G. Santostasi Bitcoin Power Law Monte Carlo Indicator

Overview:
The "G. Santostasi Bitcoin Power Law Monte Carlo" is a sophisticated TradingView indicator inspired by the Bitcoin Power Law Theory developed by physicist Giovanni Santostasi.

This theory posits that Bitcoin's price follows a power-law relationship with time, measured in days since the Bitcoin Genesis Block (January 3, 2009). The indicator leverages this framework to analyze Bitcoin's price dynamics through a normalized metric called "Daily Slopes," which captures local deviations from the long-term power-law trend. By fitting these Daily Slopes to a t-location scale distribution on a moving window, the indicator computes key parameters (mu, sigma, and nu) and plots them along with deviation bands. This allows traders to identify local minima and maxima in price action relative to the global power-law slope of approximately 5.9.Additionally, the indicator incorporates Monte Carlo simulations to project potential future price paths up to 100 days ahead, generating up to 500 randomized trajectories based on the statistical properties of the Daily Slopes. This tool is particularly useful for understanding Bitcoin's inherent diminishing returns, assessing market stability, and forecasting short-term scenarios while emphasizing the asset's long-term predictability as a self-organizing network akin to natural systems.

The indicator does not predict exponential growth but instead highlights Bitcoin's scale-invariant behavior, where returns diminish predictably over time—a feature, not a bug, of its design. It has been observed that the core metric (mu) remains stable across Bitcoin's entire history, reinforcing the power law as Bitcoin's "DNA."

Core Concept: Daily Slopes:

At the heart of the indicator is the "Daily Slopes" metric, which normalizes daily logarithmic returns to account for the diminishing nature predicted by the power-law model. This normalization reveals a stable "local slope" (n) that oscillates around a fixed global value, providing insight into Bitcoin's consistent behavior over time.

Definition and Calculation:
Daily logarithmic returns are calculated as log⁡(P2/P1)\log(P_2 / P_1)\log(P_2 / P_1), where P2P_2P_2 is the current day's closing price and P1P_1P_1 is the previous day's closing price.
According to the power-law model, if Bitcoin's price ( P(t) ) follows P(t)=c⋅tnP(t) = c \cdot t^nP(t) = c \cdot t^n

(where ( t ) is days since the Genesis Block, ( c ) is a constant, and n≈5.9n \approx 5.9n \approx 5.9
is the global slope from log-log regression), then the expected daily log return is n⋅log⁡((t+1)/t)n \cdot \log((t+1)/t)n \cdot \log((t+1)/t)
.
The Daily Slope is thus the normalized value:
Daily Slope=log⁡(P2/P1)log⁡((t+1)/t)\text{Daily Slope} = \frac{\log(P_2 / P_1)}{\log((t+1)/t)}\text{Daily Slope} = \frac{\log(P_2 / P_1)}{\log((t+1)/t)}

This normalization "stabilizes" the returns by dividing out the theoretical decay factor log⁡((t+1)/t)\log((t+1)/t)\log((t+1)/t)
, which diminishes as ( t ) increases (reflecting slower growth in mature systems).
Result: The Daily Slope represents a "local n" that should remain stable, oscillating around the global slope of ~5.9 without long-term drift. Empirical data shows this stability holds over Bitcoin's 16-year history, with oscillations but no systematic change—indicating Bitcoin has statistically "done the same thing" since inception.

Interpretation:

Positive deviations (Daily Slope > 5.9) signal bullish momentum or potential local maxima.
Negative deviations (Daily Slope < 5.9) indicate bearish pressure or local minima.
The metric adjusts for absolute volatility, which appears to decrease over time due to diminishing returns. However, when normalized via Daily Slopes, relative volatility has been constant for the last 8 years, underscoring Bitcoin's resilience to macroeconomic factors.

Distribution Fitting and Parameter Estimation:

To quantify the behavior of Daily Slopes, the indicator fits them to a t-location scale distribution (Student's t-distribution with location and scale parameters) over a user-configurable moving window (e.g., 365 days for annual analysis).

This distribution is chosen as the best empirical fit for the heavy-tailed, outlier-prone nature of Bitcoin's normalized returns, outperforming alternatives like Gaussian or Laplacian.t-Location Scale Distribution:

The distribution is parameterized by:μ (mu): Location parameter, representing the mean or "average slope." This is the most critical metric, stable around 5.9 across Bitcoin's history. It tracks the central tendency of Daily Slopes and signals overall market regime (e.g., rising mu indicates strengthening momentum).
σ (sigma): Scale parameter, akin to standard deviation, measuring the spread or volatility of slopes. It has shown slight increases in certain contexts (e.g., hash rate applications) but remains stable for price data.
ν (nu): Degrees of freedom, controlling the "tailedness" (lower ν means heavier tails, capturing extreme events like bubbles or crashes).

Fitting is performed on a rolling basis, updating μ, σ, and ν dynamically.

Plotting:

Local μ: Plotted as a central line, showing the moving average slope.
Deviation Bands: μ + σ (upper band) and μ - σ (lower band), highlighting 1-standard-deviation ranges.
These bands help identify overbought/oversold conditions by measuring deviations from the global mean of 5.9.

For example:

Crossing above μ + σ may signal a local maximum (potential sell opportunity).
Dipping below μ - σ could indicate a local minimum (buy signal).

Additional visualizations include raw Daily Slopes (oscillating series) and smoothed averages for clarity.

Stability and Insights:μ has remained remarkably stable over 16 years, oscillating without drift, validating the power law's predictive power.
Parameters may show minor trends in rolling windows (e.g., slight σ increases), but no monotonic drift is observed in price data. This stability extends to related metrics like addresses and hash rate, where Daily Slopes can be derived similarly (e.g., via log(A2/A1) / log((t+1)/t) for addresses, yielding equivalent slopes around 5.9).

Monte Carlo Simulations for Future Projections

The indicator enables short-term forecasting (up to 100 days) by reversing the normalization process and simulating paths using the fitted distribution.

Projection Mechanism:

Recover expected daily returns: Multiply the sampled Daily Slope (drawn from the t-location scale distribution with current μ, σ, ν) by log⁡((t+1)/t)\log((t+1)/t)\log((t+1)/t)
.
Generate randomized samples to create up to 500 Monte Carlo paths, incorporating the distribution's properties to model uncertainty (e.g., heavy tails for rare events).
Simulations can use the full historical dataset for broader spreads or recent windows (e.g., last 8 years) for tighter, regime-specific forecasts.
Output: Fan chart of projected prices, showing median path (based on μ), confidence intervals (e.g., ±σ bands), and extreme scenarios.

Applications and Limitations:
Useful for risk assessment, e.g., probability of reaching $200K in 2025 is low (1-2% per recent simulations).
Assumes parameters evolve minimally; if drift is detected, simulations can adjust dynamically.
Not for long-term predictions (beyond 100 days), as the power law excels in multi-year trends rather than short-term noise.
Empirical validation: Simulations align with historical backtests, where deviations (bubbles/crashes) revert to the power-law trend.

Usage Notes Inputs:
Customize moving window size, number of Monte Carlo paths (default: 500), projection horizon (up to 100 days), and global slope (default: 5.9).
Visuals: Overlay on BTCUSD log-log chart for context; bands and simulations appear in separate panels.
Caveats: This is not financial advice. The power law describes emergent behavior from network effects, not guarantees. Cycles and bubbles are secondary deviations, not core to the model.
Extensions: The concept applies beyond price (e.g., to addresses or hash rate), revealing interconnected power laws in Bitcoin's ecosystem.

This indicator transforms Santostasi's theoretical insights into a practical tool, empowering users to navigate Bitcoin's dynamics with statistical rigor.