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taming the tiger the black–scholes equation
In 2003, Roy Horn—half of the famous performing duo Siegfried & Roy—was bitten on the neck by one of his own tigers. Roy nearly died, and his performing days were over. To fans, it was a shocking reminder that tigers are still tigers. The sheer technical proficiency of Siegfried & Roy had lulled audiences into complacency. As Neil Strauss wrote in The New York Times: “Danger was still present, but it was no longer recognized as such.”
A similar thing could be said about the Black–Scholes equation. Published in 1973 by Fischer Black and future Nobel Prize winner Myron Scholes (with a big assist from another future Nobelist, Robert Merton), it seemed to take the danger out of investing. It led to an explosive growth in the market for financial products called derivatives—essentially, bets on the direction that certain other assets, such as stocks and bonds,* would move. Like the Las Vegas audiences of Siegfried & Roy, Wall Street was dazzled by the technical proficiency with which a new generation of traders, called “quants,” wielded the Black–Scholes equation. Danger—or as financial engineers call it, “risk”—seemed to be under control.
V is the market value of a financial derivative called a call option, and S is the value of the underlying asset (e.g., a stock) at maturity. ρ and σ represent the interest rate and the volatility of the stock. The equation gave economists the possibly mistaken impression that risk can be managed according to objective laws that resemble the laws of physics.
But tigers are still tigers, and financial markets are still financial markets. Not once but three times since Fischer and Scholes’ breakthrough—in 1987, 1998, and 2007—the market has turned on the people who thought they could control it. No one has been killed, but careers have been ruined and fortunes lost. The credit crisis of 2007–8, in particular, led to the most severe recession in America since the Great Depression of the 1930s.
Some people have tried to blame mathematics, or the quants, for these events. For example, an article in Wired in 2009 referred to “the formula that killed Wall Street.” But before we can pass judgment, we should first understand what Black, Scholes, and Merton accomplished.
Let’s start with a greatly simplified example of a derivative. Suppose that today’s price of one share of stock in the Well-A-Day Oil Company is $100. Suppose also that you happen to know that the stock has a 50-50 chance of rising to $101 tomorrow, and a 50-50 chance of dropping to $99. Amazingly, you can use this knowledge to turn a guaranteed profit of 50 cents per share, even though you don’t know which way the stock is going to go.
First, you call up your broker and you ask him to give you an option to buy two shares of Well-A-Day stock tomorrow for $100 each. This seems like a reasonable deal, right? After all, there is an equal chance that the price will be above $100 or below $100. (Note: If your broker agrees to this, he is a fool and will go out of business soon, but I’ll explain why in a moment.)
Now, armed with your option, you hedge it by borrowing one share of Well-A-Day from your friend Bob and selling it for $100. The next day, there is a 50-50 chance that the stock will go down to $99. In that case, you let the option expire unclaimed, but you can buy one share for $99 and return it to Bob. Your profit is $1, because you bought for $99 and sold for $100. On the other hand, if the stock goes up to $101, then you claim your option and buy two shares from your broker for $100 each. You return one to Bob, and sell the remaining share for $101, the current market price. Once again, your profit is $1, because you bought two shares for $200 and sold them for $201. Thus, no matter what happens, you earn a dollar. It doesn’t matter whether the price of the “underlying asset” goes up or down.
THIS HEDGING PRINCIPLE has been rediscovered many times over the years. Prior to 1973, the people who discovered it usually thought they had discovered the secret to getting rich. In a typical book called Beat the Market! from 1967, economist Sheen Kassouf writes about his epiphany, “I realized that an investment could be made that seemed to insure tremendous profit whether the common rose dramatically or became worthless. I would win whether the stock went up or down! It looked too good to be true.”
Of course, like all get-rich-quick schemes, it is too good to be true, for two reasons. First, your broker will get wise to this game pretty fast. In fact, the hedging principle shows that the option to buy one share of Well-a-Day for $100 tomorrow is actually worth 50 cents today. The option itself has value. However, in the early days of option trading, brokers and investors did not have a very good idea of how to price real-world options (as opposed to this made-up example). When Kassouf and his co-author Edward Thorp wrote their book, they could sift through the published prices of stock warrants (the most common type of option available then) and find mispriced warrants. A savvy investor could beat the market.
Secondly, even if we can find a broker who will sell us the option for the wrong price—say, 49 cents instead of 50 cents—we will realize our profit (which is now down to a cent per share!) only if we are absolutely right about our prediction that the stock will either go up to $101 or down to $99. Unlike normal investors, who base their investments on the direction that the stock is going to move, we don’t care about the direction. But we do care about the volatility—the size of the jump. Given the narrow margin for error, the hedging principle will only work if our estimate of the stock’s volatility is right on the money.
Above Symbols of the stock exchange: The bull and bear statues outside the stock exchange in Frankfurt, Germany.
However, this critical point was hidden by the technical virtuosity of the Black–Scholes formula. Suppose that we want to find the fair market value, V, of a call option—an option to buy Well-A-Day stock at price K (the “strike price”) at time T (the “expiration date”). We don’t want to cheat anyone; we just want to know how much this option is worth.
Clearly, the option’s value depends on the current stock price, S. The higher the current price, the more likely it is that the stock price will still be above K on the expiration date. Also, the value depends on the time remaining until expiration, T – t. Even if the option is “out of the money” today, more time gives the stock more of a chance to rise to the strike price. Finally, on the expiration date, the value of the option is either 0 (if the stock price is below K) or else it is S – K (because if the stock price is above K, we will exercise the option and buy it for K dollars, then go to the open market and sell it for S dollars). Thus at time T, the option’s value looks like the solid line in the graph below. At earlier times t, before the expiration date, the option’s value should be a little bit higher, like the dotted line in the figure. But how much higher?
Black, Scholes, and Merton proved that there is a fair market value V(S, t). This is pretty surprising, because you might think that an investor who is bullish on Well-A-Day would value the option differently from someone who is bearish. Their main idea was to use dynamic hedging to eliminate risk. This is just a more elaborate version of the hedging principle, in which the investor has to adjust his or her portfolio constantly, in accordance with the current price S and time t. The effect is the same: dynamic hedging makes it irrelevant whether we personally believe the stock will go up or down.
Thus the bulls and the bears can agree on the value of the option—provided that they agree on a model for the volatility. And that was the second ingenious stroke of Black and Scholes. They proposed a model of stock prices that was so “intuitively obvious” that hardly anyone could disagree with it. Changes in stock prices, they said, have two components: an upward or downward drift plus a random jiggle. It’s only the size of the jiggle, the volatility, which matters for the option price, and this volatility is measured by a number sigma (σ). A well-known function, called the normal distribution or the bell-shaped curve (see opposite), gives the likelihood of any particular size of fluctuation. For example, the likelihood of a “one-sigma” (or more) increase in the price is about 15.8 percent, and the likelihood of
a “two-sigma” increase is about 2.2 percent.
Above On the expiration date, the value of an option to buy stock at $650 looks like the solid line. Because of dynamic hedging strategies, the value of the option before the expiration date is always higher (dotted line).
Why was the Black–Scholes proposal so seductive? Perhaps because the bell-shaped curve is so familiar. Any stochastic process that amounts to infinitely many independent flips of a coin (even a biased coin) will produce a normal distribution. If you’ve ever watched a stock ticker, you have probably seen an endless stream of pluses and minuses scroll past, indicating upticks and downticks of the stock. It really does look like an infinite (or nearly infinite) succession of coin flips.
FINALLY, BLACK AND SCHOLES had one more stroke of genius. Unlike previous investors who saw the hedging strategy as a way to beat the market, they insisted that there is no way to beat the market. If you hedge your portfolio in a way that eliminates risk, you should get exactly the same rate of return as someone who invests in the most risk-free investment—a 30-year US Treasury bond. This argument closed the loop and gave them the Black–Scholes equation:
In this equation, the left-hand side represents the return on your investment if you buy the option and hedge it dynamically according to Black and Scholes’ prescription. The right-hand side represents the return if you simply put it in the bank (with r representing the interest rate). According to Black and Scholes, in an efficient market the two returns are equal.
Above The standard deviation, denoted σ, is a measure of the amount of spread in a classic bell-shaped curve.
Like Maxwell’s equations and the heat equation, the Black-Scholes equation is a partial differential equation, a type that physicists and mathematicians are very aware of. The same sort of equation describes the diffusion of molecules in a gas, because their motion likewise consists of innumerable tiny jiggles. For the simplest call options, Black and Scholes deduced an exact solution for the value V. However, the Black–Scholes equation also applies to all sorts of other, more “exotic” options, such as options based on more than one underlying stock or options based on mortgage defaults.
Remarkably, one mathematician had anticipated Black and Scholes by more than 70 years. In 1900, a student of Poincaré named Louis Bachelier had studied an almost identical model of option prices—including the same idea of random fluctuations—and derived an almost identical equation. But Bachelier lived at the wrong time. The discipline of mathematical economics did not yet exist. Pure mathematicians were very interested in his idea of a process consisting of infinitely many small jiggles (now called “Brownian motion”). However, they were not the least bit interested in Bachelier’s motivating example. One colleague, Paul Lévy, wrote a disparaging comment in his personal notebook: “Too much on finance!”
But by 1973, the world was ready. That year, the world’s first options exchange opened in Chicago. Over the next two decades, the Black–Scholes formula changed Wall Street. First of all, it created jobs for a whole new kind of trader—a “quant,” usually someone with a mathematics or physics background who understood differential equations. But perhaps more importantly for society, Black–Scholes created an aura of invincibility around mathematical finance. Options, once seen as a somewhat disreputable investment (a high-risk wager on the stock market), now seemed to be the exact opposite. They were an essential tool for controlling or eliminating risk. For Black, Scholes, and Merton, it was an article of faith that the world was gradually moving toward an ideal state where you truly couldn’t beat the market, and all options would be rationally priced according to the mathematical models. From this point of view, quants weren’t just making money; they were helping to make the market more efficient.
Just like von Neumann’s dream of perfect weather forecasting, the dream of perfect markets never came true. The first crack appeared in 1987, on Black Monday, when the Dow Jones Industrial average dropped by more than 22 percent, by far its biggest one-day percentage loss ever. It was widely blamed on programmed trading—the kind of automatic selling of stock that dynamic hedging requires.
Above Brownian motion: computer artwork of the small seemingly random movements of particles suspended in a fluid
The second crack, in 1998, was much more personally embarrassing to the theory’s founders. By this times, Scholes and Merton were both partners in a hedge fund called Long-Term Capital Management (LTCM). Although they were not involved in day-to-day operations, the two Nobel Prize winners lent a huge amount of prestige to the fund, which was like a laboratory experiment in risk-neutral investment.
Between 1994 and 1998, LTCM quadrupled its investors’ money and seemed to live up to the promise of guaranteed returns. Then, over a span of less than two months, it all came crashing down. A cascade of events, starting with the Russian government defaulting on its debts, pushed volatilities to stratospheric levels, far beyond where the models said they should have gone. LTCM hemorrhaged money. Finally, with the company on the brink of bankruptcy, the Federal Reserve Bank organized a bailout by a consortium of fourteen leading private banks—a move the banks agreed to, reluctantly, because they feared that LTCM was “too large to fail.” If it went down, they could fall like dominos.
FINALLY AN EVEN GREATER CRISIS rocked the financial world in 2007. This time it was a collapse in the market for credit derivatives—one of the more exotic kinds of derivatives mentioned earlier. This time, the Black–Scholes formula was more peripherally involved. For several years, banks had been offering “subprime” loans to home buyers who would not have qualified in the past. The banks were not motivated by altruism; they believed that they could control the risk of default by lumping many mortgages into one security called a collateralized debt obligation (CDO). The quants had developed a convenient formula based on the normal distribution (called the “Gaussian copula”) that gave a fair market value for the CDO’s … provided that the correlation between loan defaults was zero (or at least not too big). In other words, if a homeowner in Miami forecloses, it shouldn’t affect a homeowner in Las Vegas.
But in 2007 the housing bubble burst, and a wave of foreclosures swept the entire nation. Suddenly Miami affected Las Vegas and vice versa. During a panic, all the correlations go to one. Banks across the country did not have enough capital to cover their bets, and one after another they started failing: Bear Stearns, Washington Mutual, Lehman Brothers. Once again the government had to intervene, only in a much bigger way than before. The Secretary of the Treasury announced a $700 billion bailout, or “Troubled Asset Relief Program.” In an unprecedented move, the government actually acquired one of the “too-big-to-fail” companies—AIG, the world’s largest insurance firm. Unlike in 1998, the private sector did not have enough healthy banks left to do the job.
The common denominator in all of these debacles was the failure of mathematical models to anticipate the volatility of the markets. The normal distribution is based on the assumption of innumerable small actors making innumerable random choices, all independently of each other. But when one actor gets too big (like LTCM) or when the actors stop behaving independently (as in the panic situations of 1987 and 2007–8), the model does not apply. In fact, a small group of dissident economists has argued for years that models like Black–Scholes should never be used because they underestimate the probability of extreme events.
However, the Black–Scholes equation and the philosophy behind it are by now too ingrained to just throw them out. Instead, economists are trying to come up with refinements that better reflect how real markets behave. For example, in the “jump diffusion” model, stock prices have three components: long-term drift, short-term stochastic jiggles, and intermittent jumps due to lurches of the stock market as a whole. This certainly seems to agree better in a qualitative way with reality. Black–Scholes does work very nicely most of the time, aside from the random infrequent occasions when it doesn’t work at all. Another approach is to view the volatility σ as bei
ng given by a stochastic process itself, or by an empirical function of S and t. Unfortunately, all of these ideas have a kludgy feel to them. They seem contrived to preserve the outward appearance of the Black–Scholes formula without maintaining its internal consistency.
In the end, the question remains: Can mathematics tame the tiger that is the future? Or will it always break out of its cage just when we least expect it? I cannot pretend to answer this question; it is something for twenty-first century economists, financial engineers, and mathematicians to work on. Until they succeed, or else prove a new Impossibility Theorem, the lesson from 1987, 1998, and 2007 is definitely “buyer beware.”
* * *
* In the rest of this chapter, for convenience, I will refer to stocks, although stock options are actually far from being the most common kind of derivative.
conclusion what of the future?
The end of this book invites a question: What comes next for equations?
First, the good news. The enterprise of mathematics and science worldwide seems still to be in very healthy shape. There seems to be an upward trend in the sheer quantity of important formulas, which mirrors the growth of mathematics and science in our society. When I began working on this book, in 2008, I searched an authoritative website, Wolfram MathWorld for three terms: “equation,” “formula,” and “identity.” The search engine dutifully reported 1947 equations, 1253 formulas, and 992 identities. Three years later, the same search yields 2032 equations, 1307 formulas, and 1026 identities.
Another positive development is the Internet, which has made the sharing of scientific ideas so much easier. Remember how in the “bad old days” of the 1500s and 1600s, mathematical progress was repeatedly slowed by the reluctance of researchers to share their work. And even in the 1900s, politics prevented some of the work of Soviet mathematicians from becoming widely known in the West. Now, thanks to such websites as the e-print archive, as well as mathematical forums and blogs, the barriers to communication are lower than they ever have been.
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