What is the Black Scholes Model
The Black Scholes model, also known as the Black-Scholes-Merton model, is a model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes 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 price variation of the stock, the time value of money, the option's strike price, and the time to the option's expiry.
BREAKING DOWN Black Scholes Model
The Black Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used now. It is regarded as one of the best ways of determining fair prices of options. The Black Scholes model requires five input variables: the strike price of an option, the current stock price, the time to expiration, the risk-free rate, and the volatility. Additionally, the model assumes stock prices follow a lognormal distribution because asset prices cannot be negative. Moreover, the model assumes there are no transaction costs or taxes; the risk-free interest rate is constant for all maturities; short selling of securities with use of proceeds is permitted; and there are no riskless arbitrage opportunities.
The Black Scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation. In mathematical notation, C = S*N(d1) - Ke^(-r*T)*N(d2). Conversely, the value of a put option could be calculated using the formula: P = Ke^(-r*T)*N(-d2) - S*N(-d1). In both formulas, S is the stock price, K is the strike price, r is the risk-free interest rate and T is the time to maturity. The formula for d1 is: (ln(S/K) + (r + (annualized volatility)^2 / 2)*T) / (annualized volatility * (T^(0.5))). The formula for d2 is: d1 - (annualized volatility)*(T^(0.5)).
Limitations of the Black Scholes Model
As stated previously, the Black Scholes model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date. Moreover, the model assumes dividends and risk-free rates are constant, but this may not be true in reality. The model also assumes volatility remains constant over the option's life, which is not the case because volatility fluctuates with the level of supply and demand.