# Black-Scholes-Merton European Option Pricer with Greeks

> Black-Scholes-Merton European Option Pricer with Greeks is a paid API for AI agents from agent402.tools, paid per call via x402, $0.002/call, status unknown (last checked 2026-09-15).

Prices a European call or put option using the Black-Scholes-Merton model and returns the option price plus all five Greeks (delta, gamma, vega, theta, rho), with support for continuous dividend yield.

## Facts

- Endpoint: POST https://agent402.tools/api/black-scholes
- Price: $0.002/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-15
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/black-scholes-merton-european-option-pricer-with-greeks-cacdbd37
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_6gAAzke0DTPxPWtjKVICQ

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-merton-european-option-pricer-with-greeks-cacdbd37 -d '<json body>'
```

Example prompt: Use Black-Scholes to price a European call option on a stock trading at $150, with a strike of $155, 30% annual volatility, a risk-free rate of 5%, dividend yield of 1.5%, expiring in 0.5 years — and give me all the greeks.

## When to prefer this

Use this endpoint when you need a fast, analytically exact Black-Scholes-Merton price and full set of Greeks for a European option, especially when continuous dividend yield is relevant. Prefer this over Monte Carlo or binomial tree services when speed and closed-form accuracy matter and the option has no early exercise (American-style) features.

## Known failure modes

- Missing required fields (spot, strike, type, volatility, riskFreeRate, timeToExpiryYears) returns a 400 error
- Invalid option type string (not 'call' or 'put') returns a validation error
- Non-positive spot or strike price may cause computation errors
- Negative timeToExpiryYears is mathematically undefined
- Extreme parameter values (e.g., very high volatility) may produce numerical edge cases
- Payment failure (insufficient USDC balance or x402 wallet not configured) blocks the call

## How this service works

Price a European option (call or put) with the Black-Scholes-Merton model, plus the greeks (delta, gamma, vega, theta, rho). Continuous dividend yield supported. Greeks are per unit: theta is per year, vega per 1.00 change in volatility, rho per 1.00 change in rate. Deterministic - matches standard references to the cent.

## Output

Returns the theoretical fair value of the European option and all five Greeks: delta (price sensitivity to spot), gamma (rate of change of delta), vega (sensitivity to 1.00 change in volatility), theta (time decay per year), and rho (sensitivity to interest rate per 1% change).

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "spot": {
   "type": "number",
   "description": "Current underlying price (S)"
  },
  "type": {
   "type": "string",
   "description": "\"call\" or \"put\""
  },
  "strike": {
   "type": "number",
   "description": "Strike price (K)"
  },
  "volatility": {
   "type": "number",
   "description": "Annual volatility as decimal (sigma)"
  },
  "riskFreeRate": {
   "type": "number",
   "description": "Annual risk-free rate as decimal (r)"
  },
  "dividendYield": {
   "type": "number",
   "description": "Continuous dividend yield as decimal (q). Default 0."
  },
  "timeToExpiryYears": {
   "type": "number",
   "description": "Time to expiry in years (T)"
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "json",
 "example": {
  "d1": 0.35,
  "d2": 0.15,
  "rho": 53.2325,
  "type": "call",
  "vega": 37.524,
  "delta": 0.6368,
  "gamma": 0.0188,
  "price": 10.4506,
  "theta": -6.414
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/black-scholes-merton-european-option-pricer-with-greeks-cacdbd37/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from agent402.tools](https://www.zero.xyz/host/agent402.tools/llms.txt)