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A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models

This paper proposes a novel deep learning framework for pricing European basket options under jump-diffusion models by using an implicit-explicit minimizing movement time-stepping scheme combined with specialized neural networks and advanced quadrature rules to efficiently handle high-dimensional partial integro-differential equations.

Original authors: Emmanuil H. Georgoulis, Antonis Papapantoleon, Costas Smaragdakis

Published 2026-02-10

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CC BY 4.0

Original authors: Emmanuil H. Georgoulis, Antonis Papapantoleon, Costas Smaragdakis

Original paper licensed under CC BY 4.0 (http://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

Imagine you are a professional weather forecaster, but instead of predicting rain, you are trying to predict the price of a "basket" of groceries (like milk, bread, and eggs) several months into the future.

The problem is that grocery prices don't just drift up and down smoothly like a calm tide. Sometimes, a sudden "jump" happens—maybe a sudden drought or a supply chain crisis—that sends prices spiking instantly. In finance, this is called a Jump-Diffusion Model.

This paper presents a new, high-tech way to predict these prices using Artificial Intelligence. Here is the breakdown of how they do it, using everyday analogies.

1. The Problem: The "Curse of Dimensionality"

Imagine trying to track the price of just one item. Easy. Now imagine tracking 15 different items, all of which affect each other. If you try to use a traditional "grid" method (like a massive spreadsheet where you list every possible combination of prices), the spreadsheet becomes so large that even the world's fastest supercomputer would run out of memory. This is the "Curse of Dimensionality." It’s like trying to map every single grain of sand on a beach; the map becomes too heavy to carry.

2. The Solution: The "Deep IMEX" Method

Instead of building a massive, rigid spreadsheet, the researchers use Artificial Neural Networks (ANNs).

Think of a traditional method like a static map—it’s detailed, but if the landscape changes, the map is useless. The researchers' method is more like a smart GPS. Instead of memorizing every inch of the road, the GPS learns the patterns of how roads work. It can "guess" the price at any point by understanding the underlying logic, which saves a massive amount of computational "weight."

They use a specific strategy called "Implicit-Explicit" (IMEX).

The "Implicit" part is like a cautious driver who looks far ahead to ensure they don't crash (handling the smooth, predictable parts of price movement).
The "Explicit" part is like a driver reacting quickly to a sudden pothole (handling the sudden "jumps" in price).
By combining these, the AI stays stable even when the market gets bumpy.

3. The Secret Sauce: Three Clever Tricks

To make this AI even smarter, the authors added three "life hacks":

The "Intrinsic Value" Shortcut (Decomposition):
Imagine you are calculating the total value of a house. Instead of starting from zero every time, you first acknowledge the value of the land (which you already know) and then only use the AI to figure out the "extra" value of the house itself. By separating the "known" part from the "unknown" part, the AI doesn't have to work as hard.
The "Zoom Out" Trick (Domain Truncation):
The AI doesn't need to spend energy calculating what happens if a loaf of bread costs a billion dollars. That’s unrealistic. The researchers tell the AI to "zoom in" on the realistic price ranges and use a mathematical shortcut to "predict" the extreme ends. It’s like a weather app that gives you a detailed forecast for your city but just says "It'll be hot" for the Sahara Desert.
The "Smart Sampling" (Quadrature):
When the AI tries to account for those sudden "jumps," it has to simulate many possible scenarios. Instead of simulating every possible jump (which is exhausting), they use two methods: one that uses a mathematical "cheat sheet" (Gauss-Hermite) and another where a second, smaller AI acts as a specialized assistant to handle the math.

The Result

When they tested this against other famous AI methods, their "Smart GPS" was not only accurate but also incredibly efficient. It could handle 15 different assets at once—a task that would make older methods crumble—and it provided a complete "weather map" of prices over time, making it easy for traders to see not just the price, but the risk (the "Greeks") associated with it.

In short: They built a smarter, lighter, and faster digital brain that can predict complex, jumpy market prices without getting overwhelmed by the sheer number of moving parts.

Technical Summary: A Deep Implicit-Explicit Minimizing Movement Method for Option Pricing in Jump-Diffusion Models

1. Problem Statement

The paper addresses the challenge of pricing European basket options in high-dimensional settings where the underlying assets follow jump-diffusion dynamics (specifically the Merton model).

In multi-asset finance, the value of a basket option depends on a set of $d$ assets. As the dimension $d$ increases, traditional numerical methods—such as finite difference, finite element, and transform methods (Fourier/COS)—suffer from the "curse of dimensionality," becoming computationally infeasible for $d > 10$ . Furthermore, the presence of random jumps transforms the standard Partial Differential Equation (PDE) into a Partial Integro-Differential Equation (PIDE). The integral term in the PIDE, which accounts for the jumps, adds significant computational complexity, especially in high dimensions.

2. Methodology

The authors propose a novel deep learning framework that avoids grid-based discretization by using Artificial Neural Networks (ANNs) to approximate the solution of the PIDE. The methodology is built on several sophisticated pillars:

Deep Implicit-Explicit (IMEX) Minimizing Movement: Instead of training a single ANN to approximate the entire space-time solution (like the Deep Galerkin Method), the authors use a time-stepping approach. They discretize the PIDE using Backward Differentiation Formulae (BDF). This is formulated as a "minimizing movement" problem, where the solution at each time step is found by minimizing a convex energy cost functional. The "IMEX" aspect refers to treating the stiff differential part implicitly (for stability) and the integral part explicitly (to reduce computational cost).
Solution Decomposition: To improve training stability and accuracy, the option price $u(t, x)$ is decomposed into:
$u(t, x) = w(t, x) + v(t, x)$
where $v(t, x)$ is the known intrinsic value (a lower bound) and $w(t, x)$ is the unknown time value. This ensures the network only needs to learn the non-negative time value component.
Domain Truncation: To handle the infinite domain of asset prices, the authors project the problem onto a bounded subset of $\mathbb{R}^d_+$ . They use an asymptotic approximation to link prices inside the bounded domain to "extreme moneyness" values outside, allowing the model to remain accurate even for very high asset prices.
Integral Operator Approximation: To solve the high-dimensional integral term, they propose two methods:
1. Sparse-grid Gauss–Hermite quadrature: Uses Singular Value Decomposition (SVD) to align the quadrature axes with the jump covariance, making it more efficient.
2. ANN-based Quadrature: A secondary, specialized "quadrature ANN" that learns to estimate the integral values, which is faster for high-dimensional applications.

3. Key Contributions

Novel Algorithmic Framework: The integration of the "minimizing movement" principle with deep learning for PIDEs.
Efficiency via Warm-Starting: By using the ANN parameters from the previous time step as the starting point for the current step, the optimization process is significantly accelerated.
Handling High Dimensionality: The combination of the IMEX scheme, solution decomposition, and specialized integral approximation allows the method to scale to much higher dimensions than traditional solvers.
Architectural Innovation: The use of a modified "Deep Galerkin" (DGM) residual-type layer architecture tailored for the energy minimization functional.

4. Results

The authors validated the method using the Merton jump-diffusion model in 5-asset and 15-asset scenarios:

Accuracy: In 5-asset tests, the method achieved absolute errors on the order of $10^{-3}$ compared to Quasi-Monte Carlo (QMC) simulations.
Scalability: The method remained robust and accurate in the 15-asset case, demonstrating its ability to overcome the curse of dimensionality.
Comparative Performance:
- The ANN-based integral approximation was roughly 1.7 times faster than the Gauss–Hermite quadrature method.
- Compared to the Deep BSDE solver and Deep Galerkin Method (DGM), the proposed "Deep IMEX" method was competitive in terms of error and runtime. Crucially, unlike local solvers (Monte Carlo or BSDE), this method provides a complete space-time solution, making it much easier to calculate "Greeks" (sensitivities).

5. Significance

This research provides a powerful tool for quantitative finance. By offering a method that is both accurate and computationally efficient in high dimensions, it enables the pricing of complex, multi-asset derivatives that were previously too difficult to compute. The ability to generate a full space-time solution is particularly significant for risk management, as it allows practitioners to derive sensitivities (Greeks) across a wide range of market conditions simultaneously.