{"uid":"cap_6gAAzke0DTPxPWtjKVICQ","slug":"black-scholes-merton-european-option-pricer-with-greeks-cacdbd37","name":"Black-Scholes-Merton European Option Pricer with Greeks","description":"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.","url":"https://agent402.tools/api/black-scholes","method":"POST","headers":{},"bodySchema":{"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)"}}},"responseSchema":{"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}},"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.002","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.002/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.002","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.002","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_jIy9kWqEqZ7T3qOiJWEm-","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.002","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"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.","exampleAgentPrompt":"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.","exampleUseCases":null,"resultDescription":"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).","failureModes":["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"],"whenToPreferThis":"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.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-09-15T00:59:20.041Z","isFirstParty":false}