Speculative Bubbles in Cryptocurrencies

Capstone project for the Applied Mathematics program at Columbia University.

This project investigates the presence of speculative bubbles in cryptocurrency markets using advanced mathematical models and statistical methods. Drawing from the local martingale theory and recent academic research, we implemented quantitative techniques to test whether asset prices—such as Bitcoin—deviate from their fundamental values.

Project Objectives

• Understand the mechanics and theory behind speculative financial bubbles
• Apply the Choi & Jarrow (2020) bubble detection method to crypto assets
• Analyze cryptocurrency time series using stochastic processes and statistical testing
• Explore the role of volatility and arbitrage in pricing anomalies

Key Concepts

• Local Martingale Theory
• Risk-Neutral Measures
• Stochastic Differential Equations (SDEs)
• Convex Hull Algorithms
• Volatility Estimation (Florens-Zmirou Estimator)
• OLS Hypothesis Testing

Tools & Technologies

• Python (NumPy, pandas, matplotlib, statsmodels)
• Jupyter Notebooks
• Financial time series data
• Data visualization and statistical inference

Outcome

Using real-world cryptocurrency price data, we demonstrated how bubble detection methods can be applied to volatile, speculative assets. Our findings add to the ongoing conversation around financial stability, crypto regulation, and the risks of ungrounded market exuberance.

References

• Choi, S. H., & Jarrow, R. A. (2020). Testing the Local Martingale Theory of Bubbles using Cryptocurrencies. SSRN. https://doi.org/10.2139/ssrn.3701960
• Delbaen, F., & Shirakawa, H. (2002). No Arbitrage Condition for Positive Diffusion Price Processes. Asia-Pacific Financial Markets.

Authors

• David Kim
• Jessica Li
• Cara Chen
• Tatsuya Hiki