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87
A closedform solution for options with stochastic volatility with applications to bond and currency options
 Review of Financial Studies
, 1993
"... I use a new technique to derive a closedform solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond option ..."
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Cited by 1470 (5 self)
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I use a new technique to derive a closedform solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strikeprice biases in the BlackScholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Many plaudits have been aptly used to describe Black and Scholes ’ (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original BlackScholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns I thank Hans Knoch for computational assistance. I am grateful for the suggestions of Hyeng Keun (the referee) and for comments by participants
Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk
 THE JOURNAL OF FINANCE • VOL. LVI
, 2001
"... This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period 1962–1997 there has been a noticeable increase in firmlevel volatility relative to market volatility. Accordingly, correlations among individual stocks and the ..."
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Cited by 509 (17 self)
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This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period 1962–1997 there has been a noticeable increase in firmlevel volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. All the volatility measures move together countercyclically and help to predict GDP growth. Market volatility tends to lead the other volatility series. Factors that may be responsible for these findings are suggested.
Conditional skewness in asset pricing tests
 Journal of Finance
, 2000
"... If asset returns have systematic skewness, expected returns should include rewards for accepting this risk. We formalize this intuition with an asset pricing model that incorporates conditional skewness. Our results show that conditional skewness helps explain the crosssectional variation of expect ..."
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Cited by 323 (6 self)
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If asset returns have systematic skewness, expected returns should include rewards for accepting this risk. We formalize this intuition with an asset pricing model that incorporates conditional skewness. Our results show that conditional skewness helps explain the crosssectional variation of expected returns across assets and is significant even when factors based on size and booktomarket are included. Systematic skewness is economically important and commands a risk premium, on average, of 3.60 percent per year. Our results suggest that the momentum effect is related to systematic skewness. The low expected return momentum portfolios have higher skewness than high expected return portfolios. THE SINGLE FACTOR CAPITAL ASSET PRICING MODEL ~CAPM! of Sharpe ~1964! and Lintner ~1965! has come under recent scrutiny. Tests indicate that the crossasset variation in expected returns cannot be explained by the market beta alone. For example, a growing number of studies show that “fundamental” variables such as size, booktomarket value, and price to earnings ratios
Post'87 Crash Fears in the S&P 500 Futures Option Market
, 1998
"... Postcrash distributions inferred from S ..."
Production–Based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations
 JOURNAL OF FINANCE
, 1991
"... ..."
Optimal investment, growth options, and security returns
 Journal of Finance
, 1999
"... As a consequence of optimal investment choices, a firm’s assets and growth options change in predictable ways. Using a dynamic model, we show that this imparts predictability to changes in a firm’s systematic risk, and its expected return. Simulations show that the model simultaneously reproduces: ~ ..."
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Cited by 239 (10 self)
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As a consequence of optimal investment choices, a firm’s assets and growth options change in predictable ways. Using a dynamic model, we show that this imparts predictability to changes in a firm’s systematic risk, and its expected return. Simulations show that the model simultaneously reproduces: ~i! the timeseries relation between the booktomarket ratio and asset returns; ~ii! the crosssectional relation between booktomarket, market value, and return; ~iii! contrarian effects at short horizons; ~iv! momentum effects at longer horizons; and ~v! the inverse relation between interest rates and the market risk premium. RECENT EMPIRICAL RESEARCH IN FINANCE has focused on regularities in the cross section of expected returns that appear anomalous relative to traditional models. Stock returns are related to booktomarket, and market value. 1 Past returns have also been shown to predict relative performance, through the documented success of contrarian and momentum strategies. 2 Existing explanations for these results are that they are due to behavioral biases or risk premia for omitted state variables. 3 These competing explanations are difficult to evaluate without models that explicitly tie the characteristics of interest to risks and risk premia. For example, with respect to booktomarket, Lakonishok et al. ~1994! argue: “The point here is simple: although the returns to the B0M strategy are impressive, B0M is not a ‘clean ’ variable uniquely associated with eco
A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk
, 1997
"... This article presents a technique for nonparametrically estimating continuoustime di#usion processes which are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and di#usion of t ..."
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Cited by 207 (5 self)
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This article presents a technique for nonparametrically estimating continuoustime di#usion processes which are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and di#usion of the short rate, and the market price of interest rate risk. While the estimated di#usion is similar to that estimated by Chan, Karolyi, Longsta# and Sanders (1992), there is evidence of substantial nonlinearity in the drift. This is close to zero for low and medium interest rates, but mean reversion increases sharply at higher interest rates.
Asset pricing at the millennium
 Journal of Finance
"... This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the tradeoff between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior ..."
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Cited by 184 (0 self)
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This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the tradeoff between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and crosssectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance. This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work. Theorists develop models with testable predictions; empirical researchers document “puzzles”—stylized facts that fail to fit established theories—and this stimulates the development of new theories. Such a process is part of the normal development of any science. Asset pricing, like the rest of economics, faces the special challenge that data are generated naturally rather than experimentally, and so researchers cannot control the quantity of data or the random shocks that affect the data. A particularly interesting characteristic of the asset pricing field is that these random shocks are also the subject matter of the theory. As Campbell, Lo, and MacKinlay ~1997, Chap. 1, p. 3! put it: What distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The starting point for every financial model is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on mar* Department of Economics, Harvard University, Cambridge, Massachusetts
Continuoustime methods in finance: A review and an assessment
 Journal of Finance
, 2000
"... I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. ..."
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Cited by 51 (0 self)
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I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuoustime models. Capital market frictions and bargaining issues are being increasingly incorporated in continuoustime theory. THE ROOTS OF MODERN CONTINUOUSTIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuoustime modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.
Stock return predictability and asset pricing models
, 2001
"... Asset pricing models based on rational timevarying expected returns or on equity characteristics imply restrictions on stock return predictability. This paper develops a framework for investigating these pricing restrictions through the use of an economic metric that is based on asset allocation wi ..."
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Cited by 31 (3 self)
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Asset pricing models based on rational timevarying expected returns or on equity characteristics imply restrictions on stock return predictability. This paper develops a framework for investigating these pricing restrictions through the use of an economic metric that is based on asset allocation with estimation risk. The evidence shows that when portfolio weights are unconstrained, the deviations from the pricing models are economically significant. Incorporating constraints on leverage and short equity positions results in a sharp reduction in these deviations, which disappear in some cases, yet they remain economically significant in most cases. Finally, imposing factor model restrictions on predictive regressions generate smaller outofsample Sharpe ratios and larger mean square forecast errors. The results carry implications for various applications in financial economics using risk factors or equity characteristics as benchmarks.