The mathematical development of present-day economic and finance theory began in Lausanne, Switzerland at the end of the nineteenth century, with the development of the mathematical equilibrium theory by Leon Walras and Wilfredo Pareto.1 Shortly thereafter, at the beginning of the twentieth century, Louis Bachelier in Paris and Filip Lundberg in Uppsala (Sweden) made two seminal contributions: they developed sophisticated mathematical tools to describe uncertain price and risk processes.

These developments were well in advance of their time. Further progress was to be made only much later in the twentieth century, thanks to the development of digital computers. By making it possible to compute approximate solutions to complex problems, digital computers enabled the large-scale application of mathematics to business problems.

A first round of innovation occurred in the 1950s and 1960s. Kenneth Arrow and Georges Debreu introduced a probabilistic model of markets and the notion of contingent claims. (We discuss their contributions in Chapter 6.) In 1952, Harry Markowitz described mathematically the principles of the investment process in terms of utility optimization. In 1961, Franco Modigliani and Merton Miller clarified the nature of economic value, working out the implications of absence of arbitrage. Between 1964 and 1966, William Sharpe, John Lintner,

1 References for some of the works cited in this chapter will be provided in later chapters in this book. For an engaging description of the history of capital markets see Peter L. Bernstein, Capital Ideas (New York: The Free Press, 1992). For a history of the role of risk in business and investment management, see Peter L. Bernstein, Against the Gods (New York: John Wiley & Sons, 1996).

and Jan Mossin developed a theoretical model of market prices based on the principles of financial decision-making laid down by Markowitz. The notion of efficient markets was introduced by Paul Samuelson in 1965, and five years later, further developed by Eugene Fama.

The second round of innovation started at the end of the 1970s. In 1973, Fischer Black, Myron Scholes, and Robert Merton discovered how to determine option prices using continuous hedging. Three years later, Stephen Ross introduced arbitrage pricing theory (APT). Both were major developments that were to result in a comprehensive mathematical methodology for investment management and the valuation of derivative financial products. At about the same time, Merton introduced a continuous-time intertemporal, dynamic optimization model of asset allocation. Major refinements in the methodology of mathematical optimization and new econometric tools were to change the way investments are managed.

More recently, the diffusion of electronic transactions has made available a huge amount of empirical data. The availability of this data created the hope that economics could be given a more solid scientific grounding. A new fieldâ€”econophysicsâ€”opened with the expectation that the proven methods of the physical sciences and the newly born science of complex systems could be applied with benefit to economics. It was hypothesized that economic systems could be studied as physical systems with only minimal a priori economic assumptions. Classical econometrics is based on a similar approach; but while the scope of classical econometrics is limited to dynamic models of time series, econophysics uses all the tools of statistical physics and complex systems analysis, including the theory of interacting multiagent systems.

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