatives such as variance swaps and swaptions.
• Empirical results demonstrate a 30–50% reduction in variance, enhancing computational efficiency and suggesting applicability beyond traditional financial models.

Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate

The paper at hand explores the integration of machine learning techniques with Monte Carlo (MC) simulations, especially within the context of financial derivative pricing. The authors propose a novel framework termed Prediction-Enhanced Monte Carlo (PEMC) that leverages machine learning for variance reduction in MC simulations. This approach maintains the unbiased nature of MC while utilizing machine learning models as control variates to enhance computational efficiency in complex financial models, such as those involving exotic options pricing.

Monte Carlo simulation is a foundational technique used extensively in finance to evaluate complex derivatives and assess risk. Despite its robustness, traditional MC methods can face significant computational challenges due to their $O(1/\sqrt{n})$ convergence rate, particularly when applied to path-dependent or high-dimensional problems. Control variate (CV) methods have traditionally been used in MC simulations to improve efficiency by reducing variance. However, these methods often necessitate the availability of suitable auxiliary functions with known analytical expectations, which are not always accessible in complex financial models such as stochastic volatility models or Heath-Jarrow-Morton models for interest rates.

PEMC addresses these limitations by introducing a machine learning-based approach to construct control variates without the need for known mean expectations. In the PEMC framework, neural networks are trained to approximate conditional expectations of derivative payoffs based on feature representations extracted from the sample paths. The pre-trained model is then used to calculate control variates during the MC simulation, aiming to reduce variance across the entire simulation scheme rather than relying on per-sample variance reduction.

The paper showcases the efficacy of PEMC through applications in pricing two complex derivatives: variance swaps under both the Heston and Stochastic Local Volatility models, and swaptions under the Heath-Jarrow-Morton framework. In these scenarios, traditional variance reduction techniques would struggle due to the complicated dependencies and high-dimensional feature spaces involved. The results demonstrate substantial reductions in RMSE compared to standard MC methods, achieving approximately 30-50% variance reduction across various setups. Notably, the framework adapts well even when the volatility surface—or structure in the case of interest rate models—is complex and high-dimensional, highlighting the flexibility and robustness of the approach.

Technically, the PEMC framework consists of a training phase that pre-processes a wide parameter space via off-line simulations, generating a comprehensive dataset for training machine learning models. During evaluation, the pretrained model aids in real-time simulations, enabling fast and accurate computations. This is particularly advantageous in environments where high-frequency updates, such as derivative pricing under rapidly changing market conditions, are necessary.

The implications of PEMC extend beyond immediate gains in computational efficiency. Practically, the reduction in variance can significantly decrease the resource requirements for attaining a desired level of accuracy in MC estimates, allowing financial institutions to scale their computations without corresponding increases in computational cost. Theoretically, PEMC opens new avenues for the incorporation of advanced machine learning architectures in quantitative finance, potentially laying the groundwork for further convergence between AI-driven models and traditional financial engineering techniques.

Looking forward, this work invites further investigation into integrating PEMC with other statistical approaches, possibly marrying it with aspects of generative models or reinforcement learning to explore new financial applications. Additionally, the adaptive nature and flexibility of the PEMC framework suggest it could be extended to non-financial domains where Monte Carlo methods are prevalent, such as engineering simulations or natural sciences, thereby broadening its impact beyond finance.