Brownian Motion

As a mathematical model, the Brownian Motion is well known for being explicitly delineated by no other than Albert Einstein in one of his “Annus Mirabilis” papers of 1905. In it, the Brownian Motion played a critical role in ending the then heated debate over the existence of atoms. However, the French mathematician Louis Bachelier was, in fact, the first to model the now called Brownian Motion as part of his PhD dissertation. Unlike Einstein, Bachelier created the Brownian to value stock options and also unlike Einstein, the importance of Bachelier’s contribution was not well appreciated at the time[1].

Nowadays, the Brownian Motion (also called Wiener process) is a cornerstone of current mathematical finance. It has been widely used to model the time trajectory of the price of financial instruments such as stocks, bonds, and options. It is also a key component of the Black-Scholes-Merton’s option pricing model who won Myron Scholes and Robert C. Merton the Nobel Memorial Prize in Economic Sciences in 1997[2].

Until not too long ago, the Brownian Motion was an enigma to me. I appreciated its importance and even recognised how to apply it to some finance related problems but its characteristics, and more importantly its construction, remained puzzling and elusive. This is no doubt why I was surprised to learn that its construction, however complex, can become very intuitive one we start looking closely. One approach to building a Brownian Motion hinges on a special family of wave-like functions called wavelets. Adding a certain kind of randomness to a particular wavelet approximation produces the famous Brownian Motion. For more details have a look at Chapter 3 of Michael Steele’s “Stochastic Calculus and Financial Applications”.

In this animation, I’ve used this wavelet-based construction to simulate and visualise three different Brownian Motions on the time interval 0 to 1. (implementation code here). The way to think about these three instantiations of the Brownian is to picture nature choosing different states of the world (three in this case). For each different state of the world, there is an entire realisation of the Brownian represented by the three different lines in the graph.

However beautiful, the simplicity of this construction runs the risk of understating the importance of a mathematical model that constitutes the bedrock of an entire branch of mathematics called stochastic calculus.

[1] Nevertheless, many consider Bachelier the Father of Modern Option Pricing Theory.
[2] Fischer Black was not eligible for the prize due to his death in August 1995.

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