{"uid":"cap_xmt9rRqvGq6Vp7KfMdNxD","slug":"derivatives-pricer-black-scholes-european-option-pricing-6f5494a8","name":"Derivatives Pricer – Black-Scholes European Option Pricing","description":"When to use: you already have model inputs (S or forward mark, K, T, r, σ) and need fair value + hedge ratios — not market-premium IV. European BSM price + analytic Greeks (delta, gamma, vega, theta, rho). Works for equity spots and power/commodity forwards (use the maturity mark as spot). USDC exact on Solana/Base. Free fixed sample: GET /v1/demo/option-price.","url":"https://derivatives-pricer-production.up.railway.app/v1/option/price?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"rate":{"type":"number","examples":[0.03,0.05],"description":"Continuously compounded risk-free rate r (e.g. 0.05 = 5%). For futures-style early exercise set r≈dividendYield."},"spot":{"type":"number","examples":[100,82.5,45.5],"description":"Underlying price S (> 0). Equity spot OR power/commodity forward mark for the option's window.","exclusiveMinimum":0},"model":{"enum":["bsm","black76"],"type":"string","default":"bsm","description":"Pricing model: \"bsm\" (default European BSM on spot) or \"black76\" (forward/futures)."},"strike":{"type":"number","examples":[100,50],"description":"Strike price K (> 0) in the same units as spot.","exclusiveMinimum":0},"forward":{"type":"number","examples":[100,82.5],"description":"Forward mark F for model=black76 (preferred). If omitted, spot is used as F.","exclusiveMinimum":0},"treeSteps":{"type":"integer","maximum":500,"minimum":50,"description":"CRR steps for American (default 200 on single price)."},"expiration":{"type":"string","format":"date-time","description":"ISO-8601 end of vol/exercise window. Mutually exclusive with timeToExpiry."},"optionType":{"enum":["call","put"],"type":"string","description":"Call or put."},"volatility":{"type":"number","examples":[0.15,0.2,0.45],"description":"Annualized volatility σ as a decimal (> 0). Applies over the vol-accrual window τ.","exclusiveMinimum":0},"optionStyle":{"enum":["european","american"],"type":"string","default":"european","description":"Exercise style (default \"european\"). American uses CRR binomial with early exercise."},"forwardStart":{"type":"string","format":"date-time","description":"ISO-8601 start of vol/exercise window. If ≤ now, behaves like plain option to expiration."},"timeToExpiry":{"type":"number","minimum":0,"examples":[0.25,1],"description":"Time to expiry T in years (≥ 0). Required unless expiration is set. Mutually exclusive with expiration."},"dividendYield":{"type":"number","minimum":0,"examples":[0,0.02],"description":"Continuous dividend / convenience / yield q (≥ 0, default 0)."},"greekConvention":{"enum":["raw","trading"],"type":"string","default":"raw","description":"Greek units: \"raw\" (vega/1.0 vol, theta/year, rho/1.0 rate) or \"trading\" (vega/1%, theta/day, rho/bp)."}}},"responseSchema":{"type":"json","example":{"model":"black-scholes-merton","price":10.45057562,"units":{"rho":"dV/dr per 1.0 absolute rate (not per 1%)","vega":"dV/dσ per 1.0 absolute volatility (not per 1%)","delta":"dV/dS or dV/dF (share / forward equivalent)","gamma":"d²V/dS² or d²V/dF²","price":"option value in spot/forward currency units","theta":"dV/dT per year (not per day)"},"greeks":{"rho":53.23248343,"vega":37.52403469,"delta":0.63683059,"gamma":0.01876202,"theta":-6.41402764},"inputs":{"rate":0.05,"spot":100,"model":"bsm","strike":100,"optionType":"call","volatility":0.2,"optionStyle":"european","timeToExpiry":1,"dividendYield":0,"greekConvention":"raw"},"timeline":{"tauYears":1,"yearBasis":"ACT/365.25","expiration":"2027-01-01T06:00:00.000Z","forwardStart":"2026-01-01T00:00:00.000Z","effectiveStart":"2026-01-01T00:00:00.000Z","calendarYearsToExpiry":1},"requestId":"00000000-0000-4000-8000-000000000001","computedAt":"2026-01-01T00:00:00.000Z","optionStyle":"european","greeksQuality":"analytic","greekConvention":"raw"}},"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.01","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":"$0.01/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.01","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.01","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_QVdzXG6POcGbK2JufCMtW","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.01","costPer":"request","priority":0,"asset":"EPjFWdd5AufqSSqeM2qN1xzybapC8G4wEGGkZwyTDt1v","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Prices European call/put options using Black-Scholes-Merton and returns the option premium plus analytic Greeks (delta, gamma, theta, vega, rho), paid per-call via USDC on Solana or Base.","exampleAgentPrompt":"Price a European call option for me: the underlying spot is 100, strike is 100, annualized vol is 20%, time to expiry is 1 year, risk-free rate is 5%, and no dividend yield — give me the Black-Scholes price and all the Greeks.","exampleUseCases":[{"title":"Equity option fair value check","prompt":"What's the Black-Scholes price and Greeks for a European put on a stock trading at $82.25, with a $85 strike, 45% implied vol, 3 months to expiry, 5% risk-free rate, and a 2% dividend yield?"},{"title":"FX option delta hedging","prompt":"I need to delta-hedge a European call on EURUSD — spot is 1.08, strike is 1.10, vol is 12%, 6 months to expiry, USD rate 5.3%, EUR rate 3.8% as the dividend yield. What's the delta and full Greeks breakdown?"},{"title":"Index call for portfolio scenario","prompt":"Price a 1-year European call on the S&P 500 index at spot 5200 with a 5250 strike, 18% vol, 4.5% rate, and 1.5% dividend yield — I want the option price and all analytic Greeks including theta and vega."}],"resultDescription":"A JSON object containing the Black-Scholes-Merton option price in spot currency units, a full analytic Greeks set (delta, gamma, theta per year, vega per unit vol, rho per unit rate), the echoed input parameters, the model name, a UUID request ID, and a UTC computation timestamp.","failureModes":["Invalid or zero spot/strike/volatility — returns 400 with validation error","timeToExpiry set to 0 for an in-the-money option — returns intrinsic value with zero Greeks","Payment not processed or insufficient USDC balance — returns 402 Payment Required","Missing required fields (e.g. optionType, spot, strike) — returns 400 schema validation error","Extremely deep in/out-of-the-money with near-zero expiry — numerical edge cases may yield near-zero Greeks"],"whenToPreferThis":"Choose this endpoint when you need fast, analytic Black-Scholes pricing and exact closed-form Greeks for vanilla European options — it's ideal for real-time hedging calculations, scenario analysis, or educational tooling where an analytical solution suffices. Prefer it over Monte Carlo or finite-difference APIs when you need speed, reproducibility, and precise analytic Greeks without numerical noise. It is not suitable for American options, barrier options, or instruments requiring path-dependent simulation.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-12T00:32:48.984Z","isFirstParty":false,"canonicalSlug":"derivatives-pricer-black-scholes-european-option-pricing-6f5494a8"}