Trading Strategy

Guide for Analyzing and Modeling Time-Price Series Datasets

This guide will walk you through the process of analyzing your 3-year, 5-minute historical price series data and help you understand which price generation models are most suitable for modeling its behavior.

The application use are on this website, we analyze 3-year, 5-minute historical price series data.

1. Initial Data Preparation: From Prices to Log Returns

The first crucial step in analyzing financial time series is to transform raw prices into log returns. This is because log returns have more desirable statistical properties for modeling than raw prices, such as approximate normality and stationarity.

The log return at time t is calculated as:
\[r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)\]

where Pt​ is the price at time t and Pt−1​ is the price at the previous time step.

Why Log Returns?

  • Additivity: Log returns are additive over time, meaning the log return over multiple periods is simply the sum of the log returns for each sub-period. This simplifies multi-period analysis.
  • Symmetry: They handle percentage changes symmetrically (e.g., a 10% increase followed by a 10% decrease results in the original price).
  • Approximate Normality: While raw prices are often non-normally distributed, log returns tend to be closer to a normal distribution, which is an assumption for many statistical models.

Using the Script: The provided JavaScript code automatically calculates log returns from your uploaded price data. This series of log returns will be the foundation for all subsequent statistical analysis.

2. Uncovering Series Characteristics: Descriptive Statistics

Once you have the log returns, calculating basic descriptive statistics gives you a fundamental understanding of your data’s distribution.

  • Mean: The average log return. For short timeframes like 5-minute data, this is often very close to zero. A non-zero mean could indicate a slight drift.
  • Standard Deviation (Volatility): This is the overall volatility of your 5-minute returns over the entire 3-year period. It gives you a baseline measure of the asset’s risk.
  • Skewness: Measures the asymmetry of the return distribution.
    • Positive Skewness: More frequent small losses and a few large gains (tail on the right).
    • Negative Skewness: More frequent small gains and a few large losses (tail on the left). Financial returns often exhibit negative skewness.
  • Kurtosis: Measures the “tailedness” of the distribution.
    • Kurtosis > 3 (Excess Kurtosis > 0): Indicates “fat tails” or “leptokurtic” distribution. This means there’s a higher probability of extreme price movements (both positive and negative) than a normal distribution would suggest. This is a common characteristic of financial data.
    • Kurtosis = 3 (Excess Kurtosis = 0): Normal distribution.
    • Kurtosis < 3 (Excess Kurtosis < 0): “Thin tails” or “platykurtic” distribution.

Using the Script: The script outputs these descriptive statistics for your log returns. Pay close attention to the Kurtosis value; if it’s significantly above 3, it strongly suggests your data has fat tails, which is a critical characteristic for model selection.

3. Autocorrelation Analysis: Stationarity and Predictability

Autocorrelation measures how much a time series is correlated with its past values. This is crucial for understanding stationarity and potential predictability.

  • Autocorrelation Function (ACF) of Log Returns:
    • Purpose: To check for linear dependencies in the returns themselves.
    • Interpretation: If your log returns behave like a true random walk (Efficient Market Hypothesis), then the ACF values for all lags should be very close to zero and fall within the confidence intervals (which are not explicitly drawn by the script but can be inferred if values are consistently small). If you see significant (non-zero) autocorrelation at certain lags, it suggests some level of predictability in the series, indicating that an ARIMA model might be suitable for capturing these linear dependencies.
    • Stationarity: For most financial modeling, you want your log returns series to be stationary (mean, variance, and autocorrelation structure do not change over time). Log returns are generally stationary. If the ACF decays quickly to zero, it supports stationarity.
  • Autocorrelation Function (ACF) of Squared Log Returns:
    • Purpose: To check for conditional heteroskedasticity or volatility clustering.
    • Interpretation: If the ACF of the squared log returns shows significant values (i.e., non-zero correlations for several lags), it strongly indicates that periods of high volatility are followed by high volatility, and periods of low volatility are followed by low volatility. This is a hallmark of financial time series and a key signal that GARCH models are highly appropriate for your data.

Using the Script: The script calculates and displays the ACF for both log returns and squared log returns for a specified number of lags. Examine these values carefully. Significant values in the squared returns’ ACF are particularly important for high-frequency data.

4. Volatility Dynamics: Monthly Volatility Analysis

While the overall standard deviation gives you a single volatility figure, it’s often more insightful to see how volatility changes over different time periods.

  • Monthly Volatility: By calculating the standard deviation of log returns for each month (or any other relevant period, e.g., weekly, daily), you can observe volatility regimes.
    • Interpretation: If these monthly volatility figures fluctuate significantly (e.g., some months have very high volatility, others very low), it confirms that volatility is not constant over time. This reinforces the need for models that can capture time-varying volatility, such as GARCH models, over those that assume constant volatility (like basic GBM).

Using the Script: The script provides a list of monthly volatilities. You can visually inspect these to see the historical fluctuations in the asset’s risk level.

5. Selecting the Right Price Generation Model for Your 5-Minute Data

Based on the analysis of your 3-year, 5-minute historical price series, here’s how the models discussed in the research paper fit:

Given the high frequency (5-minute data) and typical characteristics of financial markets, your data is highly likely to exhibit:

  • Volatility Clustering: Periods of high volatility followed by high volatility, and vice versa.
  • Fat Tails: More extreme price movements than a normal distribution would predict.
  • Potentially Autocorrelation in Returns: Especially at very high frequencies, market microstructure effects can introduce short-term autocorrelation.

Therefore, the most suitable models for generating realistic price series for your 5-minute data are:

  1. GARCH Models (High Suitability):
    • Why: GARCH models are specifically designed to capture conditional heteroskedasticity (volatility clustering) and can be extended to handle fat tails (e.g., by using a Student’s t-distribution for the errors instead of a normal distribution). Since your data is 5-minute, volatility clustering is almost certainly present.
    • Variables to Set Up: You don’t “set up” variables directly; you estimate the model parameters (ω, α, β for GARCH(1,1)) from your historical log returns. Statistical software (like Python’s arch library or R’s rugarch package) will perform this estimation for you. These estimated parameters will then define the behavior of the generated price series, closely matching your historical volatility dynamics.
    • How to Use: Once the GARCH model is fitted, you can simulate future price paths by drawing random innovations from the specified error distribution (e.g., normal or Student’s t) and using the estimated conditional variance equation to generate the next return.
  2. Monte Carlo with Historical Bootstrapping (High Suitability):
    • Why: This method is non-parametric and doesn’t assume a specific distribution. By resampling directly from your 3-year 5-minute historical log returns, it inherently captures all the empirical properties of your data, including fat tails, skewness, and any observed jumps or non-linearities, without you needing to explicitly model them.
    • Variables to Set Up: Your “variables” are simply your entire historical log returns series.
    • How to Use: To generate a new price path, you would randomly select a log return from your historical dataset and apply it. To preserve volatility clustering, consider block bootstrapping, where you sample contiguous blocks of returns (e.g., 1-hour or 4-hour blocks) instead of individual 5-minute returns. This helps maintain the local dependencies in volatility.
  3. ARIMA Model (Medium Suitability, often combined):
    • Why: If your ACF of log returns shows significant autocorrelation, an ARIMA model can capture these linear dependencies. For high-frequency data, short-term autocorrelations can arise from market microstructure effects.
    • Variables to Set Up: You’ll need to determine the orders p (AR), d (differencing, likely 0 for log returns), and q (MA) by analyzing the ACF and PACF plots of your log returns.
    • How to Use: ARIMA models are often used to model the mean process of the returns, and then combined with a GARCH model for the variance process (resulting in an ARMA-GARCH model). This hybrid approach is very powerful for financial data.
  4. Geometric Brownian Motion (GBM) Model (Low Suitability as Standalone):
    • Why: While foundational for options pricing, GBM assumes constant volatility and normally distributed returns with no jumps. Your 5-minute data will almost certainly violate these assumptions due to volatility clustering and fat tails. Using GBM alone would lead to highly unrealistic simulations that underestimate risk.
    • Variables to Set Up: Drift (μ) and volatility (σ). You could estimate these from your historical data, but the constant volatility assumption makes it a poor fit for high-frequency data.
  5. Markov Series (Markov Chains) (Medium Suitability, for specific applications):
    • Why: Could be used to model regime changes (e.g., high volatility vs. low volatility states) or short-term trend transitions. However, their “memoryless” property is a strong simplification for financial markets.
    • Variables to Set Up: You would define discrete states (e.g., price up, down, flat) and then estimate the transition probabilities between these states from your historical data.
  6. Random Walk Model (Low Suitability):
    • Why: This is primarily a theoretical null hypothesis. If your data truly follows a random walk, then past prices offer no predictive power, and active trading strategies based on historical patterns are futile. Your analysis of ACF for log returns will tell you if your data deviates from a random walk. For high-frequency data, deviations are common.

Summary of Modeling Approach for Your Data

For your 3-year, 5-minute historical price series, I recommend a multi-pronged approach focusing on models that can capture the specific characteristics of high-frequency financial data:

  1. Data Transformation: Always start by converting raw prices to log returns.
  2. Exploratory Data Analysis: Use the provided script to calculate:
    • Descriptive Statistics of Log Returns: Pay attention to skewness and especially kurtosis (fat tails).
    • ACF of Log Returns: Check for any significant linear autocorrelation.
    • ACF of Squared Log Returns: This is critical for identifying volatility clustering (conditional heteroskedasticity).
    • Monthly Volatility: Observe the time-varying nature of volatility.
  3. Model Selection:
    • If you observe volatility clustering (significant ACF in squared returns) and/or fat tails (high kurtosis), GARCH models (or their asymmetric variants like EGARCH/GJR-GARCH) are your primary candidates.
    • For generating diverse scenarios that inherently capture all empirical features without strong parametric assumptions, Monte Carlo with Historical Bootstrapping (especially block bootstrapping) is highly recommended.
    • If you find significant autocorrelation in log returns, consider combining an ARIMA model with a GARCH model (e.g., ARMA-GARCH).
    • Avoid using basic GBM or Random Walk models for realistic price generation, as they will likely misrepresent the true dynamics and risk of your high-frequency data.

By following these steps, you’ll gain deep insights into your historical price series and be well-equipped to select and parameterize the most appropriate price generation models for robust trading strategy backtesting.