The equation that priced options won a Nobel, and traders still bend it.
In 1973 Fischer Black and Myron Scholes published a formula for the fair price of an option, the right to buy or sell something later at a set price. It launched a boom in options trading and won a Nobel prize. Traders still use it daily, while knowing its assumptions are wrong.
An option's value depends on an uncertain future, which made pricing it look like guesswork. Louis Bachelier had applied the mathematics of random motion to the problem as early as 1900, and several economists, including Paul Samuelson and his student Robert Merton, advanced the theory in the 1960s. In 1968 Black and Scholes realised that by constantly adjusting a portfolio of the option and the underlying share, you could cancel out the risk, so the option's price would not depend on how fast anyone expected the share to grow. Their paper appeared in 1973, and Merton published an important extension the same year.
The result was a formula that needs only a handful of inputs: the share price, the exercise price, the time left, the interest rate and the share's volatility. Only volatility cannot be observed directly. The formula gave options trading mathematical legitimacy and helped drive a boom in it. Scholes and Merton received the 1997 Nobel memorial prize in economics; Black had died in 1995 and was ineligible, though the committee mentioned his contribution.
The model's assumptions do not hold in real markets. It treats volatility and interest rates as constant, assumes trading is continuous and free, and underestimates extreme price moves, which happen far more often than it predicts. When traders work backwards from market prices to the volatility the formula implies, they get a curve, the 'volatility smile', rather than the single number the model expects. Using it anyway has been described as putting the wrong number in the wrong formula to get the right price.
Critics go further. Nassim Nicholas Taleb and Espen Haug argue that similar formulas existed earlier, and Warren Buffett has written that it can give absurd results for long-dated options. The mathematician Ian Stewart called it one ingredient in the mix behind the 2008 crisis, while stressing that the abuse, not the equation, was the problem.
Source: Wikipedia — Black–Scholes model · Text summarised from Wikipedia (CC BY-SA 4.0)
