Black-Scholes Option Pricer
European option pricing with Greeks.
Overview
The Black-Scholes Option Pricer values European call and put options under the classic 1973 model and reports the Greeks — delta, gamma, vega, theta and rho. Enter spot price, strike, volatility, risk-free rate, dividend yield and time to expiry and you get back fair-value prices and sensitivities.
It is the workhorse for finance students learning options pricing, quants prototyping a model, traders ballparking a position before opening a Bloomberg terminal and developers building derivatives tooling. The Greeks tell you how the value moves as each input changes.
How it works
Under Black-Scholes the underlying follows geometric Brownian motion. The fair value of a call with strike K, spot S, time T, risk-free rate r, dividend yield q and volatility σ is C = S * e^-qT * N(d1) - K * e^-rT * N(d2) where d1 = (ln(S/K) + (r - q + σ^2 / 2) * T) / (σ * sqrt(T)) and d2 = d1 - σ * sqrt(T).
Put-call parity gives the put: P = K * e^-rT * N(-d2) - S * e^-qT * N(-d1). Each Greek is a partial derivative of C with respect to a parameter — delta is ∂C/∂S, vega is ∂C/∂σ, theta is ∂C/∂T and so on. N denotes the standard normal CDF.
Examples
S=100, K=100, r=5%, q=0, σ=20%, T=1
→ Call ≈ 10.45, Put ≈ 5.57
→ Delta call ≈ 0.6368, Vega ≈ 0.3752
S=110, K=100, r=5%, q=0, σ=20%, T=0.5
→ Call ≈ 14.04, Put ≈ 1.57
S=50, K=55, r=2%, q=1%, σ=30%, T=0.25
→ Call ≈ 1.31, Put ≈ 6.18
FAQ
Does Black-Scholes work for American options?
Not exactly. American options allow early exercise, which raises the value of puts and dividend-paying calls. The tool prices European-style options only.
What volatility should I input?
Annualised standard deviation of log returns, expressed as a decimal. 20% is 0.20, not 20. Implied volatility from market prices is most common for pricing.
Why does theta vanish near expiry for deep in-the-money options?
Their value is dominated by intrinsic value, not time value, so additional days add little.
Is the model dimensionally consistent?
Yes — rates, dividends, time and volatility must all use the same time unit, typically years.
How accurate is it in practice?
It captures the broad shape of option prices but ignores volatility smile, jumps and stochastic volatility. Traders calibrate around it rather than blindly trusting it.