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A deep learning method for solving American double Heston model

This paper proposes a deep learning framework utilizing multiplicative ensembling of neural networks (MLA-MoS-1 and MLA-MoS-2) to simultaneously solve the American double Heston model under both constant and stochastic interest rates, effectively predicting the early exercise boundary, option values, and Greeks.

Original authors: Chinonso Nwankwo, Weizhong Dai

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Chinonso Nwankwo, Weizhong Dai

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of finance, predicting the future price of a stock is a high-stakes gamble. Investors rely on mathematical models to determine the fair value of options, which are contracts giving the holder the right, but not the obligation, to buy or sell an asset at a specific price by a certain date. Unlike standard options that can only be exercised at the very end of their life, American-style options can be bought or sold at any moment before they expire. This flexibility introduces a complex puzzle: at exactly what point in time and at what stock price should an investor choose to exercise the option to maximize profit? This decision point is known as the early exercise boundary. To solve this, mathematicians use equations that describe how asset prices and their volatility—the measure of how wildly prices swing—evolve over time. When these models become sophisticated enough to account for two different sources of volatility and even fluctuating interest rates, the equations become incredibly difficult to solve, often requiring massive computational power and making simplifying assumptions that can reduce accuracy.

A team of researchers from the University of Calgary and Louisiana Tech University has developed a new approach to tackle this specific challenge using a form of artificial intelligence known as deep learning. They focused on a complex financial model called the American double Heston model, which attempts to capture the reality that asset prices are influenced by two separate, interacting sources of volatility, and in some cases, by interest rates that change over time. Instead of trying to solve the difficult equations directly with traditional grid-based methods, which can be rigid and prone to errors when dealing with sharp changes, the researchers trained a neural network. This is a type of computer program inspired by the human brain, capable of learning patterns from data. Their goal was to teach this network to simultaneously predict three critical things: the value of the option, the precise moment and price at which it should be exercised, and the "Greeks," which are sensitive measures of how the option's price will react to small changes in the market.

The researchers designed a unique architecture for their neural network, which they call a "mixture of solutions." Rather than relying on a single, monolithic program, they built a system composed of three smaller, specialized networks working together. One part of the system focuses on the passage of time, another on the current state of the asset and its volatility, and a third on the interaction between the different volatility factors. These components are combined in a specific way that allows the network to handle the sharp, sudden changes that occur at the moment an option is exercised. A key innovation in their method is how they handle the boundaries of the problem. Traditional methods often force the solution to fit strict rules at the edges of the calculation, which can introduce errors. Instead, this new method allows the network to learn the behavior at these distant boundaries directly from the underlying mathematical rules during the training process. This flexibility removes the need for artificial constraints and lets the model discover the most accurate behavior on its own.

To ensure the network could learn effectively, the researchers employed a strategy called grid stretching. Imagine trying to map a coastline where the details are incredibly fine in some areas and broad in others; a standard map might miss the tiny coves. Similarly, the value of an option changes very rapidly near the point where it is exercised, but more slowly elsewhere. The researchers adjusted their data sampling to focus more intensely on these critical, rapidly changing areas while still covering the broader range of possibilities. They tested their method on a standard computer with a modern processor, using a dataset that was surprisingly small compared to what is typically required for such complex problems. The results showed that their neural network could predict the value of the option and the early exercise boundary with high accuracy, matching or even slightly improving upon the results of established methods used by other experts in the field.

The study also explored a more complicated scenario where interest rates are not fixed but fluctuate randomly, adding a fourth dimension to the problem. In this extended version, the researchers tested the model under two different conditions: one where the various market factors were largely independent of each other, and another where they were strongly linked. The network performed well in both cases, successfully predicting the value and the optimal exercise strategy even when the relationships between the variables were complex and non-zero. The researchers found that while the model remained robust, the accuracy was slightly higher when the market factors were less interconnected, suggesting that the complexity of the relationships does impact the precision of the prediction. However, the ability of the network to handle these dense, interconnected scenarios without breaking down is a significant step forward.

What makes this work particularly notable is not just the accuracy of the numbers, but the way the solution is presented. The neural network does not just output a single price; it generates a complete surface that shows how the option's value and the optimal exercise point change across time, asset price, and volatility. This provides a comprehensive view of the financial landscape that is difficult to obtain with older methods. The researchers also demonstrated that their approach could calculate the Greeks, the sensitivity measures that traders use to manage risk, with the same level of precision. By avoiding the rigid discretization of space and time that plagues traditional numerical methods, and by using automatic differentiation to calculate rates of change directly, the method offers a smoother and more reliable way to navigate the complexities of modern financial derivatives.

The implications of this research extend beyond a single financial model. It demonstrates that deep learning can be effectively applied to free boundary problems, a class of mathematical challenges where the boundary itself is part of the solution to be found. This is a common issue in physics and engineering, but it has been a persistent hurdle in finance. The success of this mixture-of-solutions framework suggests that similar approaches could be adapted for other complex pricing models that involve multiple sources of uncertainty. The researchers acknowledge that their work is a simulation-based proof of concept, and they plan to expand their future investigations to include a full range of parameters and correlation structures. For now, they have provided a clear, flexible, and accurate tool for understanding the American double Heston model, offering a new path for solving some of the most intricate equations in quantitative finance.

Technical Summary: A Deep Learning Method for Solving the American Double Heston Model

Problem Statement
The paper addresses the computational challenge of solving the American double Heston model, a complex financial derivative pricing problem characterized as a free boundary problem. The model incorporates two stochastic variance factors and, in its extended form, a stochastic interest rate (modeled via the Cox-Ingersoll-Ross process). The primary difficulty lies in simultaneously determining the value function, the early exercise boundary (a free boundary surface), and the Greeks (sensitivities) across a high-dimensional state space. Traditional numerical methods, such as finite difference or finite element methods, often struggle with the "curse of dimensionality" and the discretization of mixed derivative terms, which can lead to instability or oscillations. Furthermore, existing analytical or semi-analytical approaches (e.g., asymptotic expansions) often rely on specific assumptions or approximations that may limit their accuracy or applicability.

Methodology
The authors propose a deep learning framework based on a "Mixture of Solutions" (MoS) strategy, utilizing multiplicative ensembling of sub-networks. The methodology is divided into two main configurations:

Model Formulation and Transformation:
- The problem is first formulated as a system of coupled Stochastic Differential Equations (SDEs) and a corresponding three-dimensional (constant interest rate) or four-dimensional (stochastic interest rate) Partial Differential Equation (PDE).
- To handle the free boundary, the authors employ a logarithmic transformation ( $x = \ln(S/\hat{s}_f)$ ) to convert the moving boundary problem into a fixed-free boundary problem.
- The value function is normalized by the strike price $K$ , and the time domain is normalized to $[0, 1]$ .
- Crucially, the authors do not impose explicit far-field boundary conditions (e.g., at $S \to \infty$ or $\upsilon \to 0$ ). Instead, these boundary behaviors are learned implicitly through the PDE residual in the loss function.
Neural Network Architecture (MLA-MoS):
The proposed architecture, termed MLA-MoS (Modular Learning Approximant with Mixture of Solutions), consists of three interacting sub-networks combined via multiplicative ensembling:
- Sub-network 1 ( $T$ ): A time-dependent network taking $\tau$ as input, designed to enforce terminal conditions.
- Sub-network 2 ( $R$ ): A spatial network taking the state variables (time, log-price, variances, and interest rate) as input.
- Sub-network 3 ( $Q$ ): An interactive network taking variance terms ( $\upsilon_1, \upsilon_2, \upsilon_1\upsilon_2$ ) as input to model the interaction between volatility factors.
- Ensembling: The final output is a weighted sum of the sub-network outputs, modulated by an exponential term involving the third sub-network. This structure ensures non-negativity and allows the network to satisfy specific boundary conditions (Value Matching and Smooth Pasting) via a Robin boundary term in the loss function.
- Loss Function: The loss function comprises the PDE residual (enforcing the governing equation) and a boundary term that couples the value matching and smooth pasting conditions at the early exercise boundary.
Sampling and Optimization:
- To address the singularity in the Gamma and Delta near the terminal time and the early exercise boundary, the authors employ a grid stretching strategy based on the hyperbolic sine ( $\sinh$ ) function. This concentrates sampling points near critical regions (e.g., $t=0$ , $S=K$ ).
- The models are trained using the Adam optimizer with exponential learning rate decay. Automatic differentiation is used to compute the partial derivatives required for the PDE loss, avoiding the need for numerical discretization schemes.

Key Contributions
The paper highlights the following specific contributions:

Explainable Framework: The introduction of an interactive, modular deep learning framework (MLA-MoS) specifically designed for the American double Heston model as a free boundary problem.
Simultaneous Prediction: The ability of the trained networks (MLA-MoS-1 for constant rates, MLA-MoS-2 for stochastic rates) to simultaneously predict the early exercise boundary surface, the value function, and the Greeks (Delta, Gamma, Vega).
Adaptive Boundary Handling: The implementation of an adaptive exponential term acting as a "mixture of experts" to control the behavior of the option value near artificial right boundaries, eliminating the need for hard-coded far-boundary conditions.
Robin Boundary Integration: The explicit incorporation of smooth pasting and value matching conditions into the loss function as a Robin boundary term, ensuring physical consistency at the exercise boundary.
Non-Discretized Derivatives: The reliance on automatic differentiation for partial derivatives, thereby avoiding the numerical inaccuracies associated with the discretization of convective and mixed derivative terms common in mesh-based methods.

Experimental Results
The authors validate the method through numerical experiments comparing the deep learning predictions against existing methods:

Constant Interest Rate (MLA-MoS-1): Results were compared against the asymptotic expansion method (Zhang and Feng [37]) and the Least Squares Method (LSM). The MLA-MoS-1 predictions for the value function and early exercise boundary showed high consistency with these benchmarks, even when trained on relatively small, stretched datasets (e.g., $14 \times 14 \times 14 \times 14$ points).
Stochastic Interest Rate (MLA-MoS-2): The method was tested on a four-factor model with both sparse and dense correlation matrices.
- In the sparse correlation case, results were compared against the AECV algorithm (Zhang and Zhang [38]), the SWIFT method (Shoude and Guo [30]), and LSM. The MLA-MoS-2 results were found to be reasonable and consistent with these established methods.
- In the dense correlation case (where all correlation factors are non-zero), the model produced reasonable predictions, though the authors noted that the dense structure introduced more variation compared to the sparse case, suggesting that correlation structure significantly impacts solution accuracy.
Qualitative Analysis: Visualizations of the value function, Greeks, and early exercise boundaries confirmed that the surfaces behaved as theoretically expected (e.g., Delta in $[-1, 0]$ , positive Gamma and Vega). The study also demonstrated that the grid stretching strategy is critical; without it, the predictive performance degraded significantly.

Significance and Claims
The paper positions its significance in the context of solving high-dimensional free boundary problems in finance. The authors claim that their approach offers a flexible alternative to traditional numerical methods by:

Simplifying Network Design: By learning far-boundary values from the PDE loss rather than imposing them, the method reduces the complexity of boundary condition handling.
Improving Accuracy: The use of automatic differentiation and grid stretching mitigates issues related to the discretization of mixed derivatives and singularities at the exercise boundary.
Scalability: The method demonstrates the ability to handle both constant and stochastic interest rate scenarios within a unified deep learning framework.

The authors conclude modestly, noting that while the current results are promising, future work will focus on extending the framework to handle full parametrization of variables and parameters (including correlation factors) to allow for inference over a wider range of market conditions, acknowledging that this will increase network complexity.

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