How to Simulate the Short Rate Without a Computer (or: a Physicists View of Financial Models)

Since this is my first post here, I would like to introduce myself. I am Diana Hufnagl and I am fairly new to the financial world. I started working in this field one and a half years ago and there is still a lot to learn for me. My scientific origin is physics and today I would like to share my first thoughts on financial models with you. 

Consider the following stochastic differential equation (SDE):

dr = m dt + s dW

As you probably all know it can be used to model the short rate for example. The last term corresponds to its stochastic part, modeled by a Wiener process.
When I first saw this equation I instantly remembered an afternoon a couple of years ago. I spent it in front of the microscope looking at micrometer-sized particles in water. What I am referring to is an experiment we conducted in a physics course at the University, where we observed and measured Brownian motion, which is a Wiener process. 

What we did was: 
  • prepare a solution of latex particles in water,
  • put it under a microscope,
  • put some paper on the wall,
  • project the picture of the microscope onto said paper
  • and then follow one particle at a time as it floats through the water.
Every few seconds we marked the position of the particle on the paper, thereby recording its random walk. We did this for lots of different particles to collect enough independent paths. We spent hours in a room lit only by the projection of the microscope on the wall, making dots on a piece of paper. And, as weird as it might sound to non-physicists, it was amazing! Describing and discovering the laws of nature (even if it is something as simple as particles moving in water) is something I find truly fascinating. Afterwards we sat down and calculated the mean value and the variance of our recorded random walks and finally obtained the expected probability distribution. And by doing so we basically made a real life simulation of the SDE given above for m = 0. If one wants to include a drift term (m ≠ 0), one could introduce a steady flow for example, which would lead to a shift of the mean value of the distribution. 

Therefore I just gave you the means to simulate for example the short rate without the use of a computer. All you need are small particles in the correct fluid (fixes s) exhibiting a specific flow (fixes m) and you are good to go! Using a microscope you can then observe the different realizations of the short rate.