### What Is the Black Scholes Model?

The Black Scholes model, also known as the Black-Scholes-Merton (BSM) 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.

### Key Takeaways

- The Black-Scholes Merton (BSM) model is a differential equation used to solve for options prices.
- The model won the Nobel prize in economics.
- The standard BSM 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.

### The Black-Scholes Formula Is

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:

$\begin{aligned} &C = S_t N(d _1) - K e ^{-rt} N(d _2)\\ &where:\\ &d_1 = \frac{ln\frac{S_t}{K} + (r+ \frac{\sigma ^{2} _v}{2}) \ t}{\sigma_s \ \sqrt{t}}\\ &and\\ &d_2 = d _1 - \sigma_s \ \sqrt{t}\\ &\textbf{where:}\\ &C = \text{call option price}\\ &S = \text{current stock (or other underlying) price}\\ &K = \text{strike price}\\ &r = \text{risk-free interest rate}\\ &t = \text{time to maturity}\\ &N = \text{a normal distribution}\\ \end{aligned}$

#### Black-Scholes Model

### What Does the Black Scholes Model Tell You?

The Black Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fischer Black, Robert Merton, and Myron Scholes and is still widely used today. 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.

The model assumes stock prices follow a lognormal distribution because asset prices cannot be negative (they are bounded by zero). This is also known as a Gaussian distribution. Often, asset prices are observed to have significant right skewness and some degree of kurtosis (fat tails). This means high-risk downward moves often happen more often in the market than a normal distribution predicts.

The assumption of lognormal underlying asset prices should thus show that implied volatilities are similar for each strike price according to the Black-Scholes model. However, since the market crash of 1987, implied volatilities for at the money options have been lower than those further out of the money or far in the money. The reason for this phenomena is the market is pricing in a greater likelihood of a high volatility move to the downside in the markets.

This has led to the presence of the volatility skew. When the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen. Thus, the Black-Scholes model is not efficient for calculating implied volatility.

### 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.

Moreover, the model assumes that there are no transaction costs or taxes; that the risk-free interest rate is constant for all maturities; that short selling of securities with use of proceeds is permitted; and that there are no risk-less arbitrage opportunities. These assumptions can lead to prices that deviate from the real world where these factors are present.