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Pricing Multi-Asset Options Using Physics-Informed Neural Networks

This study proposes a meshfree Physics-Informed Neural Network (PINN) framework that solves the high-dimensional Black–Scholes obstacle problem to simultaneously price American options on multiple assets and identify their optimal exercise boundaries as endogenous hypersurfaces without explicit parameterization.

Original authors: Marjan Uddin, Muhammad Usman

Published 2026-10-22

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CC BY 4.0

Original authors: Marjan Uddin, Muhammad Usman

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, investors often buy contracts that give them the right to sell an asset at a specific price before a deadline. These are called American options. Unlike simpler contracts that can only be used on a single day, American options offer a crucial advantage: the holder can choose the exact moment to cash in, provided it is before the expiration date. This flexibility makes them incredibly valuable, but it also makes them notoriously difficult to price. The challenge lies in figuring out the perfect moment to exercise the option. If the market price of the asset drops too low, it might be wise to sell immediately. If the price stays high, it is better to wait. The line separating the "sell now" zone from the "wait" zone is not a fixed, known location; it is a moving boundary that shifts as time passes and as the prices of the underlying assets change. For a single asset, this boundary is a simple curve. But when the option depends on a basket of multiple assets, like a mix of different stocks, the boundary becomes a complex, multi-dimensional surface that is incredibly hard to map using traditional mathematical tools.

Researchers Marjan Uddin and Muhammad Usman tackled this problem by applying a modern computational technique known as a physics-informed neural network. Instead of relying on the old-fashioned method of breaking the market space into a rigid grid of points, which becomes impossible to manage when dealing with many assets, they trained a computer program to learn the rules of the market directly. They taught the network the fundamental laws that govern how option prices behave, specifically a set of equations that describe how value changes over time and with price fluctuations. The network was also instructed to respect the rule that an option can never be worth less than the money one would get by selling it immediately. By feeding the network thousands of random scenarios rather than a fixed grid, the researchers allowed it to discover the shape of the invisible boundary where it becomes optimal to exercise the option. This approach avoids the "curse of dimensionality," a problem where the complexity of calculations explodes as more assets are added, by letting the network sample the market space freely.

The team tested their method on a basket option involving two correlated assets. In their initial attempts, the computer struggled to find the correct answer, often producing prices that were far too high. The researchers discovered that the problem was not with the network's architecture, but with how they were teaching it. They had been using a penalty system that was too blunt; it punished the network for being wrong in ways that didn't match the actual financial rules. Specifically, the network was being forced to satisfy conditions in the "exercise" region that should only apply in the "waiting" region, creating a conflict that led to inaccurate results. Through a series of careful experiments, they refined their approach. They normalized the data to make the numbers easier to handle, adjusted how they sampled the market scenarios to focus more on the critical areas near the decision boundary, and, most importantly, redesigned the penalty function to respect the one-sided nature of the problem. They realized that in the region where it is best to hold the option, the price must follow the market laws, but in the region where it is best to sell, the price simply equals the payout, and the market laws do not need to apply in the same way.

After correcting these mathematical misunderstandings, the network produced a final result of approximately $7.87. This result represented an overestimate of about 14% compared to the benchmark price of $6.90 calculated using the Longstaff–Schwartz Monte Carlo technique. While this was the closest any configuration came during the entire inquiry and fell within the range of typical numerical-discretization error rather than pointing to a structural failure, the authors noted that the PINN approach is only competitive with classical Monte Carlo methods once its loss functional accurately encodes the underlying variational inequality. The study demonstrated that while neural networks hold great promise for pricing these complex financial instruments, their success depends entirely on encoding the correct mathematical logic into the training process. The researchers found that simply throwing more data or more computing power at the problem was not enough; the model had to be guided by a precise understanding of where the rules of the game change. By getting the underlying physics of the problem right, they showed that these networks can effectively map the invisible boundaries that dictate the value of multi-asset options, offering a powerful new tool for navigating high-dimensional financial markets.

Technical Summary: Pricing Multi-Asset Options Using Physics-Informed Neural Networks

Problem Statement
The paper addresses the valuation of American options on multiple underlying assets ( $d > 1$ ), such as basket, spread, or rainbow options. Unlike European options, American options allow early exercise, transforming the pricing problem into a free-boundary problem. The option value must satisfy the Black–Scholes partial differential equation (PDE) in a "continuation region" (where holding is optimal) while remaining constrained by the intrinsic payoff in an "exercise region" (where immediate cashing out is optimal). The boundary separating these regions is unknown a priori.

While single-asset ( $d=1$ ) problems have been effectively solved using methods like finite differences, binomial trees, and radial basis functions (RBF), extending these to high dimensions presents a "curse of dimensionality." Specifically, traditional grid-based or knot-based quasi-interpolation schemes (e.g., Hon and Schaback, 2002) suffer from exponential growth in complexity ( $O(N^d)$ ) as the number of assets increases, making them computationally intractable for $d > 1$ .

Methodology
The authors propose a Physics-Informed Neural Network (PINN) framework to approximate the option value and the optimal exercise boundary without explicit mesh generation.

Mathematical Formulation:
- The problem is formulated in log-price coordinates ( $x_i = \log S_i$ ) to symmetrize the diffusion term and convert variable coefficients to constants.
- The pricing problem is expressed as a variational inequality or a linear complementarity problem:
  $\min\left( -\frac{\partial u}{\partial t} + \mathcal{L}u, \, u - g \right) = 0$
  where $\mathcal{L}$ is the multi-asset Black–Scholes operator, $u$ is the option value, and $g$ is the payoff. This implies that either the PDE residual is zero (continuation region) or the option value equals the payoff (exercise region), with $u \geq g$ always holding.
Neural Network Architecture:
- A feed-forward network with smooth activation functions (specifically tanh) approximates the option value $u_\theta(S, t)$ .
- Automatic differentiation is used to compute the necessary first and second-order derivatives for the PDE operator, avoiding the need for fixed basis functions.
Loss Function and Training:
- The network is trained by minimizing a weighted sum of four residual terms:
  1. PDE Residual: Enforces the Black–Scholes equation in the interior.
  2. Terminal Condition: Anchors the solution to the known payoff at expiration.
  3. Boundary Conditions: Enforces behavior at asset price boundaries (e.g., zero or infinity).
  4. Obstacle/Early-Exercise Condition: Penalizes violations where $u < g$ .
- Sampling: Instead of a fixed grid, collocation points are sampled from low-discrepancy sequences (Sobol or Latin Hypercube) over the space-time domain. This allows the number of points to scale more favorably with dimension $d$ compared to tensor-product grids.
- Exercise Boundary Recovery: The optimal exercise boundary is identified a posteriori as the zero level set where the predicted value equals the intrinsic payoff ( $u_\theta - g = 0$ ), forming a $(d-1)$ -dimensional hypersurface.

Experimental Results and Diagnostic Process
The authors conducted an extensive diagnostic study on a two-asset American basket put option using MATLAB. The paper details a progression of eight experimental runs, highlighting that architectural choices alone were insufficient; the mathematical formulation of the loss function was critical.

Initial Failures (Runs 1–3): Early attempts using standard squared-hinge penalties for the obstacle condition and unnormalized inputs resulted in divergence or significant overestimation of the option price (e.g., $12.58 vs. a Monte Carlo reference of $6.90). The network tended to inflate values to satisfy the "greater than" constraint without properly balancing the PDE residual.
Sampling and Stability (Runs 4–7): Adjustments such as near-strike sampling, adaptive resampling, and varying the loss landscape (e.g., using a single variational inequality term) failed to resolve the discrepancy. In fact, adaptive resampling based on current residuals led to worse performance ($30.77), and variational inequality losses introduced instability. Run 6, which combined stable hinge-based obstacles with near-strike sampling, also failed, producing a price of $12.96 with flat Greeks.
The Breakthrough (Run 8): The successful configuration relied on correcting a theoretical misunderstanding in the loss function:
1. One-Sided PDE Penalty: The PDE residual is only penalized when negative (in the continuation region), allowing it to be positive in the exercise region where the PDE does not strictly hold.
2. Fischer–Burmeister Complementarity: Instead of a non-smooth min function or a simple hinge, a smooth Fischer–Burmeister function was used to enforce the complementarity condition ( $gap \ge 0, F \ge 0, gap \cdot F = 0$ ).
3. Normalization: Inputs and outputs were normalized by the strike price $K$ .

Final Outcome:
With the corrected loss functional (Run 8), the PINN achieved a price of $7.87 for the basket put, compared to the Longstaff–Schwartz Monte Carlo reference of $6.90. This represents an overestimate of approximately 14%. The authors note that while this is a significant improvement over previous runs, the initial comparison revealed a "clear disagreement" between the approaches, and the final result remains an overestimate requiring specific theoretical corrections to approach the reference value. The Greeks (deltas) also aligned more closely with the reference.

Significance and Claims
The paper claims that the PINN framework offers a mesh-free methodology capable of handling high-dimensional American option pricing where traditional grid-based methods fail due to the curse of dimensionality.

The primary contribution is not merely the application of neural networks, but the rigorous identification of the mathematical structure required for the loss function. The authors demonstrate that for free-boundary problems, the loss function must accurately encode the variational inequality, specifically respecting that the PDE residual and the obstacle condition do not need to vanish simultaneously everywhere. The study concludes that the PINN approach is competitive with classical Monte Carlo methods only once its loss functional accurately encodes the underlying variational inequality, and not before. This underscores that the general approach requires specific theoretical corrections to the loss structure to achieve accuracy, rather than being inherently competitive in its standard formulation.

Technical Summary: Pricing Multi-Asset Options Using Physics-Informed Neural Networks

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