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

# SABR Probability Density

> Calculate the risk-neutral probability density function implied by SABR parameters across multiple strike levels. Derived from the second derivative of call prices with respect to strike. Used for understanding market-implied distributions, calculating probabilities, and ensuring arbitrage-free surfaces. [Tier: PRO, Credits: 5]



## OpenAPI

````yaml api-specs/volatility.json post /quantlib/volatility/sabr/density
openapi: 3.1.0
info:
  title: FinceptQuantLib API - Volatility
  description: >-
    Volatility module endpoints for FinceptQuantLib API. Standard Tier module (2
    credits per request) providing comprehensive volatility surface modeling,
    SABR model calibration and analytics, and local volatility calculations for
    derivatives pricing and risk management.
  version: 3.0.0
  contact:
    name: Fincept API Support
    url: https://fincept.in
servers:
  - url: https://api.fincept.in
    description: Fincept API Production Server
security:
  - APIKeyHeader: []
tags:
  - name: quantlib-volatility
    description: >-
      Volatility surface modeling, SABR calibration, and local volatility
      calculations
    x-displayName: Volatility
paths:
  /quantlib/volatility/sabr/density:
    post:
      tags:
        - quantlib-volatility
      summary: SABR Probability Density
      description: >-
        Calculate the risk-neutral probability density function implied by SABR
        parameters across multiple strike levels. Derived from the second
        derivative of call prices with respect to strike. Used for understanding
        market-implied distributions, calculating probabilities, and ensuring
        arbitrage-free surfaces. [Tier: PRO, Credits: 5]
      operationId: sabr_density
      requestBody:
        required: true
        content:
          application/json:
            schema:
              type: object
              required:
                - forward
                - expiry
                - alpha
                - beta
                - rho
                - nu
                - strikes
              properties:
                forward:
                  type: number
                  description: Forward price of the underlying
                  example: 100
                  minimum: 0
                expiry:
                  type: number
                  description: Time to expiry in years
                  example: 1
                  minimum: 0
                alpha:
                  type: number
                  description: SABR alpha parameter
                  example: 0.25
                  minimum: 0
                beta:
                  type: number
                  description: SABR beta parameter
                  example: 0.5
                  minimum: 0
                  maximum: 1
                rho:
                  type: number
                  description: SABR rho parameter
                  example: -0.2
                  minimum: -1
                  maximum: 1
                nu:
                  type: number
                  description: SABR nu parameter
                  example: 0.3
                  minimum: 0
                strikes:
                  type: array
                  items:
                    type: number
                  description: Array of strike prices for density evaluation
                  example:
                    - 85
                    - 90
                    - 95
                    - 100
                    - 105
                    - 110
                    - 115
                  minItems: 3
            example:
              forward: 100
              expiry: 1
              alpha: 0.25
              beta: 0.5
              rho: -0.2
              nu: 0.3
              strikes:
                - 85
                - 90
                - 95
                - 100
                - 105
                - 110
                - 115
      responses:
        '200':
          description: Successful Response
          content:
            application/json:
              schema:
                type: object
                properties:
                  success:
                    type: boolean
                    example: true
                  data:
                    type: object
                    properties:
                      forward:
                        type: number
                        description: Forward price (echo of input)
                        example: 100
                      expiry:
                        type: number
                        description: Expiry time (echo of input)
                        example: 1
                      density:
                        type: array
                        items:
                          type: object
                          properties:
                            strike:
                              type: number
                              description: Strike price level
                            density:
                              type: number
                              description: Probability density at this strike
                        description: Array of strike-density pairs
                        example:
                          - strike: 85
                            density: 0.0052
                          - strike: 90
                            density: 0.0118
                          - strike: 95
                            density: 0.0215
                          - strike: 100
                            density: 0.0285
                          - strike: 105
                            density: 0.0241
                          - strike: 110
                            density: 0.0152
                          - strike: 115
                            density: 0.0078
              example:
                success: true
                data:
                  forward: 100
                  expiry: 1
                  density:
                    - strike: 85
                      density: 0.0052
                    - strike: 90
                      density: 0.0118
                    - strike: 95
                      density: 0.0215
                    - strike: 100
                      density: 0.0285
                    - strike: 105
                      density: 0.0241
                    - strike: 110
                      density: 0.0152
                    - strike: 115
                      density: 0.0078
        '401':
          $ref: '#/components/responses/UnauthorizedError'
        '402':
          $ref: '#/components/responses/InsufficientTierError'
        '422':
          $ref: '#/components/responses/ValidationError'
      security:
        - APIKeyHeader: []
components:
  responses:
    UnauthorizedError:
      description: Authentication information is missing or invalid
      content:
        application/json:
          schema:
            type: object
            properties:
              detail:
                type: string
                example: Invalid API key
    InsufficientTierError:
      description: API tier insufficient for this endpoint
      content:
        application/json:
          schema:
            type: object
            properties:
              detail:
                type: string
                example: Endpoint requires Standard tier or higher
    ValidationError:
      description: Validation Error
      content:
        application/json:
          schema:
            type: object
            properties:
              detail:
                type: array
                items:
                  type: object
                  properties:
                    loc:
                      type: array
                      items:
                        type: string
                    msg:
                      type: string
                    type:
                      type: string
          example:
            detail:
              - loc:
                  - body
                  - volatility
                msg: field required
                type: value_error.missing
  securitySchemes:
    APIKeyHeader:
      type: apiKey
      in: header
      name: X-API-Key
      description: >-
        API key for authentication. Get your key at
        https://api.fincept.in/auth/register

````