Black Scholes Option Pricing Model

Introduced in 1973, the Black-Scholes model provides a closed-form solution for pricing European-style options, which can only be exercised at expiration.

Key Features:

• Assumes constant volatility and interest rates.
• Does not account for dividends (though extensions of the model do).
• Provides a formula to calculate the theoretical price of European call and put options.

Formula: [ C = S \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2) ]

Where:

• C: Call option price
• S: Current stock price
• X: Strike price
• r: Risk-free interest rate
• T: Time to expiration
• N: Cumulative distribution function of the standard normal distribution
• [ d_1 = \frac{\ln(S/X) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} ]
• [ d_2 = d_1 - \sigma\sqrt{T} ]
• σ: Volatility of the underlying asset

Example: For a stock priced at $100, with a strike price of $100, time to expiration of 1 year, risk-free rate of 5%, and volatility of 20%, the Black-Scholes formula can be used to calculate the theoretical price of a European call option.