In my previous post, I introduced a simple random process known as Brownian motion that has widespread applications in science, economics, and finance. It is a tribute to the ingenuity of mathematicians that from this simple process, a large body of more elaborate random processes were built that have broadened the usefulness of the original Brownian motion model. Practicality aside, the equations of these more complex random processes are also often quite beautiful from a mathematical viewpoint: many are compact and elegant in form.
This post continues where I left off. It pulls one area of application that involves functions of the Brownian motion in what is known as the theory of First Passage Time. This theory is concerned with events triggered when that a random or stochastic process first encounters a threshold. The threshold can be a barrier, a boundary, or a specified state of a system. The amount of time required for the process, starting from some initial state, to encounter a threshold for the first time is referred to variously as first hitting time or first passage time. A relatable example is the first time the price of a stock, say that of Amazon, hits a certain level, which may be a target price to sell, or the initial purchase price. If the threshold is the purchase price, the first time Amazon’s stock price falls below this threshold signifies the investor’s first encounter with a loss. Knowing the probability of this happening is clearly of importance to risk-averse investors. It turns out that Brownian motion greatly simplifies the math used to estimate this probability. The result is a group of equations that are both beautiful and useful.
A Bit of History
In yesterday’s post, I mentioned that Brownian motion is also known as Wiener diffusion process, after the brilliant mathematician Norbert Wiener who studied it rigorously in the early 20th century. Fittingly, the idea of first passage time for Wiener diffusion processes first appeared in economics (in the context of financial ruin in insurance and stock investment). This happened in the early 1900s with the work of scientists like Ernst Filip Oskar Lundberg (1876-1965), a Swedish actuary, and mathematician and French mathematician, Louis Bachelier (1870-1946) who laid the foundations for modern mathematical finance. Since then, the scope and number of applications of Brownian motion (Wiener processes) have exploded, with major advances seen not only in economics and insurance but also in physics and engineering.
Without getting you bogged down in technical details, I will go straight into the equations of the applications in mind – that of estimating the probability that an investment will fall below a threshold within a specified time period. I will illustrate this problem with the help of data from the stock market, or more precisely, stock market indices that track the prices of a basket of stocks traded on different exchanges. such as the UK, the US, Europe etc. Equations based on first passage time theory will be shown without the derivations (I hope you will find them beautiful) and the empirical results from applying these equations to the data are then shown and the key insights discussed. This application of first passage time math is based on my own research during my tenure as professor of finance and quantitative methods some years ago. It appeared in a peer-reviewed journal, the Journal of Wealth Management in 2015. Instead of attaching the entire article, I have chosen to show only those parts of the paper that relate to the application of first passage time. Here are the relevant extracts: