Deep neural networks enable accurate pricing of American options under stochastic volatility

Background and Motivation

Accurately pricing American-style options, which allow early exercise at any time before expiry, remains a significant challenge in quantitative finance. This task becomes even more complex under realistic market conditions where asset volatility is not constant but fluctuates randomly, as described by stochastic volatility models like Heston's. Traditional numerical methods, often mesh-based, can be computationally intensive and struggle with high-dimensional problems. With the exponential growth of derivatives trading and the critical need for effective risk management, evidenced by billions of contracts traded annually, there is a pressing demand for more efficient and accurate pricing tools. Furthermore, markets for newer derivatives, such as those linked to real estate indices, lack reliable pricing models, creating a gap that this research aims to fill.

Methodology and Scope

This study pioneers the application of Physics-Informed Neural Networks (PINNs) and a faster variant, Physics-Informed Extreme Learning Machines (PIELMs), to solve the complex partial differential equations governing option prices. The research focuses on two critical, real-world two-factor models: the Heston stochastic volatility model for equity options and an extended Fabozzi-Shiller-Tunaru model with stochastic volatility for real estate index options. For American options, formulated as linear complementarity problems, the authors integrate a penalty method within the neural network framework. The models are trained to minimise a loss function that encodes the governing PDE, initial conditions (option payoff), and boundary conditions, using automatic differentiation to compute crucial hedging sensitivities (Greeks) efficiently.

Key Findings and Contributions

High Accuracy for Complex Models: PINNs successfully produce highly accurate prices for both European and American options under the two-factor Heston and real estate stochastic volatility models. Computed prices and Greeks (Delta, Gamma, Vega) show close agreement with established finite-difference benchmark methods. Speed vs. Accuracy Trade-off: PIELMs, with their single-hidden-layer and analytical weight calculation, train significantly faster than multi-layer PINNs—sometimes in seconds versus minutes—while maintaining comparable, though slightly lower, accuracy. This offers a practical choice for rapid pricing exercises. Effective Penalty Method for Early Exercise: The PINN framework effectively handles American option early exercise constraints using a penalty method. Notably, it achieves accuracy with smaller penalty parameter values than typically required in traditional mesh-based methods. Novel Tool for Real Estate Derivatives: The work provides a novel, reliable neural network-based pricing algorithm for American options on real estate indices, a valuable contribution given the scarcity of robust models for this asset class and the importance of real estate in the global economy.

Why It Matters

This research represents a significant step in applying modern AI techniques to solve core, realistic problems in financial engineering. By demonstrating that deep learning frameworks can accurately and efficiently price complex derivatives under stochastic volatility, it opens the door to tackling even higher-dimensional pricing problems that are intractable for traditional grid-based methods (the "curse of dimensionality"). The development of a credible model for real estate index derivatives is particularly impactful, offering investors and institutions a much-needed tool for hedging exposure to property market risks without direct physical investment.

Practical Applications

For Financial Institutions: Provides a robust alternative for front-office pricing and risk management desks to value equity and real estate derivatives, especially for quick sensitivity analysis and hedging calculations via automatic differentiation. For Quantitative Developers: Offers a blueprint for implementing PINN and PIELM frameworks for other complex derivative products beyond the two-factor models studied. For Real Estate Investors and Fund Managers: Delivers a practical model to price and hedge real estate index options, facilitating better risk management strategies for property portfolios. For Computational Finance: Highlights PIELMs as a compelling, high-speed alternative for scenarios where approximate prices are needed rapidly, potentially for real-time risk assessment or within large-scale Monte Carlo simulations.

 

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