Bayesian Volatility Modeling

Bayesian Volatility Modeling: GARCH vs Stochastic Volatility

This project applies Bayesian inference to estimate and compare two volatility models on S&P 500 (IVV ETF) weekly data. The models were trained on 2013 to 2017 data and evaluated on 2018 to present, capturing events like the COVID-19 market crash.

The Models

GARCH(1,1) models conditional variance as a deterministic function of past squared shocks and past variance:

\[\sigma_t^2 = \alpha_0 + \alpha_1 a_{t-1}^2 + \beta_1 \sigma_{t-1}^2\]

Stochastic Volatility (SV) treats log-volatility as a latent AR(1) process:

\[y_t = \epsilon_t \exp\left(\frac{h_t}{2}\right), \quad h_{t+1} = \mu + \phi(h_t - \mu) + \delta_t \sigma\]

Why Bayesian?

Instead of finding a single “best” parameter set via maximum likelihood, Bayesian inference computes the full posterior distribution of parameters using MCMC sampling (Stan). This gives more robust estimates and naturally quantifies uncertainty.

Interactive Demo

Key Findings

• The SV model achieved lower MSE on both training and test data overall
• GARCH(1,1) better captured the COVID-19 volatility spike
• SV performed better during stable periods
• Both models benefit from Bayesian estimation

Technology Stack

• Python 3.9+, Shiny for Python, Plotly, pandas, NumPy
• Stan (PyStan) for Bayesian MCMC sampling
• ArviZ for Bayesian diagnostics