{"uid":"cap_K3O67yxYNRzqF-6KJttQW","slug":"black-scholes-european-option-calculator-c0184009","name":"Black-Scholes European Option Calculator","description":"Black-Scholes European option calculator: fair price, all greeks (delta, gamma, vega, theta, rho), or implied volatility from a market price. Pure computation over your inputs.","url":"https://payai.agentstools.dev/quant/options?utm_source=zero.xyz","method":"GET","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method"],"properties":{"type":{"type":"string","const":"http"},"method":{"enum":["GET","HEAD","DELETE"],"type":"string"},"queryParams":{"type":"object","required":["op","S","K","T","r"],"properties":{"K":{"type":"number","description":"Strike price (> 0)"},"S":{"type":"number","description":"Spot price of the underlying (> 0)"},"T":{"type":"number","description":"Time to expiry in years (> 0)"},"q":{"type":"number","description":"Continuous dividend yield as a decimal, default 0"},"r":{"type":"number","description":"Risk-free rate, continuous, as a decimal"},"op":{"enum":["price","greeks","iv"],"type":"string","description":"price returns the fair value; greeks returns price plus all greeks; iv solves for implied volatility"},"type":{"enum":["call","put"],"type":"string","description":"Option type, default call"},"sigma":{"type":"number","description":"Volatility as a decimal (needed for price and greeks)"},"market_price":{"type":"number","description":"Observed option price, used for op=iv instead of sigma"}}}},"additionalProperties":false},"output":{"type":"object","required":["type"],"properties":{"type":{"type":"string"},"example":{"type":"object"}}}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","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":{"p10Cents":"0.3000","medianCents":"0.3000","p90Cents":"0.3000","minCents":"0.3000","maxCents":"0.3000","p95Cents":"0.3000","sampleCount":1,"varies":false,"failureChargeRate":1},"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_gq-Q9vdbM09r0chbSuX-S","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes Black-Scholes fair price, all Greeks (delta, gamma, vega, theta, rho), or implied volatility for European options given spot price, strike, time to expiry, rate, and volatility.","exampleAgentPrompt":"Use Black-Scholes to price a European call option where the spot price is $150, strike is $155, time to expiry is 0.5 years, risk-free rate is 5%, and volatility is 20% — give me the fair value and all the Greeks.","exampleUseCases":[{"title":"Hedge delta exposure across option positions","prompt":"I'm holding several European call and put options on the same stock. Can you run Black-Scholes calculations for each one — spot at $120, strikes ranging from $110 to $130, 3 months to expiry, 4% rate, and 25% volatility — so I can see which positions have the highest delta and need rehedging?"},{"title":"Evaluate relative value of traded options","prompt":"I just saw a European put option trading at $8.50 in the market — the underlying is at $95, strike is $100, expires in 6 months, risk-free rate is 3.5%, and I estimate volatility at 22%. What's the implied volatility from that market price so I can tell if it's overpriced or underpriced?"},{"title":"Assess interest rate sensitivity of options","prompt":"I want to understand how sensitive my European call option positions are to changes in rates. Can you compute theta, vega, and especially rho for me using Black-Scholes? Spot at $200, strike at $210, 1 year to expiry, 2.5% rate, and 18% volatility — I need to know the rho value most."}],"resultDescription":"Returns the Black-Scholes fair price and/or Greeks (delta, gamma, vega, theta, rho) for a European option, or the implied volatility solved from an observed market price, depending on the 'op' parameter chosen (price, greeks, or iv).","failureModes":["Missing required parameters (op, S, K, T, r) returns a 400 error","Sigma not provided when op=price or op=greeks returns an error","Market_price not provided when op=iv returns an error","Non-positive values for S, K, or T cause computation errors","Implied volatility solver may fail to converge for extreme or nonsensical market prices"],"whenToPreferThis":"Use this endpoint when you need fast, pure-compute Black-Scholes analytics for European options — fair pricing, Greeks, or implied volatility — without any market data dependencies. Prefer it over general financial APIs when you already have your own inputs and just need the mathematical result.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-12T00:39:51.594Z","isFirstParty":false,"canonicalSlug":"black-scholes-european-option-calculator-c0184009"}