Probability density - BackQuant API

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 returns:

{
  "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:

{
  "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.

The IV suite

The IV surface that underpins this density extraction.

Max pain

A simpler scalar measure of “where price wants to be” at expiry.

Greeks beyond delta

Vanna and charm shape how the PDF evolves between now and expiry.

​The intuition

​What you get from /v2/options/probability/density

​Confidence bands — the easiest way to use this

​Filtering

​How traders use it

​What it isn’t

​See also

​Related concepts