5.c Black Scholes Option Pricing Model
2. 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.
How can I help you today?