The predictability of an asset's returns will affect option prices on that asset, even though predictability is typically induced by the drift which does not enter the option pricing formula. For discretely-sampled data, predictability is linked to the parameters that do enter the option pricing formula. We construct an adjustment for predictability to the Black-Scholes formula and show that this adjustment can be important even for small levels of predictability, especially for longer-maturity options. We propose several continuous-time linear diffusion processes that can capture broader forms of predictability, and provide numerical examples that illustrate their importance for pricing options.

Data-snooping—finding seemingly significant but in fact spurious patterns in the data—is a serious problem in financial analysis. Although it afflicts all non-experimental sciences, data-snooping is particularly problematic for financial analysis because of the large number of empirical studies performed on the same datasets. Given enough time, enough attempts, and enough imagination, almost any pattern can be teased out of any dataset. In some cases, these spurious patterns are statistically small, almost unnoticeable in isolation. But because small effects in financial calculations can often lead to very large differences in investment performance, data-snooping biases can be surprisingly substantial. In this review article, I provide several examples of data-snooping biases, explain why it is impossible to eliminate them completely, and propose several ways to guard against the most extreme forms of data-snooping in financial analysis.

We propose a nonparametric method for estimating the pricing formula of a derivative asset using learning networks. Although not a substitute for the more traditional arbitrage-based pricing formulas, network pricing formulas may be more accurate and computationally more efficient alternatives when the underlying asset's price dynamics are unknown, or when the pricing equation associated with no-arbitrage condition cannot be solved analytically. To assess the potential value of network pricing formulas, we simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a six-month training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For purposes of comparison, we perform similar simulation experiments for four other methods of estimation: OLS, kernel regression, projection pursuit, and multilayer perceptron networks. To illustrate the practical relevance of our network pricing approach, we apply it to the pricing and delta-hedging of S&P 500 futures options from 1987 to 1991.

The non-trading or non-synchronous effect arises when time series, usually financial asset prices, are taken to be recorded at time intervals of one length when in fact they are recorded at time intervals of another, possibly irregular, lengths. For example, the daily prices of securities quoted in the financial press are usually "closing" prices, prices at which the last transaction in each of those securities occurred on the previous business day. these closing prices generally  do not occur at the same time each day, but by calling them "daily" prices, we have implicitly and incorrectly assumes that they are equally spaces at 24-hour intervals. Such an assumption can generate spurious predictability in price changes and returns even if true price changes or returns are statistically independent. The non-trading effect induces potentially serious biases in the moments and co-moments of asset returns such as their means, variances, covariances, and autocorrelation and cross-autocorrelation coefficients.

The pricing of options, certificates, and other derivatives or assets—financial assets whose payments depend on the prices of other assets—is one of the great successes of modern financial economics. Although the pricing of derivatives is computationally intensive, there is little done in terms of the traditional empirical analysis since by the very nature of the determination of prices and arbitrage there is no error term to minimize. There are, however, many issues of statistical inference that affect the pricing of options and other derivatives. This paper analyzes two of the most common issues neglected in the literature: reduced form empirical instruments for the determination of prices and how to use Monte Carlo simulations to calculate option prices depend on a path.

We estimate the conditional distribution of trade-to-trade price changes using ordered probit, a statistical model for discrete random variables. This approach recognizes that transaction price changes occur in discrete increments, typically eighths of a dollar, and occur at irregularly-spaced time intervals. Unlike existing models of discrete transaction prices, ordered probit can quantify the effects of other economic variables like volume, past price changes, and the time between trades on price changes. Using 1988 transactions data for over 100 randomly chosen U.S. stocks, we estimate the ordered probit model via maximum likelihood and use the parameter estimates to measure several transaction-related quantities, such as the price impact of the trades of a given size, the tendency towards price reversals from one transaction to the next, and the empirical significance of price discreteness.

A test for long-run memory that is robust to short-range dependence is developed. It is an extension of the "range over standard deviation" or R/S statistic, for which the relevant asymptotic sampling theory is derived via functional central limit theory. This test is applied to daily and monthly stock returns indexed over several time periods and, contrary to previous findings, there is no evidence of long-range dependence in any of the indexes over any sample period or sub-period once short-range dependence is taken into account. Illustrative Monte Carlo experiments indicate that the modified R/S test has power against at least two specific models of long-run memory, suggesting that stochastic models of short-range dependence may adequately capture the time series behavior of stock returns.

Tests of financial asset pricing models may yield misleading inferences when properties of the data are used to construct the test statistics. In particular, such tests are often based on returns to portfolios of common stock, where portfolios are constructed by sorting some empirically motivated characteristic of the securities such as market value of equity. Analytical calculations, Monte Carlo simulations, and two empirical examples show the effects of this type of data snooping can be substantial.

If returns on some stocks systematically lead or lag those of others, a portfolio strategy that sells "winners" and "losers" can produce positive expected returns, even if no stock's returns are negatively autocorrelated as virtually all models of overreaction imply. Using a particular contrarian strategy we show that, despite negative autocorrelation in individual stock returns, weekly portfolio returns are strongly positively autocorrelated and are the result of important cross-autocorrelations. We find that the returns of large stocks lead those of smaller stocks, and we present evidence against overreaction as the only source of contrarian profits.

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.

Since 1958 the number of United States cities with competing central-city newspapers has dwindled from 70 to 19. This evident drift toward monopoly has provoked public concern over the loss of independent editorial voices. Economically, it raises important questions about what cost structures can mandate a 'natural monopoly' and how rival firms in small-numbers markets behave when structural conditions favor a monopoly equilibrium in the long run. In this paper, we explore the rival behavior of two or three sellers in a market where monopoly profits may substantially exceed those of duopoly or triopoly. We develop a theoretical model of the newspaper firm and derive its econometric implications for the demand for and pricing of central-city newspaper advertising and circulation. We test these implications using data for 50 major newspapers located in 30 US metropolitan areas.

We examine the finite-sample properties of the variance ratio test of the random walk hypothesis via Monte Carlo simulations under two null and three alternative hypotheses. These results are compared to the performance of the Dickey-Fuller t and the Box-Pierce Q statistics. Under the null hypothesis of a random walk with independent and identically distributed Gaussian increments, the empirical size of all three tests are comparable. Under the heteroskedastic random walk null, the variance ration test is more reliable than either the Dickey-Fuller or Box-Pierce tests. We compute the power of these three tests against three alternatives of recent empirical interest: a stationary AR(1), the sum of this AR(1) and a random walk, and an integrated AR(1). By choosing the sampling frequency appropriately, the variance ratio test is shown to be as powerful as the Dickey-Fuller and Box-Pierce tests against the stationary alternative and is more powerful than either of the two tests against the two unit root alternatives.

This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Ito type (containing both jump and diffusion components). We derive a particular functional partial differential equation which characterizes the exact likelihood function of a discretely sampled Ito process. In addition, we show by a simple counterexample that the common approach of estimating parameters of an Ito process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consistent estimators.

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.

Upper bounds on the expected payoff of call and put options are derived. These bounds depend only on the mean and variance of the terminal stock price and not on its entire distribution, so they are termed semi-parametric. A corollary of this result is a set of upper bounds for option prices obtained by the risk-neutral valuation approach of Cox and Ross. As an example, these bounds are shown to obtain across both lognormal diffusion-jump processes for any given data set. We present an illustrative example that suggests these bounds may be of considerable practical value.