The Mathematics of Financial Modelingand Investment Management

3-Milestones Page 78 Wednesday, February 4, 2004 12:47 PM

78 The Mathematics of Financial Modeling and Investment Management

A lasting contribution of Pareto is the formulation of a law of

income distribution. Known as the Pareto law, this law states that there

is a linear relationship between the logarithm of the income I and the

number N of people that earn more than this income:

Log N = A + s log I

where A and s are appropriate constants.

The importance of the works of Walras and Pareto were not appre-

ciated at the time. Without digital computers, the equilibrium systems

they conceived were purely abstract: There was no way to compute

solutions to economic equilibrium problems. In addition, the climate at

the turn of the century did not allow a serene evaluation of the scientific

merit of their work. The idea of free markets was at the center of heated

political debates; competing systems included mercantile economies

based on trade restrictions and privileges as well as the emerging cen-

trally planned Marxist economies.

PRICE DIFFUSION: BACHELIER

In 1900, the Sorbonne University student Louis Bachelier presented a

doctoral dissertation, Théorie de la Spéculation, that was to anticipate

much of today’s work in finance theory. Bachelier’s advisor was the

great French mathematician Henri Poincaré. There were three notable

aspects in Bachelier’s thesis:

■ He argued that in a purely speculative market stock prices should be

random.

■ He developed the mathematics of Brownian motion.

■ He computed the prices of several options.

To appreciate the importance of Bachelier’s work, it should be

remarked that at the beginning of the 20th century, the notion of proba-

bility was not yet rigorous; the formal mathematical theory of probabil-

ity was developed only in the 1930s (see Chapter 6). In particular, the

precise notion of the propagation of information essential for the defini-

tion of conditional probabilities in continuous time had not yet been

formulated.

Anticipating the development of the theory of efficient markets 60

years later, the key economic idea of Bachelier was that asset prices in a

speculative market should be a fair game, that is, a martingale process

such that the expected return is zero (see Chapter 15). According to Bach-