Crystal ball reflecting stock charts with entropy waves improving financial predictions.

Beyond the Bell Curve: How Entropy Can Fix Financial Forecasting

Kai Mendoza in Business & Economy December 2025 • 4 min read.

"Geometric Brownian Motion (GBM) is a popular tool in finance, but its reliance on log-normal distributions can limit its accuracy. Discover how entropy corrections can enhance GBM and improve predictions."

In the world of finance, predicting the future is the name of the game. The Geometric Brownian Motion (GBM) has long been a go-to model for capturing the stochastic nature of fluctuating systems, particularly in forecasting diffusion processes, population dynamics, and, most notably, stock prices. GBM operates under the assumption that the logarithm of its solutions results in a normal distribution, making it analytically convenient.

However, real-world data often throws a curveball, exhibiting non-zero skewness, excess kurtosis, and fluctuating volatility, all of which deviate significantly from the idealized bell curve. These deviations can have major implications for the accuracy of traditional GBM, challenging its ability to portray extreme events or interpret the underlying dynamics of the market accurately.

Recognizing these limitations, researchers are exploring solutions that go beyond the constraints of normality. While alternative models like stochastic volatility and jump-diffusion processes have been proposed, they often call for the modification or replacement of GBM. Instead, a new study proposes a non-violent approach: employing entropy constraints to improve the predictive capabilities of GBM without sacrificing its beneficial features.

Entropy: The Key to Unlocking Better Predictions

Crystal ball reflecting stock charts with entropy waves improving financial predictions.

The core idea behind this approach stems from Shannon's information theory, where entropy plays a pivotal role as the minimum number of logical states needed to communicate a message. Essentially, a well-ordered message has lower entropy than a random one. Entropy, therefore, provides a means to judge how well GBM predicts future events by measuring the level of uncertainty around forecast data relative to the original time series. This concept is further reinforced by the ability of entropy measures to capture extreme events.

To illustrate this, consider a dice roll. In a conventional dice roll, the probability of getting any side number is equally probable, resulting in a uniform distribution. According to Shannon, the information about such a process can be quantified by entropy. However, in a biased dice roll, where the probability of getting a particular number is higher, the entropy decreases, indicating a more deterministic scenario. The more deterministic the probability distribution, the lower the entropy.

Conventional Dice Roll: Equal probability for each side, resulting in maximum entropy.
Biased Dice Roll: Unequal probabilities, leading to reduced entropy and a more deterministic outcome.
Time Series Application: Normality assumptions in GBM may be rough approximations, and entropy constraints can refine predictions.

Extending this principle to evolving time series reveals that normality assumptions may be rough approximations, particularly within a short to mid-term timeframe. Consequently, employing GBM for this purpose may prove less efficacious. The observed correlation between an augmented determinism level and decreased system entropy reinforces this viewpoint.

The Future of Financial Modeling

The study highlights the potential of entropy-corrected GBM (EC-GBM) to improve financial forecasting by addressing the limitations of traditional GBM. By incorporating entropy constraints, EC-GBM can better capture the complexities of real-world data, leading to more accurate predictions and better-informed investment decisions. This approach marks a significant step forward in refining financial modeling techniques and enhancing our understanding of market dynamics.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2403.06253,

Title: Entropy Corrected Geometric Brownian Motion

Subject: physics.data-an q-fin.st

Authors: Rishabh Gupta, Ewa A. Drzazga-Szczȩśniak, Sabre Kais, Dominik Szczȩśniak

Published: 10-03-2024

Everything You Need To Know

What is Geometric Brownian Motion (GBM) and why is it used in finance?

Geometric Brownian Motion (GBM) is a commonly used model in finance to simulate the random behavior of fluctuating systems. It is particularly useful for forecasting diffusion processes, population dynamics, and stock prices. GBM is based on the assumption that the logarithm of its solutions follows a normal distribution, which simplifies calculations and makes it analytically convenient. However, this assumption can be a limiting factor when dealing with real-world data.

What are the limitations of using Geometric Brownian Motion (GBM) for financial forecasting?

Traditional Geometric Brownian Motion (GBM) relies on the assumption of log-normal distributions. Real-world data often deviates from this ideal, exhibiting characteristics like non-zero skewness, excess kurtosis, and fluctuating volatility. These deviations challenge the accuracy of traditional GBM, particularly when portraying extreme events or interpreting the underlying dynamics of the market. Alternative models like stochastic volatility and jump-diffusion processes exist but may require significant modification or replacement of GBM.

How does incorporating entropy into the Geometric Brownian Motion (GBM) model improve financial forecasting?

Incorporating entropy, particularly through an entropy-corrected Geometric Brownian Motion (EC-GBM), helps to address the limitations of traditional GBM. Entropy, derived from Shannon's information theory, measures the level of uncertainty in forecast data relative to the original time series. By applying entropy constraints, EC-GBM can better capture the complexities of real-world data, leading to more accurate predictions without sacrificing the beneficial features of the traditional GBM.

Can you explain how the concept of entropy, using the dice roll example, applies to improving predictions in time series analysis with Geometric Brownian Motion (GBM)?

In a conventional dice roll, each side has an equal probability of being rolled, resulting in maximum entropy, signifying high uncertainty. A biased dice roll, where certain numbers are more likely, exhibits lower entropy, indicating a more deterministic outcome. Applying this to time series, the normality assumptions in Geometric Brownian Motion (GBM) may be rough approximations. Employing entropy constraints can refine predictions by capturing the level of determinism present in the time series. A higher level of determinism correlates with decreased system entropy, improving forecast accuracy.

What are the potential implications of using entropy-corrected Geometric Brownian Motion (EC-GBM) for investment decisions?

The entropy-corrected Geometric Brownian Motion (EC-GBM) has the potential to significantly improve investment decisions by providing more accurate financial forecasts. By addressing the limitations of traditional GBM and better capturing the complexities of real-world data, EC-GBM leads to more reliable predictions. This enhanced accuracy enables better-informed investment strategies and a more profound understanding of market dynamics, potentially leading to improved investment outcomes. Moreover, EC-GBM offers a non-violent approach, enhancing rather than replacing existing GBM frameworks.