Modeling Volatility and GARCH Family › Applications of GARCH Volatility

Michael BrenndoerferDecember 10, 20255 min read
Quantitative FinanceAll chapters
Part 8 of 15Econometrics and Statistical Modeling
Chapter 42Modeling Volatility and GARCH Family › Applications of GARCH Volatility
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Dynamic Value at Risk

You use GARCH to estimate time-varying Value at Risk. Rather than assuming constant volatility, you update daily VaR with the conditional variance:

VaRt,α=−μ−zα⋅σt\text{VaR}_{t,\alpha} = -\mu - z_\alpha \cdot \sigma_t

where:

  • VaRt,α\text{VaR}_{t,\alpha}: Value at Risk at confidence level α\alpha for time tt
  • μ\mu: expected daily return
  • zαz_\alpha: lower tail quantile of the distribution (typically negative)
  • σt\sigma_t: conditional volatility forecast
In[25]:
Code
# Calculate 99% VaR using GARCH volatility
confidence = 0.99
mu_est = results.params["mu"]
nu_est = results.params["nu"]

# Unit-variance Student-t quantile, matching the GARCH model's standardization
t_scale = np.sqrt(nu_est / (nu_est - 2))
z_99 = stats.t.ppf(1 - confidence, nu_est) / t_scale

# Time-varying VaR
var_99 = -(mu_est + z_99 * conditional_vol)

# For comparison: constant volatility VaR
constant_vol = returns.std()
var_99_constant = -(mu_est + z_99 * constant_vol)
Out[26]:
Visualization
Time series of daily returns with negative dynamic GARCH and constant-volatility 99 percent VaR thresholds.
Daily returns compared with dynamic GARCH and constant-volatility 99 percent Value at Risk thresholds from 2000 to 2024. The dynamic loss threshold moves farther below zero during volatile periods and closer to zero during calm periods, while the constant threshold cannot adapt to changing market conditions.

The dynamic VaR adapts to market conditions. During the 2008 crisis, GARCH VaR increases to reflect higher risk, then gradually declines as markets stabilize. By contrast, constant VaR underestimates risk during crises and overestimates during calm periods. This adaptive property makes GARCH valuable for risk management.

Out[27]:
Visualization
Scatter plot of GARCH VaR exceedances over time.
GARCH-based 99 percent Value at Risk exceedances from 2000 to 2024 mark days when actual returns fell below the dynamic loss threshold. Because the threshold responds to estimated volatility, exceedances are distributed across the sample rather than being determined by one fixed cutoff. A well-calibrated 99 percent VaR should produce approximately 1 percent exceedances on average.
Scatter plot of constant VaR exceedances over time.
Constant-volatility 99 percent Value at Risk exceedances mark days when returns fell below one fixed loss threshold. Their temporal concentration shows the limitation of a static risk measure when market volatility changes substantially over time.

The exceedance analysis reveals a critical difference between the two approaches. The constant VaR produces clustered violations during the 2008 and 2020 crisis periods, when the fixed threshold was inadequate for the high volatility. The GARCH-based VaR distributes exceedances more uniformly across time because it adapts to changing market conditions. For a 99% VaR, we expect approximately 1% of observations to exceed the threshold; the dynamic approach comes closer to achieving this uniform coverage.

Option Pricing Implications

The Black-Scholes framework assumes constant volatility, but GARCH shows volatility is predictably time-varying. This has several implications for option pricing:

  • Volatility term structure: GARCH implies that implied volatility should vary with option maturity, which is consistent with the term structure we observe in markets
  • Volatility forecasting for trading: If GARCH forecasts volatility higher (lower) than current implied volatility, options may be underpriced (overpriced)
  • GARCH option pricing models: Academic research has developed option pricing formulas under GARCH dynamics, though these are more complex than Black-Scholes

The connection between GARCH forecasts and implied volatility creates opportunities for volatility trading strategies. We'll explore regression-based approaches to this relationship in the next chapter.

Portfolio Volatility Timing

GARCH volatility forecasts can inform portfolio allocation. A simple volatility timing strategy adjusts exposure based on predicted risk:

wt=σtargetσ^t+1w_t = \frac{\sigma_{\text{target}}}{\hat{\sigma}_{t+1}}

where:

  • wtw_t: portfolio weight for the risky asset
  • σtarget\sigma_{\text{target}}: target volatility level
  • σ^t+1\hat{\sigma}_{t+1}: forecasted volatility for the next period

If predicted volatility is high, you reduce exposure to maintain constant risk. When low, you increase exposure. This approach connects to the portfolio optimization frameworks we'll develop in Part IV.