How Markets Really Move Together: Smarter Insights with Fractional Differencing & Variation of Information

Markets Move Together — But Not in Straight Lines

In finance, we often hear that two markets are “highly correlated.” But what does that really mean?

Most people rely on correlation, which assumes that relationships between assets are linear and stationary. That’s a problem—because markets are almost never either.

So I asked a better question:

What if we measured how much information two asset prices really share—regardless of whether they move in lockstep?

This led me to a combination of:

• ✅ Fractional Differencing — to make price series stationary without destroying memory
• ✅ Variation of Information (VI) — to measure structural similarity between instruments without assuming linearity

Let’s break down what these are, what they revealed about real asset relationships, and how you can use the results in practice.

Why Stationarity Matters in Financial Modeling

Stationarity means a time series has constant statistical properties—its mean and variance don’t drift over time.

Why does this matter?

Because most models (machine learning or statistical) break down when the underlying data is non-stationary.

But financial time series are almost always non-stationary.

So how do we make them usable?

The Power of Fractional Differencing

Traditional differencing (price[t] - price[t-1]) removes trends but wipes out long-term memory, which is crucial in finance.

Fractional differencing is a smarter way: it removes just enough memory to make the series stationary while preserving longer dependencies.

🧮 What is d  in Fractional Differencing?

The parameter d controls how “deep” the differencing goes:

• d = 0 → raw prices (non-stationary)
• d = 1 → classic first difference (may over-difference)
• 0 < d < 1 → just right

You can estimate d using statistical tests (like ADF) and optimize it per instrument.

👉 Learn more and see examples:

• How to Fractionally Difference Time Series
• From Commodities to Currencies: Fractional Differencing In Action

Measuring Market Structure with Variation of Information (VI)

Once we make data stationary, we want to know:

Which instruments actually move together—without assuming a linear relationship?

That’s where Variation of Information comes in.

“Variation of Information is a distance metric grounded in information theory. It measures how much information is not shared between two variables.”

— Marcos López de Prado, Machine Learning for Asset Managers

• Lower VI → strong shared structure
• Higher VI → little to no structural similarity

Unlike correlation, VI doesn’t care about direction or linearity. It simply looks at shared information content.

🔍 Results: Which Markets Move Together (And Which Don’t)?

Using fractionally differenced price data, I computed VI scores between multiple instruments.

✅ Most Structurally Similar Pairs (Lowest VI):

Takeaways:

• Brent and Crude Oil are tightly linked—no surprise given shared fundamentals.
• Currencies that share exposure to the US dollar and Eurozone (like USD/SEK, EUR/USD, and DXY) show strong structural ties.
• These pairs are ideal for co-integration, hedging, or feature engineering in ML models.

❌  Least Structurally Similar Pairs (Highest VI):

Takeaways:

• These instruments do not move together, structurally or otherwise.
• Some might look “related” by asset class (e.g. cocoa and cotton) but still have distinct pricing behaviors.
• Including these in a model as predictors of each other would likely inject noise.

🧠 Use Cases: How You Can Apply This in Trading & Modeling

1.  Feature Selection in Machine Learning

• Select low-VI features for stronger signal
• Drop high-VI inputs that add noise

2. Pairs Trading / Statistical Arbitrage

• Use low-VI pairs (e.g., Brent–Crude) for mean-reversion strategies
• VI helps identify true long-term co-movement

3. Risk Clustering / Portfolio Diversification

• Group instruments by shared structure, not just sector labels
• Helps avoid false diversification

4. Detecting Structural Breaks

• If a historically low-VI pair starts diverging → potential regime shift
• Useful for risk management and macro modeling

🎯 Final Thoughts

Most of us grew up using correlation matrices. But in modern markets, they’re not enough.

With fractional differencing, we stationarize without erasing memory.

With Variation of Information, we measure relationships without assumptions.

So instead of asking “Are these correlated?”, ask:

“How much do these markets truly share in structure?”

That’s a smarter way forward.