Modeling Stock Returns and Volatility Using Bivariate Gamma Generalized Laplace Law

1. Problem Statement

The paper addresses two primary challenges in financial modeling and statistical inference:

1. Limitations of Univariate Models: While the Generalized Asymmetric Laplace (GAL) distribution (also known as the Variance-Gamma distribution) is widely used to model asset returns due to its ability to capture heavy tails and skewness, it is typically studied in a univariate setting. In this setting, the mixing gamma variable (representing stochastic volatility or business time) is unobserved, making Maximum Likelihood Estimation (MLE) complex and computationally intensive.
2. Handling Observed Volatility: In many financial contexts, volatility indices (like the VIX) are observed alongside asset returns. Standard stochastic volatility models often treat volatility as a latent (hidden) variable, requiring complex filtering techniques (e.g., Hidden Markov Models). The authors seek a framework where both the return and the volatility proxy are modeled jointly as observable random variables.

2. Methodology: The BGGL Model

The authors introduce the Bivariate Gamma-Generalized Laplace (BGGL) distribution. This is a bivariate extension of the standard GAL model where the mixing gamma variable is observed.

• Stochastic Representation: Let $Z \sim N(0,1)$ and $X \sim \text{GAM}(\alpha, \beta)$ be independent. The random vector $(X, Y)$ is defined as:
  $$Y = \delta + \mu X + \sigma \sqrt{X} Z$$
  Here, $X$ represents the observed volatility proxy (or business time), and $Y$ represents the asset return.
• Joint Density: The joint probability density function (PDF) of $(X, Y)$ is derived explicitly. It is shown that the conditional distribution $Y|X$ is Gaussian, while the marginal distribution of $X$ is Gamma, and the marginal of $Y$ is GAL.
• Exponential Family Structure: The authors demonstrate that the BGGL family is a curved exponential family. Although it has 5 parameters ($\alpha, \beta, \delta, \mu, \sigma$), its sufficient statistics span a 6-dimensional space. This non-standard structure is crucial for understanding the behavior of the estimators.
• Fisher Information: The paper derives the Fisher Information Matrix (FIM). A key finding is that for the shape parameter $\alpha \le 1$, the FIM becomes singular (infinite) for certain parameters (specifically related to $\delta$), indicating non-standard asymptotic behavior.

3. Key Contributions

The paper makes three principal theoretical and methodological contributions:

1. Simplified Maximum Likelihood Estimation (MLE):
   Unlike the univariate GAL model where MLE requires numerical optimization of a complex likelihood function, the bivariate formulation allows for explicit, closed-form solutions for the location ($\delta$), skewness ($\mu$), and scale ($\sigma$) parameters.

   • The MLEs for $\delta$ and $\mu$ correspond exactly to the Least Squares estimators in a weighted linear regression model (specifically, a regression without an intercept where the error variance is proportional to $X$).
   • The MLEs for the gamma parameters ($\alpha, \beta$) are derived from the marginal distribution of $X$ and are available in closed form (involving the digamma function).

2. Non-Standard Asymptotic Theory:
   The authors establish that the convergence rates of the MLEs depend critically on the shape parameter $\alpha$ of the mixing gamma distribution:

   • Case $\alpha > 1$: The estimators are asymptotically normal with the standard $\sqrt{n}$ convergence rate.
   • Case $\alpha = 1$: The estimator $\hat{\delta}$ converges at a rate of $\sqrt{n \ln n}$, and the limiting distribution remains Gaussian but with a different variance structure.
   • Case $\alpha < 1$: The convergence rate for $\hat{\delta}$ accelerates to $n^{1/(2\alpha)}$ (faster than $\sqrt{n}$). Crucially, the limiting distribution is non-Gaussian, involving a stable subordinator ($\xi_\alpha$). This is a rare phenomenon where the MLE converges faster than the standard rate but loses normality.

3. Financial Application Framework:
   The paper proposes a novel way to model stock returns and volatility jointly. Instead of treating volatility as a latent state, they transform the observed volatility index (VIX) into a Gamma-distributed variable $X$ and model the pair $(X, Y)$ directly. This avoids the need for hidden Markov models.

4. Key Results

• Estimator Properties:

  • The estimators $\hat{\delta}_n$ and $\hat{\mu}_n$ are unbiased.
  • For $\alpha > 1$, the joint distribution of estimators is asymptotically multivariate normal with a block-diagonal covariance matrix, implying asymptotic independence between the gamma parameters and the regression parameters.
  • For $\alpha \le 1$, the estimators $\hat{\delta}_n$ and $\hat{\mu}_n$ become asymptotically independent, a property not present in the $\alpha > 1$ case.

• Simulation Study:
  Simulations with sample sizes $n=50$ and $n=500$ confirm the theoretical findings.

  • For $\alpha = 1$ and $\alpha = 0.25$, the Root Mean Squared Error (RMSE) for $\delta$ is very low, consistent with the faster convergence rates.
  • For $\alpha = 2$ and $\alpha = 5$, the RMSE for $\delta$ increases, reflecting the slower $\sqrt{n}$ convergence rate and the influence of the heavy tails in the mixing distribution.

• Empirical Application:
  The model was applied to weekly data (2019–2024) for three major US indices: S&P 500, DJIA, and NASDAQ 100, paired with their respective volatility indices (VIX, VXD, VXN).

  • Results: The BGGL model provided a satisfactory fit for all three indices.
  • Diagnostics: Quantile-Quantile (QQ) plots showed that the residuals (extracted from the regression) were well-approximated by a standard normal distribution, and the transformed volatility variable $X$ fit the Gamma distribution well.
  • Interpretation: The authors interpret the variable $X$ as "unexpected" or "surprise" volatility (the deviation of current volatility from the autoregressive expectation), allowing the model to capture the joint dynamics of returns and volatility shocks using IID assumptions rather than complex time-series dependencies.

5. Significance

• Computational Efficiency: By observing the mixing variable, the paper transforms a difficult non-linear optimization problem into a simple linear regression problem, making parameter estimation significantly more tractable.
• Theoretical Insight: The discovery of super-fast convergence rates ($n^{1/(2\alpha)}$) coupled with non-Gaussian limits for $\alpha < 1$ adds a significant new case to the literature on non-standard asymptotic behavior in MLE.
• Practical Financial Modeling: The approach offers a parsimonious alternative to complex Stochastic Volatility (SV) and GARCH models. By treating volatility as an observable input rather than a latent state, it simplifies the modeling of the joint distribution of returns and volatility, which is essential for risk management and derivative pricing.

In conclusion, the paper successfully bridges the gap between theoretical probability (bivariate mixtures, stable laws) and practical financial econometrics, offering a robust, analytically tractable model for asset returns driven by observed volatility.