Black-Scholes model: What it is and how it works

The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is one of the most important concepts in modern financial theory. Black-Scholes postulates that instruments, such as stocks or futures contracts, will have a normal price distribution that follows a random walk with constant drift and volatility. Let's see then what the Black Scholes model is, how it works, its formula and its main advantages and disadvantages.

What is the Black-Scholes model?

The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is one of the most important concepts in modern financial theory. This mathematical equation estimates the theoretical value of derivatives based on other investment instruments, taking into account the impact of time and other risk factors. Developed in 1973, it is still considered one of the best methods for pricing an options contract.

How the Black-Scholes model works

Black-Scholes postulates that instruments, such as stocks or futures contracts, will have a lognormal distribution of prices that follows a random walk with constant drift and volatility. Starting from this assumption and taking into account other important variables, the equation obtains the price of a European-type call option. The Black-Scholes equation requires five variables. These variables are volatility, the price of the underlying asset, the exercise price of the option, the time to expiration of the option, and the risk-free interest rate. With these variables, it is theoretically possible for option sellers to set rational prices for the options they sell. Furthermore, the model predicts that the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant variation of the stock price, the time value of money, the option's exercise price, and the time to expiration of the option.

The formula of the Black-Scholes model

The math involved in the formula is complicated and can be intimidating. Fortunately, it is not necessary to know or understand mathematics to use the Black-Scholes model in our own strategies. Options traders have access to a wide variety of online options calculators, and many of today's trading platforms have robust options analysis tools, including indicators and spreadsheets that perform the calculations and obtain fixing values. of option prices. The call option Black-Scholes formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Next, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the value resulting from the above calculation.

Benefits of the Black-Scholes model

• Allows you to manage risk: By knowing the theoretical value of an option, investors can use the Black-Scholes model to manage their risk exposure to different assets. Therefore, the Black-Scholes model is useful to investors not only in evaluating potential returns but also in understanding portfolio weaknesses and poor investment areas.
• Allows you to optimize the portfolio: The Black-Scholes model can be used to optimize portfolios by providing a measure of the expected returns and risks associated with different options. This allows investors to make smarter decisions better aligned with their risk tolerance and search for profits.
• Provides a framework: The Black-Scholes model provides a theoretical framework for pricing options. This allows investors and traders to determine the fair price of an option using a structured and defined methodology that has been tested.
• Streamline pricing: Similarly, the Black-Scholes model is widely accepted and used by professionals in the financial sector. This allows for greater consistency and comparability across markets and jurisdictions.
• Improve market efficiency: The Black-Scholes model has led to greater market efficiency and transparency as traders and investors can better price and trade options. This simplifies the pricing process as there is greater implicit understanding of how prices are derived.

Limitations of the Black-Scholes model

• Limits usefulness: As stated above, the Black-Scholes model is only used to value European options and does not take into account that US options could be exercised before the expiration date.
• Lack of cash flow flexibility: The model assumes that dividends and risk-free rates are constant, but this may not be true in reality. Therefore, the Black-Scholes model may lack the ability to truly reflect the exact future cash flow of an investment due to the rigidity of the model.
• Assume constant volatility: The model also assumes that volatility remains constant over the life of the option. In reality, this is not usually the case because volatility fluctuates with the level of supply and demand.
• Mislead other assumptions: The Black-Scholes model is also based on other assumptions. These assumptions include that there are no transaction costs or taxes, that the risk-free interest rate is constant for all maturities, that short selling of securities with use of the proceeds is permitted, and that there are no risk-free arbitrage opportunities. Each of these assumptions can result in prices that deviate from actual results.