Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Prologue

In The Beginning Was The Motion…

The development of modern mathematical techniques for financial applications can be traced back to Bachelier's work, Theory of Speculation, first published as his PhD Thesis in 1900. At that time Bachelier was studying the highly irregular movements in stock prices on the French stock market. He was aware of the earlier work of the Scottish botanist Robert Brown, in the year 1827, on the irregular movements of plant pollen when suspended in a fluid. Bachelier worked out the first mathematical model for the irregular pollen movements reported by Brown, with the intention to apply it to the analysis of irregular asset prices. This was a highly original and revolutionary approach to phenomena in finance. Since the publication of Bachelier's PhD thesis, there has been a steady progress in the modelling of financial asset prices. Few years later, in 1905, Albert Einstein formulated a more extensive theory of irregular molecular processes, already then called Brownian motion. That work was continued and extended in the 1920s by the mathematical physicist Norbert Wiener who developed a fully rigorous framework for Brownian motion processes, now generally called Wiener processes.

Other major steps that paved the way for further development of mathematical finance included the works by Kolmogorov on stochastic differential equations, Fama on efficient-market hypothesis and Samuelson on randomly fluctuating forward prices. Further important developments ...