Black–Scholes Model

Black–Scholes ModelThe Black-Scholes model is a formula that changed the stock market and is blamed for the financial crisis. Introduced in the 1970s, this mathematical model gave birth to a new financial system based on options, futures, and derivatives trading. This new system had nothing in common with the old stock markets. The phenomenal success of the formula led to the fact that Myron Scholes received the Nobel Prize in Economics in 1997 "for a new method for determining the value of derivative securities".

What Is the Black-Scholes Model?

It is a model that determines the theoretical price of options, implying that if the underlying asset is traded on the market, then the price of its option is already implicitly set by the market itself. This model is widely used in practice and, among other things, can also be used to evaluate all derivative securities and even to assess the equity of financially dependent companies.

According to the Black-Scholes model, the key element in determining the value of an option is the expected volatility of the underlying asset. Depending on the asset's fluctuation, its price increases or decreases, which proportionally affects the value of the option. Thus, if you know the value of the option, you can determine the level of expected volatility.

The Black-Scholes model is mainly used in the following cases:

  • to look for undervalued options for sale and overpriced options for purchase;

  • for portfolio hedging, which allows reducing risks in case of low volatility;

  • to evaluate market prerequisites for the future value of volatility.

History

The futures first appeared in trading in the 17th century, on the Japanese rice exchange. Traders then began to conclude futures transactions, that is, to set the price of the goods, the delivery of which will take place in the future.

By the 20th century, not only futures, but also options were used on American commodity exchanges - the same price agreement for the future, but without the obligation to make a purchase. Options were bought as an "insurance" from a sharp increase in prices. Over time, traders got a desire to resell these options, which was difficult, because no one could define the price of those options. This is where the revolutionary Black-Scholes model appeared.

The formula of the option pricing model was first introduced by Fisher Black and Myron Scholes in 1973 in the article "The Pricing of Options and Corporate Liabilities". Their research was based on the previous work of Jack Treynor, Paul Samuelson, James Bones, Sheen Kassouf, and Edward Thorp, and was developed during the rapid growth in options trading.

Myron Scholes, a finance professor at Stanford University, has been passionate about finances since childhood. At 27, he got a position at MIT and, together with his colleague Fisher Black, seriously took up the puzzle of pricing options. As already mentioned, the key to the solution was to consider the limit of market volatility. Myron Scholes says that after a year and a half of working on the formula, they saw elements of options in all the objects of the surrounding world.

To the surprise of the authors themselves, the Black-Scholes model began to be used everywhere: in 2007, the global volume of derivatives trade exceeded 1 square trillion dollars, which is ten times the value of goods produced in the entire history of human civilization.

As a result, unforeseen changes in market volatility led to unpleasant consequences for financial markets. Now, some experts call this model a "dangerous invention", which simplified such a complex thing as asset valuation. The 1998 crisis showed that a strong change in volatility happens more often than expected, and therefore all assets will have to be overestimated with new rates.

Black-Scholes Model Assumptions

To derive their option pricing model, Black and Scholes made the following assumptions:

  • Securities (base asset) are traded continuously, and their price behavior follows the geometric Brownian motion model with known parameters. These parameters are constant throughout the life of the option.

  • Dividends are not paid on the option's underlying asset until the option expires.

  • There are no transaction costs associated with the purchase or sale of a stock or option.

  • The short-term risk-free interest rate is known and constant over the life of the option.

  • Any buyer of security can receive loans at a short term risk free rate to pay for any part of its price.

  • A short sale is permitted without restriction, while the seller will immediately receive the entire cash amount for the security sold without coverage at today's price.

  • There is no possibility of arbitration.

The model is derived based on the no-risk hedging concept. By buying shares and simultaneously selling call options on these shares, the investor can design a risk-free position where the profits on the shares will exactly compensate for the losses on the options and vice versa.

A risk-free hedged position should generate income at a rate equal to the risk-free interest rate. Otherwise, there would be a possibility to take arbitration profits. As a result, investors, trying to benefit from this opportunity, would bring the price of the option to the equilibrium level determined by the model.

Black-Scholes Model Formula

The mathematical formula of the model is:

C = S N (d1) - Ke- rt N (d2), where

  • C is the theoretical price of the call option.

  • S is the current stock price.

  • t is the time remaining until the option expires.

  • r is the risk-free interest rate.

  • K is an option strike.

  • e is the base of the natural logarithm (2.71828).

Wrapping Up

The Black-Scholes model is used to estimate the options price. The model was introduced in the 1970s by Myron Scholes and Fischer Black. The model is derived based on the no-risk hedging concept. The Black-Scholes model began to be used everywhere, surprising even the authors themselves. In 2007, the global volume of derivatives trade exceeded 1 square trillion dollars. Specifically, it was ten times equivalent to the value of goods produced in the entire history of human civilization.