Risk Latte - A Hundred Years of Brownian Motion

A Hundred Years of Brownian Motion

Rahul Bhattacharya
Jan 07, 2005

Yesterday, one of my programmers asked me "what is Brownian motion?" He has been writing codes for developing derivatives pricing models and came across this term in one of the analysis papers. He wanted a one line answer in simple English and so I replied Brownian motion is the simplest stochastic process on a continuous domain. He said that wasn't any good. So I tried, Brownian motion is a random walk with a drift and saw he wasn't very satisfied with that either. I finally said that Brownian motion is a Wiener process that is Markovian in nature and where a particle position is given by a Guassian distribution. He said he will look up on the internet.

While going back home that evening it struck me that this is January of 2005 almost 100 years to date since Einstein published his famous papers in 1905, one of which explained Brownian motion. I called up one of my friends in Citi London and interrupted his trading by asking him how he felt about it if we organised an online symposium on Brownian motion on our website.

Some twenty two years ago when I was in the second year of my Bachelor's degree programme at St. Stephen's College at University of Delhi in India I fell in love with a young lady across the lawn from the Physics department. I was a Physics major and she was studying Chemistry and every morning as she passed by in front of my classroom window my heart would flutter and I could literally see the rainbow in the sky. Every morning ten minutes into the class of statistical mechanics I would look out of the window and there she would be, passing by my window and I would crane my neck to the maximum to get a glimpse of her; but she would be gone in a second. And that second, every morning, would change my world.

As it turned out that it was more of a post-teen infatuation on my part than love but I would like to call it love; it sounds better and poetic. Also, and more importantly, it turned out she had a boyfriend and she was far too busy with her credits than bothering with some Romeo who had lost his bearings. I fell in love and lost the object of my love. But she would never know.

It was the autumn of '82 the year I fell in love with Monica and it was also the year when I first learned, quite by accident, about Brownian motion, which soon became my other, surrogate object of love. I first heard about it from one of my classmates, who I still believe may one day win a Nobel in Physics and whose take on the topic was that it was quintessential to the existence of all things on earth. Many afternoons, while waiting to catch a glimpse of Monica, while she was on a way out of the Chemistry lab to the main building, I would sit on the lawns reading about the mechanics of Brownian motion or some other related paper on statistical mechanics and try to unravel this so called "quintessential" process of life; when I stripped away the math and the calculus it was simply a motion, a phenomenon where all motion was random and yet there was the possibility to detect some sort of a pattern, a drift or expectation, whatever it was; right or wrong this was what I understood of the subject in those brief afternoons in the autumn of 1982. Randomness with a purpose, chaos with a trend, a coincidence with certainty; life suddenly became a dream. This was fascinating - a seemingly random process could have an expectation built in it. Even my erudite friend was surprised to see that I was so totally smitten by an otherwise boring and unfashionable theory. It was as if many millions of particles were colliding in my heart, seemingly at random and yet with a purpose, to produce the prefect Brownian motion, the song of life. Monica had become my Brownian motion.

Twenty years later I am still in love and as much fascinated by Brownian motion as I was when I first read about it. And twenty years later I am sitting beside my programmer and trying to explain Brownian motion to him so that he can model the stock price behaviour in a Monte Carlo framework with a stochastic equation that is the result of Brownian motion.

In the last twenty years I have travelled the continents and traversed the distances that separate our hearts, I have discovered and lost a thousand Monicas and I have come to believe in certainty of the existence of God; I have levelled with my peers, doubted the probability theory embraced fuzzy logic, experimented with my professions and have lived a thousand lives of Brownian motion. And each one a life of randomness and uncertainty and in each one of these lives as if, through some invisible hand, I am being guided somewhere, being taken some place though all events appear purely coincidental and happened by chance.

In 1905, Albert Einstein published three seminal papers that would change the way we looked at the world and universe. One of them, was a detailed description of the statistical properties of a type of motion called the Brownian motion. It is difficult to say when and under what circumstances this phenomenon became part of mainstream finance, but I reckon it would be the Black-Scholes paper of 1973 on Option Pricing that formally introduced this concept in the world of stocks and bonds. Every single asset, be it stock, bond or foreign currency is assumed to follow Brownian motion and all modelling of their behaviours follow from this premise. It is anybody's guess whether our world of Black-Scholes would have happened without the notion of Brownian motion.

Anyway, as I held on to the cell phone waiting for my friend to finish his trade in London, it occurred to me that somehow, in some peculiar yet fundamental way our lives were a replica of Brownian motion - random yet with a drift. When my friend came on the line he said, "Brownian motion? What's Einstein got to do with Brownian motion? The guy's name is Robert Brown who discovered Brownian motion. Sorry mate, you've got Physics blowing up your ass!"

© Rahul Bhattacharya
This column is written by Rahul Bhattacharya and reflects his own views about life and business. It does not necessarily reflect the views and opinions of other members of Risk Latte Company Limited, Hong Kong (“the Company”) and the Company accepts no responsibility for any factual errors contained in the column and strongly advises readers not to pay much attention to it.

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