Actuarial Finance

14Brownian motion

Brownian motion is a very important stochastic process widely studied in probability and has been applied with success in many areas such as physics, economics and finance. Its name originates from botanist Robert Brown, who studied moving pollen particles in the early 1800s.

The first application of Brownian motion in finance can be traced back to Louis Bachelier in 1900 in his doctoral thesis titled Théorie de la spéculation. Then, physicist Albert Einstein and mathematician Norbert Wiener studied Brownian motion from a mathematical point of view. Later, in the 1960s and 1970s, (financial) economists Paul Samuelson, Fischer Black, Myron Scholes and Robert Merton all used this stochastic process for asset price modeling.

This part of the book (Chapters 14–20) is dedicated to the Black-Scholes-Merton model, a framework that laid the foundations of modern finance and largely contributed to the development of options markets in the early 1970s. The cornerstone of the model is a stochastic process known as Brownian motion and whose randomness drives stock prices.

Therefore, this chapter aims at providing the necessary background on Brownian motion to understand the Black-Scholes-Merton model and how to price and manage (hedge) options in that model. We also focus on simulation and estimation of this process, which are very important in practice.

Even if Brownian motion is a sophisticated mathematical object with many interesting properties, the role of this ...