{"uid":"cap_2ZlXSOJvkO-U9NvGCZPg8","slug":"saylor-innovations-data-desk-black-scholes-option-pricer-32774958","name":"Saylor Innovations Data Desk: Black-Scholes Option Pricer","description":"Black-Scholes-Merton price for European calls and puts with delta, gamma, vega, theta, rho, intrinsic and time value, moneyness and put-call parity check","url":"https://saylorinnovations.com/api/finance/black-scholes?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"],"type":"string"},"queryParams":{"type":"object","required":["s","k","t","sigma"],"properties":{"k":{"type":"string","description":"Strike price"},"r":{"type":"string","description":"optional risk-free rate decimal, default 0.04"},"s":{"type":"string","description":"Spot price"},"t":{"type":"string","description":"Years to expiry (0.5 = six months)"},"type":{"type":"string","description":"call (default) or put"},"sigma":{"type":"string","description":"Volatility as a decimal (0.25 = 25%)"}}}},"additionalProperties":false},"output":{"type":"object","required":["type"],"properties":{"type":{"type":"string","const":"json"},"example":{"type":"object","required":["inputs","price","greeks"],"properties":{"d1":{"type":["number","null"],"description":"Black-Scholes d1 term"},"d2":{"type":["number","null"],"description":"Black-Scholes d2 term"},"note":{"type":"string","description":"Caveat or context for reading the result"},"price":{"type":"number","description":"Price"},"greeks":{"type":"object","properties":{"delta":{"type":["number","null"],"description":"Option price change per 1 change in the underlying"},"gamma":{"type":["number","null"],"description":"Change in delta per 1 change in the underlying"},"theta_per_day":{"type":["number","null"],"description":"Theta per day"},"rho_per_rate_point":{"type":["number","null"],"description":"Rho per rate point"},"vega_per_vol_point":{"type":["number","null"],"description":"Vega per vol point"}},"description":"Delta, gamma, vega per volatility point, theta per day and rho per rate point"},"inputs":{"type":"object","properties":{"spot":{"type":["number","null"],"description":"Underlying price"},"type":{"type":["string","null"],"description":"Type or kind"},"strike":{"type":["number","null"],"description":"Option strike price"},"volatility":{"type":["number","null"],"description":"Annual volatility as a fraction"},"dividend_yield":{"type":["number","null"],"description":"Dividend yield"},"risk_free_rate":{"type":["number","null"],"description":"Risk free rate"},"years_to_expiry":{"type":["number","null"],"description":"Years to expiry"}},"description":"Inputs used: spot, strike, years to expiry, risk-free rate, volatility, dividend yield and option type"},"source":{"type":"string","description":"Data source the values come from"},"moneyness":{"type":["number","null"],"description":"Spot divided by strike"},"put_price":{"type":"number","description":"Put price"},"call_price":{"type":"number","description":"Call price"},"time_value":{"type":"number","description":"Time value"},"intrinsic_value":{"type":"number","description":"Intrinsic value"},"put_call_parity_check":{"type":["number","null"],"description":"Put call parity check"},"probability_in_the_money_risk_neutral":{"type":"number","description":"Probability in the money risk neutral"}}}}}}},"responseSchema":null,"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_P-AviE7JVIhX1Y3pO6jSk","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.01","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Calculates Black-Scholes-Merton theoretical price and full Greeks (delta, gamma, vega, theta, rho) for European call or put options, plus intrinsic/time value breakdown, moneyness, and put-call parity check","exampleAgentPrompt":"What's the Black-Scholes price and Greeks for a call option with a spot price of $150, strike of $155, 6 months to expiry, 25% volatility, and a 4% risk-free rate?","exampleUseCases":[{"title":"Equity option pricing before earnings","prompt":"Price a call option on a stock trading at $200 with a strike of $210, 45 days to expiry (about 0.123 years), implied volatility of 35%, and a risk-free rate of 5% — give me the full Greeks too."},{"title":"Put option hedge analysis","prompt":"I'm hedging a position with a put option — spot is $85, strike is $80, time to expiry is 3 months (0.25 years), volatility is 20%, risk-free rate 4%. What's the theoretical price, delta, and intrinsic vs time value?"},{"title":"Put-call parity verification","prompt":"Can you check put-call parity for me? The underlying is at $100, strike is $100, expiry is 1 year, volatility is 30%, and the risk-free rate is 4% — price both the call and put and confirm the parity relationship holds."}],"resultDescription":"Returns a JSON object with: the theoretical option price, Black-Scholes d1 and d2 intermediate terms, a full Greeks breakdown (delta, gamma, vega per volatility point, theta per day, rho per rate point), intrinsic value, time value, moneyness classification, a put-call parity check, and a note with contextual caveats about interpreting the result.","failureModes":["Missing required parameters (s, k, t, sigma) returns an error — all four are mandatory","Invalid or non-numeric values for spot, strike, time, or volatility cause computation failure","Time to expiry of 0 or negative values may produce undefined or null Greek values","Extremely deep in-the-money or out-of-the-money options may return null Greeks due to numerical precision limits","Risk-free rate defaults to 0.04 if omitted; forgetting to pass it explicitly may produce unexpected pricing","Option type defaults to 'call' if omitted — omitting 'type=put' will silently price a call instead"],"whenToPreferThis":"Choose this endpoint when you need a fast, cheap ($0.01), theoretically rigorous Black-Scholes-Merton price and full Greeks suite for European-style options in a single API call. It is ideal for agents performing real-time options analysis, hedging calculations, portfolio risk assessment, or educational finance tasks. Prefer it over generic financial data APIs when you need computed derivatives metrics (Greeks) rather than market-observed prices, and over building your own BSM implementation when you want a validated, hosted computation with put-call parity sanity checking included.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-18T12:56:35.807Z","isFirstParty":false,"canonicalSlug":"saylor-innovations-data-desk-black-scholes-option-pricer-32774958"}