A Brief History of Mathematical Finance

The history of mathematical finance dates back to the end of the 19th century, building on prior work and significant contributions from mathematical figures such as Vito Volterra, Paul Lévy, and Norbert Wiener.

One such pioneer was Louis Bachelier, whose groundbreaking work laid the foundation for modern financial mathematics. Bachelier's doctoral thesis, "Théorie de la spéculation" (The Theory of Speculation), was completed in 1900 under the supervision of Henri Poincaré ^[1].

In the aforementioned period, the produced scientific work was truly revolutionary. In his doctoral thesis, Bachelier proposed a modelling framework where stock prices are assumed to follow a random walk. This remarkable insight would later evolve into the theory of the application of Brownian Motion in financial markets. The idea was ahead of its time, predating Albert Einstein's famous paper, “Investigations on The Theory of Brownian Movement”, by five years. Bachelier's model assumed that price changes were independent and normally distributed—although an unrealistic simplification, such concepts remain fundamental in many financial models.

Brownian motion, also known as the Wiener process (named after Norbert Wiener), was first observed by Robert Brown when analysing the behavior of minute particles suspended in water under a microscope. Brownian motion later became a cornerstone of financial modeling. It represents the continuous-time random movement of particles, serving as an analogy for the unpredictable movements of stock prices ^[2].

The culmination of these ideas came with the Black-Scholes Model, developed by Fischer Black, Myron Scholes, and Robert Merton in the 1970s. This model, which assumes that the price of the underlying asset follows a geometric Brownian motion, revolutionized option pricing and risk management ^[3]. However, it's important to note that practitioners like Edward O. Thorp had been using similar methods under analogous assumptions for some time prior.

The evolution from Bachelier's initial ideas to the sophisticated models we use today highlights the continuous development in the field of mathematical finance. These stochastic processes and models provide the mathematical foundation for risk management, portfolio optimization, and derivative pricing.

In future posts, we'll explore more recent developments in financial mathematics, including jump processes, stochastic volatility models, and their applications in modern financial practices such as algorithmic trading. We'll also delve deeper into the contributions of other pioneers who have shaped this dynamic field.