**Understanding Bitcoin’s Predictability: A Monte Carlo Simulation of Power Law Dynamics**

Bitcoin, the world’s first cryptocurrency, has fascinated both financial experts and enthusiasts for years due to its unpredictable price movements and explosive growth. However, beneath this apparent chaos, there exists a mathematical model that can explain and predict Bitcoin’s price behavior with surprising accuracy. This model is based on a power law relationship, which we’ve found to be a key factor in Bitcoin’s long-term growth. By incorporating this power law into a Monte Carlo simulation, we can generate a model that not only explains past price movements but also predicts future price action with a high degree of confidence.

**The Power Law Governing Bitcoin**

At the heart of this model is the observation that Bitcoin follows a power law in time. This means that the price of Bitcoin has been growing in a manner that can be mathematically described as a function of time since the Genesis Block — the very first block mined by Bitcoin’s creator, Satoshi Nakamoto, in January 2009.

The power law relationship we discovered can be expressed through the returns formula:

𝑅(𝑡) = [(𝑡 + 1) / 𝑡]⁵·⁸²

Here, 𝑡 represents any given day, expressed as the number of days from the Genesis Block. The value 5.82 is the slope of the power law. This formula tells us that the returns on Bitcoin diminish over time but do so in a predictable way that follows the power law.

**Monte Carlo Simulation: A Powerful Tool for Prediction**

To further explore and validate this power law model, we employed a Monte Carlo simulation. Monte Carlo (the name is inspired by the famous Casino in that small country) methods are a class of computational algorithms that rely on repeated random sampling to obtain numerical results. In finance, these simulations are used to model the probability of different outcomes in processes that are inherently uncertain, such as the price of assets like Bitcoin.

**How Monte Carlo Simulation Works:**

**Modeling Randomness**: We know that Bitcoin’s price is influenced by a range of factors, including market sentiment, regulatory news, and technological developments. To capture this randomness, we use the observed distribution of Bitcoin’s returns, which follows a Laplacian distribution — a distribution characterized by sharp peaks and heavy tails, often used to model data with outliers or extreme deviations. This distribution does a very good work of reproducing the shape of the returns of Bitcoin both during the bubbles and the general power law behavior.

The observed (blue bars) and simulated distributions of returns. We used a Laplace distribution with parameters (average and b parameter) similar to the one observed in Bitcoin data.

**Simulating Multiple Histories**: Using the power law returns formula and the Laplacian distribution, we run thousands of simulations, each one generating a possible price history for Bitcoin. Each simulation incorporates the daily return as determined by the power law, plus a random component based on the Laplacian distribution to account for day-to-day price fluctuations. This process was done piecewise given each bubble has a particular growth rate. The decay is very uniform and similar to each other. The b parameter (related to the standard deviation) is similar both for the bubbles and the power law (a little higher for the bubbles).

Bitcoin price history is in blue, the support power law is in purple, and the general middle power-law trend is in red. The grey lines are multiple independent simulated histories (the fuzziness is due to the superposition of many histories) with similar statistical properties observed in Bitcoin. The different paths show the inherent variability of Bitcoin. The last part of the data is a project to 4 years in the future.

**Capturing the Bubbles**: Bitcoin’s history is marked by a series of bubbles — periods where the price rises rapidly in an exponential fashion, only to crash and then recover. Interestingly, our model reveals that these bubbles are themselves governed by exponential growth and decay rates. During a bubble, Bitcoin’s price might grow by 1.6% per day in the early years, but more recent bubbles have seen growth rates around 0.46% per day. The subsequent crashes exhibit a slower, but consistent, decay of about 0.25% per day.

Close-up of each bubble simulation (plus the projected path for the next cycle). The red lines represent averages of all the histories.

The bubbles basically cancel out when the price goes back to the original power law trend.

**The Surprising Role of Bubbles**

One of the most intriguing findings from our analysis is the role that bubbles play in Bitcoin’s long-term growth. While bubbles may seem disruptive, they tend to cancel each other out over time. After a bubble bursts, Bitcoin’s price typically returns to the trajectory dictated by the original power law growth model. This suggests that the dramatic up-and-down movements are not anomalies but rather integral parts of Bitcoin’s journey along a sustainable growth path.

The bubbles basically cancel out when the price goes back to the support line, in blue are the inliers (data that is close to the support line) and outliers (data that is far away from the support line).

**The Power of Prediction: Simulating Future Price Action**

With the model validated by its ability to reproduce Bitcoin’s historical price action, we can use it to make predictions about future price movements. The next step is to estimate the peak of Bitcoin’s next cycle, which we predict will occur near the end of the next four-year cycle.

Our analysis shows that the peaks of Bitcoin’s price cycles have followed a predictable exponential decay with an incredibly high correlation (R² = 0.98). This means we can reasonably predict not only when the next peak will occur but also the likely magnitude of that peak. This peak is estimated to be around $200,000 and it is supposed to occur in November 2025. To achieve this large deviation from the trend we will need to see exponential growth with about 0.33 % returns per day.

Armed with this information, we conducted a simulation to generate a complete price model that includes both the historical price action and a forecast for the next four years. This model suggests that while Bitcoin will continue to experience volatility, it will do so within the bounds set by the power law, with each new cycle contributing to the long-term growth trajectory.

**Conclusion: A Model Rooted in Empirical Evidence**

In conclusion, the power law model, combined with a Monte Carlo simulation, provides a robust framework for understanding Bitcoin’s price dynamics. The fact that this model can reproduce the detailed characteristics of Bitcoin’s price history, including the bubbles, gives us confidence in its predictive power. By following the proportional growth pattern dictated by the power law, Bitcoin has managed to grow sustainably and resiliently over the years.

As we look to the future, the model suggests that Bitcoin will continue to follow this path, experiencing both growth and corrections in a manner consistent with its historical behavior. This makes the model not only a tool for understanding Bitcoin’s past but also a valuable resource for predicting its future, helping investors and analysts make informed decisions.

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