| Black-Scholes: The Black Hole of Maths and Money |
| Saturday, 10 February 2007 | |
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The use of mathematical equations to model complex patterns plays a vital role in science, physics and economics. However, there are also dangers in putting too much trust in one simplified formula. Occasionally these dangers become a dreadful reality, as happened in 1987, when stock markets worldwide suffered one of the biggest financial crashes of the century. More recently, in 1998, thousands of investors in the Long-Term Capital Management (LTCM) hedge fund saw their monies disappear, as LTCM lost over $4.6 billion in less than four months. Who was to blame for these disastrous losses?  Credit: Agnes Becker, Kelly Neaves and Jon Heras Few mathematical equations form the subject of popular articles and books, and fewer still are credited with crashing a stock market. In the 1990s, Forbes business magazine pointed its finger directly at the Black-Scholes equation, as the formula behind the disastrous miscalculation of the stock market prices. In recognition of the importance of this equation, the Nobel committee awarded its 1997 Economics prize "for a new method to determine the value of derivatives", to Myron Scholes and Robert Merton. Fischer Black, who also gave his name to the formula, became ineligible for the award, due to premature death in 1995. The Black-Scholes equation was developed to calculate a fair price for the financial derivatives called options. These are contracts that can be bought just like insurance policies, which give you the right to buy or sell a share, at a future date and for a pre-agreed price. This practice is as ancient as trade itself; merchants from the Phoenicians to the Romans frequently traded options on their goods. In 1900, this ancient system stood on the brink of revolution driven by the application of scientific options pricing. Louis Bachelier, a PhD student of the famous mathematician Jules Henri Poincaré, decided to apply ‘random walk theory’ to options, in his thesis Théorie de la Spéculation. This was the notion that stock market prices fluctuate randomly without being affected by past price movements, simulating the same jagged price variation in options. Remarkably, his work came five years prior to Einstein’s paper that applied random walk theory to so-called ‘Brownian motion’, to describe the random movements of gas particles. Bachelier’s scientific model of pricing options presented a new approach to an old problem. Unfortunately, it was flawed in several ways. Firstly, it failed to take into account catastrophic events such as stock market crashes, because it assumed that the probability of extreme price fluctuations was negligible. Crashes may be infrequent, but they are all too real. This flaw is inherent in a model based on random walk theory, where the distribution of probabilities is said to be ‘normal’. A normal distribution with price on the horizontal axis and probability on the vertical assumes a bell shape, with a sharp ski-jump drop followed by a gentle short tail—too short, in fact, to accurately reflect the reality of extreme share price movements. Crucially, when calculating an option’s price, Bachelier’s model was also unable to calculate the risk premium, or the financial value of risk. This problem remained unsolved for many years, as the amount of risk investors are willing to accept is anyone’s guess. It was Black and Scholes, 70 years later, who realized that the risk premium calculation could be avoided altogether. Their inspiration came from gamblers who covered their losses by betting in opposite directions. In a portfolio with a mixture of shares and options, Black and Scholes saw that they could create a riskfree investment, since the movement of shares and option prices would offset each other. The underlying point is that the prices of shares in companies move completely randomly. If you try picking shares by throwing darts at the financial pages, as a group of US academics tried to do in the 1930s, you may discover that your portfolio will perform as well as any professionally managed fund. After further refinement by Robert Merton, who adapted the equation for a fast-paced market, the formula was finally unveiled in 1973. Its impact was virtually immediate, spawning a trillion-dollar market for financial derivatives such as options, as well as the growth of giant hedge funds managing share-option portfolios. When Texas Instruments began making calculators with an inbuilt Black-Scholes button to allow traders to calculate option prices, they were snapped up like candy in a playground. The world was bedazzled and, for the first time in history, it seemed that traders could play the markets risk-free. The Black-Scholes formula is obtained by solving a differential equation that closely resembles the equation describing the diffusion of heat, which emphasizes its relationship to physics. The original formula is then converted to the expression used for ‘call’ options, which allow one to buy a share at a specific date in the future. A modified expression is also used for ‘put’ options, for selling shares. The equation takes as its inputs the price of the share it underwrites, the length of time before the date on which the option can be used, the interest rate, and, significantly, the so-called ‘volatility’, or the amount by which a share price can be expected to fluctuate. It can be likened to the jitter of particles in a gas, which bang into each other ever more violently with higher temperatures. The formula relies on a number of assumptions. The most significant of these is that fluctuations in the share price follow a normal distribution, like in Bachelier’s model from 1900, and that the volatility is constant during the lifetime of the option. Neither assumption is quite true. The assumption of a normal distribution for share values fails to predict extreme price fluctuations and market crashes, while volatility itself can follow a ‘random walk’. Blindly trusting the formula to calculate option prices gives one a rosy picture of the world. Despite the warnings of Forbes magazine, this costly lesson was driven home in 1998 to thousands of investors. The formula’s inventors were directly to blame. In 1994 Robert Merton and Myron Scholes set up a company, Long Term Capital Management (LTCM), intending to play the markets using the models that would eventually earn them the Nobel Prize. LTCM was a "gigantic vacuum cleaner sucking up nickels from all over the world", according to Scholes. Cashing in on their iconic status as the fathers of modern finance, the start-up raised three billion dollars, which the company set forth to multiply, with average returns of 35% in the first three years. Then, the unpredictable happened. The economies of South-East Asia skidded to a halt in a domino effect, sparked by the collapse of property prices in Thailand in 1997. Banks went out of business and loans went unpaid worldwide. Yet this was only a foretaste. The big shock came in August 1998, when Russia suddenly announced it would cease all foreign debt payments. Pandemonium reigned in the financial world, with all indicators jittering furiously back and forth, reflecting the panic in the market. By this time, LTCM had bets topping one trillion dollars, with its models now as useful as a dud compass. Models based on the normal distribution work well on calm days, but they simply could not predict such catastrophic events, nor could they point the way out of the rut. Finally, the US Federal Reserve stepped in to bail out LTCM, preventing it from taking all of Wall Street down along with it.  Credit: Financial Times Today, the Black-Scholes formula is still used millions of times each day by derivative traders and fund managers worldwide. Astonishingly, it is used more than ever before, but with a significant difference: Black-Scholes no longer predicts option prices. Instead, it is being used in reverse. The option’s price is taken as a given, and is plugged into the equation in order to calculate the volatility of the share price. This does not address the problem that volatility itself does not follow a genuine normal distribution for large fluctuations. However, it does relieve the equation’s Achilles’ heel of direct option pricing for extreme share price fluctuations. Investment decisions can now be made with a very useful tool that complements, rather than replaces, the qualities that traders value above all—instinct, gut feeling and experience. Outside the world of finance, large companies have also found a way of adapting the Black-Scholes equation. It is used to calculate the cost of business decisions, such as closing down plants or buying new equipment, by treating them as options. The trend of modelling economics, and even certain social and historical phenomena, is booming, and looks set to spawn new fields of interdisciplinary research. Econophysics, a field barely in existence ten years ago, now features in several dedicated academic departments around the globe. Despite all this, the next time a magic formula is unleashed on us, we would do well to remember how the Black- Scholes equation, with its beauty and apparent simplicity, lulled many into an unjustified sense of confidence. Albert Einstein once said that "if you are out to describe the truth, leave elegance to the tailor". The costly lesson that Black and Scholes gave, teaching us how intractable the human element can be, is as valuable as the equation itself. Tristan Farrow is a PhD student in the Cavendish Laboratory |
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