

Here is the formula for the Black-Scholes model for pricing a call option1:
The Black-Scholes model is a mathematical formula used to calculate the theoretical price of an option. It is commonly used for pricing options contracts and helps investors determine the value of options they’re considering trading. The model takes into account factors like the option’s strike price, time until expiration, underlying stock price, interest rates, and volatility.
Let's break down the formula:
- S: Current stock price
- K: Strike price of the option
- T: Time to expiration (in years)
- r: Risk-free interest rate
- σ: Average future volatility of the underlying asset
- N(d1): Cumulative standard normal probability distribution function of d1
- N(d2): Cumulative standard normal probability distribution function of d2
- C: Theoretical price of the call option
The parameters d1 and d2 are calculated as follows:
- d1 = {ln(S/K) + (r + (σ^2)/2)T} / σ√T
- d2 = d1 - σ√T
The cumulative standard normal probability distribution function, N(x), is the probability that a random variable from a standard normal distribution falls within the interval [-∞, x]. It is a standard statistical function that can be found using tables or numerical methods.
It's important to note that the Black-Scholes model has some limitations and assumptions, such as the assumption of lognormal distribution of prices, constant volatility, and no dividends. Despite these limitations, it remains a widely used and foundational model in options pricing and risk management.
