LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A normal model of returns is a story about likelihoods, not a claim that returns are truly normal. The empirical rule puts about 68 percent of draws within one standard deviation of the mean, about 95 percent within two, and more than 99 percent within three. Fit that bell to Nvidia and the bars stick out of both tails. If the bell were true, a move beyond four standard deviations would show up about once every 125.3 years. The history already contains many. The distribution that generates the returns is moving, and volatility is one latent driver you cannot read off a quote screen.
Why it matters
Realized volatility is a proxy: a rolling window, 20 days or 60, sometimes 80, 150, or a year, around a benchmark mean. Change the window and the proxy changes. A stack of proxies can be compressed with a principal component. Implied volatility is another proxy. None of them is the latent process. In the lecture's animation a volatility near 2.6 spreads the return law, and a volatility near zero does not. Trend or momentum can move the same law. A fixed bell omits all of that and then understates the tails.
Two Nvidia windows make the split concrete. High volatility around the tariff period at the start of 2025 has a negative mean return. A calmer stretch from September through December 2024 has a positive mean. Pool them and you get one law that matches neither regime.
The idea in plain words
A Markov chain can carry the latent piece if you define the states. Rank historical volatility and cut it at the 33rd and 66th percentiles: low, mid, and high. Every state can move to every state. Paint the price path and the color changes through time. Conditional on the state, the return law changes with it. The lecture still fits a normal inside each bucket. Low has the smallest variance, mid the next, high the largest, which is partly by construction and still the dynamic the single bell missed. Simulate across the chain and the mixture has excess kurtosis of 0.687, a kurtosis of 3.687 against the normal value of 3. That is closer to the fat tails in the data. It is still three normals, not the real law. Adding a second factor, such as trend, turns three states into nine. Naming "bull" and "bear" is its own modeling choice.
A hidden Markov model skips the naming. You pick a count of states, three or five or ten, and the data assign the returns to hidden regimes that can mix volatility, trend, and anything else you did not list. You can read the mean and the spread of each state's emission. You cannot point at state 3 and say "high volatility" with the same confidence, any more than a principal component tells you its name. In the lecture's three-state fit the standard deviations are 1.48, 1.74, and 6.46. The wide one is at least behaving like a volatile regime.
How to use it
Fitting uses a forward pass and a backward pass. Their combination gives state probabilities. The Baum–Welch algorithm, which wraps those passes, updates the transitions and the emissions to raise the likelihood of the observed series until it settles. The estimates are maximum-likelihood estimates, with the large-sample properties that come with that, when the assumptions hold. The lecture's code does this for the three-state diagram. More states can fit more dynamics and overfit, and they get expensive. A simpler chain that you defined yourself can explain as much, or more, and a business that wants a sentence per regime will prefer it. ARCH and GARCH are other ways to let variance move without a hidden label.
What usually goes wrong
Quoting a 125-year waiting time from a single normal, then being surprised by a month of large Nvidia moves. Treating percentile buckets as the latent process rather than a proxy. And assuming the hidden model wins because it is harder to explain. The lecture's test is out-of-sample performance against the loss of a name for each state. Interpretability is a feature of the three-color chain. It is not free in the hidden one.
Questions people ask
What was the 125.3-year figure?
The waiting time, under the fitted normal, for a Nvidia return beyond four standard deviations. The sample already has many such returns, so the normal likelihood is far too small.
What did the three explicit regimes do to kurtosis?
Draws from the low, mid, and high normals, mixed by the chain, had kurtosis 3.687. A normal has kurtosis 3. The extra 0.687 is the fat-tail piece the mixture picked up.
What are 1.48, 1.74, and 6.46?
The standard deviations of the three hidden states in the lecture's fit. The third is wide enough to look like a stress regime. The fit does not label it.