This article provides an overview of the Black-Scholes model, also known as the Black-Scholes-Merton model, first published by Fischer Black and Myron Scholes in their 1973 paper, “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy. Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, Black predeceased the award.

### Partial differential equation: an overview

The partial differential equation of the Black-Scholes model estimates the price of the option over time. The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. The model determines the price of a European call option, which simply means that the option can only be exercised on the expiration date. This hedge, called delta hedging, implies that there is only one right price for the option, as returned by the Black–Scholes formula. Delta hedging is the basis of more complicated hedging strategies used by investment banks and hedge funds.

The Black-Scholes model follows a geometric Brownian motion with constant drift and volatility and is used to determine the theoretical value for a call or a put option based on six variables: volatility, type of option, underlying stock price, time, strike price and risk-free rate. The Black-Scholes model is used by option traders who buy options that are priced under the formula calculated value, and sell options that are priced higher than the Black-Scholes model calculated value.

The Black-Scholes model is still regarded as one of the best ways of determining fair prices of options. The model assumes stock prices follow positive interest rates since asset prices cannot be negative. The model also assumes there are no transaction costs or taxes. Yet many empirical tests have shown that the Black–Scholes price is ‘fairly close’ to the observed prices, although there are well-known discrepancies. Black and Scholes showed that “it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock”. Several assumptions of the original model have been removed in subsequent extensions, with modern versions accounting for dynamic interest rates, transaction costs and taxes and dividend payout.

### The Black-Scholes call option formula

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.

**Тhe 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

The Black-Scholes model is only used to price European options and does not take into account that American options could be exercised before the expiration date. The model also assumes dividends and risk-free rates are constant, though 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.

While in the Black–Scholes model one can perfectly hedge options by simply delta hedging, in practice there are many other sources of risk. Nevertheless, Black–Scholes pricing is widely used in practice because it is easy to calculate, a useful approximation and a robust basis for more refined models

The Black–Scholes model is robust in that it can be adjusted to deal with some of its failures. Defects that cannot be mitigated by modifying the model include tail risk and liquidity risk and these have to be managed outside the model.

### Warren Buffett’s overview

In 2008, Warren Buffett wrote a short overview of his opinion of the Black-Scholes model. He pointed out the implied volatility used in the Black-Scholes model is typically determined by how much the stock has moved around recently. He noted the Black-Scholes model probably works well for short investment horizons, but the value of the model diminishes rapidly as the horizon lengthens because recent stock volatility is not a good predictor of the long-run outcome for the underlying security. Also, the value of retained earnings of the underlying companies is not accounted for in the model. He also points out that the model’s creators probably understood these limitations of the model.