What is Black Scholes Model and How to Calculate It?

Summary

The Black-Scholes Model, developed by Fischer Black, Myron Scholes, and Robert Merton in the 1970s, is a transformative framework for estimating option prices. It treats options as financial derivatives, offering a closed-form solution that calculates theoretical option values.

In the dynamic world of finance, the Black-Scholes Model stands as a beacon of innovation. Developed by three groundbreaking economists - Fischer Black, Myron Scholes, and Robert Merton - in the early 1970s, this model revolutionized the realm of options pricing. It provides a systematic framework for estimating the fair market value of options, enabling investors, traders, and financial institutions to make informed decisions and manage risk effectively. In this blog, we will delve deep into the Black-Scholes Model, demystify its fundamental components, and walk you through the process of calculating option prices.

The need for options pricing models

Before we embark on our journey into the intricacies of the Black-Scholes Model, it's essential to understand why pricing models for options are indispensable. Options, whether they are call options (bestowing the right to buy an underlying asset) or put options (granting the right to sell), possess intricate pay-off structures influenced by several factors, including the current price of the underlying asset, time until expiration, and market volatility. Accurate pricing of options is a critical prerequisite for financial professionals, allowing them to make well-informed decisions and effectively hedge their portfolios against risk.

The inception of the Black-Scholes model

The Black-Scholes Model, named after its creators Fischer Black and Myron Scholes, and later extended by Robert Merton, emerged as a mathematical breakthrough in the early 1970s. This revolutionary model altered the landscape of finance in several profound ways:

1. Treats options as derivatives: The Black-Scholes Model was groundbreaking because it treated options as financial derivatives, explicitly connecting them to the underlying assets.
2. Closed-form solution: The model provided a closed-form solution for calculating option prices, making it highly practical for real-world applications.
3. Risk-neutral valuation: It introduced the concept of risk-neutral valuation, where the expected return on the underlying asset is assumed to be equal to the risk-free interest rate.

Key components of the Black-Scholes Model

To comprehend the Black-Scholes Model, it's vital to grasp its core components:

1. S: The current price of the underlying asset: Denoted as 'S,' this represents the prevailing market price of the underlying asset, such as a stock or index.
2. K: The strike price: 'K' signifies the strike price, which is the pre-determined price at which the option holder has the right to buy (for a call option) or sell (for a put option) the underlying asset.
3. T: Time to expiration: 'T' stands for the time remaining until the option's expiration date. It's typically measured in years but can also be expressed in fractions of a year.
4. r: Risk-free interest rate: 'r' is the risk-free interest rate, often based on government bond yields. It is employed to discount future cash flows back to their present value.
5. σ (Sigma): volatility: Volatility, denoted as 'σ,' quantifies how much the underlying asset's price is anticipated to fluctuate over time. It reflects the level of uncertainty in the market.

Steps to calculate Black-Scholes option prices

Let's now break down the steps to calculate option prices using the Black-Scholes Model:

Step 1: Calculate d1 and d2

Utilize the given values for 'S_0,' 'K,' 't,' 'r,' and 'σ' to compute 'd_1' and 'd_2' using the formulas provided earlier.

Step 2: Compute the Cumulative Distribution Function

Calculate the cumulative distribution function values 'N(d_1)' and 'N(d_2)' based on the computed values of 'd_1' and 'd_2.' You can employ statistical tables or software to obtain these values.

Step 3: Determine the Option Price

Insert the computed values of 'S_0,' 'K,' 't,' 'r,' 'N(d_1),' and 'N(d_2)' into the Black-Scholes call or put option formula, depending on the type of option under consideration.

Step 4: Evaluate the Result

The calculated value 'C' or 'P' represents the theoretical price of the option. Compare this value to the current market price to assess whether the option is undervalued or overvalued.

Limitations of the Black-Scholes Model

While the Black-Scholes Model has significantly influenced finance, it's essential to acknowledge its limitations:

1. Constant volatility assumption: The model assumes that volatility remains constant over the option's life, which may not reflect real-world market conditions accurately.

2. European-style options focus: The Black-Scholes Model primarily addresses European-style options, which can only be exercised at expiration. It may not accurately price American-style options, which permit exercise at any time before expiration.

3. Market dynamics: The model doesn't account for factors such as dividends, transaction costs, or fluctuations in interest rates, all of which can impact option prices.

4. Complex securities: It may not accurately price complex derivatives and securities.

In conclusion

The Black-Scholes Model serves as a foundational tool in finance, providing a method to estimate the theoretical prices of options. By comprehending its components and following the calculation process, you can gain valuable insights into options pricing. However, it's essential to recognize its assumptions and limitations, and to use it in conjunction with real-world market data and additional risk management strategies to make well-informed financial decisions.