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Black-Scholes Calculator

Calculate the fair value of European call and put options using the Black-Scholes pricing model. Get option prices and Greeks (Delta, Gamma, Theta, Vega, Rho) instantly.

Option Parameters

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Call Option

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Put Option

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The Greeks

Δ DeltaPrice sensitivity to stock movement
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Γ GammaRate of change of Delta
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Θ ThetaTime decay per day
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ν VegaSensitivity to volatility
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ρ RhoSensitivity to interest rate
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What is the Black-Scholes Model?

The Black-Scholes model, also known as Black-Scholes-Merton (BSM), is a mathematical model used to calculate the theoretical price of European-style options. Developed by economists Fischer Black and Myron Scholes in 1973, with contributions from Robert Merton, this model revolutionized options trading and earned Scholes and Merton the Nobel Prize in Economics in 1997.

The model assumes that stock prices follow a lognormal distribution and uses five key inputs: current stock price, strike price, time to expiration, risk-free interest rate, and volatility. It provides a closed-form solution, meaning the option price can be calculated directly without iterative methods.

The Black-Scholes Formula

The Black-Scholes equations for call and put options are:

Call: C = S₀e-qTN(d₁) - Ke-rTN(d₂)

Put: P = Ke-rTN(-d₂) - S₀e-qTN(-d₁)

d₁ = [ln(S₀/K) + (r - q + σ²/2)T] / (σ√T)

d₂ = d₁ - σ√T

Where:

  • S₀ = Current stock price
  • K = Strike price
  • T = Time to expiration (in years)
  • r = Risk-free interest rate
  • σ = Volatility of the stock
  • q = Dividend yield
  • N(x) = Cumulative standard normal distribution

Understanding the Greeks

The Greeks measure the sensitivity of the option price to various factors. They are essential for risk management and hedging strategies.

Delta (Δ)

Measures the rate of change of the option price with respect to the underlying asset price. A Delta of 0.5 means the option price moves $0.50 for every $1 move in the stock.

Gamma (Γ)

Measures the rate of change of Delta. High Gamma means Delta can change rapidly, which is important for managing delta-neutral positions.

Theta (Θ)

Measures time decay — how much value the option loses each day as it approaches expiration. Usually negative for option buyers.

Vega (ν)

Measures sensitivity to volatility. A Vega of 0.10 means the option price changes by $0.10 for every 1% change in implied volatility.

Rho (ρ)

Measures sensitivity to interest rates. Generally less significant than other Greeks but becomes important for longer-dated options.

💡 Quick Tips

  • ✓Higher volatility = higher option prices
  • ✓Time decay accelerates near expiration
  • ✓At-the-money options have highest Gamma
  • ✓Use risk-free rate of same duration as option

⚠️ Model Assumptions

  • • European options only
  • • Constant volatility
  • • Constant interest rate
  • • No transaction costs
  • • Efficient markets

Frequently Asked Questions

The Black-Scholes model is used to calculate the theoretical fair price of European-style options. It helps traders and investors determine whether an option is overpriced or underpriced in the market, enabling more informed trading decisions.

The Black-Scholes model uses a continuous-time framework and provides a closed-form solution, making it faster to compute. The Binomial model uses discrete time steps and can handle American options (early exercise). Black-Scholes is generally preferred for European options due to its simplicity and speed.

The standard Black-Scholes model is designed for European options, which can only be exercised at expiration. For American options, which allow early exercise, modifications or alternative models like the Binomial model are typically used.

Delta measures how much the option price changes for a $1 change in the underlying stock price. A Delta of 0.5 means the option price will increase by $0.50 if the stock price rises by $1. Call options have positive Delta (0 to 1), while put options have negative Delta (-1 to 0).

Volatility is one of the most critical inputs because it measures the expected price fluctuation of the underlying asset. Higher volatility increases option prices because there's a greater chance the option will end up profitable (in-the-money) at expiration.