# Black-Scholes Delta

> Black-Scholes Delta is a paid API for AI agents from api.delx.ai, paid per call via x402, $0.001/call, status unknown (last checked 2026-09-14).

Computes call and put delta for European options using the Black-Scholes closed-form formula given spot, strike, volatility, rate, and time to expiry.

## Facts

- Endpoint: POST https://api.delx.ai/api/v1/x402/bs-delta
- Price: $0.001/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-14
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/black-scholes-delta-44ae70b7
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_BLuzIHjNlfMsRIzLo_RaF

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability black-scholes-delta-44ae70b7 -d '<json body>'
```

Example prompt: What's the Black-Scholes call and put delta for an option with a spot price of 150, strike of 155, annualized vol of 25%, risk-free rate of 5%, and 45 days to expiry?

## When to prefer this

Choose this endpoint when you need a fast, deterministic, closed-form Black-Scholes delta with no API keys, no live market data dependencies, and no external RPC calls. Ideal for delta-hedging calculations, options inventory management, or any scenario where you need a reliable, reproducible BS delta for a known set of inputs at very low cost ($0.001 USDC). Prefer this over stochastic or Monte Carlo approaches when Black-Scholes assumptions are acceptable and speed is paramount.

## Known failure modes

- Missing required fields (spot, strike, vol, t_years) return a validation error
- vol <= 0 or t_years <= 0 returns an invalid parameter error
- spot or strike <= 0 returns an invalid parameter error
- Payment not included or insufficient USDC returns a 402 Payment Required response
- Extremely small t_years values near zero may produce near-degenerate delta values (0 or 1)

## How this service works

Call/put delta from BS closed form. Call when delta-hedging or option inventory needs BS delta. Returns delta for call and put as deterministic JSON for $0.001 USDC via x402 on Base. First-party local math only—no RPC, live market feed, API keys, storage, or mediagen. Advisory only; the caller owns capital, risk, and production controls.

## Output

Returns a deterministic JSON object containing the Black-Scholes delta for both the call and the put, computed from the closed-form d1 calculation using the supplied spot, strike, annualized volatility, risk-free rate, and time to expiry. No external data sources or live feeds are used.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "vol": {
   "type": "number",
   "description": "Annualized volatility as decimal (>0), e.g. 0.2 = 20%"
  },
  "rate": {
   "type": "number",
   "description": "Risk-free rate as decimal (optional, default 0)"
  },
  "spot": {
   "type": "number",
   "description": "Underlying spot price (>0)"
  },
  "strike": {
   "type": "number",
   "description": "Option strike price (>0)"
  },
  "t_years": {
   "type": "number",
   "description": "Time to expiry in years (>0)"
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "json",
 "example": {
  "schema": "delx/util-bs-delta/v1",
  "delta_put": -0.4800611941616275,
  "delta_call": 0.5199388058383725
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/black-scholes-delta-44ae70b7/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from api.delx.ai](https://www.zero.xyz/host/api.delx.ai/llms.txt)