Black Scholes Model
The Black Scholes Model, also known as the Black-Scholes-Merton model, is one of the most influential concepts in modern financial theory. Developed in 1973 by Fischer Black, Robert Merton, and Myron Scholes, the model provides a mathematical framework to calculate the fair value of options contracts. By introducing a consistent and transparent method for pricing options, the Black-Scholes Model revolutionized the options market, making derivatives trading more accessible to investors while improving risk management practices. The model estimates option prices using six key variables: volatility, option type, underlying stock price, strike price, time to expiration, and the risk-free interest rate. Rooted in the principle of hedging, it aims to minimize risks associated with the volatility of underlying assets, enabling traders and institutions to better assess fair market value and manage exposure in options trading.
BSM FORMULA FOR CALL OPTIONS
C=S⋅N(d1)−K⋅e−rt⋅N(d2)
Where:
C = Call option price
S = Current stock price
K = Strike price of the option
r = Risk-free interest rate
t = Time to option expiration (in years)
N = Cumulative distribution function of the standard normal distribution
e = Mathematical constant (~2.71828)
BLACK SCHOLES MODEL KEY ASSUMPTIONS
Efficient Markets
Asset prices are assumed to follow a geometric Brownian motion, meaning they move randomly but within a predictable mathematical framework.
Constant Volatility
The volatility of the underlying stock is assumed to remain constant throughout the life of the option. In reality, volatility often changes, especially for fast-growing or unstable companies.
No Transaction Costs or Taxes
The model assumes a frictionless market, with no fees, taxes, or costs associated with trading.
Log-Normal Distribution of Prices
Stock prices are assumed to follow a log-normal distribution, meaning prices cannot be negative and tend to move upward over time.
No-Arbitrage Condition
The model assumes there are no opportunities for risk-free arbitrage (guaranteed profits through simultaneous buying and selling).
Constant Risk-Free Interest Rate
The risk-free interest rate (such as the yield on government bonds) is assumed to be known and constant during the option’s life.
BLACK SCHOLES MODEL LIMITATIONS
European Options Only
BSM is designed for European-style options, which can only be exercised at expiration. It does not accurately price American-style options, which allow early exercise and are more common in the U.S.
Interest Rate Assumptions
The model assumes interest rates remain constant, which is rarely true in real-world markets.
In reality, transaction costs, bid-ask spreads, and taxes impact option pricing, which BSM does not account for.
No Dividends or Returns
The original model assumes underlying stocks do not pay dividends or generate additional returns, which limits its accuracy for dividend-paying equities.