> ## 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 > 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. Risk-neutral PDF/CDF derived from option prices via Breeden-Litzenberger. Optional `?confidence_band=0.68|0.95` returns interpolated price bands per expiry. ## See also ## OpenAPI ````yaml GET /options/probability/density openapi: 3.1.0 info: title: BackQuant API v2 description: >- # BackQuant API v2 Options + gamma-exposure focused public API. Built on the same data the BackQuant Pro Terminal renders — pre-computed every 30s, served from cache. ## What's in v2 * **Discovery**: `/v2/symbols` (universe + per-symbol freshness + supported endpoint list), `/v2/expiries` (active expiry tokens with DTE), `/v2/status` (per-symbol-per-category health with thresholds + overall classification). * **GEX**: composable levels (HVL / call wall / put support / max-pain / expected move / gamma-flip zones), strike profile with typed expiry filter, expiry profile, strike × expiry heatmap with downsampling, greek time-heatmap with DTE/OI/IV filters, history (cursor-paginated), Postgres-backed stress history, **per-expiry max-pain with pain curve**. * **Options**: filtered options chain with two-layer projection (top-level `?fields=` and per-contract `?include=oi,iv,greeks,bid_ask,volume,gex`) plus moneyness filter (`?moneyness_min=0.9&moneyness_max=1.1`), expiry summary table, full IV suite (surface / term structure / 25Δ-10Δ skew / curves / **single-expiry smile** / IV-RV history / VRP), expected move, **Breeden-Litzenberger probability density and surface**, typed greek profiles (delta / theta / vanna / charm / vega), strike × time charm/vega surfaces, strike × expiry 3D greek surface, OI by expiry + history, put/call ratio (intraday or daily), 0DTE & weekly premium tide, dated-futures term structure. * **Liquidation**: heatmap + leverage-tiered distribution. * **Multi**: `/v2/multi/gex/levels` — bundled multi-symbol read across the universe in one round-trip, with the same `?include=` model as the single-symbol endpoint. ## Authentication Every v2 route (except `/v2/openapi.json`, `/v2/docs`, `/v2/redoc`, and `/v2/health`) requires the `X-API-Key` header. ``` X-API-Key: bq_live_your_api_key_here ``` **Get your API key at [backquant.com/api-access](https://backquant.com/api-access).** The same key works across v1 and v2 — if you already have a v1 key, no re-issuance is needed. ## Rate limits Per-key, derived from your subscription tier: | Tier | Limit | |-------------|----------------------| | Monthly | 60 requests/min | | Yearly | 120 requests/min | | Enterprise | No limit | Headers `X-RateLimit-Limit / -Remaining / -Reset` are included in every response, and the same triple is echoed inside `meta.rate_limit` when populated by middleware. ## Response envelope ```json { "success": true, "data": { ... }, "meta": { "version": "2.0", "timestamp": "2026-04-29T12:00:00.000Z", "request_id": "req_…", "symbol": "BTCUSDT", "spot_price": 67213.5, "computed_at": "2026-04-29T11:59:48.000Z", "freshness_seconds": 12.0, "source": ["deribit","bybit","okx","binance"], "exchanges_filtered": ["deribit","bybit"] } } ``` ## Errors ```json { "success": false, "error": { "code": "NOT_FOUND", "message": "No data for BTCUSDT" }, "meta": { "version": "2.0", "timestamp": "..." } } ``` | Code | When | |-----------------------|------| | `UNAUTHORIZED` | Missing or invalid API key | | `FORBIDDEN` | Subscription doesn't allow API access | | `NOT_FOUND` | Symbol/expiry/etc. not present in cache | | `VALIDATION_ERROR` | Bad query parameters | | `RATE_LIMIT_EXCEEDED` | Per-tier limit hit | | `UPSTREAM_ERROR` | Cache or DB temporarily unreachable | | `INTERNAL_ERROR` | Anything else | version: 2.0.0 servers: [] security: [] tags: - name: Meta description: Unauthenticated metadata + endpoint catalog (planning your integration) - name: Discovery description: Symbol catalog, active expiries, and service status - name: GEX description: Gamma exposure analytics - name: Options description: Options chain, IV, greeks, probability, premium tide - name: Liquidation description: Liquidation heatmap and distribution - name: Multi description: Bundled multi-symbol endpoints (single round-trip across the universe) paths: /options/probability/density: get: tags: - Options summary: Risk-neutral PDF/CDF (per expiry) description: >- 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. operationId: get_probability_density_options_probability_density_get parameters: - name: symbol in: query required: false schema: enum: - BTCUSDT - ETHUSDT - SOLUSDT - HYPEUSDT type: string description: 'Trading symbol: BTCUSDT, ETHUSDT, SOLUSDT, or HYPEUSDT.' default: BTCUSDT title: Symbol description: 'Trading symbol: BTCUSDT, ETHUSDT, SOLUSDT, or HYPEUSDT.' - name: expiry in: query required: false schema: anyOf: - type: string - type: 'null' description: Filter to one expiry token (e.g. 28MAR25) title: Expiry description: Filter to one expiry token (e.g. 28MAR25) - name: dte_max in: query required: false schema: anyOf: - type: integer maximum: 720 minimum: 0 - type: 'null' description: Drop expiries with DTE greater than this value. title: Dte Max description: Drop expiries with DTE greater than this value. - name: confidence_band in: query required: false schema: anyOf: - type: number maximum: 0.999 minimum: 0.5 - type: 'null' description: >- Confidence level for the implied price band, e.g. `0.68` for ≈1σ or `0.95` for ≈2σ. Adds `confidence_band` per expiry. title: Confidence Band description: >- Confidence level for the implied price band, e.g. `0.68` for ≈1σ or `0.95` for ≈2σ. Adds `confidence_band` per expiry. - name: X-API-Key in: header required: false schema: anyOf: - type: string - type: 'null' title: X-Api-Key responses: '200': description: Successful Response content: application/json: schema: $ref: '#/components/schemas/ProbabilityDensityResponse' '401': description: Invalid or missing API key content: application/json: example: success: false error: code: UNAUTHORIZED message: Invalid API key meta: version: '2.0' timestamp: '2026-04-29T12:00:00Z' '404': content: application/json: schema: $ref: '#/components/schemas/V2ErrorEnvelope' description: Not Found '422': description: Validation error on query parameters content: application/json: example: success: false error: code: VALIDATION_ERROR message: Invalid request parameters details: errors: [] meta: version: '2.0' timestamp: '2026-04-29T12:00:00Z' '429': description: Rate limit exceeded content: application/json: example: success: false error: code: RATE_LIMIT_EXCEEDED message: Rate limit exceeded. Try again later. meta: version: '2.0' timestamp: '2026-04-29T12:00:00Z' security: - ApiKeyAuth: [] components: schemas: ProbabilityDensityResponse: properties: success: type: boolean const: true title: Success default: true data: $ref: '#/components/schemas/ProbabilityDensityData' meta: $ref: '#/components/schemas/V2Meta' type: object required: - data - meta title: ProbabilityDensityResponse V2ErrorEnvelope: properties: success: type: boolean const: false title: Success default: false error: $ref: '#/components/schemas/V2Error' meta: $ref: '#/components/schemas/V2Meta' type: object required: - error - meta title: V2ErrorEnvelope description: Returned for every non-2xx response. ProbabilityDensityData: properties: expiries: additionalProperties: $ref: '#/components/schemas/ProbabilityExpiryPayload' type: object title: Expiries description: Keyed by expiry token (e.g. `28MAR25`). spot_price: anyOf: - type: number - type: 'null' title: Spot Price filtered_dte_max: anyOf: - type: integer - type: 'null' title: Filtered Dte Max confidence_band_requested: anyOf: - type: number - type: 'null' title: Confidence Band Requested type: object required: - expiries title: ProbabilityDensityData V2Meta: properties: version: type: string const: '2.0' title: Version description: API version tag. timestamp: type: string title: Timestamp description: ISO 8601 wall-clock when the response was built. request_id: anyOf: - type: string - type: 'null' title: Request Id description: X-Request-ID for tracing across logs / Sentry. symbol: anyOf: - type: string - type: 'null' title: Symbol description: Symbol the response is about, if applicable. spot_price: anyOf: - type: number - type: 'null' title: Spot Price description: Current spot price for the symbol. computed_at: anyOf: - type: string - type: 'null' title: Computed At description: ISO 8601 of when the underlying cache value was last written. freshness_seconds: anyOf: - type: number - type: 'null' title: Freshness Seconds description: '`now - computed_at` in seconds; useful for staleness alarms.' source: anyOf: - items: type: string type: array - type: 'null' title: Source description: Upstream venue list the data flowed through. exchanges_filtered: anyOf: - items: type: string type: array - type: 'null' title: Exchanges Filtered description: Echo of the `?exchanges=` filter when one was applied. rate_limit: anyOf: - additionalProperties: true type: object - type: 'null' title: Rate Limit description: '`{limit, remaining, reset}` echo of the rate-limit headers.' extra: anyOf: - additionalProperties: true type: object - type: 'null' title: Extra description: Open dict for endpoint-specific metadata (filter_hash, etc.). type: object required: - version - timestamp title: V2Meta description: |- 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`. V2Error: properties: code: type: string title: Code description: Stable machine-readable error code (see /v2/meta). message: type: string title: Message description: Human-readable description. details: anyOf: - additionalProperties: true type: object - type: 'null' title: Details description: >- Optional structured payload — e.g. validation_error returns the failing field list. type: object required: - code - message title: V2Error ProbabilityExpiryPayload: properties: strikes: anyOf: - items: type: number type: array - type: 'null' title: Strikes pdf: anyOf: - items: anyOf: - type: number - type: 'null' type: array - type: 'null' title: Pdf pdf_normalized: anyOf: - items: anyOf: - type: number - type: 'null' type: array - type: 'null' title: Pdf Normalized cdf_above: anyOf: - items: anyOf: - type: number - type: 'null' type: array - type: 'null' title: Cdf Above cdf_below: anyOf: - items: anyOf: - type: number - type: 'null' type: array - type: 'null' title: Cdf Below mode: anyOf: - type: number - type: 'null' title: Mode mean: anyOf: - type: number - type: 'null' title: Mean std: anyOf: - type: number - type: 'null' title: Std skewness: anyOf: - type: number - type: 'null' title: Skewness kurtosis: anyOf: - type: number - type: 'null' title: Kurtosis prob_25: anyOf: - type: number - type: 'null' title: Prob 25 prob_50: anyOf: - type: number - type: 'null' title: Prob 50 prob_75: anyOf: - type: number - type: 'null' title: Prob 75 confidence_band: anyOf: - $ref: '#/components/schemas/ProbabilityConfidenceBand' - type: 'null' additionalProperties: true type: object title: ProbabilityExpiryPayload description: |- Per-expiry PDF/CDF data. Many fields are optional because the cache shape evolves with worker versions; SDKs treat absent fields as null. ProbabilityConfidenceBand: properties: confidence: type: number title: Confidence lower: anyOf: - type: number - type: 'null' title: Lower upper: anyOf: - type: number - type: 'null' title: Upper lower_percentile: anyOf: - type: number - type: 'null' title: Lower Percentile upper_percentile: anyOf: - type: number - type: 'null' title: Upper Percentile available: anyOf: - type: boolean - type: 'null' title: Available reason: anyOf: - type: string - type: 'null' title: Reason type: object required: - confidence title: ProbabilityConfidenceBand securitySchemes: ApiKeyAuth: type: apiKey in: header name: X-API-Key description: Your BackQuant API key (same key as v1) ````