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

# Probability density

> Risk-neutral PDF / CDF via Breeden-Litzenberger — the market's implied price distribution

**Risk-neutral probability density** is the full implied probability
distribution of the underlying's price at a future expiry, extracted
from option prices. It tells you not just where the market expects
price to land (the mean / mode) but the full *shape* of expectations —
including skew toward downside crashes or upside squeezes.

## The intuition

Breeden and Litzenberger (1978) showed that the shape of an option
chain — specifically the way call prices respond to strike — implicitly
encodes the market's probability distribution for the underlying at
expiry. We do the heavy lifting on our side and return a clean,
normalised PDF over the strike grid for every active expiry.

You get the distribution directly. No surface-fitting, no numerical
differentiation, no boundary-handling. Plot it, integrate it, sample
percentiles from it.

## What you get from `/v2/options/probability/density`

For every active expiry, the [endpoint](/api/v2/options/probability/density)
returns:

```json theme={null}
{
  "expiries": {
    "30MAY26": {
      "strikes": [60000, 62500, 65000, 67500, 70000, 72500, 75000, ...],
      "pdf": [0.001, 0.003, 0.012, 0.025, 0.018, 0.008, 0.002, ...],
      "pdf_normalized": [...],
      "cdf_below": [0.05, 0.12, 0.30, 0.55, 0.78, 0.92, 0.99, ...],
      "cdf_above": [0.95, 0.88, 0.70, 0.45, 0.22, 0.08, 0.01, ...],
      "mean": 67200,
      "mode": 67500,
      "std": 4100,
      "skewness": -0.18,
      "kurtosis": 3.6,
      "prob_25": 64500,
      "prob_50": 67500,
      "prob_75": 70200
    },
    ...
  },
  "spot_price": 67213.5
}
```

| Field               | Meaning                                                                   |
| ------------------- | ------------------------------------------------------------------------- |
| `pdf`               | Probability density per strike (un-normalized — sum × strike spacing ≈ 1) |
| `pdf_normalized`    | PDF divided by total mass, sums to 1 over the strike grid                 |
| `cdf_below`         | Cumulative probability `P(price < strike)` — monotone non-decreasing      |
| `cdf_above`         | Survival function `P(price > strike)` = 1 − cdf\_below                    |
| `mean`              | Expected price at expiry under the risk-neutral measure                   |
| `mode`              | Most likely single price (peak of the PDF)                                |
| `std`               | Standard deviation in price units                                         |
| `skewness`          | Distribution skew — negative = downside-skewed (crash risk priced in)     |
| `kurtosis`          | Tail-fatness — > 3 means fat tails, \< 3 means thin                       |
| `prob_25 / 50 / 75` | The 25th / 50th / 75th percentile prices                                  |

## Confidence bands — the easiest way to use this

Pass `?confidence_band=0.68` (≈ 1σ) or `?confidence_band=0.95` (≈ 2σ)
and each expiry payload gets an extra block:

```json theme={null}
{
  "30MAY26": {
    ...,
    "confidence_band": {
      "confidence": 0.68,
      "lower": 63100,
      "upper": 71200,
      "lower_percentile": 0.16,
      "upper_percentile": 0.84
    }
  }
}
```

`lower` and `upper` are the price levels containing 68% (or 95%) of the
implied probability mass, computed by interpolating the CDF. Use these
directly as range bounds in dashboards or mean-reversion strategies.

## Filtering

* `?expiry=28MAR25` — slice to a single expiry token (case-insensitive)
* `?dte_max=30` — drop expiries beyond N days, focus on the front

## How traders use it

**Identifying skew**: a `skewness` of -0.5 on the next monthly means the
market is paying up for downside protection. Often a contrarian buy
signal when extreme.

**Sizing risk**: a `confidence_band` at 95% gives you the implied 2σ
range. Position sizing that assumes a tighter range is taking a vol
view; sizing that assumes wider is hedged.

**Strike selection**: looking at the PDF for the next monthly, you can
see where the "shoulders" of the distribution are — strikes with
elevated PDF density that aren't the mode. These are often the strikes
where dealer hedging clusters and worth knowing for execution.

**Comparing expiries**: the term structure of `std` (per-expiry vol) and
`skewness` (per-expiry crash risk) tells you whether the market is
pricing volatility expansion or compression over the next few weeks.

## What it isn't

* **Not a forecast.** Risk-neutral probabilities differ from real-world
  probabilities by the risk premium. A 70% RN probability of upside
  doesn't mean a 70% real-world chance.
* **Not infinitely smooth.** The underlying chain can be sparse for
  far-OTM strikes, so the tails of the PDF are noisier than the body.
  Use the body for sizing decisions; treat the tails as informational.
* **Not free.** It's computationally heavy. The endpoint is cached at
  the worker layer — refreshed every 30s, not real-time.

## See also

* **3D surface**: [`/v2/options/probability/surface`](/api/v2/options/probability/surface)
  returns the same PDF aligned across all expiries on a unified strike
  grid, suitable for 3D visualisation.

## Related concepts

<CardGroup cols={2}>
  <Card title="The IV suite" icon="wave-square" href="/concepts/iv-suite">
    The IV surface that underpins this density extraction.
  </Card>

  <Card title="Max pain" icon="bullseye" href="/concepts/max-pain">
    A simpler scalar measure of "where price wants to be" at expiry.
  </Card>

  <Card title="Greeks beyond delta" icon="function" href="/concepts/greeks">
    Vanna and charm shape how the PDF evolves between now and expiry.
  </Card>
</CardGroup>