What is Black Scholes Model?

The Black-Scholes model, also known as the Black-Scholes-Merton (BSM) model, is a foundational concept in modern financial theory. Introduced in 1973, it provides a mathematical framework for estimating the theoretical value of derivatives, particularly options contracts. 

The model calculates option prices based on key variables, including time to expiration, volatility, risk-free interest rates, and the underlying asset's price. It remains one of the most widely used and influential tools in options pricing today.

History of the Black-Scholes Model

The Black-Scholes-Merton (BSM) model, developed by Fischer Black, Myron Scholes, and later expanded by Robert Merton, marked a significant milestone in financial economics when it was introduced in 1973. As the first comprehensive mathematical framework for valuing options, the model calculates the theoretical price of an option contract based on key variables including the current stock price, expected dividends, strike price, interest rates, time to expiration, and anticipated volatility. 

Black and Scholes published the original formulation in a seminal 1973 paper, and Merton subsequently contributed further theoretical refinements. In recognition of their ground-breaking work, Scholes and Merton were awarded the Nobel Memorial Prize in Economic Sciences in 1997.

How Black Scholes Model Works?

Underlying Assumption:

The Black-Scholes model assumes that the prices of financial instruments (such as stocks or futures) follow a lognormal distribution, evolving through a random walk with constant drift and volatility, as outlined in the Black-Scholes-Merton model. The model's assumptions also consider that the volatility remains constant over the life of the option.

Purpose of the Model:

The Black-Scholes Option Pricing Model was specifically designed to calculate the theoretical fair value of European-style options, including both call and put options. By providing a mathematical framework, it enables traders to price options consistently in a market where uncertainty prevails.

Key Inputs Required:

The Black-Scholes model requires the following six key variables to calculate the price of an option:

• Current price of the underlying asset
• Strike price of the option
• Time to expiration of the option
• Volatility of the underlying asset
• Risk-free interest rate
• Type of option (call or put)
  
  These inputs allow for the calculation of the option's fair value using the Black-Scholes-Merton model.

Pricing Logic:

With these key inputs, the Black-Scholes Option Pricing Model helps option sellers determine rational and market-consistent prices for the options they offer. The model uses sophisticated pricing formulas to ensure that the options are priced according to the risks and rewards associated with them.

Stochastic Process Assumption:

The Black-Scholes model posits that the price dynamics of assets, especially those that are heavily traded, follow a geometric Brownian motion. This model incorporates both the randomness and the directional drift of asset prices over time, as described in the Black-Scholes-Merton model, which plays a crucial role in understanding the behavior of financial markets.

Incorporated Financial Concepts:

The Black-Scholes formula integrates:
 

• The constant volatility of the underlying asset
• The time value of money through the risk-free rate
• The option’s strike price
• The remaining time to maturity
   

The Black Scholes Model Formula

The Black-Scholes formula calculates the theoretical price of a European call option by taking the current stock price, multiplying it by the cumulative distribution function (CDF) of the standard normal distribution, and then subtracting the present value of the strike price, also adjusted by the standard normal CDF.

Mathematically, the formula is expressed as: 

C=N (d1) St – N (d2) Ke-rt

Where,

C = call option price

N = CDF of the normal distribution

St = spot price of an asset

K = strike price

r = risk-free interest rate

t = time to maturity

σ = volatility of the asset

Assumptions Black-Scholes Model

The Black-Scholes model is based on several key assumptions:

No Dividends: The underlying asset does not pay any dividends throughout the life of the option.

Market Efficiency: Financial markets are random, implying that future market movements cannot be predicted or anticipated.

No Transaction Costs: There are no transaction fees or costs associated with buying or selling the option.

Constant Risk-Free Rate and Volatility: The risk-free rate and the volatility of the underlying asset are assumed to be known and remain constant over the life of the option.

Normal Distribution of Returns: The returns of the underlying asset follow a normal distribution.

European-Style Option: The option can only be exercised at its expiration date, rather than at any point before. 

Benefits and Limitations of the Black-Scholes Model

Conclusion

The Black-Scholes model is a widely recognized mathematical framework used to determine the fair or theoretical value of an option. It incorporates key variables such as the underlying asset's current price, the option's strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset. 

This model is essential for evaluating options in the stock market and is often integrated into advanced stock market trading app, helping investors make more informed decisions by calculating accurate option prices based on real-time market conditions.