Chapter 2: Empirical Properties of Financial Volatility#

If you look at almost any financial market over time, you’ll notice a clear pattern: calm stretches tend to be followed by more calm, and wild stretches by more wild. This chapter looks at the real‑world statistical patterns of volatility — the patterns that keep showing up across stocks, indices, currencies, and commodities. Knowing these patterns gives you solid facts for building models, forecasting risk, and making smarter trading choices.

The Big Picture#

Why should you care about how volatility actually behaves? Because if you want to measure, predict, or trade volatility, you first need to understand its real habits. Many simple models assume that volatility is constant or that returns form a neat bell curve. Real data tells a different story: volatility bunches up, pulls back toward a long‑run average, produces extreme moves more often than expected, and moves in uneven ways. This chapter walks you through the most important of these facts. By the end, you’ll have a mental toolkit to recognise what “normal” volatility looks like, and you’ll be better at spotting when markets are acting strangely.

Volatility Clustering and Autocorrelation#

If you plot daily absolute returns or squared returns for a stock index, you see a shape like a mountain range — quiet valleys and stormy peaks. Large moves are often followed by more large moves; small moves by more small moves. This pattern is called volatility clustering.

To measure how long the clustering lasts, we use autocorrelation. The autocorrelation of a time series tells us how much today’s value is linked to a past value. For returns themselves, the link is usually near zero: you can’t guess tomorrow’s direction from today’s. But for measures of volatility, like squared returns or absolute returns, the autocorrelation is positive and fades slowly — often over many days or even weeks.

Autocorrelation function (ACF): A chart that shows how strongly a series relates to its own past values. For volatility measures, the ACF often stays positive for 50 to 100 lags.

Why does clustering happen? Think of a crowd in a stadium. When everyone is seated and quiet, small noises die away quickly. But once a wave starts, people react to each other, and the wave can circle the stadium many times. In markets, news, emotions, and trading strategies feed off one another. A sharp price drop triggers stop‑loss orders, margin calls, and hedging trades, which create more price movement. That movement grabs attention and more trading, keeping volatility high until the information is fully absorbed and unsettled positions are closed.

This clustering is why a simple historical standard deviation over a fixed window can be misleading. A single large event bumps the estimate up for the whole window, then drops out suddenly, making the estimate jump around. Models that explicitly capture clustering, like Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, use past returns and past volatility to forecast future volatility. They were built to capture this autocorrelation pattern.

📝 Section Recap: Volatility clumps together in time: quiet periods follow quiet ones, turbulent periods follow turbulent ones. This shows up as slow‑decaying positive autocorrelation in squared or absolute returns.

Mean‑Reversion Toward a Long‑Term Level#

Volatility can stay high or low for long stretches, but it does not drift off to infinity or crash to zero for good. Over time, it tends to be pulled back toward a typical long‑run average. This property is called mean‑reversion.

Picture a rubber band attached to a peg. You can stretch the band far away — that’s a spike in volatility — but eventually the tension pulls it back. The longer and further it’s stretched, the stronger the pull. In volatility, that long‑term average level is not a fixed physical constant; it depends on the asset, the economy, and the time scale. For a broad equity index like the S&P 500, the long‑term annualised volatility has often been around 15–20%, but it can spend months above 30% or below 10%.

Mean‑reversion has a big impact on forecasting: if today’s volatility is unusually high, your best guess for future volatility is lower than today’s. If it’s unusually low, you should expect it to rise. The pullback isn’t instant — it can take weeks or months — but it’s a steady force. That’s why many volatility‑based strategies are built on the idea of selling volatility when it spikes and buying it back when things calm down, though the timing is tricky.

📝 Section Recap: Volatility is mean‑reverting: it tends to return to a long‑run average level after extreme moves, giving forecasters and traders a natural anchor.

Fat Tails and Negative Skew in Return Distributions#

If you plot a histogram of daily stock returns, you might expect a clean bell‑shaped curve — the normal distribution. Real returns, however, have two glaring differences: fat tails and negative skew.

Fat tails mean that extreme events happen much more often than a normal distribution would predict. Under a normal curve, a 5‑standard‑deviation daily move should appear roughly once every 7,000 years. In real equity markets, such moves show up a few times per decade. The tails are “fatter” because the chance of very large losses (and very large gains) is simply higher. The technical term for how heavy the tails are is kurtosis.

Kurtosis: A measure of tail heaviness. Higher kurtosis means more frequent extreme values compared to a normal distribution.

Negative skew means the distribution is lopsided: the left tail (losses) stretches out further or is plumper than the right tail (gains). In stock markets, big down days are both more common and more severe than big up days of the same size. One reason is that fear and forced selling — margin calls, automatic hedging — can speed up a decline, while rallies usually build more gradually.

Why does this matter for volatility? Standard deviation — the most common volatility measure — treats up moves and down moves as equal. When the distribution is negatively skewed, a simple standard deviation understates the risk of a large loss. Downside‑focused measures, such as semi‑variance or Value‑at‑Risk, try to capture this unevenness.

Fat tails and negative skew together explain why option markets price out‑of‑the‑money put options at a premium compared to call options — an effect we’ll explore deeply when we look at the volatility surface.

📝 Section Recap: Real return distributions have fat tails (many more extreme moves than a normal curve) and negative skew (larger, more frequent left‑tail losses). This unevenness challenges simple volatility measures that treat all moves the same.

The Leverage Effect: Rising Volatility When Prices Fall#

One of the most reliable relationships in equity markets is the leverage effect: when a stock’s price falls, its volatility tends to rise. The name comes from a simple corporate‑finance idea. A company’s equity is what’s left over after paying off debt. If the stock price drops, the market value of debt compared to equity rises — the company becomes more leveraged. Higher leverage makes the equity riskier, so its volatility goes up.

But the leverage effect is not only about individual stocks. It appears strongly in equity indices, too. A wider explanation is the volatility feedback idea: if investors expect future volatility to rise, they demand a higher expected return as compensation. For a given stream of future cash flows, a higher required return means a lower current price. So an expected rise in volatility can cause an immediate price drop, which then confirms the higher volatility — a feedback loop.

In the data, the correlation between daily returns and later changes in volatility is negative. That’s why volatility indices like the VIX often shoot up during market crashes. It also has a practical side: if you own a stock and the price drops, your position becomes riskier — not just because you lost money, but because the remaining position now has higher expected volatility. Hedging plans must take this into account.

The leverage effect is not symmetric — rising prices usually lower volatility, but the effect is smaller and slower. That asymmetry adds to the negative skew we already saw.

📝 Section Recap: The leverage effect is the inverse link between stock prices and volatility: falling prices drive volatility up. This happens partly from higher financial leverage and partly from a loop where expected higher volatility pushes prices down.

The Volume–Volatility Positive Correlation#

On days when trading volume jumps, volatility also tends to jump. This volume–volatility correlation is one of the most solid stylised facts across all asset classes and time scales — within a day, over a month, or anywhere in between.

Why does it happen? Several ideas exist. The mixture of distributions hypothesis says both volume and volatility are driven by an unseen flow of information. On news‑heavy days, traders disagree more about the right price, leading to more trades and bigger price changes. Another view points to market‑microstructure effects — order splitting or inventory management — that cause trades to push prices around mechanically.

Think of an earnings announcement. Before the news, volume is usually light and volatility low. The moment the news hits, trading explodes and the price often jumps sharply. Over the next few hours, volume stays high while the market digests the information, and volatility stays elevated. The same pattern repeats around economic releases, central bank decisions, and geopolitical shocks.

For a trader, volume can act as a real‑time hint about volatility. If you see volume spiking without an obvious price move, it might signal an upcoming breakout. Conversely, very low volume often goes hand‑in‑hand with low volatility — and that can make traders too comfortable. The relationship is positive but not a straight line: extremely high volume does not always mean proportionally higher volatility, because liquidity can absorb part of the impact.

📝 Section Recap: Trading volume and volatility move together. Information arrival triggers both, making volume a useful companion indicator for gauging current and future volatility.

The Distribution of Volatility Itself#

So far we’ve talked about the distribution of returns. What about volatility itself? If you take a rolling window of historical volatility — say, a 30‑day annualised standard deviation — and plot its histogram, you’ll see it is not symmetric. It is right‑skewed and roughly log‑normal.

A log‑normal distribution means that the logarithm of the variable is normally distributed. For volatility, this means very low values are common, moderate values are typical, and extremely high values happen now and then but can be staggeringly large — way above the median. The distribution has a long right tail.

This shape makes sense. Volatility cannot go below zero (a hard lower bound), so the left side is pinched. Meanwhile, during crises, volatility can explode to several times its average. The right tail captures those rare but fierce spikes.

Why does log‑normality matter? Many statistical tools and risk models assume a normal distribution. If you model volatility itself as normally distributed, you’ll badly underestimate the chance of a volatility spike. A log‑normal assumption is better, though still not perfect. In practice, the right tail of volatility can be even fatter than a log‑normal curve, especially for short‑dated measures. That has led to more advanced models that allow for jumps in volatility.

📝 Section Recap: Measured volatility over time follows a right‑skewed, roughly log‑normal shape — pinned at zero but able to shoot up in extreme spikes.

Overnight Returns and Their Contribution to Kurtosis#

When you measure daily returns from close to close, you are combining two very different chunks: the daytime trading session and the overnight gap. An interesting empirical finding is that overnight returns (from yesterday’s close to today’s open) add more than their fair share to the fat tails of the return distribution.

Think about it: most corporate earnings, economic data, and geopolitical news are released outside regular trading hours. The market can only react at the next open, often with a large gap. During the trading day, information flows more continuously, and prices adjust in many smaller steps. As a result, the distribution of overnight returns has fatter tails — higher kurtosis — than the distribution of intraday returns.

This matters for measuring volatility. If you use close‑to‑close returns, your volatility estimate will be heavily influenced by those overnight gaps. Some traders prefer to use open‑to‑close returns (ignoring overnight moves) for intraday volatility signals. Others model the overnight component separately. For option pricing, the fact that a big slice of the variance arrives overnight affects the value of options that expire at the open versus the close.

The overnight effect also links with the leverage effect. Overnight gaps are often negative during bear markets, and those large negative gaps pump up the negative skew and overall kurtosis of the return series. In other words, not all price moves are created equal — when they happen changes the shape of the distribution.

📝 Section Recap: Overnight returns have fatter tails than intraday returns because major news breaks when markets are closed. This inflates the kurtosis of close‑to‑close returns and affects both volatility measurement and option pricing.

Summary#

We’ve walked through the real‑world behaviour of financial volatility and uncovered patterns that repeat across markets and time. Volatility bunches together, gets pulled back toward an average, and rises unevenly when prices fall. Return distributions aren’t normal — they have fat tails and a leftward tilt. Trading volume and volatility rise together, and volatility itself follows a lopsided, log‑normal‑like shape. Even the overnight period adds its own special spice to extreme moves. These facts are the bedrock of modern volatility modelling and trading. They remind us that volatility is not a single number but a rich, ever‑changing process with its own personality.