{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"black-scholes"},"result":{"entity_type":"calculator","id":"black-scholes","calculator_id":"black-scholes","canonical_url":"https://tttkmbb.com/investing/black-scholes","name":"Black-Scholes Option Calculator","title":"Black-Scholes Option Pricing Calculator – Call and Put Prices with Greeks (Merton dividend yield)","category":"investing","category_name":"Investing & Real Estate","tool_name":"price_black_scholes_option","featured_mcp_tool":true,"description":"Prices a European call and put with the Black-Scholes-Merton formula from spot, strike, time to expiry, risk-free rate, volatility and a continuous dividend yield, and reports d1, d2 and the Greeks (delta, gamma, vega, theta, rho).","use_when":"You need a theoretical European option price or its sensitivities from a given volatility, or want to check put-call parity for quoted prices.","do_not_use_when":"The option is American with early exercise value, the underlying pays discrete dividends, or you need implied volatility from a market price (solve numerically instead). Informational only; not financial advice.","inputs":[{"name":"spot_price","label":"Spot price (S)","type":"number","required":true,"max":1000000000,"exclusive_min":0,"description":"Current price of the underlying asset.","example":100},{"name":"strike_price","label":"Strike price (K)","type":"number","required":true,"max":1000000000,"exclusive_min":0,"description":"Exercise price of the option.","example":100},{"name":"time_years","label":"Time to expiry","type":"number","unit":"years","required":true,"max":50,"exclusive_min":0,"description":"Time until expiration in years (e.g. 0.25 for 3 months).","example":1},{"name":"risk_free_rate_percent","label":"Risk-free rate","type":"number","unit":"% per year","required":true,"min":-20,"max":100,"description":"Continuously compounded annual risk-free interest rate in percent.","example":5},{"name":"volatility_percent","label":"Volatility (σ)","type":"number","unit":"% per year","required":true,"max":500,"exclusive_min":0,"description":"Annualized volatility of the underlying's returns in percent.","example":20},{"name":"dividend_yield_percent","label":"Dividend yield (q)","type":"number","unit":"% per year","required":false,"default":0,"min":0,"max":100,"description":"Continuous dividend yield of the underlying in percent (0 for a non-dividend stock).","example":0}],"outputs":[{"name":"call_price","label":"Call price","type":"number","decimals":4,"description":"S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2)."},{"name":"put_price","label":"Put price","type":"number","decimals":4,"description":"K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1)."},{"name":"d1","label":"d1","type":"number","decimals":4,"description":"[ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)."},{"name":"d2","label":"d2","type":"number","decimals":4,"description":"d1 − σ·√T."},{"name":"delta_call","label":"Call delta","type":"number","decimals":4,"description":"e^(−qT)·N(d1): price change per 1 unit change in S."},{"name":"delta_put","label":"Put delta","type":"number","decimals":4,"description":"−e^(−qT)·N(−d1)."},{"name":"gamma","label":"Gamma","type":"number","decimals":6,"description":"e^(−qT)·φ(d1) / (S·σ·√T): change in delta per 1 unit change in S (same for call and put)."},{"name":"vega_per_1pct","label":"Vega (per 1 % volatility)","type":"number","decimals":4,"description":"S·e^(−qT)·φ(d1)·√T / 100: price change per 1 percentage-point rise in volatility."},{"name":"theta_call_per_day","label":"Call theta (per day)","type":"number","decimals":4,"description":"Annual call theta / 365: price decay per calendar day."},{"name":"theta_put_per_day","label":"Put theta (per day)","type":"number","decimals":4,"description":"Annual put theta / 365."},{"name":"rho_call_per_1pct","label":"Call rho (per 1 % rate)","type":"number","decimals":4,"description":"K·T·e^(−rT)·N(d2) / 100: price change per 1 percentage-point rise in r."},{"name":"rho_put_per_1pct","label":"Put rho (per 1 % rate)","type":"number","decimals":4,"description":"−K·T·e^(−rT)·N(−d2) / 100."},{"name":"put_call_parity_check","label":"Put-call parity","type":"string","decimals":4,"description":"C − P compared with S·e^(−qT) − K·e^(−rT)."}],"input_schema":{"type":"object","properties":{"spot_price":{"description":"Current price of the underlying asset.","type":"number","maximum":1000000000,"exclusiveMinimum":0,"examples":[100]},"strike_price":{"description":"Exercise price of the option.","type":"number","maximum":1000000000,"exclusiveMinimum":0,"examples":[100]},"time_years":{"description":"Time until expiration in years (e.g. 0.25 for 3 months). Unit: years.","type":"number","maximum":50,"exclusiveMinimum":0,"examples":[1],"x-unit":"years"},"risk_free_rate_percent":{"description":"Continuously compounded annual risk-free interest rate in percent. Unit: % per year.","type":"number","minimum":-20,"maximum":100,"examples":[5],"x-unit":"% per year"},"volatility_percent":{"description":"Annualized volatility of the underlying's returns in percent. Unit: % per year.","type":"number","maximum":500,"exclusiveMinimum":0,"examples":[20],"x-unit":"% per year"},"dividend_yield_percent":{"description":"Continuous dividend yield of the underlying in percent (0 for a non-dividend stock). Unit: % per year.","type":"number","minimum":0,"maximum":100,"default":0,"examples":[0],"x-unit":"% per year"}},"additionalProperties":false,"required":["spot_price","strike_price","time_years","risk_free_rate_percent","volatility_percent"]},"output_schema":{"type":"object","properties":{"call_price":{"description":"S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2).","type":"number"},"put_price":{"description":"K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1).","type":"number"},"d1":{"description":"[ln(S/K) + (r − q + σ²/2)·T] / (σ·√T).","type":"number"},"d2":{"description":"d1 − σ·√T.","type":"number"},"delta_call":{"description":"e^(−qT)·N(d1): price change per 1 unit change in S.","type":"number"},"delta_put":{"description":"−e^(−qT)·N(−d1).","type":"number"},"gamma":{"description":"e^(−qT)·φ(d1) / (S·σ·√T): change in delta per 1 unit change in S (same for call and put).","type":"number"},"vega_per_1pct":{"description":"S·e^(−qT)·φ(d1)·√T / 100: price change per 1 percentage-point rise in volatility.","type":"number"},"theta_call_per_day":{"description":"Annual call theta / 365: price decay per calendar day.","type":"number"},"theta_put_per_day":{"description":"Annual put theta / 365.","type":"number"},"rho_call_per_1pct":{"description":"K·T·e^(−rT)·N(d2) / 100: price change per 1 percentage-point rise in r.","type":"number"},"rho_put_per_1pct":{"description":"−K·T·e^(−rT)·N(−d2) / 100.","type":"number"},"put_call_parity_check":{"description":"C − P compared with S·e^(−qT) − K·e^(−rT).","type":"string"}}},"formula":"d1 = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T); d2 = d1 − σ·√T; C = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2); P = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1); Δ_call = e^(−qT)·N(d1); Γ = e^(−qT)·φ(d1)/(S·σ·√T); vega = S·e^(−qT)·φ(d1)·√T; Θ_call = −S·e^(−qT)·φ(d1)·σ/(2√T) − r·K·e^(−rT)·N(d2) + q·S·e^(−qT)·N(d1); ρ_call = K·T·e^(−rT)·N(d2), with r, q, σ as decimals","method":"Black-Scholes-Merton model for European options on an asset with lognormal returns, constant volatility and rate, no transaction costs and a continuous dividend yield; N is the standard normal CDF and φ its density. Vega and rho are scaled per percentage point, theta per calendar day (÷365). Informational mathematics only; not financial advice.","sources":[{"name":"Black F. & Scholes M. (1973) The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3):637-654","url":"https://doi.org/10.1086/260062","type":"peer_reviewed","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Black–Scholes model","url":"https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Greeks (finance)","url":"https://en.wikipedia.org/wiki/Greeks_(finance)","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"S 100, K 100, 1 year, r 5 %, σ 20 %","inputs":{"spot_price":100,"strike_price":100,"time_years":1,"risk_free_rate_percent":5,"volatility_percent":20},"expected":{"call_price":10.4506,"put_price":5.5735,"d1":0.35,"d2":0.15,"delta_call":0.6368,"delta_put":-0.3632,"gamma":0.018762,"vega_per_1pct":0.3752,"theta_call_per_day":-0.0176,"rho_call_per_1pct":0.5323,"rho_put_per_1pct":-0.4189},"url":"https://tttkmbb.com/api/v1/calculate/black-scholes?spot_price=100&strike_price=100&time_years=1&risk_free_rate_percent=5&volatility_percent=20"},{"name":"Same option with a 3 % dividend yield","inputs":{"spot_price":100,"strike_price":100,"time_years":1,"risk_free_rate_percent":5,"volatility_percent":20,"dividend_yield_percent":3},"expected":{"call_price":8.6525,"put_price":6.7309,"d1":0.2,"d2":0,"delta_call":0.5621,"gamma":0.018974,"vega_per_1pct":0.3795,"theta_call_per_day":-0.0123,"rho_call_per_1pct":0.4756},"url":"https://tttkmbb.com/api/v1/calculate/black-scholes?spot_price=100&strike_price=100&time_years=1&risk_free_rate_percent=5&volatility_percent=20&dividend_yield_percent=3"}],"faq":[{"q":"Which volatility should I enter?","a":"The annualized standard deviation of log returns in percent (20 for 20 %). Implied volatility from quoted options reproduces market prices; historical volatility gives a theoretical value."},{"q":"Why is theta divided by 365?","a":"The formula gives decay per year; dividing by 365 calendar days gives the commonly quoted per-day theta. Some platforms use 252 trading days, which gives a value about 45 % larger."},{"q":"Does this work for American options?","a":"Only as an approximation. American calls on non-dividend stocks equal European calls, but American puts and options on dividend-paying assets can be worth more because of early exercise."}],"tags":["black scholes","option pricing","call option","put option","greeks","delta gamma vega theta"],"related":[{"calculator_id":"normal-distribution","reason":"The cumulative normal probabilities N(d1) and N(d2) used in the formula."},{"calculator_id":"stock-profit","reason":"Profit on the underlying shares themselves."},{"calculator_id":"compound-interest","reason":"Continuous compounding behind the e^(−rT) discount factor."}],"links":{"html":"https://tttkmbb.com/investing/black-scholes","markdown":"https://tttkmbb.com/investing/black-scholes.md","json":"https://tttkmbb.com/investing/black-scholes.json","api":"https://tttkmbb.com/api/v1/calculate/black-scholes","schema":"https://tttkmbb.com/api/v1/calculators/black-scholes","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:50:01Z","next_actions":[{"tool":"price_black_scholes_option","calculator_id":"black-scholes","reason":"Run Black-Scholes Option Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/investing/black-scholes.md"}}