The mathematics of emotions: models and dynamics in Markets

Emotions as Economic Variables

Traditionally, economic models are based on assumptions of rationality, but empirical evidence demonstrates that emotions such as fear and greed profoundly influence market dynamics. Mathematics offers powerful tools to represent these phenomena, allowing emotional variables to be integrated into traditional models. This integration enables the explanation and prediction of seemingly irrational behaviors, creating a scientific basis for more informed economic decisions.

Nonlinear Models and Volatility: The "Fear Index"

A practical example of emotional impact is the Volatility Index (VIX), often called the "fear index." During periods of instability, the VIX follows exponential growth, indicative of collective fear. Mathematically, the behavior of the VIX can be described by a differential equation:

dV/dt = αV − βV²

where:

• V represents the level of volatility.

• α describes the market's sensitivity to news.

• β represents the return to normalcy once fear dissipates.

This equation illustrates how fear can rapidly amplify volatility, generating intense and irregular market movements. Integrating historical data and analyzing the frequency of exceptional events further allows the anticipation of volatility spikes.

Prospect Theory: The Mathematics of Irrational Decisions

Daniel Kahneman and Amos Tversky introduced Prospect Theory, which explains how individuals evaluate gains and losses not in absolute terms but based on emotional perception. The utility function of the theory is:

U(x) = x^α if x ≥ 0, −λ(−x)^β if x < 0

where:

• α,β<1 represent the diminishing marginal effect of gains and losses.

• λ>1 measures the intensity of loss aversion.

This function describes how the fear of losses (more intense than the desire for gains) leads to irrational decisions, such as impulsive selling during market crashes. Incorporating these parameters into predictive models can simulate realistic scenarios and design risk mitigation strategies.

Speculative Bubbles and Mathematical Models

Collective emotions such as euphoria and panic underpin speculative bubbles, which can be modeled mathematically. Sornette's model describes the evolution of a bubble until its collapse:

P(t) = P₀ + A(tc − t)^β + Bcos[ωln(tc − t)]

where:

• P(t) is the observed price.

• tc is the critical time when the market collapses.

• β captures the accelerated price growth driven by euphoria.

• The oscillatory component represents fluctuations caused by speculation and uncertainty.

This model highlights how emotions can amplify prices beyond economic fundamentals, inevitably leading to a collapse. Analyzing the model’s dynamics can help identify early signs of bubbles, supporting preventive policy interventions.

Sentiment Analysis and Machine Learning

Advanced tools such as sentiment analysis and machine learning are revolutionizing emotional analysis in markets. A linear regression model integrating emotional variables is:

R = α⋅S + β⋅V + γ⋅T + ϵ

where:

• R is the expected return.

• S represents emotional sentiment calculated from textual data (social media, news).

• V is the current volatility.

• T represents historical trends.

Optimizing the parameters (α,β,γ) quantifies the impact of emotions on market dynamics, improving forecasts and strategies. Neural network-based approaches further capture nonlinear relationships and complex emotional patterns.

Conclusion

The mathematics of emotions provides a powerful lens for understanding irrational behaviors in markets. Models such as Prospect Theory and Sornette's model demonstrate that fear, greed, and euphoria are not merely psychological manifestations but measurable forces shaping economic dynamics.

Integrating these models with advanced tools like machine learning can help build more resilient and predictive economic systems capable of addressing the emotional complexities inherent in human decision-making.