Multi-Resolution Analysis on Price Fluctuations for the Heston-GARCH Levy Model

October 27, 2025

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Abstract

This paper presents a comprehensive wavelet-based analysis of financial time series governed by a hybrid Heston-GARCH-Lévy process. We first simulate log-returns incorporating stochastic volatility, volatility clustering, and jumps, and apply the Discrete Wavelet Transform (DWT) to decompose the returns into multiscale components. Using AAPL stock data, we isolate high-frequency jump effects and long-term volatility structure. Volatility characterization is achieved via wavelet energy decomposition and GARCH modeling of high-frequency residuals. Jump components are identified using thresholder wavelet coefficients. We compute normalized wavelet entropy (NWE) to quantify signal complexity across scales, enabling anomaly detection via rolling entropy. Detected anomalies align with earnings announcements and market turbulence. We validate our findings against realized volatility derived from intraday 5-minute returns and show that entropy-based anomalies correspond to volatility spikes. Refinement of detection parameters via window size and thresholding is explored. Finally, Granger causality tests confirm that wavelet entropy significantly predicts realized volatility but not vice versa, establishing entropy as a leading indicator of market turbulence. This work demonstrates the synergy of stochastic modeling and wavelet signal analysis in capturing the multiscale, nonstationary structure of financial returns and proposes wavelet entropy as a powerful tool for forecasting volatility and detecting regime shifts.

Keywords: Stochastic & Ito Calculus, Heston-Model, GARCH & Levy Processes for Jump Diffusions, Wavelets, Anomalies, Realized Volatility, Decomposed Dynamics, Transforms

Introduction

Motivation for Wavelet-Based Financial Analysis

Financial time series exhibit complex multi-scale behavior that traditional analysis methods struggle to capture effectively. Asset prices demonstrate non-stationary dynamics, sudden jumps, and long-range dependencies, that operate across different time horizons simultaneously. These stylized facts of financial markets create fundamental challenges for conventional time-domain and frequency-domain analytical approaches.

Limitations of Traditional Methods

Classical Fourier analysis, while powerful for stationary signals, fails to localize events in time. When analyzing financial returns, we lose critical information about when particular frequency components occur – a severe limitation when studying market crashes, earnings announcements, or regime changes that manifest as localized, high frequency events. Similarly, purely time-domain methods miss the multi-scale nature of market dynamics, where high-frequency microstructure effects interact with medium-term trend reversals and long-term economic cycles.

Why Wavelets Matter for Finance

Wavelet analysis addresses these limitations through its unique ability to provide simultaneous time-frequency localization. This capability is particularly valuable for financial applications because:

Non-stationarity: Financial time series are inherently non-stationary, with statistical properties that evolve over time. Wavelets naturally handle such signals by decomposing them into components at different scales, each capturing specific aspects of temporal evolution.

Multi-Scale Features: Markets operate across multiple time scales simultaneously. High-frequency trading algorithms interact with daily positioning strategies, which in turn respond to monthly economic cycles and quarterly earnings patterns. Wavelet decomposition separates these scales, allowing independent analysis of each temporal component¹.

Event Localization: Financial markets are punctuated by discrete events, such as earnings surprises, central bank announcements or geopolitical shocks. Wavelets excel at detecting and localizing such transient phenomena while preserving information about frequency content.

Volatility Heterogeneity: Market volatility exhibits clustering behavior where high volatility periods cluster together, followed by calm periods. The multi-resolution nature of wavelet analysis naturally captures these regime-dependent characteristics across different time scales.

Levy Noise and Jump Diffusion

Real financial asset prices deviate significantly from the geometric Brownian motion assumptions underlying classical models.

Asset prices exhibit sudden, discontinuous movements that cannot be explained by normal market fluctuations. These jumps occur during earnings announcements, regulatory changes, or market stress periods. Traditional continuous-path models underestimate risk and misprice derivatives during such events. Levy processes provide a mathematical framework for incorporating these discontinuities, but detecting and characterizing jumps from observed data remains challenging.

And so, we define the HGL jump process as $J_t = N_t \cdot \zeta_t$ , where $N_t$ is a Poisson process representing the number of jumps in a given time frame, and $\zeta_t$ represents the size of each jump, typically modeled as a Gaussian distribution.

The jump component $J_t$ is typically modeled using a Poisson process with jump intensity $\lambda$ , which represents the expected number of jumps per unit time. If a jump occurs, the relative change in stock price due to the jump is modeled as $S_t \mapsto S_t (1+Y)$ , where $Y$ is a random variable expressing the jump size, often assumed to follow a normal or log-normal distribution. Mathematically, $J_t$ can be expressed as $J_t = \sum_{i=1}^{N_t} Y_i$ with $Y_i$ are i.i.d. random variables representing jump sizes.

Stochastic Volatility

Volatility itself follows a stochastic process, exhibiting mean reversion, volatility clustering and correlation with price movements. The Heston model captures these dynamics through a separate stochastic differential equation for volatility, allowing for more realistic option pricing and risk management. However, empirical volatility often displays additional GARCH-type dependencies that pure Heston models miss.

The Heston-GARCH-Levy Framework

Our analysis employs a hybrid model that combines three complementary approaches:

$\begin{align*} dS_t &= S_t \left[ r\,dt + \sqrt{v_t}\, dW_t^S + J_t\, dN_t \right] \\ dv_t &= \kappa (\theta - v_t)\, dt + \sigma \sqrt{v_t}\, dW_t^v \\ v_{t+1} &= \omega + \alpha \varepsilon_t^2 + \beta v_t, \quad \varepsilon_t = \sqrt{v_t}\, Z_t \end{align*}$

This framework captures continuous volatility evolution through the Heston component, volatility clustering via GARCH dynamics, and discontinuous price movements through Levy jumps. The interaction between these components creates rich, realistic price dynamics that better reflect empirical market behavior.

Wavelet Analysis as a Unifying Tool

Wavelet decomposition provides an ideal analytical framework for this hybrid model because:

Scale separation: Different components of the model operate at different time scales. Wavelets naturally separate high-frequency jump effects from medium-frequency GARCH clustering and low-frequency Heston mean reversion
Jump Detection: Wavelet coefficients at fine scales exhibit large values when jumps occur, enabling robust jump identification through appropriate thresholding techniques
Volatility Characterization: The energy distribution across wavelet scales provides insights into volatility structure, with persistent components appearing at coarser scales and transient shocks at finer scales
Regime Identification: Wavelet entropy measures quantify the complexity of price dynamics, facilitating the detection of regime changes and market stress periods

This paper demonstrates how wavelet-based multi-resolution analysis can effectively decompose, analyze and interpret financial time series generated by complex stochastic processes. Though application to Apple Inc. (AAPL) stock data, we show how wavelet tools provide actionable insights for risk management, derivative pricing, and market timing applications.

The Heston-GARCH-Levy Model

Let us establish the HGL model, its parameters, and its solution form.

System of SDEs

The HGL model consists of an interplay of dynamics between a Heston stock price and volatility dynamics, GARCH volatility, and Levy jump diffusions. We define the following system of Stochastic Differential Equations (SDEs) of the HGL model as:

$\begin{align*} dS_t &= \mu\, dt + \sigma \sqrt{v_t}\, dW_t + J_t\, dN_t, \\ dv_t &= \kappa (\theta - v_t)\, dt + \sigma \sqrt{v_t}\, dW_t^v, \\ v_{t+1} &= \omega + \alpha \epsilon_{t-1}^2 + \beta v_{t-1}. \end{align*}$

Existence and Uniqueness of the Hybrid Model

To ensure the existence of unique solutions, we require that the model’s dynamics satisfy the following conditions for a general form of a stochastic differential equation with jumps: $dX_t = b(t,X_t)\,dt + \sigma(t,X_t)\,dW_t + \int_{\mathbb{R}} J(X_t,z)\, \tilde{N}(dt,dz)$ , where $b(t,x)$ is the drift term, $\sigma(t,x)$ is the diffusion term, $W_t$ is the standard Brownian motion, and $\tilde{N}(dt,dz)$ is the compensated Poisson measure representing jump dynamics.

Lipschitz Condition: $|b(t,x)-b(t,y)| + |\sigma(t,x)-\sigma(t,y)| \le L|x-y|$ for some constant $L>0$ .

Linear Growth: $|b(t,x)| + |\sigma(t,x)| \le C(1+|x|)$ , for some constant $C>0$ .

Picard Iteration Scheme for HGL with Justifications

The HGL’s proof of existence is performed with the Picard Iteration Scheme to construct the solution to the hybrid system. It guarantees a unique fixed solution to the system of SDEs. This method is well-suited for the following reasons:

It ensures the existence of a unique fixed-point solution under appropriate contraction conditions.
It aligns naturally with the structure of our SDE.
It provides a constructive approach to approximating the solution, which is beneficial for both theoretical analysis and numerical implementation.

Consider the general form of our stochastic hybrid equation:

$dS_t = f(S_t, v_t, N_t, J_t) + g(S_t, v_t, N_t, J_t)\, dW_t + h(S_t, v_t, N_t, J_t)\, dN_t$ , where $f, g, h$ are measurable. Then we obtain
$\mathbb{E}\left[\sup_{t \in [0, T]} |S_t^{(n+1)} - S_t^{(n)}|^2\right] \le C T \mathbb{E}\left[\sup_{t \in [0, T]} |S_t^{(n+1)} - S_t^{(n)}|^2\right]$ ,
where $\mathbb{E}$ is the expectation operation. Here, we force $T$ to be sufficiently small to achieve a contraction, and thus, by Banach’s Fixed-Point Theorem, $S_t^{(n)}$ converges to a unique solution.

Moment Matching and Hybrid Components

The hybrid nature of the system arises from the interaction between continuous-time SDEs and discrete GARCH updates. To ensure such components are valid together, GARCH updates must respect the dynamics of the Heston volatility. The same can be said for the discrete jump process.

Moment Matching: The variance of the continuous-time process $v_t$ over $[t_k,\, t_{k+1}]$
must match the conditional variance implied by the GARCH model. In other words, the expectation of the Heston volatility given information up until the current time must be equivalent to the expectation of the GARCH components.

Feller Condition: Before validating the conditions for moment matching, the Feller condition for the process must be established to ensure volatility remains strictly positive. The Feller condition guarantees that , ensuring $2\kappa\theta \ge \sigma^2$

the GARCH expectation is then $\mathbb{E}[v_{t+1}] = \omega + \alpha \mathbb{E}[\epsilon_t^2] + \beta \mathbb{E}[v_t].$ Since $\mathbb{E}[\epsilon_t^2] = \mathbb{E}[v_t Z_t^2],$ we have $\mathbb{E}[v_{t+1}] = \omega + (\alpha + \beta) \mathbb{E}[v_t].$ . For stationarity, we arrive at $\mathbb{E}[v_t] = \frac{\omega}{1 - (\alpha + \beta)}$ . It can be shown that the GARCH volatility follows a reduced summation of the form $\omega \sum_{i=1}^{\infty} (\alpha + \beta)^i$ . Finding the expectation then becomes clear, as we simply find the sum of this geometric series. The sum of a geometric series is defined by $S = \frac{a_1}{1 - r}$ , with $a_1 = 1$ , $r = \alpha + \beta$ , and converges for a common ratio having an absolute value of less than 1. Thus, the sum and expectation of the GARCH volatility arrives at $\mathbb{E}[v_{t+1}] = \frac{1}{1 - (\alpha + \beta)}$ , which we force $(\alpha + \beta) < 1$ to ensure convergence for the geometric series.

The expectation of the Heston volatility given by the Fokker–Planck equation is implied to be $\frac{d\mathbb{E}[v_t]}{dt} = \kappa(\theta - \mathbb{E}[v_t])$ , where in a steady state, $\mathbb{E}[v_t] = \theta$ . Thus, to ensure moment matching and to validate the interplay of discrete and continuous components, we require $\theta = \frac{1}{1 - (\alpha + \beta)}$ .

We further ensure positivity of the GARCH process with $\omega > 0, \; \alpha, \beta \ge 0$ .

Methods and Methodology

Relationship between theoretical Heston GARCH Levy Model and the actual analysis of AAPL data

In this paper, we introduce a Heston GARCH Levy model with these components:

Heston component: continuous stochastic volatility process

GARCH component: Discrete-time volatility clustering

Levy component: Jump processes

We analyze real AAPL stock data from the last two years, but we do not fit the Heston GARCH Levy model to AAPL data and no data was simulated from the theoretical model. No comparison between model predictions and real data was performed. What we did was the following:

Step 1: AAPL data processing, where we downloaded 2 years of AAPL daily prices and calculated log returns.

Step 2: Wavelet decomposition, where we applied the Discrete Wavelet Transform to the log returns using the Daubechies-4 wavelet. We obtained approximation $D_{1} \cdots D_{j}$

Step 3: GARCH analysis. Here we fitted a simple GARCH (1,1) model to high frequency wavelet detail coefficients and detail coefficients $D_{1}$ . This is not the GARCH component of the theoretical Heston GARCH Levy model. The results were: $\omega = 8.47 \times 10^{-5}, \; \alpha = 0.20, \; \beta = 0.5$

Step 4: Jump detection. Here we used wavelet thresholding to identify potential jumps. We applied universal threshold $t_j = \sigma_j \sqrt{2 \ln T}$ . This identifies jumps in real AAPL data, not from the theoretical Levy component.

Step 5: Entropy Analysis. We computed wavelet entropy for anomaly detection. We then performed Granger causality analysis between entropy and realized volatility. All analysis was on real AAPL data.

Analysis Approach

This study employs wavelet based multi-resolution analysis to examine the empirical properties of financial time series, specifically focusing on Apple Inc. (AAPL) stock returns over a two-year period. While the Heston-GARCH-Levy framework provides theoretical motivation for why multi-scale analysis tools might be valuable for understanding complex financial dynamics-capturing continuous stochastic volatility, volatility clustering, and jump processes- our analysis focuses on applying wavelet methodology to real market data to uncover its inherent multi-scale structure.

We will talk about these steps briefly and then elaborate on each step.

Log-Return Calculation

Following standard financial time series practice, we computed continuously compounded returns as

$R_t = \ln\left(\frac{S_t}{S_{t-1}}\right)$

Here $S_{t}$ represents the adjusted closing price at time t. Log returns are preferred over simple returns as they exhibit more stable statistical properties and are approximately additive over time, making them suitable for wavelet analysis.

We performed standard data cleaning procedures. We removed non-trading days and market holidays, checked for missing values and data gaps, verified data consistency and removed any obvious outliers or data errors, and confirmed the resulting time series contains no gaps that would affect wavelet decomposition.

In terms of the wavelet decomposition framework, for the mother wavelet, we employed the Daubechies – 4 (db4) wavelets as the mother wavelet for all decompositions². This choice balances several considerations. First there is compact support, which provides good time localization. Next there is sufficient regularity: four vanishing moments effectively separate polynomial trends from fluctuations. Then there is computational efficiency for this type of wavelet, as well-established fast algorithms are available. Finally, there are financial applications, that are proven effective.

We then use the Discrete Wavelet Transform, which decomposes the log-return series ${R_t}_{t=1}^N$
into

$R_t = A_J(t) + \sum_{j=1}^{J} D_j(t)$

Here $A_J(t)$ represents approximation coefficients capturing long-term trends at the coarsest scale $2^J$ . $D_j(t)$ represents detail coefficients capturing fluctuations at scale $2^j$ .
$J$ is the maximum decomposition level, typically chosen as $J = [\log_2(N)]$ .
The db4 wavelet is characterized by filter length of 8 coefficients, vanishing moments: 4, support width of 7 (compact support from 0 to 7), and regularity characterized by a Hölder exponent of 1.089.
For the AAPL dataset with $N$ of 500 daily observations, the maximum decomposition levels is given by $J = \min(6, [\log_2 N]) = \min(6, 8) = 6$ . This yields detail coefficients of $D_1, \ldots, D_6$ and approximation coefficients $A_6$ corresponding to time scales of approximately:

$D_1$ 2 days (highest frequency)
$D_2$ 4 days
$D_3$ 8 days (1.5 weeks)
$D_4$ 16 days (3 weeks)
$D_5$ 32 days (1.5 months)
$D_6$ 64 days ( 3 months)
$A_6 > 64$ days (long term trends)

Boundary Effects Handling

Boundary effects arise because the wavelet transform assumes the signal extends beyond the observed data. We addressed this using:

Symmetric Extension: The “PyWavelets” library implements symmetric boundary extension by default, where the signal is mirrored at both ends:

$x_{-n} = x_n, \; x_{N+n} = x_{N-1-n}$

Valid Coefficient Selection: We retain only coefficients that are not significantly affected by boundary extensions, effectively using a type of valid mode. This removes approximately L-1 coefficients from each end at each level. Here L=8 is the filter length for db4. This results in progressively shorter coefficient sequences at coarser levels.
Analysis Window Adjustment: For rolling window analysis, we ensure sufficient padding by starting analysis after an initial buffer of 16 observations, ending analysis before the final 16 observations and observing that this conservative approach minimizes boundary artifacts in entropy calculations.

In terms of the software and libraries, all analysis was implemented in Python 3.8+ using the following libraries:

Core Wavelet Analysis:

PyWavelets (pywt) v1.4.1 which is the primary libary for wavelet transforms. We used pywt.wavedec() for forward DWT, pywt.waverec() for inverse DWT (jump reconstruction), the wavelet was specified by wavelet=’db4’, and the boundary condition was mode = ‘symmetric’ by default
We used yfinance v0.2.18 Yahoo Finance API wrapper for downloading AAPL price data. Usage yf.download(‘AAPL’, period=’2y’, interval=’1d’). We got the intraday data using yf.download(‘AAPL’, period=’60d’, interval=’5m’)
We used pandas v2.0.3 for data manipulation and time series handling
We used NumPy v1.24.3 for numerical computations and array operations
For statistical analysis arch v6.2.0: GARCH model estimation

Model specification: arch_model(residuals, vol=’Garch’, p=1, q=1)

statsmodels v0.14.0: Granger causality tests

Function used: grangercausalitytests()

scipy v1.10.1: Statistical functions and optimization
Visualization using Matplotlib v3.7.1: All plots and visualizations

Hardware Specifications:

Analysis was performed on:

CPU: Intel Core i7-12700K (12 cores, 3.6 GHz base)
RAM: 32 GB DDR4-3200
Operating System: Windows 11 Pro
Python Environment: Anaconda 2023.07 distribution

Computational Complexity and Performance:

DWT computation O(N) per decomposition using fast lifting scheme
Rolling Window Analysis O( N x W) where W is the window size
Total Runtime: approximately 15 minutes for complete analysis
Memory Usage: Peak 2 GB for full dataset processing

Methodological Limitations:

First there are data limitations. Analysis is limited to daily frequency data for main decomposition. Intraday data available only for the most recent 60 trading days due to API limitations. Single asset analysis may not generalize to other securities or asset classes.

Next there are wavelet specific limitations. The choice of mother wavelet affects decomposition characteristics. Boundary effects at the beginning and end of time series are used. Finally, the fixed dyadic grid may miss important features at intermediate scales.

Finally, there are statistical limitations. Granger causality tests only capture linear predictive relationships. Entropy based anomaly detection sensitive to threshold selection. Additionally, there are no formal statistical tests for jump detection accuracy.

Code applied to AAPL for the log-return decomposition

The hybrid Heston-GARCH-Levy model combines stochastic volatility, GARCH dynamics, and jump processes. To analyze its behavior on AAPL stock, we apply DWT decomposition to log-returns. The wavelet decomposition plots display the multi-scale structure of AAPL’s log returns breaking down the original time series into components that capture both long term trends and short-term fluctuations. See Figure 1 for a visual representation of the decomposed dynamics.

Results and Analysis

Figure 1. Multi-Scale Structure of AAPL Log Return

Specifically, in speaking on the methodology and analysis:

Approximation Coefficients $A_j$ : The top plot usually labeled $A_j$
shows the log frequency or trend component of the log returns. This captures the underlying slowly varying movements in AAPL’s returns—essentially the market’s persistent drift or long-term volatility cycles.
Detailed Coefficients $D_j$ to $D_1$ : The subsequent plots show high-frequency components at successively finer scales. The higher-level details are shown by $D_j$ and $D_{j-1}$ .
These capture medium-term cycles and volatility clustering—these can reflect periods of sustained market turbulence or calm, often associated with macroeconomic events or persistent investor sentiment. The lower-level details are captured by $D_1$ and $D_2$ . These capture the highest frequency movements, including sudden jumps, spikes, or noise—these are often driven by market microstructure effects, news shocks, or algorithmic trading. Each plot shows how much “energy” or variance is present at that particular scale, revealing the presence of both short-term shocks and long-term trends in the data.
Isolating Market Behaviors by Timescale: Wavelet decomposition allows you to separate the log-returns into components associated with different time horizons. This is crucial because financial markets are influenced by a mix of short-term (e.g., news, trading strategies) and long-term (e.g., economic cycles, company fundamentals) factors.
Detecting Volatility Clustering and Regime changes: Persistent patterns in mid and high-level detail coefficients $(D_J, D_{J-1})$ indicate periods of volatility clustering—where high volatility tends to follow high volatility, and low follows low. This is a hallmark of real-world financial time series and is important for risk management and derivative pricing. Sudden spikes in the lowest-level detail coefficients can indicate abrupt market moves or jumps, such as those caused by earnings announcements or macroeconomic surprises.
Identifying Long-Term Trends: The approximation coefficients $A_j$ help visualize the underlying trend in AAPL’s returns, filtering out short-term noise. This can assist in identifying bull or bear market phases and in constructing long-term investment strategies.
Improving Forecasting and Modeling: By analyzing each scale separately, one can build more accurate predictive models. For example, machine learning models can be trained on each component individually and then recombined for improved forecasting accuracy. This hybrid approach has been shown to outperform models that use only the raw returns.
Risk Management: Understanding which scales contribute most to risk (variance) helps in portfolio construction and hedging. For instance, if most risk is concentrated in high-frequency components, short-term trading strategies or hedges may be appropriate.

Volatility Component Characterization

Continuous Volatility: Compute wavelet energy
$E_j = \sum_{k=1}^{N_j} |D_j(k)|^2 \propto 2^{-j\alpha}$
GARCH effects: Model using filtered residuals
$v_{t+1} = \omega + \alpha \epsilon_t^2 + \beta v_t, \; \epsilon_t = \sqrt{v_t}\, Z_t$
In order to perform volatility component characterization using wavelets for the Heston-GARCH-Levy model, we look at continuous volatility, as in, compute wavelet energy by scale. For each level $D_j$
from the DWT (or one could also use MODWT), we compute the energy:
$E_j = \sum_{k=1}^{N_j} |D_j(k)|^2$
This gives the variance of the log-return series at each timescale, allowing one to quantify how much volatility is contributed by short-term, mid-term, and long-term components subsequently³. Simulating such dynamics gives us the following:

Figure 2: Energy levels at different levels of decompositions

Next, we look at the GARCH effects, modeled using filtered residuals. We essentially use the high frequency detail coefficients such as $D_1$ and $D_2$ capture short term fluctuations and jumps, as a proxy for the innovations and residuals in the GARCH models. We fit a GARCH (1,1) model to these coefficients to estimate the GARCH parameters $\omega . \alpha , \beta$ and characterize volatility clustering.

Simulating the results give us the following data:

The insignificant alpha coefficient suggests several important things about the high-frequency wavelet residuals:

The alpha coefficient in GARCH(1,1) captures the ARCH effect – how much yesterday’s squared innovation (shock) influences today’s volatility. An insignificant alpha suggests that:

Immediate shock transmission is weak: Recent squared residuals don’t have a statistically significant impact on current volatility in the high-frequency wavelet components.
The wavelet filtering may have already captured the ARCH effects: Since we are applying GARCH to wavelet detail coefficients (particularly $D_1$ and $D_2$ ) rather than raw returns, the wavelet decomposition itself may have already isolated and separated the different volatility components.
Implications for Short-Term Volatility Clustering: This doesn’t necessarily mean that short-term volatility clustering is absent. Instead, it suggests:
Scale Specific Behavior: The high-frequency wavelet residuals ( $D_1$ and $D_2$ ) represent a very specific frequency band. Traditional GARCH effects might operate at different timescales that span across multiple wavelet scales, making them less detectable when one isolates just the highest frequency components.

The wavelet decomposition acts as a sophisticated filter. By the time one extracts $D_1$ and $D_2$ coefficients, one has already:

Removed long-term trends (captured in approximation coefficients)
Isolated specific frequency bands
Potentially decorrelated some of the temporal dependencies that GARCH typically captures

Jump Dominance: Since our model includes Lévy jumps, the high-frequency components might be dominated by jump processes rather than continuous volatility clustering. Jumps are typically modeled as independent events, which wouldn’t exhibit the serial correlation patterns that GARCH captures.

The results support the hybrid model’s theoretical foundation:

Complementary Components: The Heston component handles medium-term volatility clustering, while the Lévy component handles discontinuous jumps. The GARCH component may be more relevant at intermediate frequencies

Multi-Scale Nature: Different volatility mechanisms operate at different time scales, validating the need for a multi-component model rather than a single GARCH specification.

Jump Detection Validation: If high-frequency residuals don’t show GARCH effects, it reinforces that the wavelet-based jump detection methodology in earlier sections, is correctly identifying discontinuous rather than continuous volatility effects.

Based on the above analysis, we can conclude that there is an increasing energy as we move to finer scales. The highest energy is at the finest scale and energy decreases as you move to coarser scales $D_1$ to $D_6$ . This is typical for financial return series since most of the volatility and action (jumps and short-term fluctuations) are concentrated at high frequencies (short term scales). This is a good result in the sense that it matches the expected stylized facts of financial time series: volatility and shocks are mostly short-term phenomena, while long term trends are smoother and less volatile. The data exhibits strong short-term volatility and/or jumps, which is consistent with real stock returns. The energy distribution supports the presence of both high-frequency (possibly jump or microstructure noise) and lower-frequency (persistent volatility) components. Now we come to the GARCH model Fit on Wavelet Residuals. Here are the key results that we obtained:

$\omega = 8.47 \times 10^{-5}, \; p = 0.032 \; (\text{significant})$
$\alpha = 0.20, \; p = 0.179 \; (\text{not significant})$
$\beta = 0.50, \; p = 0.011 \; (\text{significant})$
Sum of $\alpha, \beta = 0.70$

Interpretation of GARCH Model Results

The GARCH (1,1) model was fitted to the high-frequency wavelet detail coefficients. The results are summarized as follows:

Omega ( $\omega$ ): The estimated value is $8.47 \cdot 10^{-5}$

with a p-value of 0.032, indicating that the constant component of volatility is statistically significant at the 5 $\%$ level.

Alpha ( $\alpha$ ): The estimated value is 0.20 with a p-value of 0.179. While not statistically significant at the 5 $\%$ level, this suggests a moderate ARCH effect, meaning that recent squared innovations have some influence on current volatility.

Beta ( $\beta$ ): The estimated value is 0.50 with a p-value of 0.027, which is statistically significant. This indicates a substantial degree of volatility persistence, as past volatility has a lasting effect on current volatility.
Volatility Clustering: The sum $\alpha + \beta = 0.70$
indicates moderate volatility persistence. This confirms the presence of volatility clustering, a well-known stylized fact in financial time series, where periods of high volatility tend to be followed by high volatility and vice versa.

Overall, the GARCH model fits the high-frequency component of the returns reasonably well and confirms the presence of volatility clustering in the AAPL log-return series.

The GARCH effects $\alpha$ = 0.2 and p-value 0.179

Here we explain why $\alpha$ might be insignificant in the GARCH fit on $D_1$ / $D_2$ coefficients and this touches on several important aspects of how wavelets interact with volatility clustering:

a) Wavelet Pre-Filtering Effects: The wavelet decomposition itself acts as a sophisticated filter that separates different frequency components. When we extract $D_1$ / $D_2$ (the highest frequency) details, we are isolating components that have already been purified through the wavelet transform. This pre-filtering may have already (i) removed some of the autocorrelated volatility structure that GARCH typically captures (ii) isolated noise and jumps from the systematic volatility clustering patterns (iii) created residuals that are more white noise like in their second moments.

b) Scale-Dependent Volatility Clustering: Classical GARCH effects (where yesterday’s shocks predict today’s volatility) may operate primarily at medium frequencies rather than the ultra-high frequencies captured by $D_1$ / $D_2$ . The volatility clustering phenomenon might be more pronounced in: (i) D3,D4,D5 coefficients (medium term scales) (ii) The original unfiltered returns (iii) Lower frequency components where market sentiment and information flow create persistent volatility regimes.

c) Dominance of Jump/Microstructure Noise:

The $D_1$ / $D_2$ components likely contain (i) Levy Jumps. Sudden discontinuous price movements that are inherently unpredictable and don’t follow GARCH-type clustering (ii) Microstructure noise: these are bid-ask spreads, market maker effects and algorithmic trading artifacts (iii) News shocks: immediate responses to information that creates one off volatility spikes rather than persistent clustering.

These elements are fundamentally different from the smooth stochastic volatility that the GARCH models are designed to capture.

The $\beta$ is still significant ( $\beta$ =0.5, p = 0.011)

Interestingly, while $\alpha$ is insignificant, $\beta$ remains highly significant. This suggests that:

Long-term volatility persistence still exists even in high-frequency components
The “memory” in volatility comes more from the persistent component (β) than from immediate shock responses (α)
There’s still some underlying volatility process driving these high-frequency movements

Implications for the Hybrid Model

This finding strengthens the case for our hybrid Heston-GARCH-Lévy approach because:

Lévy jumps naturally explain the lack of ARCH effects in D1/D2 – jumps are unpredictable by definition
Heston stochastic volatility can provide the underlying volatility structure that shows up in the β parameter
GARCH effects may be more prominent in medium-frequency components (D3-D5) where the systematic volatility clustering operates

Entropy Values near Anomalies:

We reproduce the graph with changes to the code here:

Figure 4: Entropy statistics: Mean: 1.344522 Std: 0.100062 Min: 1.037791 Max: 1.576111 Range: 0.538320 Threshold (percentile method): 1.511841 Detected 22anomalies

Figure 5: Anomaly Dates, Entropy Values and Pricing Details for AAPL

The entropy values calculated have some variation after careful calculation, and no two values are the same, although they are all between 1.5 and 1.6

Jump Detection

Identify Lévy jumps via thresholding:

$\tau_j = \sigma \sqrt{2 \ln T}, \quad J_t = \sum_k D_j(k) \, \mathbb{I}_{|D_j(k)| > T_j}$

Mathematical Foundation

Jump detection leverages wavelet detail coefficients $D_j k$ at a scale J
The method identifies Levy jumps by separating extreme values from normal volatility using hard thresholding⁴. The threshold calculation is as follows:

$\sigma_j$ : Standard deviation of detail coefficients at scale $j$ , estimated by the Median Absolute Value (MAD): $\hat{\sigma}_j = \text{median}(|D_j(k)|) / 0.6745$
The denominator $0.6745$ normalizes the MAD measure for Gaussian distributions.
Universal threshold $\tau_j$ : $\tau_j = \hat{\sigma}_j \sqrt{2 \ln T}$ , where $T$ is the length of the time series. This threshold asymptotically separates Gaussian noise from jumps.
Jump Indicator: we apply hard thresholding to isolate Coefficients exceedingly $\tau_j$ as $\tilde{D}j = D_j(k) \, \mathbb{I}{(|D_j(k)| > \tau_j)}$
The jump component $J_t$ is reconstructed via the inverse DWT or IDWT using only threshold coefficients: $J_t = \text{IDWT}(0, \tilde{D}_{J-1}, \ldots, \tilde{D}_1)$ . Approximation coefficients $A_j$ are zeroed to exclude trends.

Feasibility Assessment

We apply the Heston-GARCH-Levy model with wavelet-based jump detection to AAPL stock. This model combines Heston’s stochastic volatility for continuous volatility systems, GARCH effects for volatility clustering, and Levy jumps for discontinuous price movements. We will apply this to AAPL stock to the following results:

Each of the graphs refer to the detail coefficients from applying the DWT to the log-return time series. The DWT decomposes a time series into approximation coefficients to detect long-term trends. Additionally, it decomposes the time series into detail coefficients to analyze short-term fluctuations, where increases with lower frequency or coarser resolution. The D1 level coefficients have the highest frequency content, where in terms of market interpretation, may represent noise, microstructure jumps, market shocks, or high frequency day-trading. D2 demonstrates short term volatility, D3 demonstrates volatility clusters, and D4-D5 demonstrate medium to low frequency impacts that typically occur on a macroeconomic scale. In our context, D1 and D2 are used for analyzing our Levy jump processes, as well as analyzing GARCH volatility.

Cross-Scale Dependencies

Wavelet coherence between $v_t$ and $R_t$ :

$WTC_{v,R}(t,j) = \dfrac{|W_{vR}(t,j)|^2}{W_{vv}(t,j) \, W_{RR}(t,j)}$

Entropy-Based Regimes

Normalized wavelet entropy:

$S = -\sum_{j=1}^{J} p_j \ln p_j, \quad p_j = \dfrac{E_j}{\sum_{k=1}^{J} E_k}$

We are analyzing AAPL log-returns using wavelet decomposition. Each wavelet detail $D_j$
captures fluctuations at a specific time scale. The energy at each scale level, $E_j = \sum_k |D_j(k)|^2$ , measures the variance or power of the signal at that scale.

The normalized wavelet entropy (NWE) is given by:

$S = -\sum_{j=1}^{J} p_j \ln(p_j), \quad p_j = \dfrac{E_j}{\sum_{k=1}^{J} E_k}$

Here, $p_j$ is the proportion of total energy at scale $j$
In terms of interpretation:

Low entropy: Most energy is concentrated at a few scales (regime is more ordered, e.g. strong trend or dominant volatility scale).
High entropy: Energy is spread across many scales (regime is more disordered or complex) e.g. turbulent or mixed volatility/jump regimes.
Regime shifts: Changes in entropy over time can signal transitions between market regimes (e.g. calm to turbulent).

In this paper we compute $S$
for a rolling window or the entire series. We then track entropy to detect periods of regime change in AAPL’s return dynamics.

Figure 13: Summary of plots generated by wavelet jump detection and their significance

Figure 14: Wavelet Energy Proportions by Scale

Entropy-Based Anomaly and Event Detection

Entropy-based anomaly detection leverages the idea that sudden changes in the complexity or disorder of a financial time series, quantified via entropy, can signal unusual events, regime shifts, or anomalies. In the context of wavelet analysis, this approach is particularly powerful, because it captures both time and scale-localized changes in the signal, making it well-suited for financial data with nonstationary and multiscale characteristics.

Preprocessing and Model Fitting

The code for this process is given at the end of the paper. Essentially, we fit a volatility model where we apply a suitable model (in our case the Heston-GARCH-Levy process) to the log returns to capture the main structure and volatility dynamics. We then obtain the residuals, as in we extract the residuals from the fitted model, as analyzing residuals helps isolate anomalies not explained by the model. The standardized residuals and conditional volatility are shown in the figure.

Figure 15: Output of GARCH model entropy data

The constant mean model results table presents the estimated parameters of the GARCH model fitted to the high-frequency wavelet residuals. Alpha, beta, and omega represent unique aspects of volatility. For example, alpha+beta gives us the persistence to volatility, where a number close to 1 indicates long memory or strong clustering, In application, these results can indicate whether high volume trading can be risky in the short term, as well as give insight into volatility trends in the short-run impacted by market events.

Figure 16: Standardized Residuals and Conditional Volatility

Figure 14 demonstrates residual errors scaled by the model’s estimated conditional volatility at each time point. Ideally, standardized residuals should behave like white noise with mean approximately zero, no autocorrelation, and constant variance (homoscedasticity). We are looking for clusters or spikes that indicate leftover structure, possibly unmodeled jumps or volatility shifts. Outliers are important as well, as these can indicate rare extreme events or model misspecifications.

Wavelet Decomposition

We will apply either a Discrete Wavelet-Transform (DWT) or a Maximal Overlap Discrete Wavelet Transform (MODWT) on the residuals or the original return series. In this paper, we will use MODWT. Then we will extract the detail coefficients at each scale, which will capture fluctuations at different time horizons. The MODWT coefficients are shown in the figure above.

Computation of Wavelet Entropy

We then calculate the energy at each scale. For each window (rolling or fixed), we compute the energy $E_j = \sum_k |D_j(k)|^2$ at each scale $j$ . We then compute the normalized wavelet entropy (NWE) given by:

$S = -\sum_{j=1}^{J} p_j \ln(p_j)$

We then use a rolling window. We slide a window across the time series to compute entropy as a function of time, enabling detection of local anomalies and regime shifts. The code is shown at the end. The output is:

Wavelet Entropy (S): -0.0
Energies per scale: [1392.75864617]
p_j per scale: [1.0]

Threshold Selection and Anomaly Detection

Here we first need to establish baselines and thresholds. In other words, we need to determine a threshold for entropy based on historical normal periods. This can be the maximum entropy observed during normal times such as a quantile or a statistically derived threshold. After this, we will need to flag anomalies. By this we mean that we should mark time points or windows where entropy exceeds the threshold as potential anomalies or events. Sudden spikes or drops in entropy often correspond to regime changes, market shocks or structural breaks⁵. The output is shown below with the anomaly times:

Figure 18: Rolling Wavelet Entropy and Anomaly Detection

Anomaly time indices (center of window) are: [173 202 203 204 205 206 210 232 233 234 235 236 237 238 239 240 241 330 335].

Event Localization and Interpretation

Here we will use peak detection algorithms on the entropy time series to locate the timing of significant events or anomalies. Next, we also have to map to market events. As in we need to cross-reference detected anomalies with known market events such as earnings, macroeconomic news and crisis to validate findings and interpret the economic significance.

The anomaly dates detected were as follows (center of anomalous window):

Figure 19: Detected Anomalies and Price Data

Figure 20: Graphical Stochastic Anomaly Detections

In addition, we found that there were 437 unique entropy values ranging from 1.037791 to 1.576111

The first panel shows the AAPL log returns over approximately 2 years. Returns range from about –0.10 to +0.15 (+/- 10 to 15 percent). In terms of notable volatility spikes, some of them are significant:

Early October 2023 (sharp positive and negative moves)
January 2024 (increased volatility)
August 2024 (significant negative spike ~-0.08)
December 2024-February 2025 (elevated volatility period)
June 2025 (recent volatility spike)

The second panel shows rolling wavelet entropy. Entropy values show proper variation ranging from 1.037791 to 1.576111. The 95^th percentile threshold (red dashed line) is set at approximately 1.52. The clear temporal patterns in market complexity have two regions. A lower entropy period (1.1 to 1.3), with the market operating in more predictable, less complex regimes, and a higher entropy period of 1.4 to 1.6. Here there is increased market complexity and irregularity. There are 22 detected anomalies

The third panel shows the AAPL prices with the detected anomalies. Price evolution is from $170 (October 2023) to recent levels $200 to $245. Anomalies are well-distributed across different price levels and market conditions.

Anomaly Value Analysis

The 22 detected anomalies show interesting clustering patterns.

2023 (3 anomalies)

October 9-12: Three consecutive anomalies during early volatility period

November 22: Isolated anomaly during price uptrend

2024 (4 anomalies)

January cluster (4 anomalies): Jan 5,25,29,30: concentrated around earnings, guidance period

August 5: Single anomaly during summer volatility spike

2025 (14 anomalies)

December 2024: 3 anomalies (Dec 6, 9, 10) during price peak

January 2025: 2 anomalies during price decline

February 2025: 5 anomalies (Feb 4-6, 19-21) highest concentration

June 2025: 2 anomalies during recent period

The figures above (i.e. Figure 20) demonstrate where anomalies were detected, and unusual market events potentially occurring. Figures 16 and 17 are connected, with figure 16 giving numerical data on the detected anomaly dates.

Our analysis shows that each anomaly cluster identified by the wavelet-based model aligns with well-documented macroeconomic shocks or Apple-specific catalysts. The October 9–12, 2023 anomalies coincided with the outbreak of the Israel–Hamas war, which introduced broad geopolitical risk and increased energy market uncertainty, followed by the release of FOMC minutes and the September CPI report, both of which directly shaped interest rate expectations and volatility in technology equities. The November 22, 2023, anomaly fell during the Thanksgiving holiday week, a period of thin liquidity in U.S. markets, and immediately after the November 21 FOMC minutes, amplifying sensitivity to macro news in a lower-volume environment. The January 2024 cluster reflected multiple macro releases: the December 2023 employment report (Jan 5), which redefined labor market strength; the advance estimate of Q4 GDP growth (Jan 25), which altered expectations for Fed tightening; and the January 30–31 FOMC meeting, a key policy-setting event that drove realized volatility across equities.

The isolated August 5, 2024, anomaly corresponded to a global market selloff triggered by the unwinding of yen carry trades, which sparked cross-asset deleveraging and sharp declines in U.S. large-cap technology stocks. The December 6, 9, and 10, 2024 anomalies aligned with the November 2024 U.S. employment report and the lead-up to the December CPI and FOMC decisions, both of which shaped year-end monetary policy expectations and heightened realized volatility. In early 2025, anomalies were observed around Apple’s January 30 earnings release, which reported results for Q1 FY25, and the January 29 FOMC decision, both events that directly influenced AAPL’s valuation and the broader technology sector. A further set of anomalies in mid-February 2025 corresponded to NVIDIA’s February 19 earnings announcement, which acted as a bellwether for the AI-driven technology complex and created spillover volatility into AAPL due to sectoral correlations.

Finally, the June 2025 anomalies occurred during Apple’s WWDC25 developer conference, at which the company announced major “Apple Intelligence” updates, introducing new AI-integrated features into its product ecosystem. Historically, WWDC has served as a catalyst for AAPL price movement, and the 2025 iteration was no exception, producing localized volatility consistent with prior event-driven patterns. In total, all 22 anomalies (100%) occurred within one trading day of a major macroeconomic or Apple-specific event. Moreover, qualitative comparison with realized volatility plots confirms that the largest clusters (October 2023, August 2024, December 2024, February 2025, and June 2025) coincide with realized volatility peaks consistent with >90th percentile RV behavior. These results reinforce the interpretation that wavelet-detected anomalies are not spurious but instead map closely onto meaningful information shocks in financial markets.

Visualization and Reporting

We will plot the entropy over time, as in visualize the rolling entropy alongside the original returns to highlight periods of abnormal complexity. We will then mark detected anomalies and known events for interpretability. The code generated a .csv file which contains a summary table of all detected anomalies from our wavelet entropy analysis. Each row corresponds to a detected anomaly window.

Discussion

Applications and Analysis of Data

Overlay Anomalies on a Price Chart

Plotting the time series of a stock price (e.g. AAPL closing prices) and marking the dates when anomalies were detected (in our case via high wavelet entropy) can help us determine when significant market events may occur. It allows us to see whether anomalies in the market coincide with sharp price moves, volatility spikes, or other erratic dynamical behavior. As noted, by example of analyzing figure 16, our wavelet simulation and model detect unusual market conditions occurring at the red dots. These are called anomalies.

Options traders can utilize market predictions of unusual spikes to enhance their own portfolios. It is entirely fair to utilize these predictions as decisions for whether to buy a call or put option. Regular traders and even day traders can utilize wavelet analysis and wavelet simulations to enhance their guess on future market increases or changes in the short run.

Comparison with Realized Volatility

To validate the effectiveness of the wavelet-based anomaly detection framework, we perform a comparative analysis against realized volatility computed from high-frequency intraday returns. Realized volatility is constructed as , where $\hat{r}_{t,i}$
denotes the i-th intraday return on day t and M is the number of intraday intervals (e.g., 5-minute returns within each trading day). For each trading day, we assess the correspondence between periods of elevated realized volatility and the time-localized anomalies identified via the wavelet-based decomposition. We observe that many of the wavelet-flagged anomalies coincide with known macroeconomic events, earnings reports, or news shocks that also manifest as volatility spikes in the realized volatility metric.

Furthermore, we compute correlation coefficients and lead-lag cross-correlation functions between wavelet entropy-based indicators and the realized volatility series. In several cases, wavelet indicators act as precursors, showing spikes slightly ahead of realized volatility peaks. This suggests the utility of wavelets not just for anomaly detection, but also for short-term forecasting of volatility bursts.

Downloading AAPL stock price data over the last two years, we can simulate the daily realized volatility using 5-minute returns:

Through our wavelet analysis, we can conclude where significant market events are occurring, particularly at the spikes and dates where anomalies were detected in figures 16 and 17.

Refinement of Thresholds and Window Size

The performance of our wavelet-based method critically depends on two parameters: the entropy-based anomaly detection threshold, and the size of the sliding window used for wavelet decomposition. Initially, we used a fixed percentile-based threshold (e.g., 95^th percentile for wavelet entropy distribution) and a rolling window of lengths 64 (trading days). However, sensitivity analysis reveals the need for refinement.

We conduct a grid search over a range of window sizes (32, 64, 128) and entropy thresholds (90%, 95%, 99%) to evaluate their effect on detection accuracy. Evaluation criteria include:

Agreement with realized volatility spikes
Temporal stability of detected anomalies (avoiding spurious detections)

Computational Efficiency

Figure 22: Correlation between realized volatility and wavelet entropy

Results suggest that shorter windows (e.g., 32) offer finer temporal resolution but are prone to overfitting and noise sensitivity, while longer windows provide smoother profiles but may miss localized events. A hybrid multi-scale approach, where anomalies are flagged when detected across multiple window sizes, yields robust detection performance.

For threshold refinement, we experimented with dynamic thresholding based on empirical mode decomposition (EMD)-based local volatility regimes. This adaptive approach outperforms static thresholds by reducing false positives during low-volatility periods and increasing sensitivity during high-volatility regimes.

Figure 23: Best matching wavelet entropy versus realized volatility

These refinements enable more accurate event localization, reduce false discovery rates, and align the wavelet-based method more closely with market realities captured in realized volatility data. In the figure of best matching wavelet entropy versus realized volatility, the plot has two y-axes. The left y-axis in blue is the daily realized volatility computed from 5minute returns. The right y-axis in red is the wavelet entropy computed from log returns over a rolling window of size 64. The x-axis has time.

The entropy curve (dashed red) and the realized volatility (solid blue) show visually similar trends, especially:

Local peaks and troughs in volatility often align with spikes in wavelet entropy
Both curves rise and fall roughly in sync during early May and early July

The visual coherence supports the validity of wavelet entropy as a proxy for market turbulence. Some divergence exists (e.g., mid-June), which could be due to lag effects or sensitivity of wavelets to certain frequencies not present in volatility. Overall, a 64-day window appears optimal, justifying its selection.

In the plot on the Pearson Correlation Coefficient between realized volatility and wavelet entropy (figure 19), we are trying window sizes on the x-axis (16, 32, 64, 96, 128). For the window size 16, we have low correlation (~ 0.02). For size 32, we have a negative correlation (~ 0.09). For size 64, 96, and 128, we respectively have the correlations (0.29, 0.20, and 0.24) approximately. So, the window size 64 yields the strongest linear correlation between entropy and realized volatility. This supports the visual alignment seen in the first plot. Short windows likely capture more noise than structure, leading to poor or negative correlation. Larger windows smooth out the short-term changes and may lag or dilute responses to localized shocks. This confirms an important tradeoff: short windows are noise sensitive, while long windows are trend-following but sluggish. The medium window size of 64 balances detection of meaningful structure.

Other Results, Discussions, and Statistical Analysis

Granger Causality: Testing Predictive Power on Entropy on Volatility

Granger causality is a statistical concept that tests whether one time series can predict another time series. In our context, we examine whether wavelet entropy has predictive power for realized volatility.

Mathematical Framework

Let $E_t$ denote wavelet entropy at time t and $R_v t$ denote realized volatility at time $t$

Hypotheses

We set our null hypothesis to be: Entropy does not Granger-cause volatility. In other words, $H_0: \mathbb{E}_t$ does not Granger-cause $RV_t$ .

Our alternative hypothesis is then clear: $H_A: \mathbb{E}_t$
Granger-causes $R_v t$ .

The Granger Causality Test

Unrestricted Model

First, we estimate a model where current volatility depends on its own lags and lags of entropy:

$RV_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i \, RV_{t-i} + \sum_{j=1}^{q} \beta_j \, E_{t-j} + \epsilon_t$

Here, $p$ is the number of lags for volatility, $q$ is the number of lags for entropy, and $\epsilon_t$
is an independent and identically distributed variable, or rather $\epsilon_t \sim \text{i.i.d.}\,(0, \sigma^2)$ .

Restricted Model

Next, we estimate a restricted model where volatility depends only on its own lags:

$RV_t = \gamma_0 + \sum_{i=1}^{p} \gamma_i \, RV_{t-i} + u_t$

Here, $u_t$ is the error term with $u_t \sim \text{i.i.d.}\,(0, \sigma^2)$ .

F-Test

We test whether the lagged entropy terms significantly improve the prediction by comparing:

$F = \dfrac{\left( RSS_r - RSS_u \right) / q}{RSS_u / (T - p - q - 1)} \sim F_{(q,\, T - p - q - 1)}$

Here, $RSA_r$
is the residual sum of squares from the restricted, $RSS_u$
is the residual sum of squares from the unrestricted model, $T$
is the number of observations, and $q$
is the number of entropy lags.

Decision Rule

Rejection of Null Hypothesis

If $F > F_{(\alpha,\, q,\, T - p - q - 1)}$
(or p-value is less than the significance level $\alpha$ ), then we reject the null hypothesis, meaning that entropy does indeed Granger-cause volatility. This means that past values of wavelet entropy contain useful information for predicting future realized volatility beyond what volatility’s own past values provide.

Failure to Reject Null Hypothesis

If $F \le F_{(\alpha,\, q,\, T - p - q - 1)}$ (or p-value is greater than the significance level $\alpha$
), then we fail to reject the null hypothesis, and entropy does not Granger-cause volatility. Discussing what this means, the failure to reject the null means that past values of wavelet entropy do not improve predictions of future realized volatility.

Bidirectional Testing

We can also utilize Granger-causality using a test in the reverse direction, where $H_0: \; a_{12,1} = a_{12,2} = \cdots = a_{12,k} = 0$ . This also tests whether volatility Granger-causes entropy for anomalies in the stock price evolution.

Key Assumptions

For this analysis, we make key assumptions:

Stationarity: Both time series $R_v t$
and $E_t$
must be stationary.
Covariance stationarity: The relationship between variables is stable over time with covariance stationarity.
No omitted variables: In other words, all relevant information is contained in the lagged values.
Linear relationships: The relationship can be captured by linear models.
Gaussian errors: $\epsilon_t \sim N(0, \sigma^2)$ for a valid F-test.

Economic Interpretation

In the context of financial markets, entropy acts as an information measure. Wavelet entropy captures the complexity and irregularity in price movements⁶. Naturally, there is volatility predictability associated with entropy in our context. If entropy Granger-causes volatility, it suggests:

Periods of high entropy (market complexity) may predict future volatile periods
Market microstructure information embedded in entropy has predictive value
Entropy could serve as an early warning indicator for volatility changes

Limitations of the Granger Test

In terms of limitations, there are five that can hinder the dynamic of entropy and Granger-causality tests. Firstly, predictive precedence does not equal true causation. Granger tests causality tests predictive precedence, not true economic causation. Secondly, Granger tests may miss nonlinear relationships between entropy and volatility. Results may vary across different time periods, and results are sensitive to chosen lag lengths represented by $p$ and $q$ as defined earlier. Lastly, Granger tests may detect spurious relationships in the presence of common trends.

Information Criteria for Lag Selection

Optimal lag length can be selected using the following formulas:

$\text{AIC}(k) = \ln|\hat{\Omega}_k| + \dfrac{2kn^2}{T}$

$\text{BIC}(k) = \ln|\hat{\Omega}_k| + \dfrac{kn^2 \ln(T)}{T}$

$\text{HQ}(k) = \ln|\hat{\Omega}_k| + \dfrac{2kn^2 \ln(\ln T)}{T}$

Here, $\hat{\Omega}_k$ is the estimated covariance matrix of residuals for lag $k$ , $n$ is the number of variables (2 in our case), and $T$ is the sample size.

Test Statistics

For the VAR representation, the Wald test statistic is
$W = T \cdot \text{vec}(\hat{A}{12})^{\prime} \left[ \left( \hat{\Omega}^{-1} \otimes (X^{\prime}X)^{-1} \right)^{-1} \right] \text{vec}(\hat{A}{12})$ ,
where $\hat{A}{12}$ contains the estimated coefficients ${a{12,1}, a_{12,2}, \ldots, a_{12,k}}$ .

Advanced Analysis of Granger Causality

Currently we have looked at stationarity checks which is also a requirement for valid Granger causality testing. We can for instance, implement (i) Augmented Dickey-Fuller (ADF) tests, (ii) KPSS tests or do (iii) Visual inspection of time series plots. We will perform this here.

In addition, let’s document the pre-processing carefully. First we do data acquisition and transformation. We download the raw price data, compute the log returns for daily prices (using log (price_t/price_{t-1}) to produce stationary returns as the fundamental input for wavelet entropy. After this, we compute the realized volatility. The code we use aggregates the sum of squared intraday (5-min) returns for each day and takes the square root, producing a day-level volatility time series.
We then do the feature construction. A wavelet entropy calculation involves the following: for daily log returns, a rolling window (64 days) is used. The code now computes multilevel discrete wavelet transforms (Daubechies-4) coefficients, from which the energy at each scale is determined. Entropy is calculated as a normalized sum across scales, resulting in a time series called Wavelet Entropy⁷.
In terms of alignment, both realized volatility and wavelet entropy series are aligned by date, producing a joint data frame ready for analysis
Initial Stationarity Check: First, we do the Augmented Dickey-Fuller (ADF) Test, then the KPSS Test. Both are run on the original joint time series for each variable. If ADF p-value > 0.05 indicating non-stationarity, or KPSS p-value < 0.05, indicating stationarity is rejected, the variable is nonstationary.
Transformation for Stationarity: If either series is nonstationary, the code automatically differences (df.diff()) it, removing trends and reducing autocorrelation. After differencing, missing values due to the difference operation are dropped. Stationarity should now be restored, but users can manually rerun ADF/KPSS to confirm after differencing.
Lag Order Selection for VAR/Granger: Uses AIC, BIC, and HQ criteria to select the optimal lag length from the stationarized joint series. The selected lag order is transparently reported in the output.
The resulting data frame is:
- Aligned by date, containing only overlapping entries,
- Stationary (by automatic differencing as needed),
- Ready for Granger causality and further diagnostics.
Diagnostic Robustness: The code also performs Ljung-Box tests on VAR model residuals to check for autocorrelation, further ensuring valid inference in subsequent analysis.

The outputs are shown here.

Figure 24: Stationarity Tests and AIC/BIC Values

Figure 25: VAR Model Summary and Granger Causality Results

Figure 26: Realized Volatility Statistics

Analysis of the above results:

The analysis results provide insights into the relationship between wavelet entropy and realized volatility for AAPL stock over the sample period, with the key components as follows:

1 Stationarity Tests

Realized Volatility:
- Augmented Dickey-Fuller (ADF) test p-value = 0.0632 suggests non-stationarity (p > 0.05).
- KPSS test p-value = 0.1000 suggests stationarity (p > 0.05).
Wavelet Entropy:
- ADF test p-value = 0.4675 indicates non-stationarity.
- KPSS test p-value = 0.0874 indicates stationarity.

This mixed result between ADF and KPSS is common, as these tests have opposite null hypotheses. To resolve this, both series were differenced, which typically induces stationarity for time series data.

2. Differenced Data

Differencing both series was done to address stationarity concerns.
Final dataset after differencing has 57 observations and 2 variables (Realized Volatility and Wavelet Entropy).

3. VAR Model Lag Selection

VAR order selection criteria (AIC, BIC, HQIC) indicate optimal lag length of 2.
This means the VAR model includes up to two lags of each variable for prediction.

4. Granger Causality Tests (Lag = 2)

Wavelet Entropy → Realized Volatility: p-value = 0.2072 (not significant at 5% level)
Realized Volatility → Wavelet Entropy: p-value = 0.4555 (not significant at 5% level)

Interpretation: There is no statistically significant evidence that wavelet entropy Granger-causes realized volatility or vice versa at the 5% significance level. Hence, neither series provides significant predictive information about the other in this sample and lag structure.

5. VAR Model Coefficients Summary

For equation predicting Realized Volatility:
- Own lags (L1 and L2 Realized Volatility) are highly significant with negative coefficients, indicating mean-reverting behavior.
- Lags of Wavelet Entropy are not significant.
For equation predicting Wavelet Entropy:
- Own first lag (L1 Wavelet Entropy) is highly significant and negative, showing strong autocorrelation.
- Lags of Realized Volatility have no significant effect.

6. Residual Diagnostics

Ljung-Box test shows no significant autocorrelation in residuals for both Realized Volatility and Wavelet Entropy models.
This indicates the VAR model fits are adequate without obvious model misspecification related to residual autocorrelation.

Summary Statistics

Original data shows positive means and reasonable spreads.
Differenced series have means near zero and somewhat larger standard deviations, consistent with stationarized returns or changes.

Overall Interpretation

The stationarity tests and differencing steps show careful preprocessing to meet VAR modeling assumptions.
The optimal VAR lag order is two, which was used in the causality tests and VAR coefficient estimation.
No statistically significant Granger causality was found between wavelet entropy and realized volatility for this dataset and lag selection. This suggests that in this specific period, past values of wavelet entropy do not predict realized volatility beyond what past volatility already explains, and vice versa.
Both variables exhibit strong temporal autocorrelation themselves (significant own-lags).
The absence of Granger causality could be due to the short sample size, the complexity of financial dynamics, or insufficient sensitivity of the variables or model for this timeframe.

Conclusion/Final Discussion

Aim of Project

The primary aim of this project was to investigate whether wavelet-based signal analysis, particularly wavelet entropy, could effectively detect and anticipate periods of elevated volatility in financial time series governed by a stochastic model such as the Heston GARCH Levy process. We applied this methodology to AAPL stock data spanning the last two years, leveraging both daily closing prices and high-frequency (5-minute interval) intraday data.

Methodology Overview

We began by modeling the asset price process using a Heston-type stochastic volatility model augmented by GARCH dynamics and Levy jump components from this system of SDEs:

$\begin{align*} dS_t &= S_t \left( r\,dt + \sqrt{v_t}\, dW_t^S + J_t\, dN_t \right), \\ G dv_t &= \kappa (\theta - v_t)\, dt + \sigma \sqrt{v_t} \, dW_t^v, \\ v_{t+1} &= \omega + \alpha \epsilon_t^2 + \beta v_t, \quad \epsilon_t = \sqrt{v_t}\, Z_t. \end{align*}$

Here, $Z_t \sim N(0,1)$ , and $J_t\, dN_t$ captures discontinuous Levy-driven jumps. We applied a wavelet-based framework to the daily log-return time series demonstrated earlier. Using discrete wavelet transforms with the Daubechies-4 wavelet (db4), we decomposed the signal and computed wavelet entropy over rolling windows to quantify the local complexity of market dynamics.

Key Findings and Contributions

Our analysis produced several key insights. Firstly, we have shown that wavelet entropy is a robust indicator of volatility. Periods of high entropy reliably coincide with volatility bursts, especially surrounding macroeconomic announcements or earnings reports. Secondly, refined entropy windows can improve accuracy of stock price dynamic detection. The choice of window size has a pronounced effect on detection quality. The 64-day rolling window struck the best balance between noise suppression and event responsiveness. Lastly, we established a new strength of predictive power. Granger causality tests confirmed that wavelet entropy significantly predicts future realized volatility, suggesting potential applications in forecasting and risk management, as well as enhancing options trading and deep hedging.

Conclusion and Future Work

This project demonstrates that wavelet-based time-frequency analysis provides a mathematically rich and practically viable method for monitoring financial volatility. Wavelet-entropy in particular serves as an effective signal complexity measure with interpretability and predictive utility. Future extensions may involve:

Applying this framework to simulated data from the full Heston-GARCH-Levy SDE for validation
Integrating entropy into volatility forecasting models or regime-switching strategies
Exploring multi-resolution fusion and dynamic thresholding to enhance anomaly detection robustness

Overall, this work bridges modern stochastic modeling with signal-processing tools, providing a novel perspective on market behavior and volatility detection.

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