Financial Simulation Modeling in Excel

BROWNIAN MOTION AND WIENER PROCESSES

With a solid understanding of random numbers in mind, we can delve deeper into how these topics evolved in finance. The term Brownian motion is often heard in financial classrooms and occasionally in jobs that involve securities analysis. In fact the first time I thoroughly went through Brownian motion was when a managing director at my first job after graduate school was introducing his method for analyzing credit default swaps. Brownian motion is named after the 19th-century Scottish botanist Robert Brown. Brown noticed that when pollen grains were suspended in water they jiggled and moved about in seemingly random fashion. Today, this random motion is known as a Wiener process and is used as a foundation to model many financial products and attributes of financial products, the most common being stock prices and interest rates. Since most of us are comfortable with the idea that prices and rates do not fluctuate in a purely random fashion, we must first assert a few stipulations, which we will discuss later in the section, to make this framework more representative of real-world observables.

Wiener Process

Formally a Wiener process (W_n(t)), otherwise known as a random walk, is defined as follows: at any given time t_i a binomial process, a process by which there are only two possible outcomes, will take either a step forward or a step backward with a step size of 1/sqrt(n), where n is the total number of steps taken. Thus the outcome at