Mathematics has a reputation of being abstract, but you can easily apply it to real-life problems. But did you know that it is also being accused of causing the subprime mortgage crisis? Not the whole mathematics, though, just a single equation known as Black-Scholes Equation.
The Black-Scholes equation tells us how specific financial instruments called options are priced.[1] An option is a contract that gives the holder the right but not the obligation to buy (a call option) or sell (a put option) an asset at a specified price on a specified date.[2]
One exciting thing about options is buying, selling, and trading options on secondary markets before they mature. However, before Black and Scholes, nobody had a mathematically justified rational method of pricing these options traded on secondary markets. Using these equations, financial engineers can determine the correct price of a given option at any given time as a function of the price and the volatility of the asset along with the time to expiration and the risk-free rate. In 1997, Scholes won the Nobel prize for this contribution, which he shared not with Black but with another option-pricing expert Merton because Black [3].
If you are asking how and why options are necessary, consider the case of a transportation company that needs a constant supply of fuel for its vehicles. To mitigate risks, for example, to correctly price long-term contracts with customers, the company would need to fix the cost of fuel. Hence comes the option: the company buys a call option to purchase fuel at a specified price at a specified time. The company saves money if the market price of fuel is above the price bound in the option. If the market price is lower than the agreed price and the company lets the option expire without exercising, the seller makes a small profit in the amount of the option's price.
But what does this have to do with the subprime mortgage crisis of 2008? To answer this, we must first investigate how markets mitigate risks. Financial markets (commodity markets, foreign exchange markets, equity markets, or debt markets) mitigate risks using three different strategies: diversification, insurance, or hedging. Options play an essential role in many hedging strategies for various financial players. Therein also lies the trouble: like all financial instruments, options can be commodified and then can be bought and sold in the markets. These derived instruments then can be hedged and then commodified, ad infinitum. These derived instruments are called derivatives.
The Black-Scholes model makes several assumptions on modeling the commodity. The model assumes that commodity price fluctuations obey the log-normal statistical distribution, that the commodity has constant volatility, and that the markets have no friction and no arbitrage. In reality, one or all of these assumptions may fail to hold. To make matters worse, these derivative financial instruments are not well-understood, and their price fluctuations most likely do not conform to the log-normal distribution. And yet, financial engineers, through ignorance, negligence, or greed, used Black-Scholes models to price these derivative financial instruments and let them be bought and sold on markets even when the model did not work correctly. In the end, it wasn't mathematics that brought financial markets to their knees, but human greed and ignorance.
Prof. Dr. Atabey Kaygun
References:
Black, Fischer, and Myron Scholes. 1974. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (3): 637–54.
Merton, Robert C. 1973. “Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science 4 (1): 141–83.
[1]Such options are called European options. For American options, the holder may exercise the right anytime before the contract’s expiration date.
[2]Black and Scholes' work on the subject is (Black and Scholes 1974). Merton helped to edit the paper, and later that year published his work on the subject (Merton 1973).
[3]passed away in 1995.
