The Midas Formula
THE MIDAS FORMULA
NARRATOR (DILLY BARLOW): This is the story of a brilliant scientific discovery.
Prof. MERTON MILLER (University of Chicago): When I saw the formula I knew enough about it to know that this is the answer. This solved the ancient problem of risk and return in the stock market. It was recognized by the profession for what it was as a real tour de force.
NARRATOR: A beautiful and elegant mathematical formula which could do something no one had ever dreamt possible.
STAN JONAS (FIMAT Brokers): Up until the time that they came up with their insight, the world was full of uncertainty and risk. Uncontrollable and un-analyzable and then in a moment of tremendous clarity they’d realized that two risky positions taken together can effectively eliminate risk itself.
NARRATOR: But when its creators used the formula to make themselves rich, only then was its dark side revealed.
ROGER LOWENSTEIN (Wall Street Journal): They began to lose 100 million or more day after day after day to finally there was one day when they dropped half a billion dollars, 500 million in a single day and the agony just didn’t end.
NARRATOR: Since the beginning of capitalism there has been one golden rule: that if you want to make money you have to take risks. Then came one of the most improbable scientific projects of the century: the attempt to find a mathematical way to break that rule, to find a formula that would allow anyone to become unimaginably wealthy without taking any risk at all. If it worked it would bring order to the world’s financial exchanges and challenge the role of one group of people: the traders.
BEN SCHWARTZ (Trader): I’ve seen many people come and go. People come in with a good fresh attitude, but they just can’t survive, they just can’t handle the competition. It’s not for the meek, it’s not for the weak. If you can’t handle it you can’t be there and it’s simple, otherwise people just take your money and they don’t feel bad. Every day when I walk in to the Exchange I walk in with a clear mind, no emotion. The emotion builds up almost like a volcano ready to explode and when I walk in that pit it explodes.
It’s crazy. I communicate by going buy 20 from someone, sell you 20. Constant chaos, it’s constant chaos. The market’s always moving. Yelling, screaming, grabbing, whatever it takes to get people’s attention, to make the market move. Constantly going where am I, where am I, where am I, meaning what is my position, give me an exact total, give me an exact figure, what do I have to do to get out of position.
NARRATOR: The products that are traded in the financial exchanges, like here in Chicago, are strange and complex things, like interest rate swaps and currency derivatives. Their prices are constantly fluctuating as market sentiment shifts. The job of the trader is to try to guess what these prices will be next year, next week or in 10 seconds’ time and do it thousands of times a day. The risks are enormous.
BEN SCHWARTZ: In any one day you can lose 6 figures, millions and millions of dollars exchange hands in these pits every day. Me personally? I’ve lost a lot of money, you know. On any given day, I’m not giving a figure, but you know it’s not... You can lose a car, you know, put in, put, apartment, a house, whatever on any given day if you make a simple mistake.
NARRATOR: To work a formula would have to beat the ordinary traders’ instincts. It would also have to compete with the accumulated wisdom of the legendary trader. In his office high above the trading floor is the acknowledged expert.
LEO MELAMED’S CLERK: Yeah.
LEO MELAMED: You see it.
CLERK: It’s 6 even. I want to try 7 even. 6 even is last. NASDAQ is unchanged.
LEO: Right, get me out. I’ll be back, don’t hang up.
NARRATOR: Leo Melamed has been trading successfully for over 30 years.
LEO MELAMED (Chicago Mercantile Exchange): What I do is I pick up the phone to my clerk who is at the pit, she’s at the edge of the pit and she flashes in the order via a hand signal to the broker in the pit who handles my account and he then executes the order that I gave, gives her the confirmation of its execution. All of that is within seconds. Up here it’s kind of still and quiet. I get other kinds of information they don’t get, but I only get what they get which is the screaming and the noise and that noise level changes from time to time and it also gives, provides information as does the, the fear in the eyes of the, of the traders around you. That too is information.
I go in, in the morning and the day is before me and I have to figure out which direction any one of a dozen markets is going to go in. Additionally, you have to figure out from reading the newspapers for, from being a psychiatrist as it were of the public attitude which way they’re going today, what are they going to do, is this, is this information going to make them bullish, this information going to make them fear, and if I can figure that out I can beat the market.
NARRATOR: Traders like Melamed are convinced that success in the markets is all to do with human judgment and intuition, qualities that could never be reduced to a formula. However, this view has powerful opposition. An important group of academics, who study the markets mathematically, believe that such success is largely a matter of – luck.
Prof. ZVI BODIE (Boston University): In flipping a coin if you flip it long enough there may be a long run of heads which doesn’t at all imply that the person flipping it had the ability to make it come up heads. It could just be the luck of the toss.
NARRATOR: This strange view arose from an unexpected discovery. In the 1930s academics decided to find out whether traders really could predict how prices moved. They struggled to find some scientific basis for this belief, but they couldn’t find one, so finally they decided to run a series of experiments. In one of them they simply picked stocks and shares at random. They threw darts at the Wall Street Journal while blindfolded. At the end of the year this random choice out-performed the predictions of top traders. This was a revelation. This meant the prices must themselves be moving totally at random and therefore it was impossible, by definition, to predict anything about them. This led the academics to a devastating conclusion: despite all the claims of the traders, it now looked like anyone who managed to make a successful prediction in the stock market must be doing it not by skill but by chance alone.
ZVI BODIE: When some individual made a fortune in the stock market we have a tendency to assume that that was because he knew something, you know after the, and of course the individual himself is happy to reinforce that believe – yes, I was a genius, yeah, or I was very clever, I always said Microsoft was going to make me rich, but what you don’t see are the thousands, hundreds of thousands, perhaps millions of people who are going – I always said that ABC company was going to make me rich and ABC company went bust.
MERTON MILLER: If there’s 10,000 people looking at the stocks and trying to pick winners well 1 in 10,000 is going to score, by chance alone, a great coup, and that’s all that’s going on. It’s a game, it’s a chance operation and people think they are doing something purposeful but they’re really not.
NARRATOR: This attitude disgusted most traders and continues to do so today.
LEO MELAMED: Listen, academics as a rule make terrible traders, so for me to think that I’m going to listen to their theory about trading I beg to differ.
NARRATOR: The discovery of randomness outraged the traders, but it galvanized the academics, for they knew that mathematics had been used successfully to study random, unpredictable phenomena before. Everything from population growth to the weather, so now they began a quest. The equivalent in economics of the race to the Moon to find a scientific and rational way to tame the markets, to use the power of mathematics to conquer risk.
MERTON MILLER: We can deal with random series in a way that to the layman may appear as just chaos, but no, no, no, no, no, once you tell me that the series is random and you’ve got probability distributions we can use some of the apparatus of modern mathematical statistics to do analyses.
Prof. PAUL SAMUELSON (Massachusetts Institute of Technology): The hope that the mathematical theory of probability in statistics could be a skeleton key to help you understand the nature of chance perhaps to predict it better, perhaps to control it, that was born at, at that time and the rest is, as they say, history.
NARRATOR: But it was to be a chequered history. For a long time the academics tried to control risk through probability. By measuring how much prices had moved in the past they calculated the probable range within which they would fluctuate in the future, but after 15 years their predictions were little more accurate than the average weather forecast. What was needed was a much more exact way to ensure someone against risk. In 1955 one academic discovered that someone had already thought of it.
PAUL SAMUELSON: In the early 1950s I was able to locate by chance this unknown book by a French graduate student in 1900 rotting in the library of the University of Paris and when I opened it up it was as if a whole new world was laid out before me. In fact as I was reading it I arranged to get a translation in English I really wanted every precious pearl to be under, understood.
NARRATOR: In 1900 a young French student, Louis Bachelier, set out to do something no one had ever done before. Using a series of equations he created the first complete mathematical model of the markets. He too realized stock prices moved at random and that it was impossible to make exact predictions about them, but then Bachelier said he had found a solution, a wonderful way to get rid of risk, an obscure, almost magical financial contract called an option. He believed that if someone could discover a formula that would allow this rare contract to be widely used they would be able to tame the markets completely, but he died before he could find it.
PAUL SAMUELSON: After the discovery of Bachelier’s work there suddenly came to the mind of all the eager workers the notion of what the Holy Grail was. There was the next step needed. It was to get the perfect formula to evaluate and to price options.
NARRATOR: The academics returned eagerly to the markets where they began to investigate this strange contract which had so intrigued Bachelier. They discovered that options were, in theory, a miraculous form of financial insurance and they worked in a remarkable way. The risk in the stock market is that if you buy a stock today the price can drop in the future and you could lose money but if you pay for an option contract this gives you the right to wait and buy the stock if it reaches some agreed price in the future, but there’s no obligation. If the stock fails to reach that price you can opt out and you would lose only the cost of the option. In theory options are a perfect way to get rid of risk, but there was a problem. How much would someone pay for such absolute peace of mind? The value seemed to depend on the personal confidence of each investor. No one could agree on a standardized way to price options. The traders were baffled, but it was exactly the sort of bewildering problem that the academics loved and they attacked it with relish.
Throughout the 1960s the academics developed their mathematical models. They were convinced if they could somehow mathematically describe the emotional confidence of the investors they would crack the problem of how to price options. To do that they kept adding more and more symbols. They added symbols for the level of satisfaction, for reasonableness and aggressiveness. When that didn’t seem to do the trick they added more. Symbols for the guesses of other traders, for defensiveness, for safety. Soon they had a giant mathematical edifice, but the options price seemed as far away as ever.
ZVI BODIE: The mathematical models that were being developed during the 50s and 60s depended on inputs that were completely unobservable in the real world, like expectations of investors which might differ very much from one investor to another and how did you actually come up with a number, how could you come up with a number to input.
STAN JONAS: They would talk about people’s utility structure, they were talking about people’s risk aversion and this made all the models seem more like psychotherapy than real science.
NARRATOR: By the end of the 60s the academics were no nearer to pricing options than they’d ever been, nor any closer to Bachelier’s dream of a perfect mathematical model of the markets. All this was about to change.
Prof. MYRON SCHOLES (Stanford University): From an early age I was very, very fascinated by uncertainty. My parents having lived in a gold mining part of the world, in Northern Canada, would be always buying penny stocks or the family would tend to be buying stocks that had very little prices because there’d be some rumor that there’s be another gold find or a silver find and so the prices would shoot up and, or not shoot up. I mean the family wasn’t seen to be getting rich though. That got me very interested in why was it the case that these prices tend to fluctuate.
NARRATOR: In 1968 Myron Scholes and his colleague Fisher Black set out to tackle the problem of options.
MYRON SCHOLES: When I first discovered options I became very excited about the possibility that here was a contract that enabled you to only be able to take the upside of the returns and not the downside and that being able to take the upside only had value and that was really exciting.
NARRATOR: It was exciting, but a goal that had eluded the greatest minds in economics. They knew that every stock price constantly moved up and down. As it did so the value of an option on a particular stock fluctuated too, but there was no predictable relationship. What they wanted to find was a formula that would obtain the correct price of an option at any moment in time just by knowing the current price of the stock, but they couldn’t see their way through the mass of equations they’d inherited.
MYRON SCHOLES: I read the literature, try to see what others had done and I became dissatisfied with the various models because they had had assumptions that didn’t seem to make that much sense to me.
NARRATOR: But then they decided to try something different. One by one they dropped any symbol which represented something that was unmeasurable. All the things that for years the academics had been busily adding to their equations. It was a brilliant insight. The loss of these elements didn’t affect the calculations at all. All the symbols had been unnecessary. Now for the first time Black and Scholes were left with the bare bones of the problem, the elements which everyone agreed you needed to know to value an option: the stock price, its volatility, the duration of the contract, the interest rate and the level of risk. They could measure all of these things. They were all quantifiable except one – the level of risk.
MYRON SCHOLES: I could do that first part and then I, I, I got stuck.
NARRATOR: So Black and Scholes decided to think laterally. If they couldn’t measure risk exactly perhaps they could somehow make it less significant. They started with the old idea of hedging in which gamblers hedged their bets by betting from the opposite direction. The method they devised was to become one of the most significant discoveries in economics this century. They created a theoretical portfolio, a mixture of stocks and options. Then whenever either fluctuated up or down they tried to cancel the movement out by making another risky move in the opposite direction. Their aim was to keep the overall value of the portfolio in perfect balance. Since everything moved at random, that was extremely difficult and at first they could only cancel a little of the movement out, but eventually using complex algebra and a mass of calculations they finally found they could balance out a movement precisely and then another.
MYRON SCHOLES: After the fact we called this dynamic hedging, but that means dynamically hedging which you want to be able to eliminate the uncertainty of the movements in the stock.
NARRATOR: They soon discovered that dynamic hedging could balance out any movement at all. They could create a perfect equilibrium in which risks cancel themselves out.
MYRON SCHOLES: We were able to build a portfolio that was essentially riskless.
NARRATOR: They had found a theoretical way not just to reduce risk, but to eliminate it altogether. By dynamically hedging Black and Scholes were able to remove the last unmeasurable element – risk itself dropped out of their equations. And without risk they finally had a mathematical formula which could give them the price of any option.
STAN JONAS: If I have the proper recipe for dynamically hedging then that position is risk-free. What that means is since I know the price of the stock the only thing that’s left is the price of the option. Unbelievably I can fix the price of the option. If I know the price of the stock I now know the price of the option and that is the miracle and the breakthrough of the Black-Scholes price.
NARRATOR: Black and Scholes has solved the problem that had baffled generations of academics. It was a marvelous achievement. But there was a practical problem with their formula. It took time to calculate the dynamic hedging. In the time this took the fast moving markets would have moved on and their calculations would effectively be out-of-date. What was needed was a way to instantly recalculate to keep eliminating the risk continuously. Unbeknownst to Black and Scholes someone had found a way.
ZVI BODIE: He was kind of a, a wunderkind. He was just recognized from the very beginning as an extraordinary intellectual talent. His creative powers with the, the power of the analytical techniques, mathematical techniques that he was bringing to bear on some age-old questions in economics. Savings behavior, investment behavior. It was just obvious that here was a guy who was going to make intellectual history in our field.
Prof. ROBERT C. MERTON (Harvard Business School): In college I started studying the stock market. I went down to the stock exchange, watched all the activity from the visitors’ gallery, people running around, calling numbers, shouting and all the paper flying and the bells ringing and of course that was exciting and it seemed to lend itself to my analytical skills.
NARRATOR: By the early 1970s Bob Merton had developed a reputation for using exotic and abstract mathematical ways to study financial contracts like options. He was the perfect person for Black and Scholes to contact.
MYRON SCHOLES: In the fall I went over ideas that we had with Bob and spent a lot of time arguing with him about whether our results were robust and exact, whether there was flaws in our methodology.
ROBERT MERTON: So I’m going to reframe, reformulate their problem in the context of my modeling that I had developed.
NARRATOR: In constructing his own complex mathematical models, Merton had explored theories no one in finance had even heard of. One was to be the unlikely final piece of the jigsaw. Merton turned to rocket science. He had studied the theories of a Japanese mathematician, Ito, who’d faced a similar problem to Black and Scholes. In order to plot the trajectory of rockets you needed to know exactly where the missile was, not just second by second, but literally all the time. Ito had developed a way of dividing time into infinitely small parcels, smoothing it out until it became a continuum so that the trajectory could be constantly updated. Bob Merton used this idea and adapted it for the Black-Scholes formula. Using the notion of continuous time, the value of the option could be constantly recalculated and risk eliminated continually.
ROBERT MERTON: And then I discovered they were right. By following their procedure, their dynamic trading strategy in the stock and cash at least in the context of my model they could eliminate all of the risk.
MYRON SCHOLES: Bob phoned me up one Saturday morning and said that, you know, he had an alternative proof to our particular model and you know was very, he was convinced that it, that it did work. He had used technology that I hadn’t been aware of and Fischer hadn’t been aware of called Ito calculus to actually, you know, solve the problem in a more elegant way and a more robust way I think than Fischer Black and I had done.
NARRATOR: The formula that Black, Scholes and Merton unleashed on the world in 1973 was sparse and deceptively simple, yet this lean mathematical shorthand was the fulfillment of a 50-year quest.
MYRON SCHOLES: When we did get the final equation obviously that was eureka.
ROBERT MERTON: This is great. I mean it doesn’t get much better than that. When you solve a problem you really know you’ve cracked it.
NARRATOR: Here was a formula which could, it seemed, get rid of risk in the financial markets. Academics marveled at its breathtaking insights and its sheer audacity.
PAUL SAMUELSON: I know and can take my hat off to what their accomplishment was because I, I got near the North Pole, but near is no cigar.
MERTON MILLER: It exploded. Within a few years it was the most widely cited article in finance, even alas outdoing some of my own articles.
NARRATOR: But barely had the academics time to celebrate their achievement than someone else began to use the formula for real. The traders had never lost belief in their own abilities, but now it seemed the academics had invented something that could complement their intuition. Now option traders simply programmed the Black-Scholes formula into their calculators. By pressing a few buttons they could find the exact price of any option at any time.
STAN JONAS: Word of the model began to circulate, particularly amongst people in the University of Chicago and more particularly amongst the option traders and literally before the official publication of the model traders had effectively started to program the model and begin to use it to trade.
NARRATOR: Soon men and women who had never heard of Bachelier, Ito or continuous time were exploiting the academics formula to make money – lots of it.
GARY LAHEY (Chicago Board of Options Exchange): The computer takes in all the data that I give it during the day. It sends it to, to, (a) it sends it to the little beeping light, it sends it to the CBOE, it sends it to the clearing firm so that the exchange on an on-going basis knows everything I’m doing online. In addition, I know online what I’m doing because this thing calculates using a Black-Scholes model what my risk is at all times.
NARRATOR: Then other traders looked at the Black-Scholes formula and realized it could make them money too. By allowing them to hedge their risks constantly the traders could feel safe enough to conduct business on a scale they had never dreamt possible. The risks in stocks could be hedged against futures, those in futures against currency transactions and all of them hedged against a panoply of new complex financia