Here is my google doc

(pl click on the link; google sheet is given therein)

Screenshot is attached above

Insights: BankNifty Options Minute Log-Return Distribution 

Top Panel: Histogram 

The most striking feature is the extreme spike at x ≈ 0, meaning the overwhelming majority of minute-to-minute returns are near zero — options prices barely move most of the time. However, the histogram also shows very wide, flat tails extending to ±0.20 and beyond, which are the rare but large price jumps.

• The Normal curve (orange) is too wide and short : it spreads probability mass evenly, completely missing the sharp central peak and the fat tails simultaneously. It's essentially the wrong shape entirely.

• The Student-t curve (green): is also imperfect visually here, but its mathematical advantage shows up in the tail regions — it assigns higher probability to extreme moves than the Normal does.

• The annotation correctly identifies the core problem: in the left tail region around −0.10, the empirical bar heights are visibly higher than what either fitted curve predicts.

Bottom-Left Panel: CDF Comparison (full range)

In the central body (roughly −0.05 to +0.05), all three CDFs look similar. The divergence becomes clear at the extremes:

• The Normal CDF (orange dashed) rises too steeply and too early: it "uses up" its probability mass in the centre, leaving very little for the tails.

• The empirical CDF (blue) and Student-t CDF (green dashed) stay closer together throughout, confirming Student-t is the better fit.

• Notice the empirical CDF has a very steep rise near x = 0, consistent with the spike seen in the histogram.

Bottom-Right Panel: Left-Tail Zoom (x < −0.02)

This is the most important panel for risk management, and it reveals the failure of the Normal most starkly:

• At x = −0.10, the Normal CDF gives F ≈ 0.0006 (virtually zero probability), while the empirical CDF gives F = 0.0057 — nearly 10× more probability of a −10% drop than Normal predicts.

• The Student-t closely tracks the empirical line all the way into the deep left tail, confirming ν ≈ 2.56 captures the fat-tail behaviour.

• The x-axis extends to −0.35, showing returns of that magnitude actually did occur in the data: something Normal would assign essentially zero probability to.

Key Takeaway

BankNifty option returns are leptokurtic : highly peaked at zero with fat tails. Using a Normal distribution to model this data would severely underestimate tail risk, making it dangerous for any VaR or options pricing application. The Student-t with ν ≈ 2.56 is a materially better fit, especially in the left tail where large losses live.