The Black-Scholes Model

The Black-Scholes model, also known as the Black-Scholes-Merton model, is a mathematical model used to calculate the theoretical price of options, including both call and put options. This model assumes that financial markets are efficient and that the price of the underlying asset follows a geometric Brownian motion with constant volatility.

The formula for the Black-Scholes model is as follows:

For a call option:

C = S0 * N(d1) - X * e^-rt * N(d2)

For a put option:

P = X * e^-rt * N(-d2) - S0 * N(-d1)

Where:

• C = Call option price

• P = Put option price

• S0 = Current price of the underlying asset

• X = Strike price of the option

• r = Risk-free interest rate

• t = Time to expiration (in years)

• N = A function that represents a standard normal distribution

• e = Mathematical constant (approx. 2.71828)

• d1 and d2 are calculated as:

d1 = [ln(S0/X) + (r + σ^2/2) * t] / (σ * sqrt(t))

d2 = d1 - σ * sqrt(t)

Where:

• ln() is the natural logarithm function

• σ = Standard deviation (volatility) of the underlying asset's return

The Black-Scholes model makes several assumptions, including no dividends paid during the life of the option, market efficiency, no transaction costs, and the risk-free interest rate and volatility of the underlying are known and constant.

While the model has limitations and doesn't perfectly describe real-world options markets (e.g., it assumes constant volatility), it has been a foundation of financial quantitative theory and is still widely used by traders today.