Implementing Option Pricing Models When Asset Returns Are Predictable
with Jiang Wang, Journal of Finance 50 (1995), 87–129.
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.
A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks
with James Hutchinson and Tomaso Poggio, Journal of Finance 49 (1994), 851-889.
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 1992.
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.
Computational Challenges in Portfolio Management
with Martin Haugh, Computing in Science & Engineering 3 (2001), 54–59.
The financial industry is one of the fastest-growing areas of scientific computing. Two decades ago, terms such as financial engineering, computational finance, and financial mathematics did not exist in common usage. Today, these areas are distinct and enormously popular academic disciplines with their own journals, conferences, and professional societies. One explanation for this area’s remarkable growth and the impressive array of mathematicians, computer scientists, physicists, and economists that are drawn to it is the formidable intellectual challenges intrinsic to financial markets. Many of the most basic problems in financial analysis are unsolved and surprisingly resilient to the onslaught of researchers from diverse disciplines. In this article, we hope to give a sense of these challenges by describing a relatively simple problem that all investors face when managing a portfolio of financial securities over time. Such a problem becomes more complex once real-world considerations factor into its formulation. We present the basic dynamic portfolio optimization problem and then consider three aspects of it: taxes, investor preferences, and portfolio constraints. These three issues are by no means exhaustive—they merely illustrate examples of the kinds of challenges financial engineers face today. Examples of other computational issues in portfolio optimization appear elsewhere.
Asset Allocation and Derivatives
with Martin Haugh, Quantitative Finance 1 (2001), 45–72.
The fact that derivative securities are equivalent to specific dynamic trading strategies in complete markets suggests the possibility of constructing buy-and-hold portfolios of options that mimic certain dynamic investment policies, e.g., asset-allocation rules. We explore this possibility by solving the following problem: given an optimal dynamic investment policy, find a set of options at the start of the investment horizon which will come closest to the optimal dynamic investment policy. We solve this problem for several combinations of preferences, return dynamics, and optimality criteria, and show that under certain conditions, a portfolio consisting of just a few options is an excellent substitute for considerably more complex dynamic investment policies.
Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation
with Harry Mamaysky and Jiang Wang, Journal of Finance 55 (2000), 1705–1765.
Technical analysis, also known as "charting,'' has been a part of financial practice for many decades, but this discipline has not received the same level of academic scrutiny and acceptance as more traditional approaches such as fundamental analysis. One of the main obstacles is the highly subjective nature of technical analysis—the presence of geometric shapes in historical price charts is often in the eyes of the beholder. In this paper, we propose a systematic and automatic approach to technical pattern recognition using nonparametric kernel regression, and apply this method to a large number of U.S. stocks from 1962 to 1996 to evaluate the effectiveness of technical analysis. By comparing the unconditional empirical distribution of daily stock returns to the conditional distribution—conditioned on specific technical indicators such as head-and-shoulders or double-bottoms—we find that over the 31-year sample period, several technical indicators do provide incremental information and may have some practical value.
Finance: A Selective Survey
Journal of the American Statistical Association 95 (2000), 629-635.
Ever since the publication in 1565 of Girolamo Cardano's treatise on gambling, Liber de Ludo Aleae (The Book of Games of Chance), statistics and financial markets have become inextricably linked. Over the past few decades many of these links have become part of the canon of modern finance, and it is now impossible to fully appreciate the workings of financial markets without them. This selective survey covers three of the most important ideas of finance—efficient markets, the random walk hypothesis, and derivative pricing models—that illustrate the enormous research opportunities that lie at the intersection of finance and statistics.
When Is Time Continuous?
with Dimitris Bertsimas and Leonid Kogan, Journal of Financial Economics 55 (2000), 173–204.
In this paper we study the tracking error resulting from the discrete-time application of continuous-time delta-hedging procedures for European options. We characterize the asymptotic distribution of the tracking error as the number of discrete time periods increases, and its joint distribution with other assets. We introduce the notion of temporal granularity of the continuous-time stochastic model that enables us to characterize the degree to which discrete-time approximations of continuous time models track the payoff of the option. We derive closed form expressions for the granularity for a put and call option on a stock that follows a geometric Brownian motion and a mean-reverting process. These expressions offer insight into the tracking error involved in applying continuous-time delta-hedging in discrete time. We also introduce alternative measures of the tracking error and analyze their properties.
Trading Volume: Definitions, Data Analysis, and Implications of Portfolio Theory
with Jiang Wang, Review of Financial Studies 13 (2000), 257–300.
We examine the implications of portfolio theory for the cross-sectional behavior of equity trading volume. We begin by showing that a two-fund separation theorem suggests a natural definition for trading volume: share turnover. If two-fund separation holds, share turnover must be identical for all securities. If (K+1)-fund separation holds, we show that share turnover satisfies and approximate linear K-factor structure, These implications are empirically tested using weekly turnover data for NYSE and AMEX securities from 1962 to 1996. We find strong evidence against two-fund separation and an eigenvalue decomposition suggests that volume is driven by a two-factor linear model.
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.