Mathematics in Finance: The Black-Scholes-Merton Model
Did you know that approximately 20% of math graduates work in financial institutions? Quantitative analysis is an area of finance that addresses financial decision-making and strategies through applied mathematics and statistical modeling. It is used by banks, insurance companies, and hedge funds in various ways, with the primary goal of making optimal investments, loans, and insurance plans. Quantitative systems are used to determine the price a company values its products at (Kenton, 2023), the amount of aid a family in need should receive, and the loans that are given to local businesses (Supplemental Nutrition Assistance Program (SNAP) | Food and Nutrition Service, 2024). Because of these systems' wide effects on the economy, it is important to understand the basics of them.
The Black-Scholes-Merton (BSM) model is a prominent example of quantitative finance. The equation was the focus of a prominent 1973 research paper written by Fischer Black & Myron Scholes, and edited by Robert Merton. It is used to estimate the value of option contracts. An option contract grants an owner the right to buy a stock (call option), or sell a stock (put option) on a certain data, at a set price. For example, let’s hypothesize an option contract for the stock of the company Apple (AAPL). The stock is currently at $100. One could buy a contract which allows them to buy the stock at $100, for the next 90 days. If they owned an option contract for the company, and the stock went up to $200, they could exercise their contract, and buy the share for $100. Option contracts generally trade above current stock price. It is important to note that the BSM model only works on option contracts which can exclusively be exercised on their expiration date. There is a key difference between European and American option contracts, in that American option contracts can be executed any time within a certain period, while European contracts can only be executed on a certain day, after a certain period. Therefore the BSM model only properly works for European contracts.
The model relies on a few key assumptions. Firstly, stock prices have a log-normal distribution, meaning that their price cannot fall below zero and that the returns on the stock are normally distributed, allowing for predictable changes in asset price over time. This assumption is based on the concept that while stocks can experience significant changes, they are more likely to show gradual growth over time (Chen, 2022). Secondly, that stock prices follow a random walk, with constant volatility. In simpler terms, random walk refers to the assumption of the model that stock prices have an equally random chance of increasing or decreasing, and random volatility refers to the assumption that these changes occur to a constant degree. The model then requires the variables pictured above. A strike price is the price at which the option can be bought on the expiration date. This is typically near the price of the stock on the date where the option contract is bought. The risk-free interest rate is the theoretical rate of return an investor could make without risk (this is typically the current rate of US bonds, which are extremely low risk). The model itself can be broken into three parts:
D1: Calculates the theoretical price of the option
D2: Utilizing D1, calculates the probability that the option will be exercised at expiration, considering volatility, and the time remaining until expiration
C: Final theoretical value of option*
*It’s important to note that N(D1) and N(D2) represent the cumulative probability of a normal variable (a variable that follows the standard normal distribution) being less than or equal to D
While the BSM is a very useful tool for brokers, banks, and other financial institutions, it is important to understand its drawbacks. First, it assumes constant volatility. In reality, volatility can vary dramatically depending on market conditions and investor sentiment. The basic BSM model also does not account for transaction costs, dividends, or taxes. Because of this, more expansive versions have been developed, and it is often used in conjunction with other tools. And lastly, as previously mentioned the BSM model only works for European options. Despite these disadvantages, the BSM remains a powerful tool for investing. Gaining an understanding of its workings can provide deeper insights into financial modeling and market dynamics (Hayes, 2024).
References
Chen, J. (2022). European Option: Definition, Types, Versus American Options. Investopedia. https://www.investopedia.com/terms/e/europeanoption.asp
Hayes, A. (2024). Black-Scholes Model: What It Is, How It Works, and Options Formula. Investopedia. https://www.investopedia.com/terms/b/blackscholes.asp#toc-how-the-black-scholes-model-workKenton, W. (2021). Log-Normal Distribution: Definition, Uses, and How To Calculate. Investopedia. https://www.investopedia.com/terms/l/log-normal-distribution.asp
Kenton, Will. (2023) Quantitative Analysis (QA): What It Is and How It's Used in Finance. In Investopedia
Supplemental Nutrition Assistance Program (SNAP) | Food and Nutrition Service. (2022). Usda.gov. https://www.fns.usda.gov/snap/supplemental-nutrition-assistance-program