There follows a speedy, roller-coaster of a ride through " the official history of quantitative finance, passing through both the highs and lows. Where possible I give dates, name names and refer to the original sources.^{[1]}

Brown

The Scottish botanist, Robert Brown, gave his name to the random motion of small particles in a liquid. This idea of the random walk has permeated many scientific fields and is commonly used as the model mechanism behind a variety of unpredictable continuous-time processes. The lognormal random walk based on Brownian motion is the classical paradigm for the stock market. See Brown (1827).

Bachelier

Louis Bachelier was the first to quantify the concept of Brownian motion. He developed a mathematical theory for random walks, a theory rediscovered later by Einstein. He proposed a model for equity prices, a simple normal distribution, and built on it a model for pricing the almost unheard of options. His model contained many of the seeds for later work, but lay 'dormant' for many, many years. It is told that his thesis was not a great success and, naturally, Bachelier's work was not appreciated in his lifetime. See Bachelier (1995).

Einstein

Albert Einstein proposed a scientific foundation for Brownian motion in 1905. He did some other clever stuff as well. See Stachel (1990).

Richardson

Most option models result in diffusion-type equations. And often these have to be solved numerically. The two main ways of doing this are Monte Carlo and finite differences (a sophisticated version of the binomial model).

The very first use of the finite-difference method, in which a differential equation is discretized into a difference equation, was by Lewis Fry Richardson in 1911, and used to solve the diffusion equation associated with weather forecasting. See Richardson (1922). Richardson later worked on the mathematics for the causes of war. During his work on the relationship between the probability of war and the length of common borders between countries he stumbled upon the concept of fractals, observing that the length of borders depended on the length of the 'ruler.' The fractal nature of turbulence was summed up in his poem ''Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity.''

Wiener

Norbert Wiener developed a rigorous theory for Brownian motion, the mathematics of which was to become a necessary modelling device for quantitative finance decades later. The starting point for almost all financial models, the first equation written down in most technical papers, includes the Wiener process as the representation for randomness in asset prices. See Wiener (1923).

s Samuelson

The 1970 Nobel Laureate in Economics, Paul Samuelson, was responsible for setting the tone for subsequent generations of economists. Samuelson 'mathematized' both macro and micro economics. He rediscovered Bachelier's thesis and laid the foundations for later option pricing theories. His approach to derivative pricing was via expectations, real as opposed to the much later risk-neutral ones. See Samuelson (1955).

Its

Where would we be without stochastic or Ito calculus? (Some people even think finance is only about Ito calculus.) Kiyosi Ito showed the relationship between a stochastic differential equation for some independent variable and the stochastic differential equation for a function of that variable. One of the starting points for classical derivatives theory is the lognormal stochastic differential equation for the evolution of an asset. Ito's lemma tells us the stochastic differential equation for the value of an option on that asset.

In mathematical terms, if we have a Wiener process X with increments dX that are normally distributed with mean zero and variance dt, then the increment of a function F(X)is given by

This is a very loose definition of Ito's lemma but will suffice. See Ito (1951).

[1] A version of this chapter was first published in New Directions in Mathematical Finance, edited by Paul Wilmott and Henrik Rasmussen, John Wiley & Sons Ltd, 2002.

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