# Charting Market Rhythms: The Rolling Hurst Exponent in Python

Ever tried catching the rhythm of the stock market? If you have, you’d know trends aren’t just about rising and falling arrows on a graph. They’re the pulse of market sentiment, the stories of gains and losses. In such environment, navigating the stock market can be likened to sailing turbulent waters, where understanding currents — or in this case, trends — can be a game changer. But how can we transition from a qualitative understanding to a quantitative measure of these trends? Here’s where the Hurst exponent enters the spotlight.

A concept that’s often reserved for scholarly articles and niche financial discussions, the Hurst exponent offers a robust measure of a stock’s propensity to trend or mean-revert. While typically estimated as a point-in-time metric, what if we could gauge the Hurst exponent on a rolling basis?Using Python, we can implement the Rolling Hurst Exponent to analyze, visualize, and eventually leverage market trends in our decision-making process.

## 1. Basic of the Hurst Exponent

The Hurst Exponent, denoted as ** H**, is a statistical measure that gives us a window into the behavior of time series data. Whether it’s the price of a stock, the flow rate of a river, or even internet traffic,

**seeks to determine the nature of its behavior. Put simply, the interpretation of the Hurst Exponent is as follows**

*H*- If
, the process is akin to a random walk, like the flipping of a coin.*H*=0.5 - If
, it suggests a series that tends to mean-revert. In the context of stocks, this might hint at an oscillatory behavior where if the stock price rises, it’s likely to fall back to its mean and vice-versa.*H*<0.5 - If
, it indicates a trending series. This would mean that if a stock price rises, it’s likely to continue on that upward trajectory.*H*>0.5

Equation. 1: Hurst Exponent Formula

Where:

*R*(*t*) is the range of cumulative sum deviations. The broad idea behind*R*(*t*) is to determine how volatile a time series is*S*(*t*) is the standard deviation of the series. It represents the standard deviation of the series, capturing its average dispersion from the mean.

#### The Rolling Hurst Exponent:

Traditionally, the Hurst exponent offers a snapshot, a point-in-time insight. But markets are dynamic, evolving with news, policies, and global events. Thus, a rolling computation, where the Hurst exponent is continually recalculated over a moving window, offers fresher, more timely insights. It’s akin to watching a movie frame-by-frame, where each frame offers a unique perspective.

#### The Link with Fractal Dimension:

The fractal dimension is another fascinating concept, rooted in the idea of measuring the ‘roughness’ or complexity of a series. It’s intrinsically tied to the Hurst exponent and can be determined as:

Equation. 2: Fractal Dimenension Formula.

Where D is the fractal dimension. A higher D indicates a more complex, irregular series, whereas a value close to 1 suggests a smoother series.

Figure. 1: Contrasting Temporal Patterns: The left plot showcases a relatively smooth time series characterized by a lower fractal dimension, hinting at less complexity. Conversely, the right plot depicts a volatile series with higher complexity, evident from its erratic behavior and indicative of a higher fractal dimension.

A Fractal dimension close to 1 indicates the time series is relatively smooth and simple, suggesting that it can be easily modeled using conventional time series methods. Fractal dimension close to 2 on the contrary indicates that the time series is very rough and complex, suggesting that it has a higher degree of irregularity and may be challenging to model using conventional methods. Just note that the interpretation might vary depending on the context and the financial instrument being analyzed.

## 2. Rolling Hurst Exponent Python Implementation

Given the inherent dynamism of the stock market, assessing the Hurst Exponent at a single point in time might not provide a comprehensive picture. Instead, evaluating it on a rolling basis offers a frame-by-frame analysis of how a stock’s behavior evolves over time. The same goes for the fractal dimension.

The computation of both the Hurst exponent and the fractal dimension in a rolling fashion demands a blend of time series manipulation and statistical techniques. Thankfully, Python’s vast libraries make this task relatively straightforward.

**Essential Libraries & Data Acquisition**

We commence by importing the necessary libraries:

`numpy`

and`pandas`

: Fundamental for numerical and data frame operations.`yfinance`

: Enables us to fetch stock data directly from Yahoo Finance.`matplotlib`

: Our go-to tool for plotting and visualizing data.

The stock data is fetched for the ticker symbol “ASML.AS” within aspecific date range.

` ````
```import numpy as np
import pandas as pd
import yfinance as yf
import matplotlib.pyplot as plt
symbol = "ASML.AS"
start_date = "2018-01-01"
end_date = "2023-12-31"
df = yf.download(symbol, start=start_date, end=end_date)

#### Hurst and Fractal Dimension Calculations

The heart of our script is the `hurst_fd`

function, which computes both the Hurst Exponent and Fractal Dimension using the log returns of a given price series. The function uses a resampling approach for more robust estimates.

We also have `rolling_hurst`

and `rolling_fractal_dimension`

functions that employ `hurst_fd`

to calculate these metrics on a rolling basis across our dataset.

` ````
```def hurst_fd(price_series, min_window=10, max_window=100, num_windows=20, num_samples=100):
log_returns = np.diff(np.log(price_series))
window_sizes = np.linspace(min_window, max_window, num_windows, dtype=int)
R_S = []
for w in window_sizes:
R, S = [], []
for _ in range(num_samples):
start = np.random.randint(0, len(log_returns) - w)
seq = log_returns[start:start + w]
R.append(np.max(seq) - np.min(seq))
S.append(np.std(seq))
R_S.append(np.mean(R) / np.mean(S))
log_window_sizes = np.log(window_sizes)
log_R_S = np.log(R_S)
coeffs = np.polyfit(log_window_sizes, log_R_S, 1)
hurst_exponent = coeffs[0]
fractal_dimension = 2 - hurst_exponent
return hurst_exponent, fractal_dimension
def rolling_hurst(price_series, window, min_window=10, max_window=100, num_windows=20, num_samples=100):
return price_series.rolling(window=window).apply(lambda x: hurst_fd(x, min_window, max_window, num_windows, num_samples)[0], raw=True)
def rolling_fractal_dimension(price_series, window, min_window=10, max_window=100, num_windows=20, num_samples=100):
return price_series.rolling(window=window).apply(lambda x: hurst_fd(x, min_window, max_window, num_windows, num_samples)[1], raw=True)

#### Visualization & Insights

With our metrics in hand, the next step is visualization. We aim to render three plots:

- Stock Price & SMA: The stock price plotted alongside its Simple Moving Average and mean price.
- Rolling Hurst Exponent: Offers a dynamic view of the Hurst exponent over time.
- Rolling Fractal Dimension: Reveals the changing fractal nature of the stock price.

This series of plots paints a holistic picture, illustrating the relationship between price movements, market trends, and fractal characteristics.

Putting it all together, we get:

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