Call Put Option Calculator

Black-Scholes Option Pricing and Option Payoffs

Financial options are derivative contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified strike price within a set expiration period. European options can only be exercised at expiration, whereas American options can be exercised at any point prior to expiration.

Call vs Put Options

• Call Option: Gives the buyer the right to buy the underlying stock at the strike price. A call option gains value when the stock price rises above the strike price plus premium.
• Put Option: Gives the buyer the right to sell the underlying stock at the strike price. A put option gains value when the stock price falls below the strike price minus premium.

The Black-Scholes Model Formulas

The Black-Scholes formula estimates the theoretical fair value of European call and put options using stock price ($S$), strike price ($K$), risk-free rate ($r$), time to expiration ($T$), and volatility ($\sigma$):

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2) T}{\sigma \sqrt{T}}$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

$$\text{Call Price } C = S N(d_1) - K e^{-r T} N(d_2)$$

$$\text{Put Price } P = K e^{-r T} N(-d_2) - S N(-d_1)$$

Understanding Option Greeks

• Delta ($\Delta$): Measures sensitivity of option price to a $1 change in stock price.
• Gamma ($\Gamma$): Measures rate of change of Delta per $1 stock move.
• Vega ($\nu$): Measures sensitivity of option price to a 1% change in implied volatility.
• Theta ($\Theta$): Measures time decay loss in option price per calendar day.