Black Scholes

European option pricer: Black–Scholes-Merton, plus the standard Greeks.

Inputs

• Spot S and strike K: positive, up to 1e12. Risk-free rate r and dividend yield q: finite percentages, greater than −100% and up to 100%. Volatility σ: 0% to 500%. Time to expiry T: 0 to 100 years.
• Type: call or put.

Output

Price plus Δ (delta), Γ (gamma), ν (vega), Θ (theta), ρ (rho), each also shown per 1 percentage point of vol/r and per trading day for theta. At T = 0 the price is the intrinsic value and at σ = 0 it is the discounted deterministic payoff; the Greeks are not defined in either case.

Implied volatility is solved by bisection from an added market price, after checking it against the no-arbitrage price interval for the current S, K, r, q and T. A price outside that interval, exactly at its lower bound (0% implied vol), at its upper asymptote, or needing more than 500% implied vol is reported as such instead of showing a raw solver failure.