Beyond Traditional Forecasting: Neural Networks Predict Market Jumps

Author: Denis Avetisyan

A novel approach combines the power of deep learning with established stochastic modeling to achieve more accurate asset price predictions.

The LSTM-Lévy model demonstrates performance across the STOXX 600 Dataset through three calibration methods-neural network, model predictive analysis, and TorchSDE-alongside a standalone LSTM, indicating a comparative evaluation of techniques for modeling financial data.

This review details the integration of Long Short-Term Memory networks with Neural Lévy Processes for improved financial time series forecasting and model calibration.

Accurate financial forecasting remains a persistent challenge due to the inherent volatility and complexity of asset markets. This is addressed in ‘Integrating LSTM Networks with Neural Levy Processes for Financial Forecasting’, which proposes a novel hybrid framework combining the predictive power of Long Short-Term Memory networks with the stochastic modelling capabilities of Lévy-Merton jump-diffusion processes. Results demonstrate that optimizing LSTM hyperparameters with the Grey Wolf Optimizer and calibrating the jump-diffusion model via artificial neural networks significantly improves forecasting accuracy across Brent oil, the STOXX 600, and the IT40 indices. Could this approach represent a new paradigm for robust and reliable financial time-series analysis?

Beyond Predictable Paths: Modeling Market Discontinuities

Conventional financial modeling frequently falters when attempting to mirror the erratic nature of actual asset price fluctuations, largely due to its difficulty in accounting for sudden, unforeseen shifts. These models, often built on the premise of stable, predictable growth, struggle to incorporate the impact of exogenous events – geopolitical shocks, economic announcements, or even shifts in investor sentiment – that can trigger immediate and substantial price changes. The assumption of continuous price diffusion, a cornerstone of many established techniques, simply doesn’t hold true in markets susceptible to ‘jumps’ – discrete, rapid movements that defy gradual progression. Consequently, risk assessments generated by these traditional approaches can be significantly understated, leaving institutions vulnerable to unexpected losses and failing to accurately reflect the inherent volatility of contemporary financial landscapes. This limitation underscores the need for innovative methodologies capable of capturing these discontinuous dynamics and providing a more realistic representation of market behavior.

Conventional financial modeling often assumes asset prices follow a predictable, continuous path, a concept known as diffusion, and that price changes adhere to a normal distribution. However, real-world markets are frequently disrupted by unexpected events – earnings reports, geopolitical shifts, or even social media trends – causing abrupt price “jumps” that deviate significantly from these assumptions. These jumps, representing instantaneous changes rather than gradual shifts, are particularly problematic for risk management because standard models underestimate the potential for large, rapid losses. Failing to account for these discontinuous movements can lead to inaccurate valuations of derivatives, flawed hedging strategies, and an overall underestimation of systemic risk within the financial system. Therefore, incorporating mechanisms to capture these jumps is vital for building more robust and realistic financial models.

Financial modeling is evolving beyond reliance on purely continuous processes to embrace the inherent ‘jumpiness’ of real-world markets. Traditional diffusion models, while mathematically tractable, often fall short when confronted with the abrupt price shifts triggered by unforeseen events – earnings surprises, geopolitical shocks, or even shifts in investor sentiment. Contemporary research increasingly focuses on models that explicitly incorporate both continuous diffusion – representing gradual price movements – and discontinuous ‘jump’ processes, often leveraging Lévy processes or stochastic jump diffusion. These advancements allow for a more realistic depiction of asset price dynamics, acknowledging that markets don’t simply glide along predictable paths but instead experience occasional, significant jolts. By accounting for these jumps, risk managers and traders can develop more robust strategies and more accurately assess potential losses, leading to a more stable and informed financial ecosystem.

Simulations of a compound Poisson process (left) and a Merton jump diffusion process (right) demonstrate stochastic fluctuations in asset value, with the latter incorporating both continuous diffusion and discrete jumps to model sudden price changes.

Calibrating Complexity: Parameter Estimation in Jump Diffusion Models

Parameter estimation within the Merton-Lévy Jump-Diffusion model presents significant computational difficulties. The model’s non-linearity arises from the exponential jump component and the stochastic volatility, requiring iterative numerical methods for optimization. Furthermore, the model’s dimensionality is typically high, often including parameters for diffusion volatility, drift, jump frequency, and jump size distribution, substantially increasing the search space for optimal values. This combination of non-linearity and high dimensionality leads to a non-convex optimization problem prone to local optima, necessitating robust optimization algorithms and potentially global search strategies to achieve reliable and accurate parameter estimates. The likelihood function, crucial for parameter estimation, does not have a closed-form solution and usually requires computationally intensive methods such as maximum likelihood estimation or simulation-based techniques.

Wavelet decomposition preprocesses time series data for jump diffusion calibration by decomposing the signal into multiple frequency components. This decomposition results in a series of wavelet coefficients that can be analyzed and filtered to remove noise and non-stationarities. Specifically, the wavelet transform provides a time-frequency representation, allowing for the isolation and removal of high-frequency noise and the identification of transient events characteristic of jumps. By reconstructing the time series using only the significant wavelet coefficients, a smoother, more stationary representation is obtained. This stationary representation reduces the complexity of the parameter estimation problem, leading to faster convergence and improved accuracy in the calibration process. The efficiency gains are particularly noticeable in high-frequency data where jump events are frequent and noise levels are high.

Bayesian regularization addresses the overfitting common in jump-diffusion model calibration by incorporating prior beliefs about parameter values. This is achieved by treating parameters as random variables with associated probability distributions – typically Gaussian – and modifying the objective function to include a penalty term proportional to the squared distance between the parameter estimates and their prior means. Formally, the penalized likelihood function becomes $L(θ) – \lambda ||θ – μ||^2$, where $θ$ represents the parameter vector, $μ$ is the prior mean vector, and $λ$ is a regularization parameter controlling the strength of the penalty. Increasing $λ$ shrinks parameter estimates towards the prior means, reducing model complexity and variance, and thereby improving generalization performance on unseen data. The regularization parameter $λ$ is often determined through cross-validation or other model selection techniques.

Comparing calibration methods on the Brent dataset, the LSTM-Lévy model demonstrates performance comparable to standalone LSTMs across neural network, Monte Carlo particle analysis, and TorchSDE calibrations of the Merton model.

Validating Performance: Metrics for Assessing Model Accuracy

Model performance assessment utilizes both Mean Squared Error ($MSE$) and Mean Absolute Error ($MAE$) as key metrics. $MSE$ calculates the average of the squared differences between predicted and observed values, providing a measure of the overall magnitude of error, while being sensitive to outliers. $MAE$, conversely, calculates the average of the absolute differences, offering a more robust measure less affected by extreme values. These metrics are computed across the datasets of Brent Oil Prices, the STOXX 600 Index, and the IT40 Index to provide a quantifiable comparison of model accuracy and predictive power.

Model validation utilizes datasets comprised of historical asset prices – specifically, the daily closing prices of Brent Oil, the STOXX 600 Index, and the IT40 Index – to assess predictive accuracy. The discrepancy between model-predicted values and these actual observed prices is quantified to determine model performance. This comparison is performed across the entire duration of the dataset, providing a comprehensive evaluation of the model’s ability to represent the underlying asset’s price behavior. The selection of these indices provides a diverse representation of energy commodities, European equities, and Italian equities, respectively, allowing for a broad assessment of model generalizability.

Performance evaluations indicate the proposed hybrid model consistently outperforms both standard LSTM and LSTM-Fractional Heston models across the analyzed datasets. Specifically, the hybrid model achieves lower Mean Squared Error ($MSE$) values, indicating a reduced average squared difference between predicted and actual asset prices. Simultaneously, it exhibits higher R-squared ($R^2$) values, signifying a greater proportion of the variance in the dependent variable that is predictable from the independent variables. These improvements were observed consistently across the Brent Oil Prices, STOXX 600 Index, and IT40 Index datasets, demonstrating the model’s robust performance across different asset classes.

The LSTM-Fractional-Heston model, calibrated using neural networks, demonstrates consistent performance across the Brent, STOXX 600, and ITALY 40 datasets.

Deep Learning for Enhanced Forecasting: A Synergistic Approach

Long Short-Term Memory (LSTM) networks are a recurrent neural network (RNN) architecture specifically designed to address the vanishing gradient problem inherent in standard RNNs when processing long sequences. This capability makes them well-suited for time series forecasting in financial applications, where patterns and dependencies can span extended periods. Unlike traditional statistical methods which often rely on assumptions of linearity and stationarity, LSTMs can model non-linear relationships and capture complex temporal dependencies within the data. The core of an LSTM network lies in its memory cells, which regulate the flow of information, allowing the network to retain relevant historical data and discard irrelevant details. This selective memory, facilitated by input, forget, and output gates, enables LSTMs to learn long-range dependencies and improve forecasting accuracy for financial instruments like stock prices, exchange rates, and volatility indices. The network learns these dependencies by iteratively processing sequences of data and adjusting its internal parameters through backpropagation.

Combining Long Short-Term Memory (LSTM) networks with optimization algorithms such as the Grey Wolf Optimizer (GWO) addresses the challenge of hyperparameter tuning inherent in deep learning models. LSTM networks possess numerous hyperparameters – including the number of layers, the number of neurons per layer, learning rate, and batch size – which significantly impact forecasting accuracy. GWO, a metaheuristic optimization algorithm inspired by the hunting behavior of grey wolves, iteratively refines these hyperparameters by simulating the social hierarchy and hunting strategies of wolf packs. Specifically, GWO explores the hyperparameter space, evaluating different combinations based on a defined fitness function – typically a measure of forecasting error, such as Root Mean Squared Error (RMSE). Through iterative updates guided by the alpha, beta, and gamma parameters representing leadership, hunting, and prey encounter, respectively, GWO converges toward optimal hyperparameter values, resulting in demonstrably improved model performance and robustness compared to manual tuning or grid search methods.

Artificial Neural Networks (ANNs) present a viable alternative to traditional calibration methods for complex financial models such as the Merton-Lévy Jump-Diffusion Model and the Fractional Heston Model. Conventional calibration typically relies on computationally intensive optimization routines to estimate model parameters by minimizing the difference between model-implied prices and observed market prices. ANNs, after being trained on historical data of option prices and underlying asset characteristics, can rapidly approximate the inverse mapping from option prices to implied volatility surfaces and model parameters. This approach can significantly reduce calibration time, potentially by orders of magnitude, and, in certain scenarios, achieve improved accuracy in parameter estimation compared to traditional methods. The performance of ANNs is dependent on the size and quality of the training dataset, network architecture, and optimization of network weights.

The LSTM-Lévy model, calibrated using neural networks, Markov process approximation, or TorchSDE, outperforms a standalone LSTM on the IT40 dataset.

Towards Adaptive and Robust Financial Modeling: A Convergent Approach

Financial modeling often struggles to encapsulate the erratic nature of asset pricing, yet a recent convergence of techniques offers a promising solution. This approach integrates the strengths of Jump Diffusion calibration – which accounts for abrupt price shifts – with Bayesian regularization, a method that prevents overfitting and enhances model generalization. Crucially, these are coupled with Long Short-Term Memory (LSTM) networks, a deep learning architecture adept at identifying complex patterns within sequential data. By calibrating the Jump Diffusion model to observed market jumps and employing Bayesian regularization to refine its parameters, the LSTM network receives a more robust and informative input. This synergy allows the model to not only predict continuous price trends, but also to anticipate and react to the sudden, impactful events that frequently characterize financial markets, ultimately yielding a more accurate and reliable forecasting framework.

Financial markets are rarely characterized by smooth, predictable progressions; instead, asset prices often exhibit a duality of behavior, displaying both gradual, continuous trends and abrupt, discontinuous jumps. This modeling challenge necessitates a framework capable of representing both phenomena accurately. The presented approach excels in this regard, effectively capturing the underlying stochastic processes that govern asset price movements by integrating the continuous drift and diffusion components of traditional models with the jump-diffusion component that accounts for sudden, unexpected events – like geopolitical shocks or earnings surprises. This allows for a more nuanced representation of market dynamics, moving beyond models that assume constant volatility or ignore the possibility of extreme price fluctuations, ultimately providing a more realistic and reliable basis for forecasting and risk management. The ability to simultaneously model these contrasting behaviors is crucial for navigating the inherent complexities and uncertainties of financial markets.

Recent advancements in financial modeling have yielded a hybrid approach that notably surpasses traditional methodologies in both predictive accuracy and resilience. This model integrates a Lévy-Merton Jump-Diffusion process, calibrated using a neural network to effectively capture the stochastic nature of asset prices and the occurrence of sudden, impactful jumps. Further enhancing performance, a Grey Wolf Optimizer (GWO) refines a Long Short-Term Memory (LSTM) network, enabling it to learn complex temporal dependencies within financial data. The synergistic combination allows for a more nuanced understanding of market behavior, successfully addressing the limitations of models that either overlook jumps or struggle with long-term forecasting. Empirical results demonstrate a significant improvement in capturing market dynamics, suggesting this hybrid architecture represents a substantial step towards more reliable and robust financial predictions.

The pursuit of predictive accuracy in financial modeling, as demonstrated by this integration of LSTM networks and Lévy processes, reveals a principle echoing through complex systems: the effect of the whole is not always evident from the parts. Calibration, traditionally a bottleneck, becomes fluid through neural networks, allowing the model to adapt to the inherent stochasticity of asset prices. Niels Bohr observed, “Every great advance in natural knowledge begins with an intuition that contradicts common sense.” This work challenges the common sense notion that simple models suffice, embracing a more nuanced approach to capture the ‘jumps’ and drifts inherent in financial time series. The model doesn’t control the market; it influences understanding by better reflecting its underlying dynamics.

Where Do We Go From Here?

The integration of LSTM networks with established stochastic processes, as demonstrated, is not simply about achieving incremental gains in forecasting accuracy. It hints at a deeper principle: complex systems rarely respond to dictates, but evolve through the interplay of local rules. The Levy-Merton Jump-Diffusion model provides the skeletal structure, the LSTM the adaptive musculature, and the neural calibration a form of distributed learning. But this isn’t control; it’s influence, a gentle nudging of the system toward more nuanced behavior. The true challenge lies not in perfecting the forecast, but in understanding why these hybrid models perform well, and where their inherent biases lie.

Current calibrations still rely heavily on historical data, implicitly assuming the future will resemble the past – a notoriously fragile assumption. Future research should explore methods for incorporating real-time information, alternative learning paradigms (perhaps reinforcement learning, where the model ‘experiments’ with different strategies), and, crucially, a more rigorous examination of model uncertainty. The system is a living organism where every local connection matters, and treating it as a static puzzle to be solved will always be a limited approach.

Ultimately, the pursuit of perfect financial forecasting may be a category error. The market isn’t a problem to be solved, but a complex adaptive system to be understood. The goal should not be to predict the inevitable, but to build models that are robust enough to navigate an inherently unpredictable world – models that can learn, adapt, and even surprise us with their emergent behavior. Top-down control often suppresses creative adaptation, and the most successful strategies will likely be those that embrace the inherent chaos.

Original article: https://arxiv.org/pdf/2512.07860.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/