API — Probability density

Probability density

Breeden-Litzenberger probability density derived from option prices. Returns PDF, normalized PDF, CDF (above and below spot), and distribution stats (mean, std, mode, skew, kurtosis, 25/50/75 percentile prices) per expiry.

Filters:

expiry — slice to a single expiry token (case-insensitive).
dte_max — drop expiries with DTE > dte_max so near-term clients can avoid carrying the long tail.
confidence_band — when set to 0.68 or 0.95 (the most common choices), each expiry payload gets a confidence_band object with lower / upper price levels containing that much implied probability mass, computed by interpolating the CDF.

GET

options

probability

density

Risk-neutral PDF/CDF (per expiry)

curl --request GET \
  --url https://api.example.com/options/probability/density \
  --header 'X-API-Key: <api-key>'

import requests

url = "https://api.example.com/options/probability/density"

headers = {"X-API-Key": "<api-key>"}

response = requests.get(url, headers=headers)

print(response.text)

const options = {method: 'GET', headers: {'X-API-Key': '<api-key>'}};

fetch('https://api.example.com/options/probability/density', options)
  .then(res => res.json())
  .then(res => console.log(res))
  .catch(err => console.error(err));

<?php

$curl = curl_init();

curl_setopt_array($curl, [
  CURLOPT_URL => "https://api.example.com/options/probability/density",
  CURLOPT_RETURNTRANSFER => true,
  CURLOPT_ENCODING => "",
  CURLOPT_MAXREDIRS => 10,
  CURLOPT_TIMEOUT => 30,
  CURLOPT_HTTP_VERSION => CURL_HTTP_VERSION_1_1,
  CURLOPT_CUSTOMREQUEST => "GET",
  CURLOPT_HTTPHEADER => [
    "X-API-Key: <api-key>"
  ],
]);

$response = curl_exec($curl);
$err = curl_error($curl);

curl_close($curl);

if ($err) {
  echo "cURL Error #:" . $err;
} else {
  echo $response;
}

package main

import (
	"fmt"
	"net/http"
	"io"
)

func main() {

	url := "https://api.example.com/options/probability/density"

	req, _ := http.NewRequest("GET", url, nil)

	req.Header.Add("X-API-Key", "<api-key>")

	res, _ := http.DefaultClient.Do(req)

	defer res.Body.Close()
	body, _ := io.ReadAll(res.Body)

	fmt.Println(string(body))

}

HttpResponse<String> response = Unirest.get("https://api.example.com/options/probability/density")
  .header("X-API-Key", "<api-key>")
  .asString();

require 'uri'
require 'net/http'

url = URI("https://api.example.com/options/probability/density")

http = Net::HTTP.new(url.host, url.port)
http.use_ssl = true

request = Net::HTTP::Get.new(url)
request["X-API-Key"] = '<api-key>'

response = http.request(request)
puts response.read_body

{
  "data": {
    "expiries": {},
    "spot_price": 123,
    "filtered_dte_max": 123,
    "confidence_band_requested": 123
  },
  "meta": {
    "version": "<string>",
    "timestamp": "<string>",
    "request_id": "<string>",
    "symbol": "<string>",
    "spot_price": 123,
    "computed_at": "<string>",
    "freshness_seconds": 123,
    "source": [
      "<string>"
    ],
    "exchanges_filtered": [
      "<string>"
    ],
    "rate_limit": {},
    "extra": {}
  },
  "success": true
}

{
  "success": false,
  "error": {
    "code": "UNAUTHORIZED",
    "message": "Invalid API key"
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

{
  "error": {
    "code": "<string>",
    "message": "<string>",
    "details": {}
  },
  "meta": {
    "version": "<string>",
    "timestamp": "<string>",
    "request_id": "<string>",
    "symbol": "<string>",
    "spot_price": 123,
    "computed_at": "<string>",
    "freshness_seconds": 123,
    "source": [
      "<string>"
    ],
    "exchanges_filtered": [
      "<string>"
    ],
    "rate_limit": {},
    "extra": {}
  },
  "success": false
}

{
  "success": false,
  "error": {
    "code": "VALIDATION_ERROR",
    "message": "Invalid request parameters",
    "details": {
      "errors": []
    }
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

{
  "success": false,
  "error": {
    "code": "RATE_LIMIT_EXCEEDED",
    "message": "Rate limit exceeded. Try again later."
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

Risk-neutral PDF/CDF derived from option prices via Breeden-Litzenberger. Optional ?confidence_band=0.68|0.95 returns interpolated price bands per expiry.

Breeden-Litzenberger explained

Authorizations

X-API-Key

string

header

required

Your BackQuant API key (same key as v1)

Headers

X-API-Key

string | null

Query Parameters

symbol

enum<string>

default:BTCUSDT

Trading symbol: BTCUSDT, ETHUSDT, SOLUSDT, or HYPEUSDT.

Available options:

BTCUSDT,

ETHUSDT,

SOLUSDT,

HYPEUSDT

expiry

string | null

Filter to one expiry token (e.g. 28MAR25)

dte_max

integer | null

Drop expiries with DTE greater than this value.

Required range: 0 <= x <= 720

confidence_band

number | null

Confidence level for the implied price band, e.g. 0.68 for ≈1σ or 0.95 for ≈2σ. Adds confidence_band per expiry.

Required range: 0.5 <= x <= 0.999

Response

Successful Response

data

ProbabilityDensityData · object

required

Show child attributes

meta

V2Meta · object

required

The meta block returned alongside every v2 response.

Every field after version/timestamp is optional because endpoints attach different combinations — e.g. /v2/status skips symbol, the chain endpoint sets extra.filter_hash, etc. Listing them here means SDKs get a typed accessor for each instead of a generic meta: dict.

Show child attributes

success

boolean

default:true

Expected move

Probability surface

Risk-neutral PDF/CDF (per expiry)

curl --request GET \
  --url https://api.example.com/options/probability/density \
  --header 'X-API-Key: <api-key>'

import requests

url = "https://api.example.com/options/probability/density"

headers = {"X-API-Key": "<api-key>"}

response = requests.get(url, headers=headers)

print(response.text)

const options = {method: 'GET', headers: {'X-API-Key': '<api-key>'}};

fetch('https://api.example.com/options/probability/density', options)
  .then(res => res.json())
  .then(res => console.log(res))
  .catch(err => console.error(err));

<?php

$curl = curl_init();

curl_setopt_array($curl, [
  CURLOPT_URL => "https://api.example.com/options/probability/density",
  CURLOPT_RETURNTRANSFER => true,
  CURLOPT_ENCODING => "",
  CURLOPT_MAXREDIRS => 10,
  CURLOPT_TIMEOUT => 30,
  CURLOPT_HTTP_VERSION => CURL_HTTP_VERSION_1_1,
  CURLOPT_CUSTOMREQUEST => "GET",
  CURLOPT_HTTPHEADER => [
    "X-API-Key: <api-key>"
  ],
]);

$response = curl_exec($curl);
$err = curl_error($curl);

curl_close($curl);

if ($err) {
  echo "cURL Error #:" . $err;
} else {
  echo $response;
}

package main

import (
	"fmt"
	"net/http"
	"io"
)

func main() {

	url := "https://api.example.com/options/probability/density"

	req, _ := http.NewRequest("GET", url, nil)

	req.Header.Add("X-API-Key", "<api-key>")

	res, _ := http.DefaultClient.Do(req)

	defer res.Body.Close()
	body, _ := io.ReadAll(res.Body)

	fmt.Println(string(body))

}

HttpResponse<String> response = Unirest.get("https://api.example.com/options/probability/density")
  .header("X-API-Key", "<api-key>")
  .asString();

require 'uri'
require 'net/http'

url = URI("https://api.example.com/options/probability/density")

http = Net::HTTP.new(url.host, url.port)
http.use_ssl = true

request = Net::HTTP::Get.new(url)
request["X-API-Key"] = '<api-key>'

response = http.request(request)
puts response.read_body

{
  "data": {
    "expiries": {},
    "spot_price": 123,
    "filtered_dte_max": 123,
    "confidence_band_requested": 123
  },
  "meta": {
    "version": "<string>",
    "timestamp": "<string>",
    "request_id": "<string>",
    "symbol": "<string>",
    "spot_price": 123,
    "computed_at": "<string>",
    "freshness_seconds": 123,
    "source": [
      "<string>"
    ],
    "exchanges_filtered": [
      "<string>"
    ],
    "rate_limit": {},
    "extra": {}
  },
  "success": true
}

{
  "success": false,
  "error": {
    "code": "UNAUTHORIZED",
    "message": "Invalid API key"
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

{
  "error": {
    "code": "<string>",
    "message": "<string>",
    "details": {}
  },
  "meta": {
    "version": "<string>",
    "timestamp": "<string>",
    "request_id": "<string>",
    "symbol": "<string>",
    "spot_price": 123,
    "computed_at": "<string>",
    "freshness_seconds": 123,
    "source": [
      "<string>"
    ],
    "exchanges_filtered": [
      "<string>"
    ],
    "rate_limit": {},
    "extra": {}
  },
  "success": false
}

{
  "success": false,
  "error": {
    "code": "VALIDATION_ERROR",
    "message": "Invalid request parameters",
    "details": {
      "errors": []
    }
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

{
  "success": false,
  "error": {
    "code": "RATE_LIMIT_EXCEEDED",
    "message": "Rate limit exceeded. Try again later."
  },
  "meta": {
    "version": "2.0",
    "timestamp": "2026-04-29T12:00:00Z"
  }
}

​See also

Breeden-Litzenberger explained

Authorizations

Headers

Query Parameters

Response

See also