GARCH treats positive and negative returns symmetrically: a return of has exactly the same impact on next-period volatility as a return of . Equity markets disagree. Bad news amplifies volatility more than good news of the same magnitude — the leverage effect first observed by Black (1976). Symmetric GARCH systematically under-reserves for the volatility that follows a crash. This note introduces the two standard asymmetric extensions (GJR-GARCH and EGARCH), derives the multi-period volatility forecast from a fitted GARCH model, explains why the square-root-of-time rule is wrong for dependent returns, and presents filtered historical simulation (FHS) — the semi-parametric method that marries the GARCH filter with the empirical residual distribution to capture both volatility dynamics and heavy tails without committing to a parametric tail model.
Learning objectives. By the end of this note you should be able to:
- write the GJR-GARCH variance equation and identify the leverage coefficient;
- state the multi-period GARCH variance forecast recursion and its long-run limit;
- explain why the square-root-of-time scaling rule fails for GARCH returns and state the conditions under which it is an approximation;
- describe the FHS algorithm step by step and explain what it buys over plain historical simulation;
- compute one-step-ahead conditional VaR and ES from a fitted GARCH model.
In this lesson
-
- The leverage effect and asymmetric GARCH
-
- Volatility forecasting from GARCH
-
- The square-root-of-time rule and its failures
-
- Conditional VaR and ES from GARCH
-
- Filtered historical simulation
Before you start
- ARCH and GARCH Volatility Models
- Value-at-Risk: Definition, Properties, and the Three Computation Methods
- Expected Shortfall and Other Tail Risk Measures
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Quantitative Risk Management syllabus.