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HERAUSGEGEBEN VON WERNER D'INKA, BERTHOLD KOHLER, GÜNTHER NONNENMACHER, HOLGER STELTZNER
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F.A.Z.-Column by Emanuel Derman The Sublime to the Ridiculous

How do you manage to develop new financial models or discover exciting mathematical theories? The surprisingly easy advice: Just be reliable.

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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.

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