Lesson 3.5: Capstone: Modeling S&P 500 Volatility
This capstone project synthesizes our entire time series module. We will build an ARMA-GARCH model to forecast the volatility of the S&P 500, following a professional quant workflow from start to finish.
The Professional Quant's Workflow
Workflow Summary
- Data Preparation: Acquire S&P 500 data and calculate daily log returns.
- Exploration: Plot the returns to visually confirm stationarity and volatility clustering.
- Mean Model Specification (ARMA): Use ACF/PACF plots and AIC/BIC to find the best-fitting ARMA model for the returns.
- Volatility Model Specification (GARCH): Test the ARMA residuals for ARCH effects. If present, specify and fit a combined ARMA(p,q)-GARCH(1,1) model.
- Diagnostics: Check that the standardized residuals of the full model are white noise.
- Forecasting: Use the final model to forecast future conditional volatility.
Python Implementation
GARCH Modeling in Python
import numpy as np
import pandas as pd
import yfinance as yf
from arch import arch_model
# 1. Get Data
sp500 = yf.download('^GSPC', start='2010-01-01', end='2023-12-31')
returns = 100 * np.log(sp500['Adj Close']).diff().dropna()
# 2. Specify and Fit GARCH(1,1) Model
# For simplicity, we assume a constant mean model (often sufficient for daily returns)
# vol='Garch', p=1, q=1 specifies a GARCH(1,1) model.
garch_spec = arch_model(returns, vol='Garch', p=1, q=1)
garch_fit = garch_spec.fit(update_freq=5)
print(garch_fit.summary())
# 3. Forecast Future Volatility
# Forecast volatility for the next 30 trading days
forecast_horizon = 30
forecast = garch_fit.forecast(horizon=forecast_horizon)
# The output is variance, so take the square root
# We annualize by multiplying by sqrt(252 trading days)
future_volatility = np.sqrt(forecast.variance.iloc[-1])
annualized_forecast = future_volatility * np.sqrt(252)
print("\nAnnualized Volatility Forecast for next 30 days:")
print(annualized_forecast)
What's Next? Moving to Multiple Series
You have now mastered the univariate time series toolkit. The ARIMA-GARCH framework is the foundation of modern financial econometrics.
The next logical step is to expand our view from one time series to many. How do we model the dynamic, interconnected relationships between multiple assets? In the next module, we will introduce **Vector Autoregression (VAR)** models to do just that.