📈 SPY — The Benchmark of the U.S. Stock Market

Let’s begin with one of the most practical and illustrative examples: the SPY ETF (SPDR S&P 500 ETF). This exchange-traded fund closely mirrors the performance of the S&P 500 index, which includes 500 of the largest U.S. companies by market capitalization.

SPY is one of the most liquid, popular, and capital-heavy ETFs in the world. The S&P 500 itself represents around 80% of the U.S. stock market’s total value, making SPY a highly relevant proxy for analyzing broader market trends.

Let’s take a look at SPY’s price chart over the years — starting around 2007.

What a magnificent image. It’s a chart of unstoppable growth in the U.S. equity market. Veteran traders will recognize several turbulent chapters. For me, it’s a kind of meme timeline — full of memories.

🔢 Logarithmic Daily Returns
Let’s move on from price charts to something more mathematical — logarithmic daily returns, calculated using this formula:

Where:

• PtP_t​ is today’s closing price
• Pt−1P_{t-1}​ is yesterday’s closing price

Here’s how SPY’s daily log returns look over time.
Honestly, I love this kind of financial chart — they’re the heart of quantitative analysis.

🧮 Normal Distribution and Historical Volatility (HV)
These log returns form the basis of most option pricing models.

We can create a histogram of the daily returns and fit a normal distribution using the calculated mean and variance. This gives us a rough idea of return probability — and there’s already a lot to digest here, but we’ll keep that for another time.

Now, let’s compute historical volatility (HV) — one of the most important indicators for option and risk analysis.
We’ll use the classic HV formula:

Where:

• σσ is the standard deviation of the daily returns
• We use 20 trading days as our rolling window

Then we convert HV into percentage terms and overlay it with SPY price and returns:
All three charts combined: SPY price, log returns, and 20-day HV.
Feast your eyes!

🤔 But What About Future Volatility? Introducing VIX

Historical volatility tells us what has already happened, but what if we want to know what traders expect to happen?

That’s where the VIX Index comes in.

VIX (Volatility Index) measures the market’s expectation of future volatility in the S&P 500, based on real-time option prices. It’s calculated by the CBOE (Chicago Board Options Exchange) using a complex formula involving options with various strikes and expirations near 30 days.

To be honest, the calculation is quite esoteric — involving weighted averages, extrapolations, and, metaphorically speaking, an ancient Native American drum discovered in 1889 near the Great Lakes. But we’ll skip the mysticism.

📊 VIX vs. Historical Volatility

Let’s compare the VIX with the 20-day historical volatility (HV) of SPY.
Here’s what we can observe:

• In major crises (e.g., 2008, COVID), VIX spikes higher than HV.
• During milder market tremors, HV is often equal to or slightly above VIX.
• In calm markets, VIX consistently stays below HV.
• Both are strongly correlated in trend, but with different meanings.

⚖️ HV vs. VIX: Two Volatility Perspectives


HV shows realized volatility — what already occurred.

VIX expresses expected volatility — what traders think is likely.

Together, they form a complementary pair, but applying them in real-world crypto trading is still a gray area. Most existing examples are post-factum explanations, not actionable systems.

📉 Translating This to Crypto Markets

Applying HV and IV logic to crypto is tricky. Here's why:

• Crypto assets are young and lack long price history.
• Volatility is decreasing as the market matures, so historical patterns may be less useful.
• There’s no VIX equivalent for crypto derivatives — at least not publicly available.
• IV data from crypto options exchanges (like AE) is often unavailable or not archived.
• Many HV formulas from traditional finance don’t work well in crypto; we need new models or revert to basics.

🧩 Final Thoughts

Studying volatility — both realized (HV) and implied (IV/VIX) — helps us understand risk, expectations, and trader sentiment.

But in crypto, we’re still explorers. Until we have better tools, we need to rely on a mix of classical methods, thoughtful interpretation, and cautious experimentation.