MokaMage beta v0.1
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S&P 500 Index (SPX) Price Probability Distribution

Production Data Close-to-Close Session 2026-07-20
Anchor Price $7,443.28 Daily Data

Probability Range Windows

Choose the coverage window shown in the range and chart.
Selected 50-55% Range $7,414.31 - $7,492.06 Computed at 52.50% target coverage. Educational model output, not a recommendation.
Below selected range Selected range Above selected range
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Color guide

Colors compare each bin midpoint with the currently selected probability range. They are reading aids, not directional signals.

Reference line

The dashed line shows where the Anchor Price sits on the chart. Use it as an orientation marker: bars to the left are lower price ranges, and bars to the right are higher price ranges. It is not a target price, prediction, or recommendation.

Chart notes: x-axis shows bin midpoints · hover/tap bars for full range · USD

Model details
AIC 6,186.862 BIC 6,221.807 Log Likelihood -3,087.431 Next Sigma 0.830504 Annualized Sigma 13.183848

Model reading guide

How to read model details

AIC

Helps compare which model is better after adding a penalty for using too many moving parts. Lower is better, but only when the models are fitted to the same data. It rewards a model for explaining the data well, while discouraging overfitting. Example: AIC 5,778 beats AIC 5,790 by 12 points for the same dataset. Treat it as a model-ranking score, not as a probability, confidence level, or price forecast.

BIC

Similar to AIC, but it gives a stronger penalty when a model uses extra moving parts. Lower is better among models tested on the same data. Because it is stricter, BIC often favors simpler models than AIC. Example: BIC 5,813 beats BIC 5,830 for the same group of candidate models. Use it to compare models, not as a standalone “good” or “bad” quality score.

Log Likelihood

Shows how well the model explains the data it was fitted on. Higher is better before adding AIC/BIC penalties. When the numbers are negative, the value closer to zero is better. Example: –2,883 is better than –2,895 on the same data. The number grows with dataset size, so it is most useful for comparison, not as an intuitive score by itself.

Next Sigma

Estimates the likely size of the next-period move, usually from the model’s next volatility estimate. Higher sigma means the model expects a wider movement range. Example: if next sigma is 0.80% and the anchor price is $6,900, a rough one-sigma move is about $55. This gives the expected size of movement, not the direction. It does not say whether price is more likely to go up or down.

Annualized Sigma

Converts short-period volatility into a yearly-style number so it is easier to compare. For daily volatility, the common shortcut is daily sigma × sqrt(252). Example: 0.80% × sqrt(252) ≈ 12.7%. This helps compare volatility across assets, models, or time periods. It is a standard comparison measure, not a promise of what future volatility will actually be.

Detailed Statistics

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Why realized volatility?

Think of it as reading the market’s recent footsteps instead of guessing its mood. Here, it helps the probability ranges stay tied to actual price movement rather than a story about what prices “should” do.

What is a probability band?

Think of it like a weather forecast for price, where each range comes with its own chance of happening. Here, it helps you compare the more likely paths with the less likely ones before the next session begins.

Why not a normal curve?

A normal curve assumes extreme moves are rarer and tidier than markets usually are. Here, we leave more room for outsized swings so the chart better reflects the messy way real prices can behave.

Glossary

Volatility

A simple way to describe how much price tends to move around. Bigger and faster swings usually mean a less predictable market.

Realized volatility

Volatility measured from recent market moves that actually happened, rather than a forecast or opinion. It helps keep this chart grounded in observed behavior.

Fat tails

A reminder that unusually large market moves happen more often than a neat textbook model would suggest. In practice, it means tail risks deserve real attention.

Sigma band

A range built around the current price with a chosen probability in mind. It gives you a practical way to think about how wide the market’s likely move could be.

Cumulative probability

The running total of probability as you add ranges together. It helps show how much of the overall distribution is covered once you move across multiple price bands.