Black Scholes Calculator
Calculate European call and put option prices using the Black-Scholes model. This Nobel Prize-winning model is the foundation of modern options pricing theory.
Option Parameters
Option Prices
Call Option Price:
$0.00
Put Option Price:
$0.00
Put-Call Parity:
N/A
Option Greeks
Delta (Call):
0.000
Delta (Put):
0.000
Gamma:
0.000
Risk Metrics
Theta (Call):
0.000
Vega:
0.000
Rho (Call):
0.000
Understanding the Black-Scholes Model
The Black-Scholes model, developed by Fischer Black and Myron Scholes in 1973, revolutionized options pricing by providing a mathematical framework for valuing European-style options. This Nobel Prize-winning model assumes that stock prices follow a geometric Brownian motion and that markets are efficient.
Black-Scholes Formula
Call Option
- C = S × N(d1) - K × e^(-rT) × N(d2)
- d1 = [ln(S/K) + (r + s²/2)T] / (svT)
- d2 = d1 - svT
- N(x) = Cumulative normal distribution
Put Option
- P = K × e^(-rT) × N(-d2) - S × N(-d1)
- Uses same d1 and d2 as call
- Put-call parity relationship
- C - P = S - K × e^(-rT)
Model Assumptions
Key Assumptions
Foundations of the Black-Scholes model
Market Assumptions
- No arbitrage opportunities
- Frictionless markets (no transaction costs)
- Continuous trading possible
- Short selling allowed
Asset Assumptions
- Stock pays no dividends
- Log-normal stock price distribution
- Constant volatility
- Constant risk-free rate
Option Assumptions
- European exercise style
- Exercise only at expiration
- Fixed strike price
- Fixed expiration date
Key Variables
| Variable | Symbol | Description | Impact on Option Price |
|---|---|---|---|
| Stock Price | S | Current price of underlying stock | Higher S increases call value, decreases put value |
| Strike Price | K | Exercise price of option | Higher K decreases call value, increases put value |
| Time to Expiration | T | Time remaining until expiration | More time increases option value |
| Risk-Free Rate | r | Current risk-free interest rate | Higher r increases call value, decreases put value |
| Volatility | s | Expected volatility of stock price | Higher volatility increases both call and put values |
Option Greeks
Delta (?)
- Rate of change of option price with respect to stock price
- Call delta: 0 to 1, Put delta: -1 to 0
- Hedging ratio for delta-neutral strategies
- Probability of finishing in-the-money
Gamma (G)
- Rate of change of delta with respect to stock price
- Measures convexity of option price
- Higher for at-the-money options
- Important for dynamic hedging
Theta (T)
- Rate of change of option price with respect to time
- Time decay - always negative for long positions
- Accelerates as expiration approaches
- Calendar spread strategies
Vega (V)
- Rate of change of option price with respect to volatility
- Always positive for both calls and puts
- Higher for longer-dated options
- Volatility trading strategies
Model Applications
Options Pricing
- Fair value calculation
- Risk-neutral valuation
- Market efficiency assessment
- Arbitrage opportunity identification
Risk Management
- Portfolio hedging
- Delta hedging strategies
- Volatility management
- Value at risk calculations
Model Limitations
Practical Issues
- Assumes constant volatility
- No transaction costs
- Continuous trading assumption
- European options only
Market Realities
- Volatility smiles and skews
- Jumps in asset prices
- Early exercise features
- Market frictions
Key Takeaways for Black-Scholes Calculator
- The Black-Scholes model calculates theoretical prices for European call and put options
- Key inputs include stock price, strike price, time to expiration, risk-free rate, and volatility
- Higher volatility increases option prices for both calls and puts
- More time to expiration increases option values due to greater uncertainty
- The model assumes log-normal stock price distribution and efficient markets
- Option Greeks (delta, gamma, theta, vega) help understand risk sensitivities
- The calculator provides theoretical prices that may differ from market prices due to real-world factors
- Use the calculator for educational purposes and to understand options pricing dynamics