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Black-Scholes Options Calculator

Price European call and put options using the Black-Scholes model. Get option price, Delta, Gamma, Vega, Theta, and Rho instantly.

Input Parameters

Call Price

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Put Price

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Option Greeks

d1

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d2

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Delta (Call)

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Delta (Put)

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Gamma

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Vega

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Theta (Call)

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Theta (Put)

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Rho (Call)

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About This Calculator

The Black-Scholes model prices European-style options assuming log-normal asset prices, constant volatility, and no dividends. The formula uses five inputs: spot price (S), strike price (K), time to expiry (T), risk-free rate (r), and volatility (sigma). Greeks measure option sensitivity to these parameters.

Understanding the Black-Scholes Model

The Black-Scholes Formula

Developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, this model revolutionized options pricing. The formula for a European call option is:

C = S · N(d1) − K · e−rT · N(d2)
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T)

Where N() is the standard normal CDF, S = spot price, K = strike, r = risk-free rate, T = time to expiry, σ = volatility

Key Assumptions

  • 1.Log-normal prices: Stock prices follow a geometric Brownian motion with constant drift and volatility.
  • 2.No dividends: The underlying pays no dividends during the option life (can be adjusted).
  • 3.European exercise: Options can only be exercised at expiration, not before.
  • 4.Constant volatility: Volatility remains the same over the option's life (a major simplification).
  • 5.Efficient markets: No arbitrage opportunities, and frictionless trading with no transaction costs.

What the Greeks Tell You

Delta (Δ): How much the option price moves per $1 change in the underlying. Call deltas range 0 to 1; put deltas range -1 to 0.
Gamma (Γ): The rate of change of Delta. High gamma means delta is unstable and the position needs frequent rebalancing.
Vega (V): Sensitivity to a 1% change in implied volatility. Long options benefit from rising vol; short options benefit from falling vol.
Theta (Θ): Daily time decay. Options lose value as expiry approaches, all else equal. This is the "cost" of holding a long option position.
Rho (ρ): Sensitivity to interest rate changes. Usually the least important Greek for short-dated options, but matters for LEAPS.
d1 and d2: Intermediate values in the formula. N(d2) approximates the risk-neutral probability the option expires in-the-money.

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