EGARCH and Asymmetric Volatility

A plain GARCH model treats a 3% price drop and a 3% price rise as generating exactly the same amount of future volatility, because it only sees the square of the return, which erases the sign. But equity markets don't behave this way in practice: a sharp drop tends to trigger a much bigger jump in subsequent volatility than an equally sized rise does — panic and forced deleveraging spread faster than calm optimism. This is the well-known "leverage effect" (partly because a falling stock price mechanically raises a company's leverage ratio, making its equity riskier). EGARCH (Exponential GARCH) is built specifically to let bad news and good news have different-sized effects on future volatility, while also fixing a technical annoyance in plain GARCH: keeping variance positive without needing to constrain parameters.

An analogy: a nervous crowd versus a calm crowd

Picture a crowded room reacting to news. A sudden piece of bad news (a fire alarm) spreads panic and jostling almost instantly and disproportionately — everyone's on edge, bumping into each other, for a while after. A piece of good news of similar "size" (someone announces free food) causes some excited chatter but nowhere near the same lasting disruption to the room's overall commotion level. EGARCH encodes exactly this asymmetry into how it updates its volatility forecast: bad news (negative returns) gets a bigger, more lasting bump to predicted future volatility than good news (positive returns) of the same magnitude.

The mechanics, one symbol at a time

EGARCH models the log of variance (rather than variance directly) as evolving with both the size and the sign of the standardized past shock:

$$\ln(\sigma_t^2) = \omega + \beta \ln(\sigma_{t-1}^2) + \alpha \left[ |z_{t-1}| - \mathbb{E}|z_{t-1}| \right] + \gamma\, z_{t-1},$$

where $\sigma_t^2$ is today's conditional variance, $z_{t-1} = \varepsilon_{t-1}/\sigma_{t-1}$ is yesterday's standardized shock (the residual divided by its own volatility, so it's on a comparable, roughly unit scale). In plain English: today's log-variance depends on yesterday's log-variance ($\beta$, persistence), the magnitude of yesterday's shock beyond what's typically expected ($\alpha$, standard volatility clustering), and — crucially — the sign of yesterday's shock through the $\gamma z_{t-1}$ term. If $\gamma$ is negative (the typical empirical finding for equities), a negative shock ($z_{t-1} < 0$) adds a positive amount to log-variance (since negative times negative is positive), pushing future volatility up, while a positive shock of the same size adds a negative amount, pushing future volatility down — the asymmetric, leverage-style response. Because variance is modeled through its log, $\sigma_t^2 = \exp(\ldots)$ is automatically positive no matter what values the parameters take, unlike plain GARCH, which needs explicit positivity constraints on its coefficients.

Worked example 1: bad news versus good news, same size

Suppose fitted parameters are $\omega = 0.01$, $\beta = 0.95$, $\alpha = 0.10$, $\gamma = -0.08$, and yesterday $\ln(\sigma_{t-1}^2) = -1.0$ with $\mathbb{E}|z| \approx 0.8$ (typical for standard normal-ish shocks). For a bad-news shock $z_{t-1} = -2$ (a 2-standard-deviation drop):

$$\ln(\sigma_t^2) = 0.01 + 0.95(-1.0) + 0.10(2 - 0.8) + (-0.08)(-2) = 0.01 - 0.95 + 0.12 + 0.16 = -0.66.$$

For a good-news shock of the same size, $z_{t-1} = +2$:

$$\ln(\sigma_t^2) = 0.01 - 0.95 + 0.10(2-0.8) + (-0.08)(2) = 0.01 - 0.95 + 0.12 - 0.16 = -0.98.$$

The bad-news case gives $\ln(\sigma_t^2) = -0.66$ versus the good-news case's $-0.98$ — exponentiating, $\sigma_t^2 = e^{-0.66} \approx 0.517$ versus $e^{-0.98} \approx 0.375$. A 2-standard-deviation drop pushes forecast variance about 38% higher than an equally-sized rise ($0.517/0.375 \approx 1.38$) — the asymmetry made concrete.

Worked example 2: forecasting one day of variance

Continuing from the bad-news case, $\sigma_t^2 \approx 0.517$ (in daily variance units, roughly a 71.9% annualized vol equivalent if this were in annualized terms, though here it's just illustrative daily units). If instead yesterday's shock had been calm ($z_{t-1} = 0$, no news either way): $\ln(\sigma_t^2) = 0.01 - 0.95 + 0.10(0 - 0.8) + 0 = 0.01 - 0.95 - 0.08 = -1.02$, giving $\sigma_t^2 = e^{-1.02} \approx 0.361$ — noticeably lower than after the bad-news shock, confirming the model expects volatility to stay elevated specifically because of the sign and size of what happened, not just because something happened.

What this means in practice

EGARCH (and its close cousin GJR-GARCH, see GJR-GARCH and the Leverage Effect) is standard in equity volatility modeling and option pricing precisely because ignoring the leverage effect causes a plain GARCH model to understate near-term volatility risk after a sell-off and overstate it after a rally — a systematic, directional bias that matters for VaR forecasts, options hedging, and any volatility-targeting strategy.