The Mathematics of Financial Modelingand Investment Management

3-Milestones Page 90 Wednesday, February 4, 2004 12:47 PM

90 The Mathematics of Financial Modeling and Investment Management

Harrison and David Kreps^24 and Michael Harrison and Stanley Pliska^25

in the late 1970s and early 1980s. Not all price processes can be

reduced in this way: if price processes do not behave sufficiently well

(i.e., if the risk does not vanish with the vanishing time interval), then

replicating portfolios cannot be found. In these cases, risk can be mini-

mized but not hedged.

SUMMARY

■ The development of mathematical finance began at the end of the nine-

teenth century with work on general equilibrium theory by Walras and

Pareto.

■ At the beginning of the twentieth century, Bachelier and Lundberg

made a seminal contribution, introducing respectively Brownian

motion price processes and Markov Poisson processes for collective

risk events.

■ The advent of digital computers enabled the large-scale application of

advanced mathematics to finance theory, ushering in optimization and

simulation.

■ In 1952, Markowitz introduced the theory of portfolio optimization

which advocates the strategy of portfolio diversification.

■ In 1961, Modigliani and Miller argued that the value of a company is

based not on its dividends and capital structure, but on its earnings;

their formulation was to be called the Modigliani-Miller theorem.

■ In the 1960s, major developments include the efficient market hypothe-

sis (Samuelson and Fama), the capital asset pricing model (Sharpe,

Lintner, and Mossin), and the multifactor CAPM (Merton).

■ In the 1970s, major developments include the arbitrage pricing theory

(Ross) that lead to multifactor models and option pricing formulas

(Black, Scholes, and Merton) based on replicating portfolios which are

used to price derivatives if the underlying price processes are known.

(^24) J. Michael Harrison and David M. Kreps, “Martingale and Arbitrage in Multipe-
riod Securities Markets,” Journal of Economic Theory 20 (1979), pp. 381–408.
(^25) Michael Harrison and Stanley Pliska, “Martingales and Stochastic Integrals in the
Theory of Continuous Trading,” Stochastic Processes and Their Applications
(1981), pp. 313–316.