Quantifying & Plotting the “Speed” of Bitcoin Price Over the Last 12 Years

1) Choose the Right “Position” Variable Use log price

Let price be \(P(t)\). Define the position-like variable as:

• Returns become additive in time
• Changes are scale-invariant (a move from \$1k→\$2k compares to \$30k→\$60k)
• Plays nicely with diffusion-style market models

2) Define Velocity (Directional “Speed”) log-return

The discrete daily velocity is the daily log return:

A less noisy operational definition uses a rolling window \(\Delta t\):

3) Define Speed (Magnitude Only) ignore direction

If you want true “speed” (how fast regardless of up/down), take the magnitude:

• Spikes during blow-off tops and capitulation drops
• Useful for detecting “eventfulness” independent of trend direction

4) Define Acceleration (Regime Change Detector) 2nd difference

Acceleration is the change in velocity. Daily version:

Equivalent second-difference form:

• Highlights cycle tops, panic bottoms, and turning points
• Pairs nicely with your “phase transition / halving” narrative

5) What to Plot (Practical Dashboard) 4 core plots

6) “Market Kinetic Energy” (Optional, Physics-Consistent) energy proxy

If you want a single scalar “how intense is the motion?” measure:

This pairs naturally with a volatility-as-temperature view (\(\sigma^2\)) and helps build a coherent analogy:
position → velocity → kinetic energy → temperature.

7) Python: Compute & Plot (12 Years) copy/paste

This script downloads BTC-USD, computes log-price, velocity, speed, acceleration, and plots them:

import yfinance as yf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Download ~12+ years of daily BTC data
btc = yf.download("BTC-USD", start="2013-01-01", auto_adjust=True)

# Use Close (auto_adjust=True returns adjusted series where applicable)
btc["log_price"] = np.log(btc["Close"])

# Daily velocity (log return)
btc["velocity_daily"] = btc["log_price"].diff()

# Rolling velocities (trend speed over horizons)
btc["velocity_30d"]  = btc["log_price"].diff(30)  / 30
btc["velocity_90d"]  = btc["log_price"].diff(90)  / 90
btc["velocity_365d"] = btc["log_price"].diff(365) / 365

# Speed (magnitude)
btc["speed_365d"] = btc["velocity_365d"].abs()

# Acceleration (change in daily velocity)
btc["acceleration"] = btc["velocity_daily"].diff()

# --- Plot (no seaborn, no forced colors) ---
plt.figure(figsize=(14, 10))

plt.subplot(4, 1, 1)
plt.plot(btc.index, btc["log_price"])
plt.title("BTC: Log Price  ln(P)")

plt.subplot(4, 1, 2)
plt.plot(btc.index, btc["velocity_365d"])
plt.title("BTC: 365-Day Velocity  v^(365) = (ln P_t - ln P_{t-365}) / 365")

plt.subplot(4, 1, 3)
plt.plot(btc.index, btc["speed_365d"])
plt.title("BTC: 365-Day Speed  |v^(365)|")

plt.subplot(4, 1, 4)
plt.plot(btc.index, btc["acceleration"])
plt.title("BTC: Acceleration (Δ daily velocity)")

plt.tight_layout()
plt.show()

8) Interpretation: What You’ll Typically See cycle structure

• Velocity tracks regime drift (bull vs bear trend strength).
• Speed spikes at manic tops and panic bottoms (event intensity).
• Acceleration is a sharp turning-point detector (phase-transition-ish moments).
• Peak speeds often decline over time as liquidity deepens (a “maturing market” signature).