Long-Term Memory in Stock Market Prices
Econometrica 59 (1991), 1279–1313.
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.
A Large-Sample Chow Test for the Single Linear Simultaneous Equation
with Whitney Newey, Economics Letters 19 (1986), 351-353.
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.
Optimal Control of Execution Costs for Portfolios
with Dimitris Bertsimas and Paul Hummel, Computing in Science & Engineering 1 (2000), 40–53.
The dramatic growth in institutionally managed assets, coupled with the advent of internet trading and electronic brokerage for retail investors, has led to a surge in the size and volume of trading. At the same time, competition in the asset management industry has increased to the point where fractions of a percent in performance can separate the top funds from those in the next tier. In this environment, portfolio managers have begun to explore active management of trading costs as a means of boosting returns. Controlling execution cost can be viewed as a stochastic dynamic optimization problem because trading takes time, stock prices exhibit random fluctuations, and execution prices depend on trade size, order flow, and market conditions. In this paper, we apply stochastic dynamic programming to derive trading strategies that minimize the expected cost of executing a portfolio of securities over a fixed period of time, i.e., we derive the optimal sequence of trades as a function of prices, quantitites, and other market conditions. To illustrate the practical relevance of our methods, we apply them to a hypothetical portfolio of 25 stocks by estimating their price-impact functions using historical trade data from 1996 and deriving the optimal execution strategies. We also perform several Monte Carlo simulation experiments to compare the performance of the optimal strategy to several alternatives.
The Psychophysiology of Real-Time Financial Risk Processing
with Dmitry V. Repin, Journal of Cognitive Neuroscience 14 (2002), 323–339.
A longstanding controversy in economics and finance is whether financial markets are governed by rational forces or by emotional responses. We study the importance of emotion in the decisionmaking process of professional securities traders by measuring their physiological characteristics, e.g., skin conductance, blood volume pulse, etc., during live trading sessions while simultaneously capturing real-time prices from which market events can be detected. In a sample of 10 traders, we find significant correlation between electrodermal responses and transient market events, and between changes in cardiovascular variables and market volatility. We also observe differences in these correlations among the 10 traders which may be systematically related to the traders' levels of experience.
Nonparametric Risk Management and Implied Risk Aversion
with Yacine Ait-Sahalia, Journal of Econometrics 94 (2000), 9–51.
Typical value-at-risk (VAR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VAR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VAR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on option prices and underlying asset value.
Econometric Models of Limit-Order Executions
with Craig MacKinlay and June Zhang, Journal of Financial Economics 65 (2002), 31–71.
Limit orders incur no price impact, however, their execution time is uncertain. We develop an econometric model of limit-order execution times using survival analysis, and estimate it with actual limit-order data. We estimate versions for time-to-first-fill and time-to-completion, and for limit-sells and limit-buys, and incorporate the effects of explanatory variables such as the limit price, the limit size, the bid/offer spread, and market volatility. We find that execution times are very sensitive to limit price and several other explanatory variables, but not sensitive to limit size. We also show that hypothetical limit-order executions, constructed either theoretically from first-passage times or empirically from transactions data, are very poor proxies for actual limit-order executions.
Frontiers of Finance: Evolution and Efficient Markets
with J. Doyne Farmer, Proceedings of the National Academy of Sciences 96 (1999), 9991–9992.
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.
The Sources and Nature of Long-Term Dependence in the Business Cycle
with Joseph Haubrich, Federal Reserve Bank of Cleveland Economic Review 37 (2001), 15–30.
This paper examines the stochastic properties of aggregate macroeconomic time series from the standpoint of fractionally integrated models, and focuses on the persistence of economic shocks. We develop a simple macroeconomic model that exhibits long-term dependence, a consequence of aggregation in the presence of real business cycles. We derive the relation between properties of fractionally integrated macroeconomic time series and those of microeconomic data, and discuss how fiscal policy may alter their stochastic behavior. To implement these results empirically, we employ a test for fractionally integrated time series based on the Hurst-Mandelbrot rescaled range. This test is robust to short-term dependence, and is applied to quarterly and annual real GNP to determine the sources and nature of long-term dependence in the business cycle.
The Three P’s of Total Risk Management
Financial Analysts Journal 55 (1999), 13–26.
Current risk-management practices are based on probabilities of extreme dollar losses (e.g., measures like Value at Risk), but these measures capture only part of the story. Any complete risk-management system must address two other important factors: prices and preferences. Together with probabilities, these comprise the three P's of Total Risk Management. This article describes how the three Ps interact to determine sensible risk profiles for corporations and for individuals, guidelines for how much risk to bear and how much to hedge. By synthesizing existing research in economics, psychology, and decision sciences, and through an ambitious research agenda to extend this synthesis into other disciplines, a complete and systematic approach to rational decision-making in an uncertain world is within reach.
Hedging Derivative Securities and Incomplete Markets: An Epsilon-Arbitrage Approach
with Dimitris Bertsimas and Leonid Kogan, Operations Research 49 (2001), 372–397.
Given a European derivative security with an arbitrary payoff function and a corresponding set of underlying securities on which the derivative security is based, we solve the dynamic replication problem: find a self-financing dynamic portfolio strategy—involving only the underlying securities—that most closely approximates the payoff function at maturity. By applying stochastic dynamic programming to the minimization of a mean-squared-error loss function under Markov state-dynamics, we derive recursive expressions for the optimal-replication strategy that are readily implemented in practice. The approximation error or "epsilon" of the optimal-replication strategy is also given recursively and may be used to quantify the "degree" of market incompleteness. To investigate the practical significance of these epsilon-arbitrage strategies, we consider several numerical examples including path-dependent options and options on assets with stochastic volatility and jumps.