Black–Scholes model

The Black–Scholes model, also known as the Black–Scholes–Merton model, is a mathematical framework for financial markets. It focuses on the dynamics of derivative investment instruments. A derivative is a security whose value depends on an underlying asset, such as a stock. The model uses a parabolic partial differential equation, called the Black–Scholes equation, to determine the theoretical price of European-style options.

These options allow investors to buy or sell assets at a specific price in the future. By providing a unique price based on risk and expected return, the model brings mathematical order to complex trading environments.

The core mechanism of the model is a strategy known as hedging. Specifically, it uses a method called continuously revised delta hedging to eliminate risk. To do this, an investor buys and sells the underlying asset in a specific way. This process creates a hedged position consisting of a long position in the stock and a short position in the option.

When done correctly, the value of this combined position does not depend on the price of the stock. This allows the model to use a risk-neutral rate instead of the security's actual expected return. This breakthrough separates the option's value from the inherent risk of the underlying security.

The model relies on several fundamental hypotheses about how markets function. First, it assumes the market contains at least one risky asset, like a stock, and one riskless asset, like a bond or cash. The risk-free interest rate is assumed to be constant. Second, the stock price follows a geometric Brownian motion. This means the stock's price moves in an infinitesimal random walk with constant drift and volatility. Third, the model assumes a frictionless market. This implies there are no transaction fees, no taxes, and no costs for buying or selling. It also assumes investors can borrow or lend any amount of cash at the risk-free rate. Finally, the original model assumes the stock does not pay any dividends.

History shows that this discovery was a gradual process involving many researchers. Louis Bachelier published the earliest work applying Brownian motion to derivative pricing in 1900. However, his work had limitations for modern markets and had little impact for many years. In the 1960s, several thinkers made important improvements to option pricing theory. These included Case Sprenkle, James Boness, Paul Samuelson, and Robert C. Merton. Fischer Black and Myron Scholes demonstrated the risk-neutral argument in 1968. They based their work on previous research by practitioners like Sheen Kassouf and Edward O. Thorp. Black and Scholes actually attempted to apply their formula to real markets but suffered financial losses due to poor risk management.

They returned to academia before finally publishing their work in 1973.

The published article, "The Pricing of Options and Corporate Liabilities," changed the financial world. It provided mathematical legitimacy to the Chicago Board Options Exchange and other global markets. This led to a massive boom in options trading. The formula is highly significant because it contains only one parameter that cannot be directly observed: the average future volatility of the asset. Because the option value increases as volatility increases, traders can invert the formula to create a "volatility surface." This surface helps calibrate other models used for complex over-the-counter derivatives. Even when an explicit formula is not possible, the Black–Scholes equation allows for pricing using numerical methods.

There are different ways to interpret the components of the Black–Scholes formula. One way is to view a call option as a combination of two binary options. It is the difference between an asset-or-nothing call and a cash-or-nothing call. An asset-or-nothing call yields the asset without any cash exchange. A cash-or-nothing call yields cash without any asset exchange. The formula accounts for the present value of these components by using a discount factor. While some use a simple "probability times value" interpretation, this is often technically incorrect. The value of the asset at expiry and the probability of it being "in the money" are not independent variables.

Today, the model serves as a foundation for much more complex financial systems. While the original assumptions were strict, many modern versions have relaxed them. Some versions now account for dynamic interest rates, as proposed by Merton in 1976. Others include transaction costs, taxes, or dividend payouts. These extensions have led to a plethora of models used for modern risk management. In 1997, Myron Scholes and Robert C. Merton received the Nobel Memorial Prize in Economic Sciences. They were honored for discovering the risk-neutral dynamic revision. Although Fischer Black passed away in 1995 and was ineligible for the prize, the Swedish Academy recognized him as a vital contributor.