{"uid":"cap_BLuzIHjNlfMsRIzLo_RaF","slug":"black-scholes-delta-44ae70b7","name":"Black-Scholes Delta","description":"Call/put delta from BS closed form. Call when delta-hedging or option inventory needs BS delta. Returns delta for call and put as deterministic JSON for $0.001 USDC via x402 on Base. First-party local math only—no RPC, live market feed, API keys, storage, or mediagen. Advisory only; the caller owns capital, risk, and production controls.","url":"https://api.delx.ai/api/v1/x402/bs-delta","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"vol":{"type":"number","description":"Annualized volatility as decimal (>0), e.g. 0.2 = 20%"},"rate":{"type":"number","description":"Risk-free rate as decimal (optional, default 0)"},"spot":{"type":"number","description":"Underlying spot price (>0)"},"strike":{"type":"number","description":"Option strike price (>0)"},"t_years":{"type":"number","description":"Time to expiry in years (>0)"}}},"responseSchema":{"type":"json","example":{"schema":"delx/util-bs-delta/v1","delta_put":-0.4800611941616275,"delta_call":0.5199388058383725}},"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.001","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":"$0.001/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.001","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.001","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_bkOIXMTYY76aeAWay94ei","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.001","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes call and put delta for European options using the Black-Scholes closed-form formula given spot, strike, volatility, rate, and time to expiry.","exampleAgentPrompt":"What's the Black-Scholes call and put delta for an option with a spot price of 150, strike of 155, annualized vol of 25%, risk-free rate of 5%, and 45 days to expiry?","exampleUseCases":[{"title":"Delta-hedge a short call position","prompt":"I'm short a call with spot at 200, strike at 210, vol of 30%, risk-free rate of 4%, and 60 days to expiry — give me the Black-Scholes delta so I know how many shares to buy to hedge."},{"title":"Options inventory delta tracking","prompt":"I hold a mix of puts with spot 100, strike 95, vol 20%, rate 2%, and 30 days to expiry — calculate the BS put delta so I can track my net delta exposure."},{"title":"ATM straddle delta check before rebalancing","prompt":"Can you compute Black-Scholes call and put delta for a straddle where spot is 500, strike is 500, vol is 18%, rate is 5.5%, and time to expiry is 0.25 years?"}],"resultDescription":"Returns a deterministic JSON object containing the Black-Scholes delta for both the call and the put, computed from the closed-form d1 calculation using the supplied spot, strike, annualized volatility, risk-free rate, and time to expiry. No external data sources or live feeds are used.","failureModes":["Missing required fields (spot, strike, vol, t_years) return a validation error","vol <= 0 or t_years <= 0 returns an invalid parameter error","spot or strike <= 0 returns an invalid parameter error","Payment not included or insufficient USDC returns a 402 Payment Required response","Extremely small t_years values near zero may produce near-degenerate delta values (0 or 1)"],"whenToPreferThis":"Choose this endpoint when you need a fast, deterministic, closed-form Black-Scholes delta with no API keys, no live market data dependencies, and no external RPC calls. Ideal for delta-hedging calculations, options inventory management, or any scenario where you need a reliable, reproducible BS delta for a known set of inputs at very low cost ($0.001 USDC). Prefer this over stochastic or Monte Carlo approaches when Black-Scholes assumptions are acceptable and speed is paramount.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-09-14T17:53:10.167Z","isFirstParty":false}