Probability, Complexity and Emanuel Derman's Blog
Rahul Bhattacharya
April 1, 2007

I recently came across Emanuel Derman's blog on the Wilmott site. There he mentioned that he was intrigued when he learned that the absolute value squared of the Schrodinger wave function represents a probability; to him, in his mind , probability was is a human concept. His view is that one cannot imagine probability without imagining an ensemble of identical experiments or coins, and then thinking of probability as some limiting ratio. So to him, no humans, no probability and if the Schrodinger wave function measures probability, then it's only valid when human beings are present .

This point of view is interesting. I'll come to the wave equation in a minute, but first a word or two about the notion of probability. The whole concept of probability, and the theory behind it, is counterintuitive. Long ago, I had read Bart Kosko and his book on Fuzzy systems and since then I have become a bit suspicious of the whole notion of probability.

Consider the following experiment: you hold a coin in one of your folded palms and ask your friend to predict what the probability of finding the coin in your right palm is. He will obviously answer 50%. He is right as far as he is concerned. However, to you the probability is either 100% or 0% because you know with complete certainty as to which of your folded palms hold the coin. So for each of you the probability measure is different. Or consider another example. You dive into a parking lot with only two parking slots - # 1 and # 2. You are supposed to park your car in one of the two slots, naturally. Therefore, obviously the probability of finding your car in slot # 1 is 50% and the same goes for slot # 2. However, say, you park your car in such a way that there is a bit of an overlap on both the slots - it can easily happen in real life. Say, the overlap is such that 10% of your car is in slot # 1 and 90% of the car is on slot # 2 (you are a bad driver!). Now what is the probability of finding your car in slot # 1 or slot # 2?

Now consider Schrodinger's wave equation in Quantum Mechanics. The wave equation is a differential equation - we have differential equations in finance as well - and its solution is called a wave function. All observed physical quantities, such as the electrons in an atom or the vibrations of a string, can be calculated using the wave equation and they turn out to be real. But the interesting thing is that the wave function is complex rather than real ("complex" in the mathematical sense, whereby it contains terms like square root of minus one, which cannot be calculated). And being mathematically complex the wave function cannot be directly observed or measured*.

In fact, all the real world quantities that are predicted by the wave functions appear as probabilities. If we take the complex wave function and multiply it by its complex conjugate we get the probability of locating a quantum particle in a region of space. (If you take a complex number as where one is the real part and is the imaginary part, with then the complex conjugate of this complex number would be which is just a reflection of this number around the real axis. Now if you multiply, with its complex conjugate we will get a real number 5.) In that sense once can say that "probability" is a complex function - at least, within the context of quantum mechanics. But the outcome of that complex function is always real. The existence of electrons in an atom is real, the vibrations of a string are real, the light ray from the sun is real and yet at the deepest and the most elementary level these real quantities are explained by complex probabilities.

Emanuel Derman's reference above to the absolute value squared of the Schrodinger's wave function is actually the wave function multiplied by its complex conjugate.

Even in finance we observe complexities. Asset prices have fractal characteristics where once again we observe complex numbers. Ito's calculus, Black-Scholes option pricing formulas, stochastic differential equations, partial differential equations, are all fairly complex, if not strictly in the mathematical sense, but in their functional form. There is enormous complexity hidden in the world of financial derivatives and assets. And that complexity is manifest in the form of probabilities - probability of a bond or a company defaulting, probability of that an option finishes in the money, probability of an asset hitting a barrier and so on and so forth. And no matter which way you look at them these probabilities are complex to understand. And yet day in and day out, while working in the financial markets we observe real and simple prices of these assets and derivatives on our screens and the entire mechanism of buying and selling securities is done by investors and agents effortlessly, as if it was all very trivial.

The world, as we observe, is simple and real and yet there is enormous complexity hidden beneath the surface.

Some of us, such as Emanuel Derman and other experts in the field of finance, are on a journey to unravel those complexities and some of us, like me, are happy, or helpless enough, to be ravaged by it.

http://www.wilmott.com/blogs/eman/

* Superstrings and the Search for The Theory of Everhthing by F. David Peat (Contemporary Books)

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