Stock Option Calculator
Price stock options using the Black-Scholes model and calculate all Greeks.
The Black-Scholes Model
The Black-Scholes model provides a theoretical estimate of European-style option prices. It was developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s.
P = K · e^(-rT) · N(-d2) – S · e^(-qT) · N(-d1)
d1 = [ln(S/K) + (r – q + σ²/2) · T] / (σ · √T)
d2 = d1 – σ · √T
Where S = stock price, K = strike price, T = time to expiration, r = risk-free rate, q = dividend yield, σ = volatility, and N(x) = cumulative standard normal distribution.
Understanding the Greeks
- Delta (Δ): Rate of change in option price per $1 change in stock price. Call delta: 0 to 1, Put delta: -1 to 0.
- Gamma (Γ): Rate of change of delta per $1 change in stock price. Same for calls and puts. Highest for at-the-money options.
- Theta (Θ): Time decay: how much the option loses per day. Always negative for long options (shown per day).
- Vega (V): Sensitivity to a 1% change in volatility. Same for calls and puts. Higher for longer-dated options.
- Rho (Ρ): Sensitivity to a 1% change in the risk-free rate. Positive for calls, negative for puts.
Model Assumptions & Limitations
- European options only: Black-Scholes prices European options. American options (early exercise) may differ.
- Constant volatility: The model assumes volatility stays the same, but real markets show volatility smiles and skews.
- Dividends: This calculator uses the generalized Black-Scholes model with continuous dividend yield. Enter 0% for non-dividend stocks.
- Log-normal distribution: Assumes stock returns follow a log-normal distribution, which underestimates tail risks.
- Frictionless markets: Assumes no transaction costs, taxes, or bid-ask spreads.
Frequently Asked Questions
What is the Black-Scholes formula for a call option?
Call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r − q + σ²/2)T] / (σ√T) and d2 = d1 − σ√T. S is stock price, K is strike, T is time in years, r is risk-free rate, σ is implied volatility, q is continuous dividend yield, and N() is the cumulative standard normal distribution. Put price follows from put-call parity.
What are the five Greeks and what do they measure?
Delta measures how much the option price changes per $1 move in the stock. Gamma measures Delta’s rate of change. Theta measures daily time decay (options lose value as expiration approaches). Vega measures sensitivity to a 1% change in implied volatility. Rho measures sensitivity to a 1% change in the risk-free interest rate.
What is implied volatility (IV) and how does the calculator find it?
Implied volatility is the market’s expectation of future price movement, back-solved from the observed market price of an option. The calculator’s Find IV mode takes your entered market price and uses Newton-Raphson iteration (up to 100 iterations, tolerance 0.0001, starting at 20% as an initial guess) to solve for the σ that matches that price.
Does the Black-Scholes model work for American options?
No — Black-Scholes prices European options only, meaning options exercisable only at expiration. American options (exercisable any time before expiration) require different models such as the Binomial Tree or Barone-Adesi Whaley approximation. Most US equity options are American-style, so treat Black-Scholes results as approximations for those.
What is the difference between intrinsic value and time value in an option?
Intrinsic value is how much in-the-money the option is: max(S−K, 0) for a call and max(K−S, 0) for a put. Time value is the option premium above intrinsic value, reflecting the possibility the option will move further in-the-money before expiration. An at-the-money option has zero intrinsic value but maximum time value. The calculator displays both components.