LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
ARCH and GARCH model realized variance, not the implied volatility backed out of option prices. Variance is written as a function of recent squared return shocks, so it can cluster, revert, and fatten the tails. Tim Bollerslev's GARCH adds lagged variance. A GARCH(1,1) can stand in for an ARCH model with infinitely many lags.
Why it matters
Volatility is a deviation from an expected return you do not observe. A rolling calculation is one proxy. The lecture's 30-day Apple series tracks a brokerage series of the same length. A shorter window is jumpier. None of the paths is a forecast. Implied volatility is the other object: invert Black–Scholes so the model price matches the market. Point readings in the lecture include about 29 percent, 18 percent, and 23 percent. Implied and realized series often move together, and a common pattern is that implied volatility is rich relative to what subsequently realizes.
A regression of a short forward average of realized volatility on implied volatility has slope below the 45-degree line. Split at the crossing and the high-volatility regime slopes even more. The lecture's displayed regression uses a five-day forward average. The models below are fit to realized variance anyway. Implied volatility is already available from the option market. The missing piece is a forward-looking model of realized variance that can carry the stylized facts.
The idea in plain words
Equity returns have fatter tails than a normal bell. Kurtosis of 10, 12, 15, 18, or above 20 shows up. A classroom value-at-risk with constant volatility then understates both tails. In one comparison the two thresholds differ by almost 20 percent. A "six-sigma" headline is often that thin-tailed benchmark.
Other facts: volatile stretches cluster, extremes fade toward a long-run level, and negative returns lift volatility more than same-sized gains lower it. One window shows a change near minus 0.06 after a gain and near plus 0.08 after a loss. Whether very old volatility still matters is described as disputed.
Robert Engle's 1982 ARCH model writes the return as a mean plus a shock. The shock is a time-varying scale times an innovation with mean zero and variance one, often normal, sometimes heavier. The scale squared is an intercept plus coefficients on lagged squared shocks, out to lag q. ARCH(1) keeps one lag. Conditional on the past, the variance equals that scale squared, so it is not a constant. Unconditional variance is a function of the alphas. Kurtosis exceeds the normal value of 3 when the formula is defined and positive in the way the lecture computes, which is the fat-tail result.
How to use it
The search is not a contest. An exponentially weighted average, which only chases lags, has root-mean-square error near 42 percent and R-squared of 5.12 percent. An ARCH(1) cuts the error to about 25 percent and raises the R-squared. The point of the comparison is the dynamics, not a claim that the search was unique.
Bollerslev's 1986 GARCH keeps the same return equation and adds lagged conditional variance, the beta terms. An ARCH model of infinite order can be rearranged into a GARCH(1,1): multiply the variance equation by beta, subtract, and collect terms. The lecture's GARCH(1,1) has out-of-sample error comparable to ARCH(1) and a higher R-squared. On single-stock samples the explained share stays low, about 5 percent for ARCH and about 8 percent for GARCH, both under 10 percent. The lecture cites literature in which intraday returns, such as five-minute increments used to forecast daily variance, let this family explain on the order of 40 to 60 percent of the variation.
A 5 percent value-at-risk threshold makes the risk point. Under constant variance the lecture counts exceedances 40 percent of the time. Under GARCH the count is 9.74 percent, nearer the 5 percent the threshold named. That is one horse race, and the lecture says it can be improved.
What usually goes wrong
Fitting these equations to implied volatility answers a question the option market already prices. Using a normal, constant-volatility value-at-risk ignores the tails and the clustering the models were built for. Reading a single-stock R-squared under 10 percent as a failure misses the lecture's distinction: daily equity variance is hard, and the intraday result is the stronger documented case. An infinite ARCH is also the wrong operational form once a GARCH(1,1) carries the same idea with two lags.
Questions people ask
Why is the conditional variance not a constant?
It is defined as the scale squared, which depends on recent squared shocks and, in GARCH, on recent variance. Multiply that scale by a unit-variance innovation and the shock's spread changes with time.
What did the moving-average race show?
The weighted average had error near 42 percent and R-squared of 5.12 percent. ARCH(1) had error near 25 percent and a higher R-squared, on that one search.
Why did GARCH miss a 5 percent value-at-risk less badly?
Constant variance produced exceedances 40 percent of the time. GARCH produced 9.74 percent. Both missed 5 percent. The second miss was smaller.