In this review article, we explore several recent advances in the quantitative modeling of financial markets. We begin with the Efficient Markets Hypothesis and describe how this controversial idea has stimulated a number of new directions of research, some focusing on more elaborate mathematical models that are capable of rationalizing the empirical facts, others taking a completely different tack in rejecting rationality altogether. One of the most promising directions is to view financial markets from a biological perspective and, specifically, within an evolutionary framework in which markets, instruments, institutions, and investors interact and evolve dynamically according to the "law" of economic selection. Under this view, financial agents compete and adapt, but they do not necessarily do so in an optimal fashion. Evolutionary and ecological models of financial markets is truly a new frontier whose exploration has just begun.

In the ’50s and ’60s, just as the era of the professional portfolio manager was dawning, financial economists were telling anyone who would listen that active management was probably a big mistake—a waste of time and money. Their research demonstrated that historical prices were of little use in helping to predict where future prices would go. Prices simply took a “random walk.” The better part of wisdom, they advised, was to be a passive investor. At first, not too many of the people who influence the way money is managed (those who select managers of large portfolios) listened. But as time went on, it became apparent that they should have. Because of fees and turnover, the managers they picked typically underperformed the market. And the worse an active manager did relative to a market index, the more attractive seemed the low cost alternative of buying and holding the index itself. But as luck would have it, just as indexing was gaining ground, a new wave of academic research was being published that weakened some of the results of the earlier research and thereby undercut part of the justification for indexing. It didn’t obviate all the reasons for indexing (indexing was still a low-cost way to create diversification for an entire fund or as part of an active/passive strategy), but it did tend to silence the index-because-you-can’t-do better school.

The practice of risk management starts with an understanding of the statistical behavior of financial asset prices over time. Models such as the random walk hypothesis, the martingale model, and geometric Brownian motion are fundamental to any analysis of financial risks and rewards, particularly for longer investment horizons. Recent empirical evidence has cast doubt on some of these models, and this article provides an overview of such evidence. I begin with a review of the random walk hypothesis and related models, including a discussion of why such models perform so poorly, and then turn to some current research on alternative models such as long-term memory models and stable distributions.

While financial economics is still in its infancy when compared to the mathematical and natural sciences, it has enjoyed a spectacular period of growth over the past three decades, thanks in part to the mathematical machinery that Wiener, Ito, and others pioneered. In this review article, I shall present a survey of some recent research in this exciting area—more specifically, in empirical finance and financial econometrics—including a discussion of the random walk hypothesis, long-term memory in stock market prices, performance evaluation, and the statistical estimation of diffusion processes. It is my hope that such a survey will serve both as a tribute to the amazing reach of Nobert Wiener's research, and as an enticement to those in the "hard" sciences to take on some of the challenges of modern finance.

Although they are only one of the many types of statistical tools for modeling nonlinear relationships, neural networks seem to be surrounded by a great deal of mystique and, sometimes, misunderstanding. Because they have their roots in neurophysiology and the cognitive sciences, neural networks are often assumed to have brain-like qualities: learning capacity, problem-solving abilities, and ultimately, cognition and self-awareness. Alternatively, neural networks are often viewed as "black boxes" that can yield accurate predictions with little modeling effort. In this review paper, I hope to remove some of the mystique and misunderstandings about neural networks by providing some simple examples of what they are, what they can and cannot do, and where neural nets might be profitably applied in financial contexts.

We develop a stochastic model of nonsynchronous asset prices based on sampling with random censoring. In addition to generalizing existing models of nontrading, our framework allows the explicit calculation of the effects of infrequent trading on the time series properties of asset returns. These are empirically testable implications for the variance, autocorrelations, and cross-autocorrelations of returns to individual stocks as well as to portfolios. We construct estimators to quantify the magnitude of nontrading effects in commonly used stock returns data bases, and show the extent to which this phenomenon is responsible for the recent rejections of the random walk hypothesis.

In this article we test the random walk hypothesis for weekly stock market returns by comparing variance estimators derived from data sampled at difference frequencies. The random walk model is strongly rejected for the entire sample period (1962-1985) and for all subperiods for a variety of aggregate returns indexes and size-sorted portfolios. Although the rejections are due largely to behavior of small stocks, they cannot be attributed completely to the effects of infrequent trading or time-varying volatilities. Moreover, the rejection of the random walk for weekly returns does not support a mean-reverting model of asset prices.