How do you manage to develop new financial models or discover exciting mathematical theories? The surprisingly easy advice: Just be reliable.
Once, twenty years ago, a colleague and I had what seemed like a good idea for a financial model. We wanted to CAT-scan the DAX options market to see what it thought about the future. I am speaking metaphorically, of course.
When technicians do a real CAT scan of your abdomen, they shoot X-rays from all angles through you and collect the scattered rays. From the way the X-rays scatter off your insides, they can reconstruct the interior of your body. How does that apply to the DAX?
The Value of a DAX option
Many investors and speculators buy options on the DAX, betting that it will rise above some higher level or fall below some lower one. The value of the DAX, the Deutsche Bourse German stock index, fluctuates every day, and the magnitude of those fluctuations is called the DAX’s volatility. Since your chance of winning a bet on the DAX is better if the fluctuations are larger, the value of a DAX option depends on the stock’s imagined future volatility. My colleague and I thought of a method to deduce other people’s idea of future DAX volatility by looking at the prices of all options on the DAX (i.e. shooting X-rays from all angles) and backing out from them what people thought its volatility (its interior) would be.
This is how most financial models work. In physics, models predict the future, moving forward in time. In finance, models extract the future as imagined by other people from the prices that other people are willing to pay for securities today. Having extracted other people’s imaginings and examined them, then, if you think they are wrong, you can bet against them by buying or selling things that will profit if they do turn out to be wrong.
Riding home on the subway that day twenty years ago with a friend, I briefly told him what we conjectured and how we had proved it. My friend was very smart and quick. After listening to me, he snorted. „That’s impossible,“ he exclaimed. Then, a minute or so later: „No, you’re right. It’s obvious. Actually, it’s totally trivial!“
Nothing is Really Obvious
The past obviousness of anything you never knew is a delusion. Many things seem clear only once they have been taught to you, once all the prejudices, confusion and competing theories have been omitted. Every iota of discovery comes at the cost of long immersion, hard labor, and struggle. I learned this most dramatically in physics graduate school many years ago, when Prof. Friedberg, a slightly spacy and disheveled but very clever man, taught us Einstein’s 1905 theory of relativity.
Prior to Einstein, there were two sets of laws that governed the universe as seen by an observer on earth: For matter, Newton’s 17th Century laws described the motion of particles. For light, Maxwell’s 19th Century laws described the oscillations of waves.
What Einstein Discovered
Einstein postulated that all laws must be universal. The same laws that work on earth must work everywhere, for everyone; they must look exactly the same, even if the person using them is moving and not stationary on our earth. But Einstein noticed a contradiction in the region where Newton and Maxwell overlapped, when light and matter interacted with each other. If Newton’s laws of matter were universal, then Maxwell’s laws for light could not be. And vice versa.
To make an inaccurate analogy: imagine that women habitually walk fast and men always stroll. When they form a couple and take a walk, one of them has to change. Einstein realized that Newton’s laws had to change in order to keep things universal. The result was that though the laws were universal, space and time became relative.
In 1905, that was obvious to no one. Professor Friedberg made us read the famous Dutch physicist Lorentz’s 1906 book, The Theory of Electrons, written before he himself had understood Einstein. The book was an account of Lorentz’s heroic struggle to avoid the clash of the laws of light and matter by sculpting, step by painful step, a clumsy ingeniously contrived model for the Newtonian electrons that emitted Maxwellian light. With his bold intuition, Einstein relegated Lorentz’s model to the interesting footnotes of history. Ever since I’ve had great respect for even the smallest breakthroughs, for the unbridgeable gap between creator and disciple. No discovery is as obvious as it seems to the people who read the textbooks written even a few years later. Everything looks simple once you’ve been taught it; before there were only disconnected confusing facts.
How To Be Reliable
I recalled all this again during the past two years when supervising young students tackling straightforward financial research using computers. All of them know a great deal of canned theory. But not all of them had learned how to go about investigating something where there wasn’t a canned answer. To do that you have to be conscious, self-conscious, and reliable. Those are things you can learn.
You must not simply rely on the results of computers you have programmed -- it’s too easy to make mistakes. You must keep observing yourself, doubting your answers. You have to think how to verify and understand, by other means, the numbers and patterns that the computer spits out. Is there a way to understand the answer without the computer? You must be eclectic about methods. Use algebra and arithmetic to check your answers. Find at least two ways to do everything and see that they agree.
One good strategy is to check complicated methods by, in a few specific cases, reducing them to something simpler. Suppose, to take a trivial example, you write a program to add a large bunch of numbers. How do you know the program works correctly? First test it when it adds only two integers and you know the answer by hand. Then try it for two decimals. Then let one of the numbers be negative. Then give it thousands of numbers, but let all of them be zero. Then thousands but all of them 1. Then add 1 + 1/2 + 1/4 + 1/8 + .... and make sure the answer converges to 2. This is a toy problem, but I hope you get the idea: Whatever method you use to solve your problem, think about another way to get an approximate answer.
Do What You Can
It sounds simple, but it’s surprising how often people forget this, and how well it works. „Improvement makes straight roads, but the crooked roads without improvement are the roads of genius,“ wrote William Blake several hundred years ago.
When I did research, in physics or finance, my colleagues and I dreamed of finding crooked roads. Then, after a few years of witnessing people we knew make discoveries we could never have made, most of us began to realize that we might never find the crooked roads. There were times when we had sudden small exciting epiphanies in our work, glimpses of what it might feel like to stumble on a real discovery. But we also learned that we were lucky even to be „mere“ improvers, and thankful for it.
No one can tell you how to find crooked roads. If children grew up according to early indications, we should have nothing but geniuses, wrote Goethe. You can buy all the Baby Einstein DVDs for your child that you can afford; genius and inspiration are still a mystery. But improving the roads that have been discovered is a skill that can be acquired, and is worthwhile and satisfying too. And once in a blue moon, you may find an unexpected pothole, and, if you try to understand whether it’s fact or artifact, you may discover something no one around you knew.
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