What is an Option?
To understand the Black-Scholes model, it is important to understand what it models: the price of a European option. An option gives someone the right, but not the obligation, to buy (a call option) or sell (a put option) an asset for a pre-specified price [1]. The European option specifies that the option to buy or sell can only be exercised on the expiration date of the contract between buyer and seller, whereas the American option can be exercised at any point within the length of the contract.
Imagine that we want to invest in “Company X”, who’s share price is currently £100. We predict that over the course of the year the share price is going to increase. We therefore pay £10 to enter a contract that allows us to buy a share of Company X for £100 in a years’ time, regardless of the share price at that time. After a year we find the share price of Company X has increased to £120. We exercise our option and sell our share immediately for £120, making a £10 profit. This would be a successful use of a European call option, but the share price will not necessarily increase. If the share price of Company X had decreased to £80, we would not have chosen to exercise our call option and would have made a £10 loss over the course of the transaction, the cost of entering the contract.
In the present-day options are an attractive form of investment, but before the development of the Black-Scholes model, there was no standardised way to value an option, with haggling between buyers and sellers the preferred method of arriving at a fair price for both parties. Before the Black-Scholes model, other mathematicians had attempted to model the price of an option, the most notable of which was Louis Bachelier.
Key figures in the development of the Black-Scholes model
Over seventy years before the development of the Black-Scholes model, French mathematician Louis Bachelier found work at the Bourse, the Paris Stock Exchange, to facilitate his post-graduate studies [2]. At the turn of the 20th century Brownian motion had been discovered and some of its properties described, but it had yet to receive the rigorous explanation provided by Einstein utilising statistical mechanics during his annus mirabilis (wonderful year) of 1905. At the Bourse, Bachelier observed the chaos of the markets and saw a connection between the fluctuation in stock prices and the “drunken” random walks taken by particles undergoing Brownian motion.
In his PhD thesis Théorie de la Spéculation (The Theory of Speculation) Bachelier modelled the price of an option as a real-valued continuous-time stochastic process, now called a Weiner process, which also describes simple Brownian motion. Bachelier’s model was visionary, recognising a relationship between the worlds of finance, probability and physics decades before these connections became widely accepted. Despite its successes, Bachelier’s model was limited in its applications to real markets. The model’s foundation upon simple Brownian motion meant it failed to describe with accuracy the behaviour of the stock market, which obeys geometric Brownian motion. Consequently, Bachelier’s model was also considered unremarkable by both economists and physicists and his work drifted into obscurity for over half a century, before being adopted and refined by three economists: Fischer Black, Myron Scholes and Robert C. Merton.
Fischer Black had majored in physics at undergraduate level, but after dabbling in mathematics and computer science came to work on economics and monetary policy at the Massachusetts Institute of Technology (MIT). There he met celebrated economist Myron Scholes [3]. Both Black and Scholes were interested in the development of options pricing, which in the 1960’s had begun to interest other eminent economists of the day such as Case Sprenkle, Paul Samuelson and Samuelson’s student Robert C. Merton, who had studied engineering and mathematics as an undergraduate.
After several years of experimentation, in 1973, Black and Scholes published a paper called The Pricing of Options and Corporate Liabilities, in which they introduced a partial differential equation for measuring the price of an option over time. The Black-Scholes equation was founded upon the work of Bachelier, modelling the price of a stock as a stochastic process but this time as geometric Brownian motion to better reflect the behaviour of real asset prices over time.
Myron Scholes and Robert C. Merton, who’s subsequent papers formalised the Black-Scholes equation with rigorous mathematical justifications, were awarded Nobel Memorial Prizes in Economic Science in 1997 for their work on options pricing and the Black-Scholes equation [4]. Fischer Black, who was ineligible for the award due to his death due to throat cancer in 1995, was commended as a key contributor to the model’s success.