> ## Documentation Index
> Fetch the complete documentation index at: https://docs.open-lemma.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Probability layer

> How OIPD turns fitted volatility into market-implied probability distributions.

The probability layer turns fitted volatility into probabilities over future prices.

It answers questions such as: what probability does the options market imply that a stock will be above \$100 at next week's expiry?

## Risk-neutral probabilities

OIPD returns **risk-neutral** probabilities.

They are not real-world forecasts. They are probabilities consistent with option prices after discounting and the chosen pricing assumptions.

## Curve or surface

`ProbCurve` is for one expiry.

`ProbSurface` is for many expiries.

You can build probability objects directly from chains:

```python theme={null}
from oipd import ProbCurve

prob = ProbCurve.from_chain(chain, market)
print(prob.prob_below(100))
```

Or you can fit volatility first and convert it:

```python theme={null}
from oipd import VolCurve

vol = VolCurve().fit(chain, market)
prob = vol.implied_distribution()
```

The second path is useful when you want to inspect the volatility fit before using the probabilities.

## PDF and CDF

The probability layer exposes two standard views: the PDF and the CDF.

The **PDF** is the probability density function. It shows where probability is concentrated across possible future prices. A higher PDF value around $100 means more probability mass is sitting near $100 than near prices with lower density.

The **CDF** is the cumulative distribution function. It answers a direct threshold question: what is the probability that the future price ends below this level?

```python theme={null}
prob.pdf(100)
prob.prob_below(100)
prob.prob_between(90, 110)
prob.quantile(0.5)
```

`quantile(0.5)` returns the median price under the fitted risk-neutral distribution.

## Across time

`ProbSurface` lets you ask the same questions at different maturities:

```python theme={null}
from oipd import ProbSurface

surface = ProbSurface.from_chain(chain, market)
prob_45d = surface.cdf(100, t=45 / 365)
```

You can also extract one date as a `ProbCurve`:

```python theme={null}
curve = surface.slice(surface.expiries[0])
curve.prob_below(100)
```

## Diagnostics

Probability calculations can expose numerical issues, especially around CDF monotonicity.

By default, OIPD uses `cdf_violation_policy="warn"`: it repairs material monotonicity issues and records warnings.

Use `cdf_violation_policy="raise"` when you prefer the calculation to fail instead.

Inspect `.warning_diagnostics` on fitted probability objects when results need review.