Black Scholes Calculator
Calculate fair market price for stock options using the Black-Scholes-Merton model
Black-Scholes Option Pricing
Current market price of the underlying stock
Exercise price of the option contract
Time until option expires (in years)
Annual risk-free interest rate (Treasury bills)
Annualized volatility of the stock
Expected annual dividend yield
Option Pricing Results
Formula Parameters
Example Calculation
Apple (AAPL) Options Example
Current Stock Price: $400.00
Strike Price: $350.00
Time to Maturity: 1 year
Risk-Free Rate: 3%
Volatility: 20%
Dividend Yield: 1%
Expected Results
Call Option Price: ~$65.67
Put Option Price: ~$9.30
Call Delta: ~0.829 (82.9% of stock movement)
Put Delta: ~-0.161 (-16.1% of stock movement)
Option Types
Call Option
Right to buy at strike price
Bullish strategy - profits when price rises
Put Option
Right to sell at strike price
Bearish strategy - profits when price falls
Black-Scholes Assumptions
European-style options only
Constant volatility and interest rates
No transaction costs or taxes
Continuous trading and liquidity
Log-normal price distribution
Trading Tips
Higher volatility increases option prices
Options lose value as expiration approaches
Consider the Greeks for risk management
Use theoretical price as fair value guide
Understanding the Black-Scholes Model
What is the Black-Scholes Model?
The Black-Scholes model is a mathematical framework for pricing European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, it provides a theoretical estimate of options prices and revolutionized options trading.
Key Components
- •Current Stock Price (S₀): Market price of underlying asset
- •Strike Price (K): Exercise price of the option
- •Time to Expiration (T): Time until option expires
- •Volatility (σ): Expected price fluctuation
- •Risk-Free Rate (r): Treasury bill rate
Black-Scholes Formulas
Call Option:
C = S₀e⁻ᵠᵀN(d₁) - Ke⁻ʳᵀN(d₂)
Put Option:
P = Ke⁻ʳᵀN(-d₂) - S₀e⁻ᵠᵀN(-d₁)
Where:
d₁ = [ln(S₀/K) + (r-q+σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
N(d): Cumulative standard normal distribution function
e: Mathematical constant (≈ 2.71828)
ln: Natural logarithm
Model Limitations
Assumptions
- • Constant volatility and interest rates
- • No dividends during option life
- • European exercise only
- • Perfect market liquidity
Real-World Challenges
- • Volatility changes over time
- • Transaction costs exist
- • Early exercise may be optimal
- • Interest rates fluctuate